The Structure of Time and Space

The constant speed of light

Why do all observers see the same speed of light? This was the big problem facing physicists at the start of the 20th century.

This surprising, but experimentally verifiable fact, has not yet been explained.

All observers, having a relative velocity to each other less than the speed of light, will measure the same speed of light, regardless of their own velocity relative to the source of the light.

If I were to throw a ball to you, and you catch it, it will cover the distance between us in a certain time, and this distance and time would allow is to calculate the ball’s velocity. If I were to move away from you with the same velocity that I throw the ball back towards you, what will be the result? We might expect that the ball’s velocity leaving my hand, and my velocity in the opposite direction, will cancel each other out, and the ball will simply drop to the ground.

This is exactly what appears to happen, at low velocities like balls, planes, trains, and automobiles, to a very high degree of experimental accuracy.

Nineteenth century physicists were happy with their definitions of space, time, and velocity, but James Clerk Maxwell and Michael Faraday started a revolution in physics, with their discoveries concerning electromagnetism, that forced physicists to change their ideas about space, time, and motion.

Faraday was a brilliant experimenter who investigated the effects of electricity and magnetism. These discoveries were quantified, and cast into an elegant mathematical form by James Maxwell.

Maxwell’s equations showed that light could be explained as an electromagnetic wave. Changing electric fields produced magnetic fields, and changing magnetic fields produced electric fields. This was what Faraday had discovered.

Maxwell’s equations showed that suitably changing electric and magnetic fields could be made to generate each other, and the result was a wave that traveled through space; apparently in a similar way to the way that water waves traveled on water, or sound waves traveled through air.

The equations predicted the speed of these waves in terms of the electric and magnetic constants that determined the strength of these fields. This speed was the same as the speed of light, as near as could be measured, and it was clear that light was, at least in some sense, one and the same as Maxwell’s predicted electromagnetic waves.

In the next few decades the invention of radio showed that electromagnetic waves could, indeed, be created and used, and they did travel at the speed of light.

The surprising problem

The problem this presented was that the picture of velocity, and the way it had been assumed to add and subtract, as described in the ball throwing experiment above, was not compatible with the way Maxwell’s and Faraday’s light waves behaved.

If physicists imagined a beam of light being substituted for the ball, and they imagined that it was shone backwards by someone moving away at the speed of light, they might expect that the two equal and opposite velocities should cancel out.

They might imagine that light in this situation would have no resultant velocity, and drop to the floor in the same way that a ball would, but careful experiments confirmed that this was not the case.

Light left its source, and arrived at its destination, at the same constant speed. It did not matter if the source of the light was moving away, or moving nearer; it was always detected arriving at the same speed.

Another way of saying this is that electric and magnetic fields are always stationary with respect to an observer.

Electromagnetic waves and light are the same thing

Maxwell’s equations didn’t specify velocities, relative to the source and the receiver, for the emission and reception of a beam of light. According to the equations, the speed of light seemed to be a fixed constant. The equations describing the effects of electric and magnetic fields could be mathematically arranged to show a wave equation in which the velocity of the wave matched the velocty of light. This led Maxwell to the realisation that his electromagnetic wave and light must be one and the same thing.

Experiments, such as those done by Albert Michelson and Edward Morley, confirmed that the velocity of light was independent of the relative motion of the emitter and the receiver.

Space and time must change

Albert Einstein realised that this meant that the space and time measurements used to calculate velocities must not be the same for observers moving relative to each other. Hendrik Lorentz had shown that there was a way to express the way space and time must change mathematically for light to always have the same value, and Einstein, in his Special Theory of Relativity, suggested that space and time actually did change in this way when observers were in relative motion.

The Lorentz transformations show that observers in inertial reference frames will see a uniform speed of light, but they only apply in idealised inertial reference frames, and not in our Universe full of matter, and gravitational fields. No physical explanation of what causes the necessary space and time transformations in our Universe, that are implied by the Lorentz transformations, has been put forward until now.

That is what this account is going to show.

Hermann Minkowski, one of Einstein’s teachers, went further, and showed that the fundamental geometry of space must be different to that of the familiar Euclidian geometry when time was included.

Minkowski’s spacetime

Minkowski defined an “interval”, in space and time coordinates, that was invariant in the same way that a length was invariant in the three dimensional space of Euclid and Pythagoras. Although space and time measurements, for observers that were moving with a constant velocity relative to each other, did not stay the same, this “interval” did.

The “interval” defines the shape that Minkowski spacetime must have for all observers to see the same speed of light. In the same way Pythagoras’ theorem shows that the hypotenuse of a right angled triangle defines a length in a flat plane that is the same whatever measurement axes at right angles to each other are used to measure it. See “The Minkowski spacetime metric” below.

Minkowski’s formula, referred to here as the Minkowski metric, or Minkowski spacetime, describing this interval is a particular case of a Riemannian differential geometry. It applies in an idealised spacetime in which there is no mass or energy. Our Universe must have a similar structure to that of the Minkowski metric for us to experience the behaviour of light, and the velocity of material objects, that we do.

Minkowski’s spacetime geometry was based on the principle that two points in it could not both exist if the distance between them in space, and their separation in time, implied that they could only be connected by a signal travelling faster than light.

A speed limit in spacetime

It is the structure of Minkowski spacetime that creates a limit to the maximum velocity that any material object in it can have. It has some surprising consequences.

A distance and time between two points that implies a velocity greater than the speed of light results in an interval that is not a real number. This means that it is not possible to go faster than the speed of light in Minkowski spacetime.

Hendrik Lorentz developed a set of coordinate transformatons that preserved a uniform speed of light for all observers travelling with a constant relative velocity to each other. The Lorentz equations showed that length contracted in the direction of a moving observer’s motion relative to another, stationary, observer, and time dilated. As the speed of light was approached, length contracted to zero, and time dilated to infinity.

It is necessary for this length contraction and time dilation to occur for all observers in inertial frames to see the same speed of light.

At the same time the moving observers saw themselves as stationary in their own Minkowski metric. When they measured the speed of light in their Minkowski spacetime, they found it to be the same constant value in all directions. Light behaved as if the moving observer, and the original stationary observer were both stationary relative to its own speed.

In particular, if a flash of light was emitted at the moment when, and at the point where, two observers in relative motion passed each other, they would both see themselves as being at the center of an expanding sphere of light.

Time slows down for a moving observer

There would be one important difference with particular relevance to this discussion. The moving observers time would be passing more slowly, and so the speed of light would be slower in the moving observer’s Minkowski spacetime.

The dilation of time is particularly important because it has permanent effects. If a clock with a moving observer is then placed next to a clock that has remained stationary, after some time has passed for both clocks, the moving observers clock will have recorded less time passing than the stationary clock.

The implication is that we cannot view the motion as symmetric between the moving clock and the stationary clock. We cannot simply reverse the sign of the velocity and reverse the time dilation effect, as some accounts would have. We cannot choose which of the two clocks to consider to be stationary, and which to be moving, and thereby change which clock shows less time having passed, and which shows more time having passed.

The clock with less time recorded as having passed is the clock that was moving, and the clock with more time having passed is the stationary clock.

There is only one observer at the present time

An observer in Minkowski spacetime would have to be at a point where light can arrive from a point where it is emitted. This emission point will have a space interval with respect to the observer equal to the distance that would be covered by light in the interval of time between emission and arrival.

Points with a greater separation in space from the observer than could be covered by light, in the interval of time between emission and arrival, cannot exist in this observers spacetime. The invariant intervals, for all such points, are calculated by the Minkowski metric to be imaginary. Mathematically they are represented by an imaginary number. Physically they can’t exist.

It might be uncomfortable for some people to appreciate this, but this implies that there can only be one observer at the present time in a Minkowski metric, since all other points in their spacetime must be in that observers past, or their future. There are no other points in Minkowski spacetime with the same time coordinate as an observer.

Calling the Minkowski spacetime interval “spacelike”, for points with the same time coordinate as the stationary observer, does not mean that such points can exist. It is merely putting a name to wishful thinking. Calling a white horse, with a horn on its forehead, a Unicorn, doesn’t mean that Unicorns exist.

Light appears to travel at a fixed speed, c, from a point in the observers past, and from a distance away, to the point where the observer is in the present. Points in Minkowski spacetime, that could be connected by a ray of light, have an invariant interval, between them and an observer, equal to zero.

We can no longer define the present moment as the set of all points with a time interval of zero between them and an observer. We will have to be more careful, and define the present moment, in a Minkowski spacetime, as the set of all points in the spacetime for which time has proceeded to its furthest extent. In other words there is no future for these points.

Points in the present moment of a stationary observer in Minkowski spacetime are not points with the same time coordinate, or in other words a time interval of zero. Points in the present moment of an observer in Minkowski spacetime exist on what is referred to as the “light cone”. This is a set of points that have a Minkowski spacetime interval of zero between them, and the observer.

It is often assumed that the point where the light is emitted is moving forward in time at the same rate as the place where it is received, but this is a misreading of the situation in an attempt to hang on to our Euclidean view of space.

If the space time of our Universe has an overall equivalence to a Minkowski metric, then an observer’s Universe is all in the observer’s past. We will see below how this can be accommodated in the cosmological model proposed.

The speed of light in our universe

We will see here that the uniformity of the speed of light for all observers can be fully explained in our Universe full of matter. We will see that we can use Birkhoff’s theorems to modify the Schwarzschild metric so that it describes a spacetime that gives all observers a uniform speed of light in the same way that the Minkowski metric does.

It is well understood that the Minkowski spacetime metric does not apply to our Universe. The Minkowski metric has no mass or energy in it. The Minkowski metric is used with the Lorentz transformations to show that the speed of light is uniform in all directions for all observers in inertial reference frames. There is, as yet, no explanation of why the length contraction, and time dilation, necessary for this observation to be true, occurs in our Universe full of matter.

No explanation of why this necessary length contraction, and time dilation, occurs in an empty Minkowski metric space, has been suggested either. The Lorentz equations are not implied by the Minkowski metric. Lorentz proposes his space and time transformations as a mathematical “fix” that gives the required uniform speed of light when measured by all inertial observers having constant relative velocity to each other. Just what causes the required changes to space and time is not addressed by Minkowski, or Lorentz.

We do know, however, that Einstein’s General Relativity predicts that mass determines the length contraction, and time dilation, outside a spherical mass distribution. Birkhoff’s theorems¹, and the Schwarzschild metric, describe this length contraction, and time dilation.

We will see how the Schwarzschild spacetime metric may be used to construct a model Universe. David Birkoff’s theorems¹ state that the empty space outside a spherically symmetric mass distribution can be modeled by the Schwarzschild metric, even if the constituent mass is in motion. Birkoff’s theorems¹ are used, with the Schwarzschild metric, to build an expanding model Universe with spherical symmetry

Birkhoff’s theorems¹ and the Schwarzschild metric are only applied to empty space at each step of building the model, but the result is a model that has a defined mass density throughout its extent in space and time. Those who say this can’t be done because Birkhoff’s theorem’s only apply to empty space are invited to read on.

David Birkhoff’s theorems¹ are used here, with the Schwarzschild solution of the equations of General Relativity, to investigate the internal structure of a Universe with spherical symmetry. It will be shown here that the length contraction, and time dilation, associated with the Lorentz equations, may be attributed to the effect of mass on length and time in the proposed model.

It will be shown how the effect of mass on length and time, in a Universe with a uniform mass density, results in an expanding Universe where all observers see a uniform speed of light.

Differential geometry simplified

The Pythagoras metric

A differential in differential geometry is an infinitely small coordinate element. In the cases we investigate here, the coordinate elements are infinitesimally small displacements, or differences, in space and time measurements.

For those who are new to differential geometry it is insightful to start with Pythagoras’ theorem. We can use it to see how infinitesimal length elements dx and dy, on the x and y axes, can define a line element, ds, in a flat plane. These infinitesimal length elements are differentials.

This idea can be extended to three coordinate axes of length. Hermann Minkowski further extends it to include time as a fourth component. In the geometry of space and time these coordinate elements are infintely small differences of length, or time.

Here c is the hypotenuse of a right angled triangle with sides a and b on a two dimensional flat plane.

Pythaoras’ theorem states that the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides. This is the basis of differential geometry. On the infinitely small scale of the differentials all “surfaces” in any number of dimensions are “flat”, and Pythagoras’ theorem applies.

Any space can be represented by an equation for infinitesimal intervals, ds, which together combine to specify the geometrical space. The equations define how the intervals link together.

We can consider two points, (x₁, y₁) and (x₂, y₂) infinitely close together on a plane measured with orthogonal x, and y, axes. The length intervals dx = x₂ – x₁, and dy = y₂ – y₁. The theorem of Pythagoras may be used to define the distance, ds, between the two points as follows.

ds^{2}= dx^{2}+dy^{2}

Here x and y are orthogonal, (at right angles), coordinate axes, and dx and dy are infinitesimal increments in x and y.

This equation is an example of Riemannian differential geometry. The infintesimal lengths dx and dy, and Pythagoras’ equation define a flat, two dimensional, plane. We could call this equation “The Pythagoras metric”

Equivalently we could write

ds^{2}= dp^{2}+dq^{2}

where p and q are different orthogonal coordinate axes.

The infinitesimal length, ds, is

ds= (dx^{2}+dy^{2})^\frac{1}{2}= (dp^{2}+dq^{2})^\frac{1}{2}

We can see above that different orthogonal axes p and q can be used, and the length ds is measured to be the same. The infinitesimal length, ds, is said to be invariant under a change of coordinate axes.

We see here that ds is constant between any two infinitely close points in a plane no matter where we choose to draw our x and y axes. We can define a set of points making up a plane by the Pythagorean metric above, and the plane is determined independently of the choice of coordinate axes x, and y.

A flat plane is defined by the Pythagoras metric giving all the possible values for ds between pairs of points infintely close together.

A single point on the plane is defined by ds = 0.

Three space dimensions

This is extended easily to three space dimensions, x, y, and z.

ds^{2}= dx^{2}+dy^{2}+dz^{2}

The length, ds, is

ds= (dx^{2}+dy^{2}+dz^{2})^\frac{1}{2}

Spherical coordinates

A coordinate change to spherical coordinates r, θ, and ϕ results in

ds^2=dr^2+r^2d\theta^2+r^2sin^2\theta d\phi^2

The length, ds, is

ds=(dr^2+r^2d\theta^2+r^2sin^2\theta d\phi^2)^\frac{1}{2}

The two dimensional surface of a sphere

The power of differential geometry lies in its ability to represent curved spaces. The two dimensional surface of a sphere of radius, R, is described by

ds^2=R^2d\theta^2+R^2sin^2\theta d\phi^2

Where θ, and ϕ are spherical coordinates as above. Here the radial coordinate, r, has become the constant, R, which is the radius of the sphere, so dr is always zero, and there is no dr² component.

On the infinitely small scale of the infinitesimal coordinate intervals, the surface of the sphere is effectively flat, and the equation of Pythagoras still holds true. We can see that it has the same form when we write it as

ds^2=(Rd\theta)^2+(Rsin\theta d\phi)^2

The length, ds, is

ds=((Rd\theta)^2+(Rsin\theta d\phi)^2)^\frac{1}{2}

Rdθ, and Rsinθdϕ are infinitesimal orthogonal lengths on the surface of the sphere, and form an infinitesimal right angled triangle with ds as the hypotenuse. On the diagram below we need to imagine that the right angled triangle formed by Rdθ, Rsinθdϕ, and ds is infinitely small, and effectively all at the point P.

The Minkowski spacetime metric

Adding the dimension of time, so that we can describe the space time we live in, proves to be a little more challenging. Hermann Minkowski, incorporated Einstein’s theory of Special Relativity into a model of space and time.

Minkowski made the experimental observation, that the speed of light was the same for all observers, a defining part of the structure of space and time. He proposed a spacetime metric of three space components, and a time component, that described a spacetime in which the time component was proportional to the space components, with a constant of proportionality, c.

This spacetime applied to inertial reference frames. That means frames of reference in which bodies are stationary or move at a constant velocity.

Minkowski’s spacetime metric, in Cartesian coordinates including time, is

ds^2=c^2dt^2-dx^2-dy^2-dz^2

Or equivalently

ds^2=c^2dt^2-(dx^2+dy^2+dz^2)

Here we are considering three space axes x, y, and z , together with one time axis t. A pair of points infinitely close together with coordinates, (x₁, y₁, z₁, t₁), and (x₂, y₂, z₂, t₂), will have intervals dx = x₂-x₁, dy = y₂-y₁ , dz = z₂-z₁, and dt = t₂–t₁. c is a constant which in our Universe is identified with the velocity of light. The interval, ds, is an invarient quantity that is defined by the Minkowski spacetime equation above. Like the example of the Pythagoras metric, it is independent of a change of coordinates.

Nothing can travel faster than light

This is the form of a spacetime in which no material body in it can travel faster than light. Here the constant, c, is the velocity of light.

The fact that the velocity of light appeared to be a fixed constant independent of the velocity of observers had been determined by experiments like the Michaelson Morley experiment, and established as a necessary part of physics by Maxwell’s equations, and Einstein’s theory of Special Relativity.

The structure of Minkowski’s spacetime means that, if it is taken to represent the Universe we experience, a speed faster than light is impossible.

Here c²dt² has the dimensions of length squared, and three dimensional Pythagorean lengths are squared, and subtracted from it. We can say that this subtraction, of length squared from time squared multiplied by the speed of light squared, is responsible for much of the strangeness of Special Relativity.

The Minkowski spacetime metric defines a three dimensional space together with time. It is empty of mass and energy, so it can’t be a complete description of spacetime in our Universe, but our spacetime must share some of the important properties of Minkowski spacetime.

The Lorentz transformations

We will now look at the behaviour of unaccelerated frames of reference in the Minkowski spacetime that can have relative velocities to each other up to, but not including, the speed of light. We will see that all observers in such frames see themselves in the same type of Minkowski metric, with the same values for ds between infinitely close points.

We consider a second reference frame, with coordinates x^/ y^/ z^/ and t^/ that is moving with a fixed velocity, v, relative to the the x, y, z, and t frame,

Here we will use rectangular coordinate frames, x, y, z, and t to represent a stationary, or rest, frame, and t^/,x^/,y^/,z^/, to represent coordinates of a frame in relative motion to this rest frame. At the end of the section we see how the Minkowski metric may be written in spherical coordinates, t^/, r^/, θ^/, ϕ^/

To show mathematically how length and time coordinates must change for each observer to see a uniform speed of light, regardless of their relative motion, Hendrik Lorentz designed a system of coordinate transformations that had this property.

The Lorentz transformations to coordinates of a coordinate frame moving with a relative velocity, v, with respect to the original x, y, z, and t frame, for infinitesimal Cartesian coordinate elements, are

dt{^/}=(1-\frac{v^2}{c^2})^{-\frac 1 2} (dt-\frac{vdx}{c^2})

dx{^/}=(1-\frac{v^2}{c^2})^{-\frac 1 2}(dx-vdt)

dy{^/}=dy

dz{^/}=dz

where dt^/ , dx^/ ,dy^/ , and dz^/ are infinitesimal coordinate elements stationary in a frame moving with a velocity, v, with respect to the x, y, z, and t frame, as seen by an observer stationary with respect to these coordinates, when the x axis, and x^/ axis of the two frames are aligned.

The need for the Lorentz transformations to be in the form they are is because the speed of light is measured to be the same in the x, y, z, and t, or rest frame, and a moving frame with coordinates x^/ y^/ z^/ and t^/. This is a confirmed experimental result.

It is conventional to refer to the original x, y, z, and t frame as the “rest” frame, but we need to remember that there is no absolute zero velocity in physics. A frame of reference is nominated as the rest frame, and frames with a velocity relative to it are referred to a moving frames. All reference frames are equivalent and we could, in principle, nominate any inertial frame as the rest frame.

If the speed of light is the same in a moving frame, and the original x, y, z, and t frame at rest, the lengths and times we measure it with must change when we compare one frame with the other.

The wavefront of an expanding sphere of light

Consider two frames moving with their x, and x^/ axes aligned, and with a constant relative velocity, v, in their x, and x^/ directions. A flash of light with velocity, c, is initiated at the origin of both frames just as their origins coincide. The velocity of an expanding sphere of light, as measured by an observer stationary in a frame moving at a velocity, v, with respect to , and starting from the origin, can be shown, in Cartesian coordinates, as

c=\frac{(dx{^/}^2+dy{^/}^2+dz{^/}^2)^{\frac 1 2}}{dt{^/}}

so

c^2=\frac{dx{^/}^2+dy{^/}^2+dz{^/}^2}{dt{^/}^2}

and

c^2dt{^/}^2=dx{^/}^2+dy{^/}^2+dz{^/}^2

Substituting into the equation above with the Lorentz transformations we get

c^2((1-\frac{v^2}{c^2})^{-\frac 1 2} (dt-\frac{vdx}{c^2})) ^2=((1-\frac{v^2}{c^2})^{-\frac 1 2}(dx-vdt))^2+dy^2+dz^2

c^2(1-\frac{v^2}{c^2})^{- 1 } (dt-\frac{vdx}{c^2}) ^2=(1-\frac{v^2}{c^2})^{- 1}(dx-vdt)^2+dy^2+dz^2

c^2(1-\frac{v^2}{c^2})^{- 1 } (dt^2-\frac{vdxdt}{c^2}+\frac{v^2dx^2}{c^4})=(1-\frac{v^2}{c^2})^{- 1}(dx^2-vdtdx+v^2dt^2)+dy^2+dz^2

(1-\frac{v^2}{c^2})^{- 1 } (c^2dt^2-vdxdt+\frac{v^2dx^2}{c^2})=(1-\frac{v^2}{c^2})^{- 1}(dx^2-vdtdx+v^2dt^2)+dy^2+dz^2

(1-\frac{v^2}{c^2})^{- 1 } (c^2dt^2+\frac{v^2dx^2}{c^2})=(1-\frac{v^2}{c^2})^{- 1}(dx^2+v^2dt^2)+dy^2+dz^2

(1-\frac{v^2}{c^2})^{- 1 }(c^2dt^2-v^2dt^2)=(1-\frac{v^2}{c^2})^{- 1} (dx^2-\frac{v^2dx^2}{c^2})+dy^2+dz^2

(1-\frac{v^2}{c^2})^{- 1}c^2dt^2(1-\frac{v^2}{c^2})=(1-\frac{v^2}{c^2})^{- 1 }dx^2 (1-\frac{v^2}{c^2})+dy^2+dz^2

c^2dt^2=dx^2+dy^2+dz^2

c^2=\frac{(dx^2+dy^2+dz^2)}{dt^2}

and

c=\frac{(dx^2+dy^2+dz^2)^{\frac 1 2}}{dt}

We see that an observer stationary in the original x, y, z, and t frame also measures a velocity, c, for a spherical expanding wavefront starting at the origin, just as the observer stationary in the moving x^/, y^/, z^/, and t^/ frame does. The Lorentz transformation was designed to achieve this result.

The two postulates that Einstein based his Special Theory of Relativity on are as follows¹²:

Postulate 1: The principle of relativity; that the laws of physics are the same in all
inertial frames.

Postulate 2: The speed of light in vacuum is the same in all inertial frames.

The Lorentz transformations on the Minkowski metric have been shown above to conform to the second of these two principles.

This second principle could actually be considered to be contained within the first. Maxwell’s theory of electromagnetism predicts that electric and magnetic fields can produce a wave travelling with the velocity of light. This wave travels at this velocity, c, relative to the electric and magnetic fields which are treated as stationary relative to all observers. Maxwell’s theory of electromagnetism implies that the velocity of light is fixed relative to all observers because the electric and magnetic fields are stationary .

The conventional Euclidean space and Newtonian time of three space dimensions extended to infinity, and a separate independent dimension of time, do not conform to these principles, and consequently have to be abandoned.

In a frame of reference in Euclidean space and Newtonian time we do not get velocities combining in this way. It would even be possible to conceive of an observer, stationary in a moving frame, that moved ahead of the expanding wavefront at a velocity greater than light. They could potentially leave the light behind them.

This is not what happens in our Universe. This means that we must abandon completely the idea that we live in Euclidean space and Newtonian time.

We can see above that an expanding sphere of light, expressed in moving x^/ y^/ z^/ and t^/ coordinates, will have the same equation for the expanding sphere of light when it is expressed in x, y, z, and t coordinates after the Lorentz transformation.

If a flash of light is created at a point in our spacetime, between two observers, at the moment when they move past each other, it will be seen by each of them to be an expanding sphere with themselves at the centre. They may each see themselves as stationary at the origin of a frame of reference at this moment. If the light is emitted at the origins of the two frames, at x^/ = 0, and at x = 0, at the moment these two origins coincide, and observers stay at the origins of their respective reference frames, they will each see the same wavefront moving away from them at the speed of light.

This result means that we cannot be living in a Universe in which space is Euclidean, and time is Newtonian.

The Lorentz transformations tell us how space and time must change between inertial frames, moving with a relative velocity to each other, to have this result. If the speed of light is the same for the two frames, length intervals and time intervals must be different.

Our spacetime must have a structure like Minkowski spacetime. It can’t be Minkowski spacetime because it contains mass and energy, but it must have the same properties when it comes to the behaviour of light. We will be developing, in this account, a model for our Universe that does have these properties.

The interval for a moving frame

We saw above that Minkowski defined an interval, ds, such that

ds^2=c^2dt^2-dx^2-dy^2-dz^2

Where the intervals dx, dy, dz, and dt are coordinate intervals measured by an observer stationary in the rest frame.

The possible values of ds define all the points in Minkowski’s spacetime.

We can see that the same algebra above, using the Lorentz transformations, will show that an observer, stationary in a moving reference frame, will also define the same interval, ds, in the coordinates of any frame moving relative to the first with a velocity, v.

Substituting for dt^/ , dx^/ ,dy^/ , and dz^/, using the Lorentz transformations as above, shows that

c^2dt{^/}^2-dx{^/}^2-dy{^/}^2-dz{^/}^2= c^2dt^2-dx^2-dy^2-dz^2

so, under the Lorentz transformations, the Minkowski metric in moving frame coordinates, x^/,y^/, z^/, and t^/ , below

ds^2=c^2dt{^/}^2-dx{^/}^2-dy{^/}^2-dz{^/}^2

becomes the following Minkowski metric in rest frame x, y, z, and t coordinates

ds^2=c^2dt^2-dx^2-dy^2-dz^2

Here we are starting with the Minkowski metric for a moving frame, and substituting the moving frame coordinate intervals for the original x, y, z, and t frame coordinate intervals. We find that the Minkowski metric for the moving frame transforms into a Minkowski metric for the x, y, z, and t frame. This is equivalent to the statement that observers in unaccelerated inertial frames in relative motion will all measure the same speed of light. We will discover below that they are not the same spacetime although it has been generally assumed that they are.

The Lorentz transformations were designed to have this effect. The Lorentz transformations have the effect that the interval, ds, and the speed of light, do stay the same after the transformations, and the length and time intervals must change to achieve this. The Lorentz transformations specify the changes of length and time that are needed.

“Common sense”, developed in a situation where the magnitude of most velocities are much less than the speed of light, would suggest that measured velocities depend on the relative velocities of the observers making the measurements. The discovery, that the speed of light does not depend on the relative velocities of the observers, forces us to revise our “common sense” notion of the way that different velocities combine. We must abandon our belief that lengths and times must be the same for observers with relative velocities.

The Minkowski intervals, ds, are equal to each other when the moving frame coordinates are viewed by an observer stationary in the moving frame, and the corresponding x, y, z, and t coordinates are viewed by an observer stationary in the rest frame. All observers moving with constant relative velocity to each other will see themselves in a Minkowski metric with the same values for ds after a Lorentz transformation.

The metric in moving frame coordinates is a Minkowski metric, but it is not the same Minkowski metric as the Minkowski metric in rest frame coordinates. The interval remains the same, but the shape of spacetime does not. This fact has not been generally appreciated.

The speed of light cannot be exceeded in inertial reference frames

If a body is moving in an inertial reference frame, a second inertial reference frame may always be found in which that body is stationary. This is implied by the first Principle of Special Relativity. The same laws of physics must apply to the moving body as they do for the stationary body, and all inertial frames are equivalent.

So if an observer moves between two infinitely close points in a Minkowski spacetime written in x, y, z, and t coordinates, that observer can also be considered to be stationary in a reference frame moving with them as they move between the two points. The points will have moving frame coordinates (t₁^/,x₁^/, y₁^/, z₁^/ ), and (t₂^/,x₂^/, y₂^/, z₂^/ ).

We may imagine that this observer is in a Minkowski spacetime, and that this spacetime can be expressed in these moving coordinates. We are considering a change of coordinates for the same pair of points, so we expect that the value for ds will be the same between the points in rest frame coordinates of the original Minkowski spacetime as it is between the points in the moving frame of the Minkowski spacetime in moving coordinates.

This Minkowski metric in moving coordinates is

ds^2=c^2dt{^/}^2-dx{^/}^2-dy{^/}^2-dz{^/}^2

The observer is stationary in the moving frame so the x^/, y^/, and z^/ coordinates do not change. We will have

(x_2^/-x_1^/)=dx{^/}^2=0

(y_2^/-y_1^/)=dy{^/}^2=0

(z_2^/-z_1^/)=dz{^/}^2=0

(t_2^/-t_1^/)=dt{^/}^2=dτ^2

Here dτ is known as proper time. It is the name given for this particular situation where a clock is stationary in a frame moving between two points, so that the space coordinates do not change as above.

For an observer stationary in a frame in this moving Minkowski spacetime, and looking at a pair of points stationary in the moving frame, the Minkowski metric for the moving frame coordinates reduces to

ds^2=c^2dτ^2

Proper time, dτ, and the speed of light, c, are real numbers. A real number squared must be a positive real number, so we must have

c^2dτ^2\geq0

This implies that we must have

ds^2\geq0

This means that ds for the moving frame coordinates must be a real number with units of length.

If the moving Minkowski metric has ds² greater than or equal to zero between a pair of points stationary in the moving frame, then so must the corresponding points in the x, y, z, and t metric.

The laws of physics must be followed in one inertial frame just as they are in the other, so the laws of physics must be followed in one Minkowski spacetime, written in inertial frame coordinates, just as they are in the other. The laws of physics acting on any system of massive particles must be independent of the inertial reference frame they are viewed in.

The first Principle of Special Relativity requires that we are able to transform Minkowski metrics, as seen by an observer in one inertial frame, to one seen by an observer in another inertial frame moving with a constant relative velocity.

All pairs of points in the original rest frame Minkowski metric may be expressed as moving frame points stationary in a moving Minkowski metric. This must be possible if all inertial frames of reference are equivalent.

The requirement that the speed of light is the same in the moving frame and the rest frame means that ds is the same for corresponding pairs of points in the rest frame and the moving frame.

Since all pairs of points in the original metric in rest frame coordinates can be written as a pair of points stationary in a Minkowski metric in moving coordinates, all pairs of infinitely close points in it must have ds² greater than or equal to zero.

ds^2\geq0

Another way of stating this is that a Minkowski spacetime metric, in which all points satisfy the first Special Principle of Relativity, must have a real value for ds between all infinitely close pairs of points.

The mathematics means that if the interval squared is less than zero, the interval itself is an imaginary number. The requirement of the physics, in the form of the Special Principle of Relativity, means that if the interval is less than zero, the pair of points cannot exist in the spacetime.

We must have for the x, y, z, and t Minkowski metric.

 c^2dt^2-(dx^2+dy^2+dz^2)\geq 0

and so

\frac{(dx^2+dy^2+dz^2)^{\frac 1 2}}{dt}\leq c

There can be no pairs of points in the original Minkowski metric with a separation in space and time that implies a velocity greater than light.

If an observer moves between these two points with a constant velocity, v, then that velocity will be given by

\frac{(dx^2+dy^2+dz^2)^{\frac 1 2}}{dt}=v

and so

v\leq c

This tells us that points with coordinates, (x₁, y₁, z₁, t₁), and (x₂, y₂, z₂, t₂), in this Minkowski spacetime, with coordinates defined in this way, cannot have a separation in space and time that implies a velocity greater than c for anything to proceed from one point to the other.

Another way to state this is that the speed of light cannot be exceeded for any observer. It is an absolute speed limit. The structure of Minkowski spacetime is what causes this speed limit. There are no pairs of points in any Minkowski spacetime in which the Special Principle of Relativity holds, and for which ds² is less than zero.

This is more than simply stating that intervals for which ds² is less than zero can’t have any massive body move between them. It is saying that such intervals do not exist in Minkowski spacetime, or the spacetime we live in. There cannot be a massive body at one point, and another massive body at another point, in the same Minkowski spacetime, for which the value of ds² between the points is less than zero.

When ds = 0 in this Minkowski spacetime we get

0=c^2dt^2-(dx^2+dy^2+dz^2)

so

c^2=\frac{(dx^2+dy^2+dz^2)}{dt^2}

and

c=\frac{(dx^2+dy^2+dz^2)^{\frac 1 2}}{dt}

This is the equation of an expanding wavefront travelling at a velocity, c. Experiments show that light behaves like this in our Universe.

Minkowski spacetime doesn’t have anything to do with light, however. Minkowski spacetime doesn’t have light in it. The equations above are about the shape of Minkowski spacetime. It is saying that the coordinate elements, dt, dx, dy, and dz have this relationship with each other when ds =0.

The shape of spacetime in our Universe must have the same properties as Minkowski spacetime, but our spacetime is full of mass and energy. Albert Einstein developed his theory of Special Relativity to address how energy and mass were related to the new discoveries about light, and later developed and extended these ideas to include gravity in his theory of General Relativity. In particular he discovered the relationship between mass and energy in his equation for total energy.

Energy and mass

Is light a wave or a particle?

In 1905 Albert Einstein published his paper on the photoelectric effect in which he developed the relations between energy, E, and frequency, ν, for light quanta. Light energy being transmitted as lumps, or quanta, of energy had also been proposed by Max Planck to explain the observed distribution of energy in black body radiation. The energy of a quantum of light, known as a photon, is

E=h ν

Where h is Planck’s constant and ν is the frequency of the light.

The momentum, p, of a photon of light is also quantised, and is given by

p=\frac{h}{λ}

Where λ is the wavelength of the light.

The constant velocity of light, c, is given by

c=λν

so

ν= \frac{c}{λ}

From the photon’s momentum above we have

h=pλ

Substituting for h and ν we have

E=\frac{pλc}{λ}

so for photons we have

E=pc

Momentum, p, for massive particles is given by

p=mv

Where m is the particles mass, and v is its velocity. Einstein retains this classical definition of momentum in Special Relativity. It is the mass, m, that is redefined along with energy, E.

The conservation of momentum observed between photons, and charged massive particles, implies that photons have a quantity equivalent to mass, m. Light is observed to always travel with the velocity, c, so a photon’s momentum can be written as

p=mc

Combining this equation with the equation for a photon’s energy above gives Einstein’s equation for total energy, E, of a photon as

E=mc^2

The quantity, m, has units of mass, but, since c is constant, m must vary with the frequency and the wavelength of the light.

Since it is always travelling at the speed of light, a photon is essentially different to the massive particles like the electrons protons and neutrons that make up massive bodies. It cannot even be said to exist at a particular point in space at a particular time the way a massive body can.

It is because we measure their energy and momentum in discrete amounts, or quanta, that we think of them as particles. In fact their mass is always measured as part of their momentum or their energy; we never measure their mass directly on its own. They do not have what we call the rest mass of a particle. They are never at rest. Light is always moving at the speed of light

We infer the existence of photons from their interactions with matter. We never measure them directly.

Richard Feynman points out, in the Auckland Lectures², that light does not bounce off a surface when it is reflected; instead it is absorbed and re-emitted. When a photon interacts with an electron, a photon’s worth of energy is absorbed by the electron, along with a corresponding change in its momentum. On the emission of a photon, a further change of energy and momentum occur to the electron. If this happens sequentially, the effect is the same as if a discrete particle that we call a photon has bounced off the electron.

Quantum Mechanics, and Maxwell’s theory of Electromagnetism, tells us that it is re-emitted as a wave travelling in all directions. This wave is not localised in space. Only when it encounters another charged particle does the photon’s wavefunction “collapse”, with its energy appearing at a defined place once again.

If we are considering electrons emitting and absorbing photons, we only measure changes in the corresponding properties of the electrons involved. We can’t measure a photon’s separate existence as it “moves” between its source and its destination. It does not make sense to consider a photon to be a particle with a defined position in space, and with a mass, moving along like a little ball. It is only after absorption that a photon’s energy, and momentum, appears at a definite place. It is never measured as a discrete particle in its own right.

It does not really make sense to think of photons as particles at all. Rather than thinking of a photon as a discrete particle with a mass, it is better to think of light as the way that energy and momentum are transferred. This energy and momentum are exchanged in defined amounts, or quanta, but light also behaves like a wave that is not localised in space. A photon has a wavelength and a frequency. These are properties of waves, not particles.

The double slit experiment

It has been suggested that a photon can be imagined as a wavepacket; a wave that has an amplitude that has a value in a local region, but is zero everywhere else. This idea cannot account for observed behavior of light in the famous double slit experiment.

In this experiment light behaves as a plane wave as shown in the diagram below. This wave interferes with itself, and the intensity of the energy of the light is measured as maximum and minimum numbers of photons arriving at a screen, or other detector, in the directions where constructive and destructive interference occurs.

In the diagrams below the black lines are wavefronts. They show where the maximum amplitude of the waves are. Where these lines intersect the waves are in phase, and constructive interference occurs.

A single photon’s worth of energy can produce interference patterns, built up over time, that show that a single photon’s worth of energy has acted as a wave passing through both slits. This happens even if there is only one photon’s worth of energy in the experimental apparatus at any one time.

The diagram above could be showing one single photon. It would be spread out over the entire diagram, (not all the wavefronts are shown). We can imagine this single photon spread out over all points in the entire spacetime between the point of emission and the point of absorption.

While it is between its source, and its destination, each photon is behaving as if it is a wave going, as Richard Feynman puts it in his Auckland lectures², “in every direction it can”. It interferes with itself as it does so. The idea that a photon is a wavepacket that has any sort of localisation in spacetime has to be mistaken. When it is between emission and absorption it is a wave. When it arrives it is a defined quantity of energy and momentum.

In the Copenhagen interpretation of the wavefunction, Maxwell’s electromagnetic wave tells us the probability of finding a photon. This was the result of attempts, notably by Niels Bohr, but also Werner Heisenberg, and Max Born in Copenhagen, Denmark, to make sense of the new Quantum Mechanics that had been formulated by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 as Matrix Mechanics. Shortly afterwards Erwin Schroedinger developed his wave equation for matter waves which has been shown to be equivalent to Heisenberg’s Matrix Mechcanics.

In this interpretation the square of the amplititude of the wave function is taken to give the probability density of finding the corresponding energy of the “particle”. In the case of a light wave, this is the photon. The problem was that no one could suggest a process, or mechanism, for the way in which a wave turned into a particle.

There is no particular point where we can expect the energy to appear. We can’t know that in advance. It is a matter of probability, but this probability is not like the probability of taking the Queen of Hearts from the top of a shuffled pack. That probability of one chance in fifty two is because we lack the knowledge that the Queen is where she is. The reality of drawing the Queen is already fixed. There is only one future, and it is fixed in advance of us picking up the top card. The Queen is already there.

The probability of detecting a photon’s worth of energy in any area of the detection screen is not already fixed. The experimental apparatus does not contain the energy of the photon at a future time in a place already established. This realisation led Hugh Everett 3rd⁶ to suppose that each measurement made on a system leads to infinite possible futures in which every possible outcome is realised. This idea has become known as “The Many Worlds Hypothesis”.

Everett supposes that each possible future is equally valid. There is no hidden, as yet undiscovered, mechanism that uses information from the square of the amplitude of Maxwell’s and Schroedinger’s waves to create one particular future out of all the possible ones. Instead Everett suggests that all possibilities are realised. This neatly solves the problem of finding a hidden mechanism at the small cost of conjuring up an infinity of new Universes every moment.

If there is just one photon’s worth of energy in the apparatus there is no way to predict where that photon will be detected. Only the probability can be calculated from the square of the amplitude of Maxwell’s wave. The intensity pattern becomes the probability for where the energy of each single photon will end up. If the process is repeated for many single photon events, the distribution of points where a photon’s worth of energy has arrived matches the interference pattern.

If one slit is blocked off, the interference pattern disappears.

When one slit is blocked off the maximum intensity of the energy, and the maximum number of photons detected, is directly opposite the open slit, and no interference occurs. There are no constructive interference maximums to either side of the central maximum.

This is a clear demonstration of the fundamental misunderstanding we have when we imagine a photon as a particle. Does it arrive at the two slits, and then choose which one to go through? Could the direction of its subsequent travel to the detector be determined by whether the other slit is opened or closed?

In Minkowski spacetime, if photons could exist in it, the path of every photon would have ds equal to zero. The implication is that the photon leaves its source and arrives at its destination in the same moment. In Minkowski spacetime “at the same moment” means that ds equals zero, not dt equals zero. At this “moment”, when ds equals zero, a photon appears to be existing as a wave throughout what we are perceiving as time and space.

Is it meaningful to talk about time passing while a photon is “travelling” between the slits and the detector in the double slit experiment? Is it even meaningful to talk about a photon moving?

For the photon there is no time. In the photon’s frame of reference from our point of view, time has dilated to infinity. A clock moving along from the point of emission to the point of absorption would be stopped; it would show no time passing. In the photon’s moving frame, when viewed by a stationary observer, its emission and absorption are simultaneous. See “Length contraction and time dilation in the Minkowski spacetime” in “Moving frames in the Minkowski spacetime below”.

Energy is equivalent to mass

Einstein suggested that the equation, E = mc² for photons, also applies to for all massive bodies with velocities, v, less than the speed of light, c. He suggested that

E=mc^2

where

m=\frac {m_{rest}} {(1-\frac{v^2}{c^2})^{\frac 1 2}}

The rest mass, m_rest, is the mass when the relative velocity, v, is zero. The mass, m, giving the total energy, E, is greater than the rest mass, m_rest, by an amount we refer to as kinetic energy.

We can write this as

E=m_{rest}c^2(1-\frac{v^2}{c^2})^{-\frac{1}{2}}

To see why the two equations above must be true for massive particles, we can imagine an electron being accelerated in a uniform electric field, and gaining an amount of energy dE in a time dt . Energy, E, is defined as the work done when a massive body is acted on by a force, F, and it moves through a distance, x. So the increase in the electrons energy, dE, will be equal to the force, F, over a distance dx, as follows.

dE= Fdx

and we have Newton’s second law. The force, F, is the rate of change of momentum, p, so

F=\frac{dp}{dt}

so

dE=\frac{dp}{dt}dx

The rate of increase in energy over time will be given by

\frac{dE}{dt}=\frac{dx}{dt}\frac{dp}{dt}

Writing the energy of the electron, using the same equation as for photons, the energy E = mc². Also writing the momentum, p = mv , and dx/dt = v, gives

\frac{d(mc^2)}{dt}= v\frac{d(mv)}{dt}

Here we need to remember that the mass, m, in E = mc², when applied to photons, is variable, and we will assume, as Einstein did, that the same is true for massive particles when this equation is applied to them, so the mass, m, in the equations above is variable. The speed of light, c, is constant, and can be written outside the differential.

c^2\frac{dm}{dt}= v\frac{d(mv)}{dt}

We are about to discover exactly how mass varies with velocity in Einstein’s equation, E = mc².

We now use some calculus, and a bit of clever algebra, thanks to Richard Feynman¹⁰.

Multiplying both sides by 2m gives

 2mc^2\frac{dm}{dt}= 2mv\frac{d(mv)}{dt}

We know that the chain rule from calculus gives

\frac{dm^2}{dm}\frac{dm}{dt}=2m\frac{dm}{dt}

so

2m\frac{dm}{dt}=\frac{dm^2}{dt}

and

\frac{d(mv)^2}{d(mv)}\frac{d(mv)}{dt}=2mv\frac{d(mv)}{dt}

so

2mv\frac{d(mv)}{dt}=\frac{d(mv)^2}{dt}

so we can substitute into the fifth equation above, and we have

 c^2\frac{dm^2}{dt}=\frac{d(mv)^2}{dt}

Integrating both sides with respect to time

 \int c^2\frac{dm^2}{dt}dt= \int\frac{d(mv)^2}{dt}dt

 c^2\int dm^2= \int d(mv)^2

c^2(m^2+constant)=(mv)^2+constant

Combining constants gives

(mc)^2=(mv)^2+constant

When the velocity v equals zero, we know that the mass, m, equals the rest mass, m_rest, of the electron, and so

(m_{rest}c)^2= 0+constant

so

(mc)^2=(mv)^2+(m_{rest}c)^2

m^2=m^2\frac{v^2}{c^2}+m_{rest}^2

m^2-m^2\frac{v^2}{c^2}=m_{rest}^2

m^2(1-\frac{v^2}{c^2})=m_{rest}^2

m=\frac{m_{rest}}{(1-\frac{v^2}{c^2})^{\frac{1}{2}}}

This is the definition of relativistic mass, m. It has been derived above from Einstein’s expression for relativistic energy, E = mc², and relativistic momentum p = mv, together with the definition of energy as force multiplied by distance.

We assumed, as did Einstein, that E = mc² applies to all massive bodies. If an electron emits a photon with an energy, E = hν, it will lose a corresponding amount of energy, ΔE = Δmc², as a direct consequence, where Δm is the decrease in the mass of the electron.

We discover that mass must increase with velocity. There is only the variable relativistic mass, m, and the rest mass, m_rest, together with the Lorentz factor, in the equation above. It is telling us that mass really does increase with velocity.

From E = mc² and the definition of relativistic mass above we get

E=\frac{m_{rest}c^2}{(1-\frac{v^2}{c^2})^{\frac{1}{2}}}

This tells us that E , and m, both go to infinitely large as v tends to c. Einstein’s equation for total energy tells us that nothing can have a relative velocity to any observer faster than light. This is also the case in Minkowski spacetime.

Einstein’s equation for total energy above shows that all material massive bodies must have a velocity less than the velocity of light relative to an observer moving with a relative velocity v.

We saw above that Minkowski’s spacetime metric, together with the Lorentz transformations, shows that no two points can exist that can allow velocities greater than light.

Together with Einsteins equation for total energy above this strongly suggests that our spacetime must be similar to Minkowski’s spacetime. Einstein’s equation, and the equation for relativistic mass, along with the observed constancy of the speed of light in all inertial frames of reference, are telling us that we must look for a description of our spacetime that has the same structure as Minkowski spacetime, but incorporates mass and energy.

Energy and mass depend on the relative velocity of the observer

The equation for relativistic mass is disturbing. It is telling us that energy is equivalent to mass. It is telling us that as the kinetic energy of a massive body increases, the mass will increase. What is hard to accept is that the velocity, v, in the equation is relative to an observer. The implication is that the amount of energy a body has depends on who is observing it. Einstein’s equation for total energy is suggesting that an observer with a higher relative velocity to a massive body will observe a greater energy, and a greater mass for that body, than an observer with a lower relative velocity.

Looking at the way relativistic mass has been treated over the last century it becomes apparent that a great deal of effort has been put into avoiding looking at this clear implication of Einstein’s equation for total energy. Without a doubt, E = mc² is the most famous equation in physics, but most physicists don’t seem to want to engage with it. Nobody likes the idea that mass is dependent on the relative velocity of whomever is observing it.

It means, on the face of it, that two different observers with different relative velocities will observe different masses for the same massive body. The massive body will have different total amounts of energy.

Consider two observers in two inertial reference frames with a relative velocity to each other. They are each observing a proton stationary next to them. We know the rest mass of a proton very accurately, and the rest mass of all protons is expected to be the same.

Each of these observers should measure the same rest mass of the proton next to them, but, when they observe the proton next to the other observer, they see that it is moving, and they calculate a relativistic mass that is greater than a proton’s rest mass. They can’t both see their own proton with rest mass, and the other proton with a greater mass.

The problem is that the concept of rest mass itself seems to imply a unique rest frame with zero velocity. According to Einstein’s Principle of Special Relativity¹², however, there should be no special inertial frames of reference. They should all be equivalent.

We could in principle, measure the mass of subatomic particles, that we believe to have a constant rest mass, and use this measurement to determine a rest frame with an absolute zero of velocity. We could make measurements of the mass of subatomic particles at different velocities until we found a frame in which all the measurements of the mass of fundamental particles stationary in the frame were a minimum. That frame would be our absolute zero velocity frame.

If the laws of Physics are the same in all inertial reference frames, there should be no special ones in which subatomic particles have rest mass, and other inertial reference frames in which they don’t.

This gedanken experiment is clearly pointing out a fundamental gap in our understanding.

Richard Feynman points out that it should not be possible to conduct an experiment that can detect constant inertial motion. He says, in The Feynman lectures on Physics¹³

“whether or not one can define absolute velocity is the same as the problem of whether or not one can detect in an experiment, without looking outside, whether he is moving.”

The experiment described above could, in principle, detect whether someone is moving, without looking outside, by measuring the masses of subatomic particles.

As scientists we make experiments, and invent theories to explain what we see. We test our ideas against reality. As Richard Feynman says², “It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.”

Does this mean that we have to abandon Einstein’s Principle of Relativity ?¹² No we don’t. There is an assumption in the gedanken experiment above.

It is not obvious, but it turns out that we can’t have two inertial frames next to each other, and with different velocities. One, at least, of them will be displaced relative to the rest of the mass of the Universe, and will not be an inertial reference frame.

Inertial reference frames in our Universe

There are seemingly special inertial frames in our Universe. They are the frames in which the expanding galaxies have zero total momentum. They are the frames in which an observer sees the expansion as movement away from themselves. These observers are referred to as co-moving observers in this account.

The expanding Universe has all its points comoving with the expansion moving apart from each other, in the same way that currents in a current cake do as it is rising in an oven. The expansion can be viewed from any point as a stationary point, with all other co-moving bodies moving away from it. All co-moving points will see themselves as being at this central stationary point. If a second observer moves past one of these co-moving observers, however, they will not see the rest of the Universe apparently moving away from them. The point from which the expansion seems to be moving away from has moved ahead of them.

We can think of this second observer as moving through the expansion. Co-moving points next to this second observer are now moving past this observer, and the only point that is stationary relative to them is now a point ahead of them. Such bodies, moving through the expansion, could, in principle, be moving at any velocity less than the speed of light. They will then see themselves closer to a spherical boundary, defined by light speed, with a co-moving observer at the centre that is stationary relative to themselves. The whole sphere of the universe, defined by the expansion, and the boundary formed by the speed of light, will have moved ahead of them.

This boundary will have moved up behind the observer moving through the expansion. The central co-moving observer, the observer seeing the matter of the Universe moving away from them, will now be ahead of them. The central co-moving observer will still be at the centre of a sphere of matter, and this sphere of matter will be ahead of the moving observer. Consequently, the moving observer will be in a gravitational field, and so they will not be in an inertial frame of reference.

This saves Einstein’s Principle of Special Relativity. One of the two protons in the gedanken experiment above, where two protons in relative motion are each observed by observers next to them, will not be in an inertial reference frame.

See “Moving through the expansion” in the section“The equivalence of inertial and gravitational mass”, below for more on this idea.

Only co-moving observers, seeing themselves as central to the expansion, can say that they are in an inertial reference frame.

Choosing the center of mass frame

Einstein’s equation, E = mc² leads to disconcerting consequences unless the situation it is applied to is carefully chosen. If an inertial frame is selected in which the centre of mass of a system is stationary, Einstein’s equation can be applied to the massive bodies moving within it.

This is the case for frames of reference moving with co-moving observers. It is not the case, in general, for all frames moving with a constant relative velocity to co-moving observers.

It is possibly for this reason that most physicists seem to prefer this equivalent expression to E = mc²

E^2=p^2c^2+m_{rest}^2c^4

Here E is the total energy, p is the momentum, and m_rest is the rest mass.

Here the increase of mass is disguised as an increase of momentum, p. When applying this equation it would be natural to choose a frame where the centre of mass has zero velocity.

In a system of massive bodies, viewed from a frame of reference stationary with respect to their centre of mass, their total momentum adds to zero. We could, in principle, see all their kinetic energy taken away in inelastic collisions. We would be left with all the bodies, stationary in that frame, and with rest mass.

In a centre of mass frame we can expect the equation above to be valid.

In the equation above, if the momentum of each massive body, p, is zero, the total energy they each have is given by

E^2=m_{rest}^2c^4

so

E=m_{rest}c^2

In this closed system, with the restriction that the reference frame is chosen as the one moving with the centre of mass of the entire system, we can apply E = mc², but Einstein’s equation is assumed to apply universally in all inertial frames of reference.

Einstein’s Principle of Special Relativity does not state that the laws of physics are the same in all inertial frames moving with the centre of mass of a closed system.

It states that that the laws of physics are the same in all inertial frames.

In Special Relativity the momentum of a massive body keeps the same form as in Newtonian Mechanics, but m is now variable, as we saw above, The mass, m, in this equation defining momentum, p, is still the relativistic mass that depends on velocity.

p=mv

and

m=\frac {m_{rest}} {(1-\frac{v^2}{c^2})^{\frac 1 2}}

so

p=\frac {m_{rest}v} {(1-\frac{v^2}{c^2})^{\frac 1 2}}

This shows that, as the mass of a massive body increases with velocity, the momentum, p, also increases in step with the mass.

The equation above for total energy, E, momentum, p, and rest mass, m_rest, can be written

E^2-p^2c^2=m_{rest}^2c^4

Substituting for E and p gives

m^2c^4-m^2v^2c^2=m_{rest}^2c^4

We should note here that m is the relativistic mass that increases with relative velocity. Rearranging this gives

m^2c^4(1-\frac{v^2}{c^2})=m_{rest}^2c^4

m^2c^4=m_{rest}^2c^4 (1-\frac{v^2}{c^2})^{-1}

mc^2=m_{rest}c^2 (1-\frac{v^2}{c^2})^{-\frac{1}{2}}

E=m_{rest}c^2 (1-\frac{v^2}{c^2})^{-\frac{1}{2}}

We see now that this is the original equation for total energy given to us by Einstein.

E=mc^2 

Where

m=\frac {m_{rest}} {(1-\frac{v^2}{c^2})^{\frac 1 2}}

The factor

 (1-\frac{v^2}{c^2})^{-\frac{1}{2}}

goes to infinitely large as v tends to c, so any mass, m, gets infinitely large as v tends to c.

Although the increase of mass is disguised as an increase in momentum, we can see that mass is still increasing with relative velocity, just as Einstein’s equation for total energy is telling us that it does.

The mass, m, is relativistic mass. It increases with relative velocity, v. The relative velocity, v, between an observer and the mass, m, determines the size of the mass m.

Which massive body is moving?

If a charged particle is accelerated, in a laboratory on the surface of the earth, to a velocity where the increase in its mass becomes noticable, it is natural to select a reference frame in which the laboratory is stationary, and the charged particle is moving.

It might be acknowledged that the laboratory is not quite stationary, and not quite an inertial frame, but it is assumed that the consequenses will be minimal when compared to the velocity of the charged particle. The velocity of the laboratory through space due to the Earth’s rotation, and its orbit round the Sun; together with the Sun’s velocity round the galactic centre, and through the local cluster of galaxies, is negligible compared to the velocity of the charged particle in the accelerator.

There is no reason to select the frame of reference centred on the laboratory, however. There is nothing about Einstein’s equation for total energy that requires this selection, and indeed, his Principle of Special Relativity requires that we could view the situation from an inertial frame in which the particle is at rest, and it is the laboratory that is moving.

In that case we would expect the laboratory, the Earth itself, and the rest of the Universe, to have its mass increase, and the particle to have its rest mass.

We saw that the preferred equation above, for relativistic energy, momentum, and rest mass, is actually the same as Einstein’s equation for total energy. The problem of explaining mass increase with velocity has not gone away, and it is not resolved by carefully selecting the reference frame applied to the situation being examined.

The last hundred years has seen a concerted effort not to engage with with this uncomfortable fact. This is a pity because it is precisely the most difficult things to understand that are usually the things that are challenging our assumptions. It is the things that make us feel uncomfortable that we need to focus our attention on. They are exactly the things we must not avoid looking at. They are exactly where the next paradigm shift is waiting to happen.

How can two observers, each with a different relative velocity to a massive body, see the same massive body, in the same Universe, at the same time, but having a different mass? This is what upsets physicists, and we should be upset. Mass is supposed to be an intrinsic property of a body. Why is it dependent on the velocity of an observer?

The velocity of an observer being included in the most famous equation of them all, E = mc², is going against the notion that physics is observer independent.

Kinetic energy in our Universe

If two massive bodies start together with zero relative velocity, and one is accelerated, we can expect the one that has undergone a change in momentum to have its mass increase. The observer is considered to be with the unaccelerated massive body. We can tell which body has been accelerated because the rate of change of momentum multiplied by the distance will take the amount of energy we expect to appear as an increase in mass.

We can’t use that argument in the case of massive bodies moving relative to each other due to the expansion of the Universe. In that case how do we decide which body has rest mass, and which body has an increase in rest mass due to velocity? We can’t use the energy argument above.

The two bodies we are considering might be two galaxies many light years apart, and moving with a relative velocity to each other. They are in an equivalent situation. Each may consider themselves to be at rest at the centre of the expansion.

The Universe is expanding, and the expansion means that the distant galaxies are moving away from anywhere they are observed from. The velocity that the galaxies are receding with increases with distance according to Hubble’s law.

v = H r

Where v is the relative velocity, H is Hubble’s constant, and r is radial distance from an observer.

Hubble’s law tells us that there is a distance away, R, that corresponds to the speed of light, c.

c = HR

So Einstein’s equation for total energy predicts infinite mass for these objects, and also predicts that they will have infinite energy.

In Euclidean space and Newtonian time, bodies further away than R will have a velocity greater than the speed of light, and their energy cannot be defined.

If we apply Einstein’s equation for total energy, together with Hubble’s law, to a Universe existing in Euclidean space and Newtonian time we get these impossible infinities.

Infinities like these appearing in physics are often indicative that our theories need to be carefully examined for fundamental flaws. In this case it is our belief in Euclidean space, expressed in the rectangular coordinates x, y, and z, going to infinity in all directions, together with Newtonian time, t, flowing at each of these points at a steady rate without reference to anything else, that we need to look at more carefully.

Hermann Minkowski’s revolutionary new type of space and time accommodated the prediction of Einstein’s total energy equation that massive bodies could not have a relative velocity greater than the speed of light. We have seen above that it was based on the proposition that all observers, stationary in an intertial reference frame, should see the same speed of light regardless of their relative velocity.

Einstein’s total energy equation is telling us that massive bodies with a relative velocity to an observer equal to the speed of light would have infinite energy. There is no limit to the maximum velocity in Euclidean space and Newtonian time, but there is a limit in Minkowskian spacetime, as we see above.

The problem that the expanding Universe presents us with is that, in Euclidean space and Newtonian time, we can imagine massive bodies, like galaxies, continuing for ever. In a Universe that is expanding in the way we perceive it to be, there would be an infinite number of massive bodies moving faster than the speed of light. This is clearly in contradiction to Einstein’s total energy equation. We can’t have Einstein’s total energy equation, E = mc², and keep Euclidean space and Newtonian time.

We can’t be living in a Minkowski spacetime either. In Minkowski spacetime we can’t have mass or energy, but even if we could, there would be the problem of massive bodies in the expanding Universe approaching the speed of light and consequently having infinte mass and energy.

The Minkowski metric in our Universe

The intervals, ds, make up the entire space time

The whole of a Minkowski spacetime, satisfying the two Principles of Special Relativity, may be created, starting from one point, and finding all the other possible points for which ds² is greater than, or equal to, zero. If an observer is at this starting point, there will be nothing in their spacetime that can be moving faster than light. There is no spacetime for anything moving faster than light to exist in.

We saw above that the the interval, ds, cannot be an imaginary number in any Minkowski spacetime in which the Principle of Special Relativity holds. There cannot be a space interval between two points that doesn’t also have a time interval that gives the quantity ds a real value.

This fits our observation that all massive bodies in our Universe are moving at less than the speed of light, and light itself moves at the speed of light.

It is the shape of Minkowski spacetime that determines the limiting velocity of light. For Einstein’s equation for total energy to be valid in our Universe, our spacetime must have an equivalent structure to a Minkowski spacetime, for which ds² is greater than, or equal to, zero.

It is the intervals, ds, that collectively make up the entire spacetime. We have seen that two points can only coexist in Minkowski spacetime if they are separated by an interval in space and time that is a real number. This implies that the two points are separated by a distance and time that correspond to a velocity less than the speed of light.

An observer with a real massive body in our spacetime can only move between two points if they are moving at less than the speed of light. This suggests that the structure of our spacetime corresponds to Minkowski spacetime. It cannot actually be Minkowski spacetime, though. Minkowski spacetime doesn’t have energy, mass, or gravitational fields, in it.

Until Minkowsk proposed his spacetime it was generally assumed that space was Euclidean, by which it was meant that all space could be measured using three coordinate axes, x, y, and z. These axes extend at right angles to each other in all directions. Until the development of differential geometry by Riemann and others, it was assumed that this was the only sort of three dimensional space there could be.

Riemann developed a differential geometry in which a space of any number of dimensions could be represented. These spaces could be curved in higher dimensions. A two dimensional plane is not necessarily flat. An example of a two dimensional surface curved to make it into a sphere in three dimensions is shown above in the section “Differential geometry simplified”.

Until the advent of differential geometry, it would have been imagined that three dimensional space could be represented by three coordinates, eg x, y, and z, and we would still be imagining that the three dimensional Pythagorean metric below applies to our Universe, as follows.

ds^{2}= dx^{2}+dy^{2}+dz^{2}

With a quite separate time axis, t, in which time flows uniformly, and without reference to anything else, as Newton thought it did. This three dimensional space is flat in the same sense that a two dimensional table top is flat. It is not curved in a higher dimension.

In this space there is no apparent limit to the size of a relative velocity. Any distance, ds, could be combined with any time, dt, to give a velocity, v, of any size.

\frac{ds}{dt}=v

The time axis is unconnected with the space axis. We may imagine all the points in an infinite space existing without time. Time is then added to this concept. There is no reason why there should not be two different space coordinates with the same time coordinate. This description of an infinite space with a separate time does not rule that possibility out.

In fact the coventional idea of a present moment, that is shared by all observers, is imagining that it is possible for there to be a complete set of space coordinates, going to infinity, all with the same time coordinate. All observers would agree that their clocks were showing the same time. This is what the present moment is for most people.

Any real length, ds, is possible with this view of space and time. All these lengths may have dt = 0 between their end points. When we imagine dt = 0, we are imagining a particular moment in time that is the same for all the infinite number of space points.

This does not seem unusual. It is the way physicists had been imagining space and time for centuries until Minkowski realised that a new description of space and time was needed to explain Einstein’s relation between mass and energy, and the observed behaviour of light.

The discovery that light had a fixed speed that was the same for all observers, together with Einstein’s equation for total energy meant that using Euclidean space and Newtonian time to describe the new ideas, and experimental observations, in physics had become untenable.

Minkowski included the time axis in his metric, as shown above. The infinitesimal coordinate interval, ds, now included infinitesimal coordinate intervals of time, dt. These time intervals were multiplied by a constant, c, which in our physical world was the speed of light. The length cdt appeared in Minkowski’s differential geometric equation. This is the distance light would cover in a time interval dt.

The intervals, ds, that collectively made up the entire space and time, or spacetime as it was now referred to, was a measure of a “length” between two points, but these “lengths” contained the infinitesimal time intervals, dt, mathematically combined with with the conventional length intervals dx, dy, and dz. The very concept of how length and time were to be measured had changed.

Perhaps even more importantly, space and time were recognised by Einstein and Minkowski to be more than the canvas on which reality was painted. They were not a purely mathematical construction to aid in the description of real things. They were, themselves, part of the reality of the Universe. Space and time were intimately bound up with mass, and energy, and they were intimately connected with each other.

It is important to realise the magnitude of this change. It is the intervals, ds, that define, and make up, the spacetime. Lengths and times are no longer separate entities. Referring to one without the other is no longer meaningful.

On 21st September 1908 Minkowski presented his paper Raum und Zeit, (Space and Time), at the 80^th Assembly of German Natural Scientists and Physicians . In it he said:

“Gentlemen! The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”⁹

At the speed of light the infinitesimal length intervals, ds, equal zero, and, for intervals that would imply motion greater than the speed of light, ds is imaginary. This implies that there are no other points except the ones for which ds² is greater than, or equal to, zero. A major consequence is that there can be no such thing as a pure length in Minkowski spacetime.

A metre rule no longer measures a metre. To mark its two endpoints in Minkowski spacetime, it is necessary to use a clock as well as the rule. Meter rules, with the same time coordinate at both ends, cannot exist in a Minkowski metric. Every measurement of space must involve a measurement of time as well.

We know that the Minkowski metric is not a full, or complete, description of our spacetime, but we do believe that it is the basis for a description of our spacetime. It is the basis of Schwarzschild’s spacetime. The Schwarzschild spacetime is Minkowskian in the limit at an infinite distance from a gravitating mass. At infinity it becomes Minkowskian, not Euclidean.

We do not yet have a solution of Einstein’s field equations for more than one mass. We will see below however, that Schwarzschild’s spacetime equation can be modified to show how a spherical distribution of mass can create a model of the spacetime structure of our universe.

This model will show how a Universe full of matter can give the same structure of spacetime that the Minkowski spacetime, together with the Lorentz transformations, show are necessary for all observers to see a uniform speed of light.

We will see that the behaviour of light we observe determines the structure of the entire Universe. The implications presently being discussed with reference to the Minkowski metric will be seen to apply to the Universe. We will see that the existence of the Universe depends on it being observed.

There is only one observer in the present moment

Physics had not included a present moment until Minkowski proposed his spacetime metric which gave time an equivalent status to the three dimensions of space. Minkowski’s spacetime still did not address the way time appeared to flow to an observer, but it did have a distinctive place that could be called the present moment. It is the place where an observer receives light that has been emitted from a different point in the Minkowski spacetime.

In Minkowski spacetime ds² must be greater than, or equal to, zero. A particularly disturbing, or perhaps enlightening, consequence, is that there can be no other points, or observers, at the same time, and at a different place in space, as an observer in the present moment. “At the same time” would imply dt equals zero, and an inspection of the Minkowski metric above will show that, if dt² = 0, then so must (dx² +dy²+dz²) = 0. This implies that if the time interval dt = 0, then the length interval (dx² +dy²+dz²)^1/2 = 0

For an observer in a Minkowski metric, all other points in the metric are in their past, or their future. Only one observer can exist at the present moment in a Minkowski metric.

We can go further and say that if there is a point in Minkowski spacetime there can be no other points with the same time coordinate. We have to abandon the picture of a Universe full of matter that all exists at the same time. That conceptual picture is of a Euclidean space with a separate time proceeding independently. It doesn’t work as a picture of reality.

The light cone

The “light cone” refers to a graphical representation of light paths in Euclidean space and Newtonian time. It may be used to discuss how Minkowski spacetime differs from Euclidean space and Newtonian time.

Euclidean space is considered to continue to an infinte distance in all directions, and may be measured by three coordinate axes perpendicular to each other. Newtonian time in his own words is described as follows “Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration. “

The three space dimensions of Euclidean space can be represented as a two dimensional horizontal plane, with the vertical axis used to represent Newtonian time. An expanding sphere of light is represented as a circle on this plane. This forms a “cone” with it’s apex being a point where the light is supposed to have originated. We can call this the future light cone. An inverted “cone” is taken to show the path of light rays travelling to the point at the apex. We can call this the past light cone.

The “past light cone” is actually spheres of points in three dimensional space, continuing to an infinite distance in the r direction, together with a single time coordinate attached to each one. We can imagine concentric spheres of space coordinates, with the time coordinate getting less with increasing radial distance, r.

This is often further simplified with all three dimensions of space represented as a single horizontal axis. The vertical axis is time as before. This forms an “X” on a space and time graph, with the arms of the “X” representing the velocity of light.

The “X” of the conventional spacetime diagram has an observer at the cross over point. 

Points in the left and right hand regions, “>” and “<“, imply that a signal between these points, and the central observer, would have to travel faster than light. It is shown above that the Minkowski spacetime implies that there is an imaginary value for ds between these points, and the central observer.

We do not live in a Euclidian geometry

We need to remember, at this point, that we do not live in a Euclidian geometry. The space-time diagram drawn above is assuming Euclidian geometry. Space and time are shown unconnected with each other, and go on to infinity in all directions. The conventional space-time diagram cannot be representing our space-time, and we should not base our understanding on it.

The Minkowski spacetime cannot be represented by Euclidian space and Newtonian time. It cannot be superimposed onto Euclidian and Newtonian space and time. In particular, because the interval, ds, is an imaginary number in the left and right hand regions, “>” and “<“, Minkowski spacetime doesn’t include these regions. Points in these regions can’t exist.

If the Minkowski metric is a complete description of all possible points in space and time then there are no points in the immediate future of any point on the past light cone except for the single point at its apex.

The implication is that the observer at the apex of the light cone in Minkowski spacetime has proceeded in time further than all other points. We can think of this point in spacetime as the present moment.

If the other points on the light cone are in the past of the observer at the apex of the light cone, then less time has passed for them. If less time has passed then time must be passing more slowly. If time is passing more slowly, then time is dilated from the point of view of the observer at the apex of the light cone. If time is dilated in our Universe, then we can expect that either the point on the past light cone is moving, or it is in a gravitational field. These are the only causes we know of for time dilation to occur.

We will see below that we can attribute this time dilation both to velocity, and to the distribution of mass in our Universe. We will see that, for the overall structure of our Universe to be the same as the Minkowski metric, have a uniform speed of light for all observers, and have the speed of light being the maximum relative velocity possible, the distribution of mass in spacetime does indeed have a structure that reproduces the same conditions as the Minkowski metric.

Points in space and time that can’t exist

If our Universe has the same form as Minkowski spacetime, it is not the case that an electron, next to a second observer at a different point on the light cone, could emit a photon in the direction of the first observer, and then carry on moving forward in time while that photon travels to the first observer. They cannot be at that same distance at the same time that the photon arrives the first observer. “At the same time as” implies dt = 0, so the length interval (dx² +dy²+dz²)^1/2 = 0. There cannot be another point at any distance away from an observer at the same time. There are no other points in Minkowski spacetime at the same time as either observer.

The spacetime diagram below, like the one above, cannot be a true representation. We cannot suppose, from looking at the diagram, that there is an observer where the light signal is emitted, and an observer where it is received, and that these two observers are both in a Universe that can be measured with Euclidian space and Newtonian time. The space and time axes shown are Euclidean and Newtonian.

The lightcone diagrams shown here are more useful to show what Minkowski spacetime is not like, than what it is like.

There are no spacetime coordinates in the future of the emitting electron from the viewpoint of the first observer. There is no Minkowski spacetime in the future of the emitting electron for it to exist in. At the moment that the first observer receives the photon, the second observer cannot have moved forward into the region where signals would have to be moving faster than the speed of light to reach the first observer. Points in such regions do not exist at the point in time that the first observer receives the photon.

These regions are sometimes called “spacelike”, but naming something doesn’t mean that it is understood. In this case the name “spacelike” is a barrier to understanding. Points in this region are not spacelike, they are not real; they can’t exist; they are not possible.

It is clear that time is not the same as space. It can be measured in seconds in a way similar to the way space can be measured in metres, but it has a distinctive difference. It has a present moment that appears to progress along the time axis at a steady rate. This present moment always progresses in the same direction from the past to the future. It is clearly connected with our present awareness, but no one seems to want to acknowledge this. No one seems to want consciousness to make an appearance in physics. The present moment did not feature in Newtonian physics at all.

The collapse of the wave function

In Quantum Mechanics the “collapse of the wavefunction” is associated with the measurement of a real quantity possessed by a body. This could be it’s position, it’s time, it’s momentum, or it’s energy. It is when Schroedinger’s wave, that is spread out over space and time, is taken to give the probability of a particular measurement. The wave disappears, or “collapses”, as the measurement is made. No one has been able to explain why this should happen when a measurement, or observation, is made.

Quantum mechanics, with its unexplained “collapse of the wavefunction”, seems to be implying a present moment in which this happens; a point in time at which the definite past ends, and the future stops being a probability, predicted by the evolving Schroedinger wave function, and becomes an established observation.

If this measurement is an observation of a body in our Universe it will involve the transmission of light, from the place and time the measurement is being made, to the observer making the measurement. If our Universe has the same structure as Minkowski spacetime, there will be an interval, ds, between the observer, and what they are measuring, equal to zero.

The interval is zero between the observer and all points on the light cone, so the observation, and the measurement, that is associated with the collapse of the wave function, is all points on the observer’s light cone.

In other words the collapse of the wavefunction is synonymous with the entire light cone as it is perceived, and measured, in the present moment by an observer in that moment.

We saw above that if there is an observer at a point (t₁,x₁, y₁, z₁ ), then a different point (t₂,x₂, y₂, z₂ ) cannot exist for which t₂ – t₁ = 0.

Other points can exist on the past light cone. These are points from which a light signal could travel to the observer. These points are all in the observers past.

This observer has spacetime coordinates on their future light cone. They are all the points that could be connected by a light signal. That light signal could proceed to any point on the future light cone, but at the present moment it hasn’t yet done so. There is just the possibility of it leaving the present observer, and travelling off in all directions.

All that actually exists, that can be measured and observed at any one moment, is the observer, and all the other points on their past light cone.

Until light arrives, as a photon of energy, at its interaction with another particle, and is absorbed, along with the “collapse” of its wave function, it is the electromagnetic wave as described by Maxwell and Faraday. In Quantum Mechanics, Maxwell’s electromagnetic wave is taken to be its Schroedinger wave function.

Is it possible that a photon can interact with an electron without an observation of it being made? Quantum Mechanics is clearly suggesting that it isn’t. Is it possible that our Universe can exist without being observed? Minkowski spacetime is suggesting that it isn’t.

While it is between emission as a wave going in all directions from its source, and its arrival as a photon of energy at a point of space and time in the future, the value of ds is zero. ds is equal to zero between the present moment, and all points on the “surface” of the light cone.

Only the point at the apex of the light cone has a point in its immediate future. As this point moves into its future by an amount of time, dt, space and time must appear for matter on the light cone to move into. It appears that spacetime is being created as an observer at the apex of the light cone moves forward in time.

The Copenhagen interpretation of Quantum Mechanics suggests that all particles are Quantum Mechanical probability waves before the “collapse of the wave function” associated with a “measurement”. This “measurement” could be taken to be the observer’s observation of the entire Universe on the past light cone.

We can imagine that this “collapse of the wave function” has just happened on the past light cone as we make an observation of the entire Universe. Information about this collapse on the past light cone travels to the observer at the apex of the light cone, apparently at the velocity of light, but actually all with an interval, ds, equal to zero.

The Minkowski metric is dependent on an observer

The existence of a point, where we are observing from, that is ahead in time compared with all other points where an observer could exist, makes the entire Minkowski metric dependent on that observer. As that observer moves into the future, the points of space and time that comprise the Minkowski metric change. In effect a new metric is created for each infinitesimal move forward in time that the observer makes.

If our Universe behaves like Minkowski space time, the past light cone is in the present moment of time in which the entire Universe appears to the observer. The present moment is not all points when dt equals zero, it is all points when ds equals zero.

For the observer at the apex of the past light cone, the spacelike regions do not exist, and neither does the future timelike region. All that exists for that observer are the points on the past light cone.

It is important to realise that this is not just an apparent effect due to a point of view. If our Universe is like Minkowski spacetime, it is a complete description of the observer’s actual reality at that moment.

The present moment moving in time

We can imagine an observer, in the present moment in Minkowski space time, moving to a point with a time coordinate an amount dt further ahead in time than the point in space and time they are presently at. As the observer in the present moment, here and now, proceeds forward in time, all of the other points on its past light cone will also have points appear in the immediate future ahead of them in time.

A new light cone comes into existence with points a time interval, dt, ahead of the points on the previous moments light cone.

The Universe we perceive as progressing in time seems to be a sequence of past light cones, with only the present one having reality for an observer in the present.

As a metric space described by differential geometry, the Minkowski space time is a description of all the points in the space and time of that particular differential geometry at the present time of the observer. It is a complete description. There are no other points.

As the observer moves forward in time, a complete new Minkowski spacetime must be generated at each infinitesimal increment of time, dt. The Minkowski spacetime itself gives no clues about this process.

In what has been presented above we are assuming that our Universe must have the same properties as Minkowski spacetime, but we need to remember that we do not live in Minkowski spacetime. Minkowski space time does not contain light, energy, mass, or gravitational fields.

As a description of the space and time we live in, Minkowski spacetime has properties that make it a better description than the Euclidian space, and separate absolute time of Newton. It is clearly not a complete description of our actual space and time.

What actually exists?

For us, it does not appear that matter ahead of us in time exists yet. It does not appear that past events still exist. Points in the so called spacelike regions can’t exist. The “surface” of the past light cone does, in some respects, correspond to our perceived reality in the present moment, but it does not show us how time flows.

If there were another observer on the past light cone of the first observer, then that observer, on the past light cone, could not have the first observer in its Minkowski metric. The first observer cannot have proceeded through a region of spacetime in the second observers Minkowski metric for which ds² < 0, to arrive in the second observer’s future. The implication is that they cannot be in the same Minkowski metric.

Nothing can get through a region of spacetime for which ds² < 0 because there are no points of space and time for it to exist in while it does so. There is no region of Minkowski spacetime for which ds² < 0. There appears to be such a region on the conventional Euclidean spacetime diagram above, showing the light cone, but it doesn’t exist in Minkowski spacetime. This means that there can be no matter in the future of the sole observer in a Minkowski spacetime. The future does not exist. This will be true for all observers.

We need to remember at this point that we have discarded the Euclidian, and Newtonian, concepts of space and time. They are clearly inadequate to describe the Universe we live in. Minkowski spacetime is better, but we have yet to include gravitation. That was what Einstein’s General Relativity set out to do. We can no longer use Euclidean and Newtonian concepts to imagine how our Universe actually behaves. We cannot point to a region of the light cone diagram above, and talk meaningfully about any event happening in regions that are “spacelike”.

If our Universe behaves like Minkowski spacetime, then no material object can have proceeded further in time than the “surface” of the past light cone.

The emission and reception of light are simultaneous

When we look out in space, we are looking back in time. What we see, at a distance, r, away, is in our past. It has not, however, proceeded into our present moment, as we have proceeded through time. For anything we see, time has progressed for it up to the point where it emitted the light wave.

In a Universe with the structure of Minkowski spacetime, an electron emitting the light can’t progress forward in time as the light proceeds from emission to reception. Such movement through time would be in the spacelike regions relative to the receiving electron. Time cannot be moving forward for emitter and the reciever at the same rate for both, and without reference to either, as Newton proposed.

Time has proceeded to to its furthest extent at the emitting electron and at the receiving electron. We will see below in “Moving frames in the Minkowski spacetime” that an observer at the point of emission, and an observer at the point of reception will both see time stopped in a reference frame moving at the speed of light. This suggests that Emission and reception happen simultaneously. It is the shape of the light cone in spacetime, that it appears to travel on, that results in a distance travelled, and a duration of time that it travels in.

The shape of Minkowski spacetime is determined by the constant, c. That shape gives light the appearance of taking time to leave its source, and arrive at its destination. Light is not travelling at the speed, c, through Euclidian space because it is not travelling in Euclidean space.

It is not that nothing is “allowed” to go faster than light in the Minkowski metric. The limitation of the speed of light is about the structure of spacetime itself. The interval, ds, is an infinitesimal length in the spacetime. Intervals of time in the Minkowski metric are multiplied by the speed of light to make them into lengths. We have seen that the value of ds² that the Minkowski metric calculates cannot be negative because lengths cannot be imaginary numbers.

The impossibility for anything to move between two points faster than light is because two such points can’t exist in the spacetime. If a point exists in the Minkowski metric, a second point cannot exist in the metric if ds² between the two points would be calculated to be less than zero.

The future is not mapped out

If we stop to think about this, it matches our personal experience of time better than a picture of a Universe in which the future is all mapped out ahead of us. It fits the requirement of Quantum Mechanics that the future is a matter of probablility, not certainty. It suggests that the probabilities of Quantum Mechanics are not a lack of knowledge, they are intrinsic. The future doesn’t exist until it happens in the present moment, and the present moment may contain any of the spacetime points on the past light cone, together with the point at the apex.

In “A photon is the smallest tick of a clock” below it is suggested that one photon travelling between emission and absorption is the smallest amount of change there can be. In that case what we conventionally think of a the light cone, may only contain one photon at any particular moment of time.

The conventional view of most physicists seems to still be of Newton’s three dimensions of space, in which real particles, with mass and charge, navigate their way from particular points in the past to other particular points in the future, with Quantum Mechanics to tell us the probability of a particular outcome instead of the certainty suggested by Newtonian mechanics.

With this view the future exists as a predetermined certainty, but we can only make probabilistic predictions about it, and these probabilities are provided for us by Shroedinger’s waves. The difficulty with this view is that the waves do seem to have independent reality in their own right. In the case of photons the waves are made from real electric and magnetic fields.

No one has, as yet, proposed a reason, or a mechanism, for waves to transform into particles.

The equations of General Relativity are still expected to describe space and time in the past, the present, and the future, as if time is flowing independently along, in the same way for all observers, the way that Newton thought it did. Everyone agrees that the new idea’s of Einstein, Minkowski, Schroedinger, and Schwarzschild have changed everything, but they mostly seem to still think of the structure of the Universe as if they haven’t.

An example of this is the popular use of the term “spacelike” to describe the regions in the old Euclidean space where the interval in the Minkowski metric is imaginary. The Euclidean view of space is accepted to be untenable, but by calling this region “spacelike” it is imbued with a legitimate reality. Having given it this label, many physicists continue to treat it as if real matter can exist in it. It is very modern, and sophisticated, to say “spacelike” and “timelike”, as if using these terms conveys some sort of esoteric wisdom, but, for those who ignore the real implications of the Minkowski and Schwarzschild spacetimes, and continue to think of space continuing in all directions forever, with time flowing independently at each point, that sophistication is an illusion.

Light cones can’t be superimposed onto Euclidean space and Newtonian time

If the spacetime of our Universe has the overall properties of Minkowski spacetime with respect to the behaviour of light, then it does appear that each observer is on their own in their own Minkowski spacetime. We are not sharing that space time with other observers.

It appears that each moment for each observer is a separate, distinct, Minkowski spacetime. At each moment every observer will see themselves as being at the apex of the past light cone in that spacetime, but it is not the case that there are many observers in the same Universe with each perceiving themselves to be at the apex of a light cone, as is shown in the diagram below. This is not what the Minkowski metric is telling us.

The Minkowski metric and the Schwarzschild metric are depicting the spacetime of an entire Universe. These Riemannian geometries are mathematical prescriptions for generating all the points in a spacetime. In the case of the Minkowski metric it is a Universe devoid of mass and energy. In the case of the Schwarzschild metric described below, it is a Universe with only one gravitating mass in it. It is clear that neither are complete descriptions of our Universe full of mass and energy.

They are approximations of our Universe, and they are better approximations than a Euclidian space with separate Newtonian time. The Euclidean space and Newtonian time model does not work at all. It cannot give the experimental and theoretical result of a Universe where there is a uniform speed of light for all observers, and an absolute velocity limit of the speed of light for the relative velocities of massive bodies.

The Universe we live in must have the overall structure of a Minkowski metric, and include the effects on space and time that the Schwarzschild metric predicts for a single mass. It is not Euclidean space and Newtonian time with Minkowski spacetime applied for observers at every point, and Schwarzschild spacetime applied near a massive body. That attempt to force the new ideas into the old simply will not work.

Little light cones are often depicted on spacetime diagrams as if each point of the old Euclidian space, and Newtonian time, had one of these little light cones attached to it. Space and time axes of Euclidian space, and Newtonian time, are drawn, extending to infinity in all directions. Lots of light cones, all over the spacetime “surface” are depicted, having these Minkowski metrics attached to each point. These light cones are shown in the past, and in the future of Newtonian time.

This has to be a misrepresentation. There is no Euclidian space, and Newtonian time, to attach these Minkowski spacetimes to. Minkowski spacetime is not just valid in infinitely small regions of Euclidian space, and Newtonian time surrounding an observer; it is valid at every point that the Minkowski metric predicts that a point will exist. Perhaps more importantly it is not valid at points that the Minkowski metric predicts that a point cannot exist. The Minkowski spacetime consists of all such space and time points. It doesn’t consist of any points in Euclidian space, and Newtonian time.

Minkowski spacetime, as a Riemannian spacetime metric, is describing all there is. It is not a restricted point of view of a greater Euclidian space, and Newtonian time. To understand what Minkowski and Einstein are telling us we have to give up the notions of Euclidian space, and Newtonian time. They do not work. They are wrong.

There are no other observers in the Universe at this moment

There cannot be an observer at every point in Euclidian space, and Newtonian time, each seeing themselves as the one observer that has proceeded furthest in time. We have seen that there are no points in time and space that exist at the same moment in time as the observer in the present moment. Two points in Minkowski spacetime cannot have the same time coordinate.

If our Universe has the structure of Minkowski spacetime, then there are no other observers in our Universe at this moment in time. There is only one observer at the tip of a lightcone. The fact that this is hard to accept doesn’t mean that we don’t need to accept it, along with all its implications. The implication is that each observer is in their own Universe, and each Universe has only one observer in it. If there is only one Universe, then there is only one observer. If more than one observer exists, then there is more than one Universe.

It also appears that, for this sole observer, there is nothing in their future. It is not already mapped out. It seems that a better view is of a future that is being created in the present of the observer as time proceeds forward. The probabilistic nature of Quantum Mechanics is not because we don’t know a future that already exists. There are only probabilities, in the present, of the future existing at a future point in time. Our future does not yet exist.

Maxwell’s electromagnetic wave exists in its own right. It is composed of real electromagnetic fields. It is not a purely mathematical device giving probabilities of something else that does exist in its own right, that we call particles. It would seem to be more accurate to say that the wave is reality, and the particles that we are aware of are effects of the wave.

An electromagnetic wave is emitted as a spherical wavefront moving out in all directions from the tip of a light cone. It arrives as a photon of energy and momentum on the tip of another light cone, having come from one particular direction. This whole spherical electromagnetic wave may consist of just one photon.

In conventional Euclidean space and Newtonian time, this whole sphere of real, oscillating, electromagnetic fields, potentially extending over many lightyears, must somehow coalesce into a single photon at a particular place and time. This is the “collapse of the wavefunction” on a grand scale. It is clear that this picture doesn’t represent reality.

Hugh Everett 3rd has suggested a plausible alternative in his “Many Worlds” hypothesis. He suggests, as this account does, that the electromagnetic wave emitted by an electron eventually interacts with a second electron, (or another charged particle) , in an infinite number of Universes. These separate Universes differ from each other by just that one photon interaction. Every different direction the energy and momentum can go is realised in a slightly different Universe. In this way, all future possibilities are realised.

Mathematically sine waves can be combined to create “wave packets”. These “wave packets” are waves that can have a large amplitude in a local region, and a much lower amplitude everywhere else. This branch of mathematics is known as Fourier analysis. Photons of light are often described as being “wave packets”.

However the idea of a photon as being a wave packet localised in three dimensional Euclidean space, with separate time, does not lend itself to understanding phenomena like the two slit experiment in which a single photon of energy appears to go through both slits at the same time.

At the moment we appear to lack a self consistent view of the world that our equations, and experimental results, are presenting to us. Photons, and massive particles, are both described in Quantum Mechanics as waves. These waves are extended over space to an infinite distance. Somehow, when photons and massive particles interact, they do so at particular places, and at particular times. This is called “the collapse of the wavefunction”, but that is not an explanation; it is actually a summary of the problem. This account will attempt to show that there is a way forward.

Photons and electrons can be represented as plane waves. The shape of Minkowski spacetime discussed here does have a singular point at which phenomena like the exchange of momentum and energy can occur.

The way that photons and electrons can travel as plane waves, and interact by transferring momentum, while remaining as plane waves, is shown in the section on “Quantum mechanics” below. We can envisage this process happening instantaneously at a defined moment in space and time on the tip of the light cone.

The Minkowski metric doesn’t show how time flows

The Minkowski metric itself doesn’t show how time flows. It is a map of possible points with three space coordinates, and a time coordinate. For an observer in the present moment at the apex of the light cone, these points are fixed.

It seems that, in some sense, what we are is the past becoming the future in our present moment. If we are “the observer”, then we have the experience of moving forward through time.

For us a time, dt, can pass. When it does, we are in a new Minkowski metric. The point at the tip of the light cone where we were is now in our past, and we are at a new tip of a new light cone. We will have moved forward, in any direction, at any velocity we have up to the speed of light, to a point at the tip of a light cone in the future.

All the points that were on the previous light cone now have points in the space time ahead of them. Material objects can utilise these new points to move forward in time to the new point with the velocity they have up to the speed of light. The observer at the apex of the new light cone will still see matter on the past light cone, but it can’t have been there in advance of being seen there by the observer at the tip of the light cone.

The light cone’s surface is a point in Minkowski spacetime

In the description of the Pythagorean metric above, ds = 0, represented a single point on a flat two dimensional plane. The entire two dimensional plane is the set of all points for which ds = 0. When ds = 0 in the Minkowski metric, it describes the whole “surface” of the light cone.

Just as we can consider the entire two dimensional Pythagorean plane to be made up of all the possible points for which ds = 0, we could define an entire Minkowski spacetime as all possible light cones for which ds = 0. Such a definition must include an observer at the point on the tip of the light cone that has proceeded furthest in time.

All observers would be at the apex of a light cone. For each of them all that exists, at that moment, is their past light cone. All the other observers on their light cone are a point where ds = 0. All these light cones together constitute an extended Minkowski spacetime.

The light cone, that has been considered to represent the entire Minkowski spacetime, is actually a subset of this extended spacetime. We can view it as a single “point” of this extended Minkowski spacetime. As time flows for an observer, they are continually moving forward into their future by infinitesimal increments of time, and as they do so they are at a new point where ds equals zero for the new light cone they see. The lightcone that they were at the tip of is now in their past, and all its points have ds greater than zero with respect to the observer at the tip of the new light cone.

Instead of imagining that observers have “world lines” showing their path through the same spacetime, we can imagine instead that a new Minkowski spacetime comes into existence for each observer, where they exist momentarily at the apex of their own lightcone.

Minkowski spacetime matches your personal experience

For any of these observers the future doesn’t exist. Every observer is at the furthest point time has yet reached. That statement may seem strange, but consider this; it matches your personal experience better than a spacetime in which the past, and the future is all mapped out for an infinite number of points in Euclidean space and Newtonian time.

If your future is predetermined, why don’t you jump in front of a bus? You are going to do it anyway.

The point where light is emitted is next to an observer at the apex of a light cone, with the future point of absorption at a, yet to be determined, point on its future light cone. It is absorbed as a photon at another “point”, also next to an observer at the apex of a different light cone.

Minkowski space time in spherical coordinates

The Minkowski space time can also be written in spherical coordinates, t, r, θ, ϕ, as

ds^2=c^2dt^2-dr^2-r^2d\theta^2-r^2sin^2\theta d\phi^2

In the same way that the change from coordinate axes x and y, to p and q above didn’t change the infinitesimal lengths, ds, in the Pythagoras metric above, this coordinate change does not change the spacetime described.

Moving frames in the Minkowski spacetime

The Minkowski spacetime metric for moving coordinates

The Minkowski spacetime metric for moving coordinates is

ds^2=c^2dt{^/}^2-dx{^/}^2-dy{^/}^2-dz{^/}^2

We have seen that substituting for dt^/ , dx^/ ,dy^/ , and dz^/, using the Lorentz transformations, we get

ds^2=c^2dt^2-dx^2-dy^2-dz^2

These spacetime metrics appear the same because in each case the Minkowski spacetime, the reference frame, and the observer, are all stationary with respect to each other. In fact, the moving frame has actually contracted in the x direction, and the moving observers time has dilated, when measured by the observer stationary in the x, y, z, and t frame.

The observer stationary in the moving frame will not notice this. All their measurements of space and time are contracted and dilated, but so are their measuring instruments, and so are they themselves. The moving frame observer is measuring everything with moving frame coordinates. The result is that the moving frame observer sees the same spherical wavefront in moving coordinates as the rest frame observer sees in rest frame coordinates.

Each observer will calculate their own Minkowski spacetime to have the mathematical form above. This form is the shape of the spacetime as viewed by an observer at rest in it. The observer in the moving frame is at rest in a moving spacetime, and the observer in the rest frame is at rest in a spacetime at rest. They both see the same shape of spacetime.

Things look very different when the moving frame is measured by the rest frame observer in rest frame coordinates.

The x, y, z, and t , or rest frame, observer sees very clearly that the moving observer, the moving frame, and the moving spacetime are not the same because the moving frame intervals of length and time are not the same size as the rest frame intervals of length and time when they are measured by the rest frame observer. The moving frame intervals of length have contracted, and the moving frame intervals of time have dilated. This contraction and dilation are needed to give both the moving frame observers, and the rest frame observers, a speed of light that is the same in all directions, as shown above.

The Lorentz transformations not only transform the moving frame points to rest frame points, they transform the reference frames and everything in them, including all observers together with their rulers and clocks. They also transform the spacetime metric itself. They do a complete job.

It can’t be detected from within the transformed spacetime. The length contracted and time dilated observers see length contracted and time dilated bodies that they measure with length contracted and time dilated rulers and clocks in a length contracted and time dilated spacetime. They have no way of telling that there is any length contraction, or time dilation going on.

The Minkowski spacetime written with coordinate intervals dt^/ , dx^/ ,dy^/ , and dz^/, can be written in the dx, dy, dz, and dt coordinate intervals of the rest frame. This is shown in “Moving frames in rest frame coordinates” below. The rest frame observer will see the entire moving spacetime have its length in the direction of its motion contracted, and its time dilated.

It is only when the moving frame, together with observers and their rulers and clocks are observed from the rest frame that they can be seen and measured to have their lengths contracted and times dilated.

The changes in lengths and times are quite real, however. Moving bodies really do shrink, and moving clocks really do run more slowly. The Lorentz transformations cannot be performed the other way round.

Observers stationary in the moving frame will see bodies stationary next to them as having uncontracted lengths and undilated times. The observer stationary in the rest frame is moving relative to them. That observer, the one stationary in the rest frame, will see the same bodies’ lengths as contracted, and their times as dilated.

Length contraction and time dilation in the Minkowski spacetime

We will now look at how the Lorentz transformations result in length contraction and time dilation.

If we consider the transformation between dx^/ and dx we have

dx{^/}=(1-\frac{v^2}{c^2})^{-\frac 1 2}(dx-vdt)

In order to measure the end points of an interval dx^/ that is stationary in the moving frame, we need to measure them at the same time, or simultaneously, in the rest frame, so we would have

dt=0

so, for a length, dx^/, stationary in the moving frame

dx{^/}=(1-\frac{v^2}{c^2})^{-\frac 1 2}dx

and

dx=(1-\frac{v^2}{c^2})^{\frac 1 2}dx{^/}

This is the Lorentz length contraction. A reference frame moving relative to a rest frame, and a body stationary in it, will contract, by the Lorentz factor, in the direction it is moving relative to an observer stationary in the rest frame.

Considering the transformation between dt^/ and dt we have

dt{^/}=(1-\frac{v^2}{c^2})^{-\frac 1 2} (dt-\frac{vdx}{c^2})

dt{^/}^2=(1-\frac{v^2}{c^2})^{-1} (dt-\frac{vdx}{c^2})^2

dt{^/}^2=(1-\frac{v^2}{c^2})^{-1} (dt^2-\frac{2vdxdt}{c^2}+\frac{v^2dx^2}{c^4})

dt{^/}^2=(1-\frac{v^2}{c^2})^{-1} dt^2(1-\frac{2vdx}{c^2dt}+\frac{v^2dx^2}{c^4dt^2})

For a point at rest in the moving frame, the observer in the rest frame will see a change in distance dx in a duration of time dt where

\frac{dx}{dt}=v

so, for an observer stationary in the moving frame

dt{^/}^2=(1-\frac{v^2}{c^2})^{-1} dt^2(1-\frac{2v^2}{c^2}+\frac{v^4}{c^4})

dt{^/}^2=(1-\frac{v^2}{c^2})^{-1} dt^2(1-\frac{v^2}{c^2})^2

dt{^/}^2=(1-\frac{v^2}{c^2}) dt^2

dt=(1-\frac{v^2}{c^2})^{-\frac 1 2}  dt{^/}

This is the Lorentz time dilation. A duration of time for a moving body dilates, by the Lorentz factor, relative to time in a rest frame.

Any physical object, stationary with respect to the moving frame, will experience the same length contraction and time dilation from the point of view of an observer stationary in the rest frame.

Proper time and rest length

The lengths and times measured for bodies stationary in the moving frame are what are termed rest lengths and proper times. Rest lengths and proper times are lengths and times measured by observers stationary next to whatever they are measuring. They will always be the same measurements whatever reference frame the observer and the body being measured are in. They will be the same measurements if the bodies are stationary in the rest frame, and measured by observers stationary next to them, as they will be if the same bodies are measured when they are stationary in the moving frame, and measured by observers stationary next to them.

If an observer, stationary in the rest frame, looks at a moving body, they can imagine that body stationary in a moving frame. It will no longer have the length it would have if it was stationary next to the rest frame observer.

If they use their rest frame coordinates to measure the length of the moving body they will measure a length that is contracted in the x direction. It will be a shorter measurement than the measurement of the same body measured by an observer stationary next to it in the moving frame.

It will also be shorter than the rest frame observer would measure if the same body was stationary next to them in the rest frame.

An observer stationary in the moving frame will measure an infinitesimal length along the x^/ axis, that is also stationary in the moving frame, as dx^/. This is a rest length. That same length will be moving when measured by an observer stationary in the rest frame, and that observer will measure a contracted length, dx.

An observer stationary in the moving frame will measure a duration of time, with a clock that is also stationary in the moving frame, as dt^/. This is a proper time. That same duration of time on the clock in the moving frame, when measured by an observer and a clock stationary in the rest frame, will measure a dilated time duration, dt. The clocks will not be running at the same rate.

The moving observer does not notice the length contraction, and the time dilation, of their coordinate intervals because they themselves, and their measuring instruments, are contracted in the x^/ direction, and dilated in time, to the same degree as the coordinate intervals they are measuring.

The rest frame itself is just one of the possible frames with different relative velocities. When discussing length contraction and time dilation, the rest frame is the frame in which an observer is at rest, or stationary, in that frame, and measuring the length contraction and time dilation in a frame that is moving relative to them. Any frame that has such an observer at rest in it could be considered to be the rest frame.

When that observer, stationary in the rest frame, sees a moving observer next to a moving body, they will consider them both to be stationary in a moving frame. The moving observer will be measuring this body’s length as uncontracted, and they will be measuring undilated time on a clock, also stationary and next to them. The observer, stationary in the rest frame, sees the moving observer next to the moving body as having contracted length and dilated time.

It is important to understand that rest lengths and proper times are the same measurements of lengths and times, but they are not the actual, or real, lengths and times. The actual, or real, lengths and times depend on the relative velocity of the observer in the rest frame.

They are the same measurements because the clocks and the rulers change along with what they are measuring. It takes a clock and a ruler moving with a different velocity to detect the length contraction and time dilation due to velocity.

So which clock and ruler will shrink, and which clock and and ruler will measure rest lengths and proper times? It depends on which frame the observer is in. It is the frame moving relative to the observer that is length contracted and time dilated.

We will see in this account how, in the model spacetime presented for our Universe, this observer is further advanced in time than all other moving frames they observe. We cannot simply choose which clock and ruler we make the rest frame, and which we make the moving frame.

To see what the Universe looks like from the point of view of an observer in the moving frame, we, as an observer, must physically move from the rest frame we are initially in, to be at rest in what was initially a moving frame. As we do so, the structure of spacetime will change.

All moving observers, with different relative velocities, will measure their own lengths and times as rest lengths and proper times, but they will all be different to each other, and different to an observer measuring them from a rest frame.

It is tempting to think that the rest lengths and proper times are the actual, or real lengths and times, and that the different lengths and times measured by an observer in the rest frame, are only apparent differences due to the different point of view. That is a widely held belief in the physics community. It is a way of not facing the contradictions that a world view still based on a fundamental belief in Euclidean space and Newtonian time presents.

If a body spends some time moving, and then returns to its starting point, it is hard to tell if its rest length really changed while it was in motion, since it will have reverted to its original length. We can tell if its time dilates, however. A clock that has spent some time moving, and returns to its starting point at a later time, will show less time having passed than an equivalent clock that remained behind. The difference in time shown by the moving clock will be a permanent record of the time dilation it experienced.

This experiment has been done. Originally it was done by Joseph Hafele and Richard Keating. In 1971 Hafele and Keating flew four atomic clocks twice around the world, first eastward, then westward aboard commercial airliners. They compared the clocks in motion to stationary clocks at the United States Naval Observatory. They demonstrated that the time dilation was a real, actual, effect, and not just apparent. More accurate experiments done since then have confirmed this.

There is no one defined “rest frame” that the velocity of all other frames may be compared to. The “rest frame” just happens to be the one that has been picked to observe the velocity of the others from.

It appears that any of them could have been chosen as the rest frame, but there is a hidden assumption when we do this. Are we justified in assuming that nothing else changes as the observer moves from one inertial frame of reference to another?

We assume that the laws of physics don’t change, and that the observed speed of light remains the same. That doesn’t necessarily mean that everything stays the same.

The “observer” is ourself. To observe what the Universe looks like from another place, and at another velocity, we must change our own position and velocity. Doing so involves accelerations and time. It involves changes in energy, and momentum. Changing where we consider the Universe to be observed from involves more than just changing our point of view.

The challenge we face as physicists is to create a self consistent physics that accommodates the contraction of space, and the dilation of time, along with the proposition that there is no preferred frame of reference. This account is going to show how this can be done.

In “The Minkowski spacetime and our model Universe compared” below we will see that length contraction and time dilation must happen in our Universe full of mass and gravitational fields, and may be measured by observers viewing the expansion from any point.

If infinitesimal lengths are stationary in the rest frame, and measured by an observer stationary next to them, we can say that

dx=dσ

Where we refer to dσ as a rest length. We can consider an observer next to a physical body with this length.

If infinitesimal time durations are measured by an observer, and a clock, stationary in the rest frame we can say that

dt=dτ 

Where we refer to dτ as a proper time. We can consider an observer next to a physical clock showing this duration.

If we now consider that the same body with an infinitesimal length, dσ, is stationary in the moving frame, or in other words moving relative to the rest frame, and measured by an observer stationary next to it. We can say that

dσ=dx{^/}

but the infinitesimal length, dx, will no longer be equal to the rest length dσ. When viewed from the rest frame, the body with length, dσ, is now moving along with the moving frame, and its length, dx, measured in the rest frame, has been contracted according to the Lorentz length contraction

dx=(1-\frac{v^2}{c^2})^{\frac 1 2}dσ

Also, if we now consider that the observer and clock are stationary in the moving frame, or in other words moving relative to the rest frame, and measuring the same infinitesimal duration dτ, we can say that

dτ=dt{^/}

but the infinitesimal duration, dt, will no longer be equal to the proper time dτ. When viewed from the rest frame, the duration, dτ, is now the time in the moving frame, and the same time, dt, measured in the rest frame, has been dilated, according to the Lorentz time dilation

dt=(1-\frac{v^2}{c^2})^{-\frac 1 2} dτ

Light is instantaneous

The apparent speed of light is because we are thinking of it moving in Euclidian space, not Minkowski spacetime. As we saw in the previous section, the apparent speed of light is due to the shape of Minkowski spacetime, and this shape is defined by the interval ds.

If we define simultaneous as two points that have each reached their furthest point in their flow of time, instead of two points that have the same time coordinate, as we would in Euclidean space and Newtonian time, then two points in Minkowski spacetime are simultaneous if time has not proceeded further at each point.

The emission and reception of light, as it travels from a point on the apex of a light cone to the observer at the apex of the observers light cone, occurs in an interval ds = 0. The place where the light was emitted cannot have moved forward into the “spacelike” region of the receivers light cone. Time has not proceeded from the point of emission, or from the point of reception. “At the same time” cannot mean that dt = 0 between the emitting point and the observing point.

We must abandon the idea that the present moment is all points at the same time.

The “surface” of the light cone is the complete set of intervals, for which ds = 0, between the observer, and other points. A central co-moving observer sees all the other points on the light cone, but they are all in that observer’s past when light arrives where they are at the apex of the light cone. If time has not moved forward from any of the points on the past light cone, then all points on the past light cone, for which ds = 0, are simultaneous in the present moment of the central co-moving observer.

Time dilation means that, at the speed of light, time has dilated to a stop from the point of view of the observer at the apex of the light cone. In the limit, as the speed of light is approached, time slows to a stop.

The apparent speed of light is observed because we are thinking of it moving in Euclidian space, not Minkowski spacetime. The speed of light, c, determines the shape of Minkowski spacetime, and this shape is defined by the interval ds. The emission and reception of light, as it travels from a point on the light cone to the observer at its apex, occurs in an interval, ds = 0. Time has not proceeded from the point of emission, or from the point of reception .

If we define simultaneous as two points that have each reached their furthest point in their flow of time, instead of two points that have the same time coordinate, as we would in Euclidean space and Newtonian time, then two points in Minkowski spacetime are simultaneous if time has not proceeded further at each point. This means that, in Minkowski spacetime, with this definition of simultaneous, light is instantaneous.

We need to remember that the discussion above, of Minkowski metrics, is hypothetical. A Universe with a Minkowski metric cannot contain massive bodies that undergo length contraction and time dilation, and cannot contain light energy.

The moving observer

The Minkowski metric for the four moving frame Cartesian coordinates, x^/, y^/, z^/, and t^/, is

ds^2=c^2dt{^/}^2-dx{^/}^2-dy{^/}^2-dz{^/}^2

We know that

ds^2\geq0

so

c^2dt{^/}^2-(dx{^/}^2+dy{^/}^2+dz{^/}^2)\geq0

So, if dt^/ = 0 then (dx^/2 + dy^/2 +dz^/2)^1/2=0 . There can be no point at a distance, (dx^/2 + dy^/2 +dz^/2)^1/2 > 0, away from any point with the same time coodinate. This implies that, in the moving observer’s Minkowski metric, no two observers can exist at the same time, and at different points in space. In other words there is only one observer at the present time in the moving observer’s Minkowski metric.

We now have a description of Minkowski spacetime in which there can only be one observer at the apex of a light cone in the present moment.

We may imagine a moving observer on the past light cone of a rest frame observer, as seen by the rest frame observer, with the rest frame observer at the apex of the light cone. We may imagine light passing from the moving observer to that observer in the rest frame.

In Minkowski spacetime ds = 0 gives the set of all possible points on the light cone with a single point, (x₁, y₁, z₁, t₁) at its apex. If a second observer is on, or in, the light cone of the first, in a moving frame with coordiates (x₂^/, y₂^/, z₂^/, t₂^/ ), we saw above that this second observer is in their own Minkowski metric. They will see themselves at the apex of their own light cone.

They can’t see themselves on the surface of the lightcone that the first observer sees them on. That “place” on the surface of the light cone is actually a sphere of points at a distance and time given by the speed of light away from the first observer. All these points would have the same time coordinate, t₂^/, and no two points with the same time coordinate can exist in a Minkowski spacetime.

We need to remember again that the light cone is a picture based on Euclidean space and Newtonian time. It is a set of points that can’t exist in Minkowski spacetime. The mathematical structure of Minkowski spacetime allows for a single pair of points to exist momentarily. From the point of view of one of these two points, another point can exist in its past in time, and at a distance away, given by the value of the constant c.

There is no light in Mikowski spacetime, but we suppose that, in our Universe, Maxwell’s electromagnetic wave can connect these two points.

We may imagine a collection of such pairs, all with ds = 0. They will all be at different moments. The electromagnetic wave is effectively being emitted, and is arriving, in a single moment of time.

The picture of the light cone suggests that a photon exists over a period of time in which it travels to its destination. It would seem to be possible for any number of events to occur in the time it takes to do this. This picture is based on Euclidean space and Newtonian time.

It may be imagined, in this picture, that a pair of electrons, one at the point of emission, and the other at the point of absorption, of the photon have vertical world lines, or paths, through spacetime, and they are connected by a diagonal line representing the path of the photon of light.

The problem with this view is that each of the worldlines would be in the “spacelike” region of the other electron. The emitting electron cannot continue to exist in the Minkowski spacetime of the absorbing electron, and the absorbing electron did not exist in the Minkowski spacetime of the emitting electron.

It appears instead that there are two different Minkowski spacetimes, momentarily connected by a “photon”. Each of these spacetimes has a single electron in it at the furthest point time has reached. These spacetimes do not exist in a Euclidean space and Newtonian time. One Rimannian spacetime metric cannot exist in another. Each one is complete and entire in itself.

There is no Euclidean space and Newtonian time. In our Universe there is something like Minkowski spacetime, but containing mass and energy. The Minkowski spacetime metric is telling us about the relationship between pairs of points with coordinates of space and time. At these points we imagine particles interacting with photons carrying energy and momentum, and we also imagine observers able to sample these photons. We observe something we call a spacetime full of matter and energy.

There is a moment in this spacetime where we are observing it at a particular place and time. There is a moment when the interaction between electrons and photons appears to be taking place. This moment is in a sense the entire spacetime. In our Universe it contains just one observer, or just one pair of electrons and a photon.

The first observer at (x₁, y₁, z₁, t₁ ) is supposedly in the future of the observer at (x₂^/, y₂^/, z₂^/, t₂^/ ), but the observer at (x₂^/, y₂^/, z₂^/, t₂^/ ) is at the apex of their own light cone. At the point in space and time, (x₂^/, y₂^/, z₂^/, t₂^/ ) , the future has not yet happened. t₂^/ is as far as time has gone in the moving observer’s Minkowski metric.

The only possible conclusion is that the original observer at (x₁, y₁, z₁, t₁ ) in the first Minkowski metric does not exist in the moving observers Minkowski spacetime. The moving observers Minkowski spacetime is contained within the original observer’s spacetime, but it is not the same Minkowski spacetime. It doesn’t contain all the points that the original observer’s spacetime does.

All moving observers with any velocity less than the speed of light will be at the apex of their own light cone, and there cannot be any observers further ahead of them in time. All observers are as far as they can proceed in time. There is no future for any observer in a Minkowski space time.

We can say that the moving observer will be in a spacetime that is a subset of the spacetime of the rest frame observer. All the points in the moving observers spacetime can be in the rest frame observer’s spacetime, but the rest frame observer, and many other points in the rest frame observer’s Minkowski spacetime, are not in the moving observers Minkowski spacetime. It is clear that they are not equivalent spacetimes.

Light and Quantum Mechanics

This actually fits well with the description of the transmission of light given by Quantum Mechanics. Light is a wave of electromagnetic fields that propagates in all directions. The wave front forms a sphere with the source of the wave at the centre. We saw above that both the rest frame, and a frame moving with a relative velocity, have an observer that will see themselves as being at the centre of an expanding sphere of light.

When the light is received, it is received as a photon with a defined amount of energy. The Electromagnetic wave described by Maxwell’s equations is taken to be the Shroedinger probability wave of Quantum Mechanics giving the probablility of the photon’s quantum of energy arriving.

Between emission and absorption, the wave has all the reality of electric and magnetic fields. It exists as oscillating electromagnetic fields until it is received as a photon. At this time, the probability of the photon arriving is given by the square of the amplitude of the wavefunction, normalised to be unity over the whole spherical surface reached by the electromagentic wave front. Until this absorption occurs, the future is not determined.

The future is not determined for all observers in a Minkowski spacetime. Since a moving observer must have time passing more slowly than an observer at rest, the observer in the rest frame is in a future that is not determined for an observer in a moving frame. We can say that the rest frame observer does not exist in the Minkowski spacetime of the moving observer.

We see that Quantum Mechanics does model the situation described above in which the observer in a rest frame is not in the same Minkowski spacetime of an observer stationary in a moving frame.

The Lorentz transformaton is not reversible

We now get a picture of an infinite number of Minkowski metrics, each with one possible observer, in their present moment, at the apex of their light cone. For each observer, at a point in space, and at a moment of time, there are a set of other points at earlier points in time, and at distances corresponding to ds = 0.

This set of points is their personal light cone. For every observer the past light cone exists, but the future does not. This implies that two observers with a relative velocity cannot exist in the same Minkowski spacetime. Every observer is alone in their present moment of time, and in their own Minkowski spacetime.

We saw above that the metric with moving frame coordinates may be transformed using the Lorentz transformations into a metric with stationary frame coordinates. This means that all the intervals calculated with moving frame coordinates can be expressed with stationary frame coordinates.

We saw that this does not mean that all the intervals calculated with rest frame coordinates can be expressed in moving frame coordinates. There will be intervals in the rest frame metric that involve points that do not exist in the moving frame metric.

We can say that the metric experienced by the observer stationary in the moving frame is contained within the metric experienced by the observer stationary in the rest frame using rest frame coordinates. The reverse is not true.

We saw that the interval, ds, is invarient under a Lorentz transformation from moving frame points measured by an observer stationary in the moving frame, to stationary frame points measured by an observer stationary in a stationary frame.

Moving frame points can be transformed into stationary frame points. The entire set of points in the moving Minkowski metric can be transformed into a set of points in the stationary Minkowski metric. There is not a one to one correspondence between all the points in the moving Minkowski spacetime and all the points in the stationary Minkowski spacetime, however.

There will be points in the spacetime of the observer stationary in the stationary frame that do not exist in the spacetime of the observer stationary in the moving frame. In particular, if there is an observer, stationary in the moving frame, at the apex of the light cone of its Minkowski spacetime, then the original observer in the stationary frame does not exist for the moving observer. There is no point in space and time for it to exist in.

The moving frame has had its lengths contracted and its times dilated when viewed from, and compared with, the rest frames lengths and times. We can now view the moving frame as a new rest frame, and the original rest frame as a frame moving with the opposite velocity. The original rest frame is the new moving frame, and will have its lengths contracted, and its times dilated when compared with and viewed from what was the moving frame, and is now the new rest frame.

The original rest frame lengths were contracted, and the original rest frame times were dilated. Then these contracted lengths and dilated times are further contracted and dilated. The original spacetime metric of the rest frame is not restored.

Invarience to coordinate transformations implies that the coordinate transformations can be performed both ways, leaving the metric that can be constucted from the intervals, ds, the same. In that case we can say that the underlying metric is unchanged by the coordinate transformations. It is the same metric. The original metric in the original coordinates can be transformed to new coordinates, and back again. The original metric will comprise the same set of points as before. The transformation can be performed both ways.

This was the case with the Pythagoras metric dicussed above in which we saw that we could use x and y axes, or p and q axes. The p and q axes have a purely mathematical relationship with the x and y axes. p and q points may be mathematiclally transformed into x and y points, and vice versa. This is a two way transformation.

We have seen above that the Lorentz transformation is not a two way transformation. Assuming that it is the same has been the source of much confusion that this account seeks to dispel.

The Lorentz transformation is not a purely mathematical coordinate transformation. It involves a physical quantity; namely velocity, and a physical constant; namely the speed of light. The ability to move from one point to another is not implied by a Riemannian metric space. It is not a property of the space. It is a physical quality of the world we live in. It is interrelated with all the other physical qualities of the world we observe like momentum, and energy. It is a property of massive bodies in a Universe full of other massive bodies; it is not a property of points in a metric space.

The Lorentz transfomation is not a purely mathematical transformation; it is a physical transformation.

The same values for ds after a reversable coordinate transformation imply the same metric space. The same values for ds, after the non-reversible Lorentz transformation, do not imply the same metric space. This has been the case since 1908 when Minkowski first proposed his spacetime to the 80^th Assembly of German Natural Scientists and Physicians.

It has been assumed that a moving Minkowski metric spacetime is transformed by the Lorentz transformations into an identical one at rest for over a hundred years. It is not true.

The moving Minkowski metric is length contracted and time dilated.

When it is, in turn, treated as the spacetime metric at rest, and clocks at rest in the original rest frame are viewed from the moving frame, these clocks will have a proper time in the original rest frame.

From the moving frame they will be moving, and their proper times will be dilated. We are now treating the original moving frame as the new rest frame, and the original rest frame as the new moving frame.

The original rest frame observers saw the proper time of a clock at rest in the original moving frame as dilated. Time is passing slower on the moving frame’s clock. If an observer next to this clock, measuring dilated time in the moving frame, looks at a clock at rest in the original rest frame, it will be moving relative to them. If they compare it with the clock next to them, they will see the clock in the rest frame’s time passing slower than theirs.

There is a glaring inconsistency here. The clock in the rest frame was originally ticking faster than the clock in the moving frame, and now it is ticking slower. The assumption that the moving frame Minkowski metric, and the rest frame Minkowski metric, are the same is untenable. Applying the Lorentz transformation backwards does not restore the original rest frame metric spacetime.

The Lorentz transformations find a yet more length contracted, and more time dilated metric spacetime. They do not reverse the original length contraction and time dilation.

Moving frames in rest frame coordinates

We can cast the Minkowski metric, as expressed with moving coordinates, and as it appears for an observer stationary with those moving coordinates, into the form it takes expressed with stationary coordinates by the observer in the rest frame, while still being the metric as experienced by the observer stationary in the moving frame.

A stationary frame in a Minkowski metric will have many frames in relative motion to it that are equivalent in the sense that they have the same calculated values for ds under the Lorentz transformation. Observers stationary in these moving frames will all see themselves as being in a Minkowski metric with the same invarient values for ds.

We can consider frames moving with any velocity magnitude, v, relative to a rest frame, up to the speed of light. We can write Minkowski metric spaces, defining ds for all these moving frames, in moving coordinates using the Lorentz transformations.

All these moving frame metric spaces may also be written in the stationary coordinates of an observer stationary in a stationary frame, using the length contraction and time dilation equations derived from the Lorentz transformations.

Using the length contraction, and time dilation equations, we can substitute into the Minkowski metric in moving coordinates with stationary, or rest frame, coordinates.

We saw that we can write the time dilation of a moving frame as seen from the rest frame as

dt=(1-\frac{v^2}{c^2})^{-\frac 1 2}  dt{^/}

so

dt{^/}=(1-\frac{v^2}{c^2})^{\frac 1 2}  dt

and

dt{^/}^2=(1-\frac{v^2}{c^2})  dt^2

We saw that we can write the length contraction of a moving frame as seen from the rest frame as

dx=(1-\frac{v^2}{c^2})^{\frac 1 2}dx{^/}

so

dx{^/}=(1-\frac{v^2}{c^2})^{-\frac 1 2}dx

and

dx{^/}^2=(1-\frac{v^2}{c^2})^{-1}dx^2

Lengths in the y and z directions are unaffected by the transformation so

dy{^/}^2=dy^2

and

dz{^/}^2=dz^2

The Minkowski spacetime metric for all moving frames is

ds^2=c^2dt{^/}^2-dx{^/}^2-dy{^/}^2-dz{^/}^2

Here dx^/, dy^/, dz^/, and dt^/ are moving frame coordinate intervals measured by an observer stationary in the moving frame. We see that all such observers stationary in a moving frame, will see themselves in their own Minkowski metric. They each see proper times and rest lengths, and they will each see all other lengths and times contracted and dilated according to that frame’s relative velocity.

The moving Minkowski metric is not the same as the Minkowski metric at rest. The moving Minkowski metric, expressed in moving frame coordinates, has the same form as the Minkowsk metric at rest expressed in rest frame coordinates, but it is not the same as the moving Minkowski metric expressed in rest frame coordinates. We are now going to write the moving Minkowski metric in rest frame coordinates. We get an equation for all the possible moving Minkowski metrics expressed in rest frame coordinates.

Substituting the equations above into the Minkowski metric in moving frame coordinates we get, for all infinitesimal lengths and infinitesimal durations stationary in a moving Minkowski metric, as measured in the x,y,z, and t coordinates of the rest frame.

\color{red}{ds^2=(1-\frac{v^2}{c^2})c^2dt^2-(1-\frac{v^2}{c^2})^{- 1 }dx^2-dy^2-dz^2}

These are moving Minkowski spacetime metrics for the various possible values of the relative velocity, v, equivalent to the one expressed in moving coordinates above, but they are expressed in the coordinates of the rest frame. They are moving Minkowski spacetime metrics, as seen by an observer stationary in a rest frame.

This metric is not commonly seen. The reason is probably because it is demonstrating clearly that the Lorentz transformation is not reversable.

There is nothing wrong with it mathematically. It is a straight forward substitution of the length contracted, and time dilated, differentials into the Minkowski metric in moving coordinates.

It is, however, clearly telling us that the Minkowski metric for the moving frame is not the same as the metric for the rest frame. This means that it does matter which frame you consider to be moving, and which frame you consider to be at rest. The moving frame is length contracted and time dilated when compared with the frame at rest. Two observers in relative motion are not equivalent.

This seems to contradict the hypothesis that all inertial frames will have the same laws of Physics. The special principle of relativity states that physical laws should be the same in every inertial frame of reference, but that they may vary across non-inertial ones.

This equation makes it clear that a moving Minkowski metric is not the same as a Minkowski metric at rest. The length contraction and time dilation necessary for observers with relative motion to see a uniform speed of light means that the metrics must be different. The metric spacetimes are constructed with different sized length and time intervals. The Lorentz length contraction and time dilation equations tell us exactly how these length and time intervals must change.

The twins paradox

This leaves us with a famous paradox known as “the twins paradox”. If two identical twins separate from each other with a relative velocity, v, the moving frame twin must age less than the rest frame twin. The paradox arises because either one of the twins could be considered to be in the rest frame, with the other one in the moving frame.

Which twin has more time pass for it, and ages more as a result? One twin will end up older than the other, but which one? If the twins are reunited, and their clocks are compared, which clock will show more time elapsed?

If it is protested that there must be acceleration involved in returning the moving clock to be stationary along side the stationary clock in this gedanken experiment, and so the two refrence frames are not inertial, we can imagine a more sophisticated gedanken experiment.

This time two identical clocks are synchronised as a moving clock passes a stationary one. The moving clock travels off, and, later on, passes another identical clock, moving with an equal and opposite velocity. These two clocks are synchronised, at the moment of passing each other, to show the time on the outbound clock. The clock with the opposite velocity returns to the stationary clock, and, at the moment of passing the stationary clock, the times shown by the two clocks are compared.

There are no accelerations in this gedanken experiment. The reference frames are all inertial. Times on the clocks are read when the clocks are momentarily next to each other.

The Lorentz equations clearly state that the outbound clock, and the returning clock, will both have time passing more slowly than the stationary clock. This will be shown as a difference between the time measured on the returning clock and the stationary clock. This difference will be a permanent record that allows the experimenter to determine which of the two clocks was moving, and which was stationary.

This gedanken experiment is the twins paradox revisited, with clocks instead of the twins. This account will resove the paradox. We will see in this account that motion needs to be measured next to the overall structure of the Universe, and its expansion in time.

Time dilation in our Universe

The problem is more acute in our expanding Universe. An observer looking at the distant galaxies sees them all moving away with a velocity proportional to their distance. This is Hubble’s law.

The galaxy where the observer is has time moving faster than any of the other galaxies. We assume that the Universe is the same from all points of view, however. We can pick any galaxy to view the others from, and we should get the same overall view.

There is something called the Cosmological Principle. This states that the universe is uniformly isotropic and homogeneous when viewed on a large enough scale. Our Universe does not appear to be homogenous in the flow of time.

Time dilation has implications for all co-moving observers in an expanding universe. If time dilation affects all co-moving points in our Universe, all other co-moving points will have time passing more slowly than a point considered to be at the center of the expansion. We are referring to such a point, in this account, as the central co-moving observer.

If we consider that points on the past light cone are moving away from the central co-moving observer, as we saw in the last section, we know that time is passing more slowly for such points. Their time will be dilated when measured by the central co-moving observer. We also know that less time has passed in total for these points by the amount of time it would take light to travel to the central co-moving observer.

An observer is, by definition, at the place and time where light is received. There is only one place in Minkowski spacetime for such an observer to be. That is the point we think of as being at the tip of the light cone.

In our expanding Universe, we can imagine light being emitted from a point, stationary in a moving frame, on the past light cone, and received by the central co-moving observer at the tip of the light cone. The frame of the central co-moving observer is the rest frame. The interval, ds, is zero between all points on the past light cone and the central co-moving observer.

There is a big problem with this analysis. We have assumed that all co-moving points are equivalent. All of them are able to consider themselves as the central co-moving observer. How can all other co-moving points be in the past of all co-moving points?

For the speed of light to be uniform for all observers in our Universe, which experiments have shown it to be, time dilation and length contraction must be operating in the way that Lorentz, Minkowski, and Einstein, have predicted.

Time dilation is real. This means that we must accept that all co-moving observers are at the tip of their own lightcone, and all other co-moving observers have time passing more slowly.

If time is passing more slowly for the other co-moving observers, then this explains why less time has passed for them in total from the point of view of the central co-moving observer, and why they have not progressed through time to be at the same time as the central co-moving observer. It explains why there is nothing in the “spacelike” regions. It explains why there are no “spacelike” regions.

Time is not proceeding at the same rate at all points in our Universe if it has the structure of Minkowski spacetime. Time is clearly not proceeding uniformly, and without reference to anything else, as Newton supposed it did.

We have seen above that the central co-moving observer has the future lightcone of its Minkowski metric containing points that could be reached by light. These points cannot be populated by massive bodies that have proceeded through the non existent “spacelike” region. The future lightcone does not yet have any light in it. It is empty of mass and energy. The central co-moving observer is about to move forward in time into this region.

With this view other co-moving observers are also on the point of moving forward into one of the possible futures open to them, and what we are seeing, on the surface of our light cone, is how they were at the moment they are about to move into their future light cone. All points on the central co-moving observer’s light cone could have an observer at them, and all these observers will be about to move on into their own Universe.

We, who are observing them on our past light cone, are just one of this infinite number of possible futures. For them, none of these futures exist yet. For them, we do not exist. We are in their empty future light cone.

We can imagine that our light cone, at this present moment, is built from an infinite collection of younger light cones. Observers at the tip of each of these younger lightcones see themselves as the senior light cone, also built of an infinite collection of younger light cones. Each observer is on the point of moving forward into one of infinitely many possible futures. All possible futures spring from all points on all possible lightcones. All the possible outcomes predicted by Quantum Mechanics actually come into existence. This process is happening right now. Every possibility is realised. This is the vision of Hugh Everett 3rd, and his “many worlds” hypothesis.

The Minkowski metric is a complete description of the points in space and time that exist at any one moment for this observer. Other points with time coordinates and space coordinates are allowed, but the only one that is able to receive a light signal is at the apex of the light cone. This is not just a point of view. We have seen that other points with the same time coordinate do not exist in this spacetime. There can only be this one observer at any one moment.

All these hypothetical observers are equivalent, but there is one that stands out from the rest. It is the observer that we, ourselves, are. It is the observer in the present moment. The present moment is both unique and ubiquitous at the same time.

The co-moving observer that is referenced in this account is more than just the point at which light is received. It is where we are aware of the Universe we perceive. It is our conscious awareness that is proceeding through time. This moment is on the point of becoming the next moment. This must be the case for all observers at the tip of their respective light cones in their own Minkowski spacetimes. All observers are about to proceed into their own future.

There is no reason that consciousness should not be a legitimate study for physicists except that science somehow split off from religion some four hundred years ago in western civilisation. The co-moving observer, that is an intimate part of this account, is consciousness by another name.

Time dilation leaves a permanent record

Length contracton is hard to pin down, but time dilation leaves a permanent record. In the example of the twins paradox above, the clocks that are moving show less time having passed than the clock that remains at rest. They will always show this discrepency. Clocks not only show the time now, they show how much cumulative time has passed.

Some physicists maintain that time dilation is just an apparent effect, and not real. The engineers who build and maintain the GPS satellite navigation system do not believe that time dilation is just apparent, and not real. This is a good thing because if they didn’t take account of time dialation, ships and aeroplanes would be crashing every day. Commerce would grind to a halt, and a great many insurance companies would go bust.

The Lorentz transformations make it clear that moving frames have time passing more slowly, but how can it be that a rest frame metric will have all other frames in relative motion have their time passing more slowly? Any one of these frames could presumably be chosen to be the rest frame.

The measured speed of light will be the same, but time dilation means that the actual speed of light must be less, when measured by an observer stationary in the rest frame, than it is when measured by an observer stationary in the moving frame. It must be less because if time slows down, all clocks, including light clocks, will show time passing more slowly.

The constant, c, appearing in the moving Minkowski spacetime, and the Minkowski spacetime at rest is about the structure of the spacetime. It stays the same under the Lorentz transformations. The path of light in a Minkowski spacetime at rest, and in a Minkowski spacetime in relative motion, both have ds = 0, and they both measure the speed of light to be c. The measured speed will be the same, but the actual speed will be less in the movng frame.

Space intervals and time intervals need to change for observers stationary in a moving frame, and using moving coordinates, to see the same speed of light. They no longer have the same reference frame as the observer in the rest frame. It may look like the same Minkowski metric, but the length intervals and the time intervals it is composed of are different sizes.

The equation above gives all the Minkowski metrics for moving frames that have a velocity of v<c relative to the rest frame. They are not the same metrics as the rest frame metric. They will give the same values of ds, but they are different metrics. The sizes of dt^/, dx^/, dy^/, and dz^/ are not the same as the sizes of dt, dx, dy, and dz.

In effect the equation above, for all moving frames, restores the transformed values dt and dx to the values for dt^/ and dx^/. The values for dt^/ and dx^/ are rest lengths, dσ, and proper times, dτ, as measured by an observer stationary in each of the moving frames described by the equation of the Minkowski metric above. They will all be the same measurements of lengths on real bodies, and times on real clocks, in every moving frame, but they are all different actual sizes from the point of view of an observer in the rest frame.

The equation above gives all the possible Minkowski metrics with velocities up to the speed of light relative to an observer stationary in a rest frame. They are moving Minkowski metrics in rest frame coordinates. Observers stationary in these moving frames, are moving with respect to the observer stationary in the rest frame.

The observers stationary in the moving frame see their measurements as rest lengths and proper times. The observer stationary in the rest frame sees the observers stationary in the moving frame having these rest lengths contracted, and these proper times dilated.

The rest, frame itself can now be seen to be the special case of the set of frames with various velocities, v, along the x axis, for which v = 0. When v = 0, in the last equation above, we have

\color{red}{ds^2=c^2dt_{v=0}^2-dx_{v=0}^2-dy_{v=0}^2-dz_{v=0}^2}

ds² is the same in both the rest frame and the moving frame. We can say that ds is invarient under the Lorentz transformation.

We saw above that all these moving frames may be expressed in their own moving coordinates, and observers stationary with respect to these moving coordinates will see themselves in their own Minkowski spacetime.

In the coordinates of the rest frame, however, these moving frames will have length contraction, and time dilation, determined by their relative velocity to the original rest frame. As this relative velocity approaches the speed of light, length elements, dx, will shrink to zero, and time elements, dt, will dilate to infinity, in other words, time passes infinitely slowly.

This will be true for all frames, moving with respect to the rest frame, with all relative velocities up to the speed of light.

The Minkowski metrics look the same, and will have the same values for ds, but the entire coordinate frame of the moving frame has shrunk in the x direction, and all the durations of time have dilated in the moving frame, when measured from the rest frame.

In the moving frame everything appears normal. The contraction in the x direction, and the dilation of time, is not noticed because all measuring rods have also contracted, and all clocks have slowed. The observers themselves will have length contracted and time dilated. All distances in the x direction will be contracted, and all durations of time, will be dilated.

No one has yet proposed any reason why relative motion should produce length contraction and time dilation in our Universe. It must happen, or we would not see the same speed of light in frames moving with a velocity relative to each other. Neither Lorentz, nor Minkowski, nor Einstein were able to explain why length contraction, and time dilation, occurred in moving frames of reference.

The length contraction and time dilation is quite real. Physical objects must contract in the direction they move in, and their time must run slow relative to a stationary observer. The Lorentz transformations are physical, not purely mathematical. We must look for physical reasons why length contraction and time dialtion occur. We know that it happens as a result of velocity, and we know that it happens in gravitational fields. We might suppose that these two causes are linked, and it turns out that they are.

We will see below in “The Minkowski metric and our model Universe compared” how this can happen in an expanding Universe full of matter.

Moving frames in spherical coordinates

The Minkowski spacetime can be written in spherical coordinates, t, r, θ, ϕ, as

ds^2=c^2dt^2-dr^2-r^2d\theta^2-r^2sin^2\theta d\phi^2

Mukul Das, and Rampada Misra, ⁵ have shown that the Minkowski metric transforms, under the Lorentz transformations, to moving coordinates, t^/, r^/, θ^/, ϕ^/ , of a frame moving with velocity, v, as

ds^2=c^2dt{^/}^2-dr{^/}^2-r{^/}^2d\theta{^/}^2-r{^/}^2sin^2\theta{^/} d\phi{^/}^2

The velocity, v, of the moving frame is taken to be along the r direction. Time will be dilated, as with the Cartesian coordinate case, and the length contraction will now be in the r direction. So

dt=(1-\frac{v^2}{c^2})^{-\frac 1 2}dt{^/}

and

dt{^/}^2=(1-\frac{v^2}{c^2})dt^2

and

dr=(1-\frac{v^2}{c^2})^{\frac 1 2}dr{^/}

so

dr{^/}^2=(1-\frac{v^2}{c^2})^{-1}dr^2

Infinitesimal lengths perpendicular to r will be unchanged so

r{^/}^2d\theta{^/}^2= r^2d\theta^2

and

r{^/}^2sin^2\theta{^/} d\phi{^/}^2=r^2sin^2\theta d\phi^2

Substituting into the equation for moving coordinates above, we get, for all points stationary in the moving frame

\color{red}{ds^2=(1-\frac{v^2}{c^2})c^2dt^2-(1-\frac{v^2}{c^2})^{-1}dr^2-r^2d\theta^2-r^2sin^2\theta d\phi^2}

This will be true for an infinitesimal volume of space and time at a distance r, with the velocity, v, in the r direction. It will give the same value of ds² for all frames with a velocity, v, up to the speed of light in the coordinates of the original rest frame.

The Schwarzschild spacetime metric

Albert Einstein used differential geometry to further extend the description of space and time to include the effect that mass would have on the structure of space and time. The Schwarzschild spacetime metric is the solution of Einsteins field equations of General Relativity that describes the space and time round a single mass.

Outside a spherical mass distribution, in otherwise empty space and time, the Schwarzschild spacetime metric is related to mass, M, radial length, r, and time, t, in spherical coordinates as

ds^2=(1- \frac{2GM}{{rc} ^{2}})c^2dt^2-(1- \frac{2GM}{{rc} ^{2}})^{-1}dr^2-r^2dθ^2-r^2sin^2θdφ^2

Where G is the gravitational constant, M is the mass, c is the velocity of light, and dt, dr, dθ, and dφ are infinitesimal coordinate elements, in spherical coordinates, as measured by an observer at an infinite distance from the mass. In what follows “metric”, and “spacetime”, and “spacetime metric” are used interchangeably.

The interval defines the shape of space and time round a mass

The interval, ds, is invariant under coordinate transformations between any two infinitely close points in the spacetime metric. It effectively defines the shape of space and time in a Universe containing a single spherical body with mass, M.

As with the Minkowski metric, we still have the subtraction of length squared from time squared multiplied by the speed of light squared, but now there are factors in front of the terms in dt² and dr². We will see below that these terms cause time dilation, and length contraction, in the region round a mass.

As r tends to infinity, the factor in brackets tends to one, and the Schwarzschild spacetime, becomes

ds^2=c^2dt_{(r=∞)}^2-dr^2_{(r=∞)}-r^2dθ^2_{(r=∞)}-r^2sin^2θdφ^2_{(r=∞)}

This is the same form as the Minkowski spacetime in spherical coordinates. It is the Schwarzschild spacetime written for when r equals infinity. We can say that the Schwarzschild spacetime is equivalent to the Minkowski spacetime at an infinite distance from the centre of the gravitating mass. This was one of the assumptions that Schwarzschild used when he derived his solution of Einstein’s equation.

We saw above in “The Minkowski spacetime metric” that the Minkowski metric for a moving frame transforms into a Minkowski metric for a frame at rest using the Lorentz transformations. We saw that a moving frame could be chosen so that a stationary observer in that frame was observing points also stationary in it. This implies that dr, dθ, and dφ are zero, and the moving metric for pairs of such points reduces to

ds^2=c^2dτ^2

where dτ is the proper time measured by an observer moving between two points.

The speed of light, c, and the proper time, dτ, are real numbers so

c^2dτ^2\geq0

and

ds^2\geq0

This implies that ds is a real number in all such moving frames, and it also implies that ds is a real number in the Schwarzschild metric at infinity, and in the Schwarzschild metric itself.

This in turn implies that, as with the Minkowski metric, there can be no such thing as a pure length in the Schwarzschild metric. Any separation in space between two points must also include a separation in time.

The Schwarzschild spacetime describes how space and time must change round a single mass in an otherwise empty Universe. It was tested by Arthur Eddington who measured the bending of starlight as it passed close to the Sun during an eclipse. Eddington successfully predicted how much the star’s light would bend in the Sun’s gravitational field.

The Schwarzschild metric differs from the Minkowski metric in spherical coordinates by the factors in brackets. These Schwarzschild factors show that space is contracted in a radial direction, and time is dilated, round a mass, M.

Proper time

When dr, dθ, and dφ equal zero, and for all r, we can write

ds^2= (1- \frac{2GM}{{rc} ^{2}})c^2dt^2

and when r= ∞

ds^2=c^2dt_{(r=∞)}^2

An infinitesimal duration of proper time, dτ, is the time measured by an observer next to, and stationary with respect to, a clock. An infinitesimal rest length, dσ, is the length of an object that is next to, and stationary with respect to, an observer that measures it with a ruler, or measuring rod, as Einstein called them.

The Schwarzschild metric is defined by infinitesimal coordinate elements as measured by an observer at an infinite distance. So, for an observer at infinity in the Schwarzschild spacetime, dt_(r=∞) is equal to the proper time, dτ, between points infinitely close in time, and stationary in the spacetime at r = ∞.

 dt_{(r=∞)} = dτ

The proper time, dτ, is the time measured next to an observer anywhere in the metric. If an observer at infinity boils an egg, and it takes five minutes on the clock that they have next to them, they will still find that it takes five minutes on their clock if they take the stove, the pot, the boiling water, and the egg, together with the clock, to a point at a radius, r.

The interval, ds, is equal to the five minute boiling time, multiplied by the speed of light, when the egg is boiled at infinity, and it is equal to the five minute boiling time, multiplied by the speed of light, when the egg is cooked at a distance , r. So we can write for all r > 2GM/c², and when dr, dθ, and dφ equal zero,

ds^2=c^2dτ^2

The velocity of light, c, is a constant, so the interval, ds, is proportional to the proper time. It is the distance travelled by light in the proper time, dτ.

We can imagine a clock, and an observer, initially together at infinity. An identical clock together with another observer, also initially at infinity, is synchronised with the first clock, so that they are both measuring the same time at the same rate. This second observer with their clock then go to a radial distance, r, where this observer, that is still next to their clock, measures an infinitely small proper time, dτ. The interval, ds, for this duration of proper time, dτ, will be as given in the equation above.

The two observers will no longer have their clocks running at the same rate, however. The observer at infinity will see the clock of the observer at r running slow. The observer at r will not notice this because all physical processes in their local region will be running slower. All physical processes running slower means that time itself is running slower. If they look at the clock with the observer at infinity, they will see it running faster than theirs.

Time dilation

The first observer remaining with their clock at infinity will measure a greater coordinate time, dt, as the observer with the clock at the radius, r, measures a proper time dτ. The clocks are not running at the same rate anymore. The observer remaining at an infinite distance will measure more than five minutes on their identical clock, as they watch the observer at a radius, r, boil their egg. The Schwarzschild metric gives ds for this measurement dt, when, dr, dθ, and dφ equal zero, and for all r > 2GM/c², as

 ds^2=(1- \frac{2GM}{{rc} ^{2}})c^2dt^2

The Schwarzschild metric allows us to calculate the amount by which time is running slower for the the observer at r when compared with the same time measured by a clock at an infinite distance. While an observer at a radius, r, measures their local time, dτ, the Schwarzschild metric compares this with the time, dt, measured on a clock with an observer at infinity. The measurements are different. The clock at infinity is ticking faster than the clock at r.

Time itself is flowing faster at an infinite distance than it is at a distance, r. At all different radial distances time will be flowing at a different rate. There is no one correct time; there is no one proper time. One rate of flow of time is not “right”, and all the others “wrong”. It is not like having two clocks side by side with one keeping “good” time, and the other running slow and keeping “bad” time. Time is running slow at any radius less than infinitely large. Perfectly good, identical, clocks will tick at different rates at different radial distances in the Schwarzschild spacetime.

Time flows fastest at an infinite distance from the mass, M; the source of the Schwarzschild metric. It is this time that the Schwarzschild metric compares to all the various “proper” times at the various distances, r, in the spacetime.

The Schwarzschild factor in brackets in the equation above exactly compensates for the amount by which the measured time, dt, at infinity is greater than the measured time dτ at r, so as to keep the interval, ds, constant. This allows us to compare one observer’s time measurements with another’s, and calculate the differences. At the radius, r, a clock with an observer at r will measure a proper time, dτ, and the same time will be measured by an observer at infinity as dt, where

c^2dτ^2=(1- \frac{2GM}{{rc^2} })c^2dt^2

This is telling is that a clock with an observer at a distance, r, will show less time elapsed than a clock with an observer at infinity. This is known as time dilation. We can write

dτ=(1- \frac{2GM}{{rc^2} })^\frac{1}{2}dt

and

dt=(1- \frac{2GM}{{rc} ^{2}})^{-\frac{1}{2}}dτ

This tells us that time at r slows down, from the point of view of an observer at infinity. The factor multiplying the interval , dτ, is greater than one for all r > 2GM/c². The measurement, dt, will be greater than dτ. The observer at an infinite distance will see more time passing for the same amount of activity at r, than the observer at r. All the activity at r will be slowed down.

In Chile, during the rule of a ruthless regime, the citizens made a very clever and effective protest. If they demonstrated in the street, they would be arrested and killed. They knew this so they didn’t dare to protest in the conventional way, and become targets of the regime.

Instead they had a “Go slow”. One day everyone started to move more slowly. Perhaps they knew about the Schwarzschild metric, and it inspired them.

The slow movements of some were almost imperceptible. Others joined in. At the start no one, and particularly none of the security forces, could be sure who was going slow, or even if the protest had started. After a while it was obvious, but they couldn’t arrest everyone.

The point of the story is that, if everyone goes slow, it can’t be detected. Only the security guards, at an infinite moral distance from the protesters, can see it happening. For the protesters they can claim that it is just life going on as usual. No one is singled out by being out on the street protesting. Without an external reference for time, either the subjective time of the security forces, or their measurements on their wristwatches, no one can say that the go slow is happening at all. The people can all claim that they weren’t protesting; they were just going about their usual routine.

The event horizon of a black hole

As we approach a distance, r, where

r=\frac{2GM}{c^2}

we see that the quantity in the brackets becomes equal to zero.

This value of r is known as the event horizon of a black hole. At the event horizon the Schwarzschild factor equals zero, so a time interval, dt, as seen by the observer at infinity goes to infinitely large, and time at r passes infinitely slowly. As the event horizon is approached, the measurement dt, tends to infinitely large when compared to dτ. The measurement dτ stays the same.

This doesn’t mean that dτ is the same amount of time. If a process like boiling an egg is measured at different distances, r, and compared with the time measured by a clock at an infinite distance, dt, that clock at infinity will measure different amounts of time for each distance r. Next to the egg an observer will always measure dτ, but dt measured at infinity will be different for them all. The closer the egg is to the mass M, the longer it will take to boil compared with eggs further away.

Proper time isn’t proper

The Schwarzschild metric uses the measurement, dt, known as coordinate time, at an infinite distance, as a reference measurement with which to compare all other measurements, dτ, at other distances. There is nothing unique or special about a measurement of proper time, dτ. They are just measurements of the the locally measured time. The same process taking the time, dτ, will all have different amounts of the reference time, dt, for measurements of dτ made at different distances.

The Schwarzschild spacetime equation gives us an overview of how lengths and times compare with other lengths and times at different places in the spacetime metric. We can use it to show that time is passing more slowly here, than there, or that lengths are shorter here, than there. These differences are quite real. They don’t disappear when measurements are made next to the body being measured; they just can’t be detected by any locally made measurements of space and time.

These names, “proper time”, and “rest length”, sometimes also called “proper length”, are just popular names, like “spacelike” and “timelike”. They are often used to convey some sort of meaning that they don’t have. Proper conveys the meaning “legitimate”, “correct”, or “real”. But it is a mistake to think that it is the real time, and other measurements of time are illusions, based on the name alone.

If you were asked to choose which of two cats you would like to play with, Fluffykins, or Mauler; which one would you choose? If you found out that Mauler was a tabby housecat kitten with a timid, but loving nature, and Fluffykins was a known man eating tiger, would you change your mind?

The proper time needed to boil an egg will always be five minutes when that time is measured by a clock next to it, but, if someone boils an egg near to the event horizon, it might take a thousand years of coordinate time, dt, compared to the time it takes to boil an egg at an infinite distance. The boiling of the egg takes the same amount of what is known as “proper” time wherever it is boiled.

The measurement of that time next to the egg is the same everywhere in a gravitational field, but it takes longer to boil the egg the nearer it is done to an event horizon. Proper time is not proper in the sense that it is the true amount of time that is the same everywhere. The measurement is the same, but the actual amount of time is different.

The observer, the egg, the boiling water, and the clock are all affected by the local rate of flow of time. If time started flowing at half the rate, the heat would be supplied to the water at half the rate, the water would boil away at half the rate, the egg would cook at half the rate, the observer would think at half the rate, his or her heart would beat at half the rate, and the clock would tick at half the rate.

It would take twice as long to boil the egg. It would take twice as long to have the same number of thoughts, experience the same number of heartbeats, and count the same number of ticks of the clock.

If everything is going at half the rate there would be nothing local that would not be affected. There would be nothing local that the observer could use to tell that time itself was flowing at half the rate.

The coordinate time intervals, dt, are defined for all t, r, θ, and φ by the Schwarzschild metric. The Schwarzschild metric tells us that dt varies with r. The time, dτ, known as “proper time” is only defined in its immediate, infinitely small, local region next to an observer. It always appears to be the same because any changes to it, due to the distance from the mass, M, also affect the observer making the measurement, along with whatever is being measured. The egg, the clock, and the observer, all share the same rate of flow of time at that particular distance in the Schwarzschild spacetime.

It would be better, perhaps, if what is now called “proper time” had its name changed to “local time”, and what is now called “coordinate time” had its name changed to “reference time”. This would help all those who chose Fluffykins as a playmate, assuming they survived the encounter.

Time flows at different rates in different places in a gravitational field, and at different velocities. An observer next to a clock who compares it to another identical clock, also next to them, will see that they tick at the same rate. They will call their time proper time. Another observer with an identical clock some radial distance away, or traveling with a relative velocity, will call their measurement, on their clock, proper time, but it will not agree with the first observer’s proper time.

Wherever they are in a gravitational field, or however fast they are moving, all observers will call their own time “proper time”. That just means that for them, all their clocks, and all the physical processes they observe next to them, are going at the same local rate of flow of time. This includes their bodies physical processes. An observer can expect to live longer if they move nearer to a massive object. They actually will live longer, but they won’t notice it, and they can’t measure it. Only a second observer with a clock nearer the mass, or further away, will notice a difference in the time that the first observer experiences.

An observer a great distance away from a mass might boil a thousand eggs one after the other, and measure five thousand minutes of proper time on their clock next to them while they do so. Meanwhile, an observer closer to the mass only has time to boil one egg, and measure five minutes of proper time on their clock next to them. The clocks will show different amounts of elapsed time. They will both call their measurements “proper time”.

If the cook with one boiled egg takes their clock up to the observer with a thousand boiled eggs, and compares their clock, side by side, with the one at the greater distance, the clock next to the cook with a thousand boiled eggs will be ahead of the one brought up by the one egg cook. If instead the cook at the greater distance takes their clock down to the cook with one egg, their clock will still be ahead of the clock with the one egg cook . Which one’s time is correct?

The answer is, “neither”. Clocks at different distances from the event horizon go at different rates. One is faster than the other, and one is slower than the other. The time shown by a particular clock will depend on its history. The clock used to time a thousand eggs will be ahead of the clock used to time one egg wherever they are brought together. It will be a permanent record of the dilation of time.

The Schwarzschild metric uses the rate time is theoretically passing at infinity to compare with all the other rates of flow of time at other distances, r. At an infinite distance from a mass the Schwarzschild metric becomes the same as the Minkowski metric. This was one of the preconditions that Schwarzschild used to find his expression for space and time round a mass. Essentially he was assuming that the mass existed in otherwise empty Minkowski spacetime. The presence of a mass in space and time is being compared to it not being there. The presence of a mass affects the space and time everywhere.

Using the time measured at an infinite distance to compare with time at a distance, r, in the Schwarzschild spacetime allows the Schwarzschild metric to be used to calculate the differences in the rate of flow of time at various distances compared to the rate of flow at an infinite distance. All these rates of flow will be slower than the rate at an infinite distance.

An observer falling in towards an event horizon will see time on their clocks passing normally, however. The time shown on their own clocks, measuring proper time, will seem to passing at the usual rate when compared with their bodily processes, their thoughts and their pulse for example, but it will actually be infinitely dilated as they approach the event horizon. If everything slows down, there is no way for them to detect that slowing.

Time dilation is real

It is important to note that this is not an apparent slowing down of time; it is real. If two identical clocks are synchronised at an infinite distance from a mass, M, and one of them is taken to a distance, r, that clock will tick more slowly than the one left at infinity.

If, after some time has passed, the two clocks are then reunited at infinity, the clock that spent some time at a distance, r, will show less time having passed than the one that stayed at an infinite distance. The difference in elapsed time is demonstrably real.

If gravitational time dilation was not taken into account, the geo-positioning satellite system would not work. If time dilation due to relative velocity didn’t happen, muons created by cosmic ray collisions in the upper atmosphere would not reach the surface of the earth in the numbers they do.

The Hafele Keating experiment, in which atomic clocks were flown round the earth in opposite directions, demonstrated time dilation due to gravitation, and due to velocity. The synchronised clocks did not stay synchronised. Their experiments have been repeated with more accuracy, and confirmed. Time dilation is real.

It is not possible to pass through an event horizon

Anyone who thinks that it is possible to reach, and go through, an event horizon needs to think again about the experimentally proven truth that time dilation is real. At the event horizon time has stopped. An event horizon cannot be reached in a finite amount of time, and it can’t be moved through if time is not passing where it is.

As shown above, when dr, dθ, and dφ equal zero

ds^2=c^2dτ^2=(1- \frac{2GM}{{rc^2} })c^2dt^2

The proper time, dτ, is a duration of time as measured by an observer with a clock at any radius, r. Proper time is a fixed measurement of time; it is not an absolute actual duration of time.

The actual duration of time varies with the distance from a mass, as given by the Schwarzschild metric. It is quantified by the “coordinate time”, dt, which is the measurement of a duration of time at a distance, r, made at an infinite distance from the mass.

The equation above, comes from the Schwarzschild equation when dr, dθ, and dφ equal zero. It gives the relationship between the “coordinate time” at the distance, r, and the “proper time” at the distance, r. A clock stationary at this distance measures a time interval, dτ, and that same time interval is measured as dt by the clock at an infinite distance.

This measurement of time, dτ, will be the same at all distances, r. It will be a measurement of five minutes for boiling an egg, for example. This is when time is measured at the egg. This does not mean that the time taken to boil an egg is the same at all distances, r. More eggs can be boiled at a greater distance, than can be boiled at a lesser distance. More time has passed at the greater distance, than at a lesser distance.

At the event horizon of a black hole no time has passed. The event horizon of a black hole is the beginning of time, an infinitely long time ago.

There is no measurement of absolute time that is valid everywhere. All we can do is compare the measurement of the rate of flow of time in one place with the measurement of the rate of flow of time in another place. We can do that using the Schwarzschild equation above. Two “proper times” of the same sequence of events at different radial distances can be compared by converting them both to coordinate time. They will be different for the same sequence of events, boiling an egg for example, at different distances, r.

The term in dt² in the Schwarzschild metric must have dt varying with r in order that ds, and dτ, should be constant for observers at all distances, r. This is how the Schwarzschild metric determines the way that dt changes with r. The Schwarzschild factor in brackets in the term for dt² above tells us that the coordinate time, dt, is infinitely greater than the proper time when r is 2GM/c². When r is infinitely large, the coordinate time is equal to the proper time.

If the proper time, dτ, is measured by an observer as they approach the event horizon, this proper time ticks by, for that observer, at the same rate for them, as measured by any clock they have with them. They themselves, all their clocks measuring dτ , and all physical processes at their current distance, slow down as measured by a clock with an observer at an infinite distance away.

They may claim that time is passing normally for them, and for them, it is. All other observers at a greater, or lesser, distance, r, will disagree. The definitive time is the coordinate time given by the Schwarzschild metric, measured at an infinite distance. The definitive, or reference, time is dt, not dτ.

The closer an observer gets to the event horizon the longer each of their proper time seconds will take to pass. At the event horizon time is passing infinitely slowly. Infinitely slowly means stopped. If time has stopped, nothing can move, so nothing can move across an event horizon.

It is not possible to pass through an event horizon.

As well as time dilation there is also length contraction in the Schwarzschild metric, as discussed below. The effect of this length contraction creates an infinite amount of space for an observer to pass through as they approach the event horizon. Length tends towards being infinitely contracted, along with any observers in it, and their time tends towards being infinitely dilated, as they approach the event horizon.

At the event horzon length has contracted to zero, and time has stopped passing.

This means that, to reach the event horizon, an infinitely large number of contracted metres must be traversed by equally contracted observers.

Lengths, and times, appear to be unchanged because the observer, and their measuring rods and clocks, are themselves contracted and dilated to the same degree that the local space and time next to them are contracted and dilated. Your proper time seems to pass normally, but, as you get close, every second of your local proper time, dτ, is actually taking thousands of years of reference time, dt. You, and everything in your locality, is going slow, so nothing that is nearby will show you that time has slowed at all.

It is a limiting process, and you can never reach the limit. As an observer gets nearer, their time goes slower and slower, but they will never reach a point where time has stopped. To reach such a point takes time, and if time hasn’t stopped, you haven’t got there.

It is not possible to reach the event horizon of a black hole. Why then does nearly everyone in the academic world seem to think that it is?

Truth in physics

Physics is a search for the truth. That truth is not obvious and straightforward; it is often difficult and mysterious. It takes many years for new ideas in physics to become accepted. It is frequently harder to unlearn past ideas that are incorrect than it is to learn new ones.

The truth in physics is a quest for understanding, and ultimately that understanding is personal. It isn’t, in the end, about satisfying someone else that we have understood, it is about convincing ourselves that we have understood, and that is not always as straightforward as it might seem.

It seems that, if the rules of mathematics are learned, a set of phrases, or word forms, used often by others, are memorised, and these word forms are copied, and repeated; a great many people believe that by doing so, they have understood physics. It is quite possible to pass exams in physics using these principles. I have done so, and realised afterwards that I didn’t really understand what I succeeded in showing that I did understand, and actually believed that I had understood at the time. Understanding in physics is not an achievement; it is an ongoing journey. I am still on it.

If everyone around you, whose opinions you respect, is saying that an event horizon can be crossed in proper time, you don’t really need to think it through for yourself, do you? If you do feel a little uneasy about the proposition, you can reassure yourself that it must be your thinking that is wrong. What are the odds after all? Which is more likely; that you are mistaken in your thinking about time dilation and length contraction, or that all your esteemed colleagues, who must have thought it through for themselves, and not relied on the opinions of others, have all got it wrong? Surely you can safely rely on their universally agreed opinion. You don’t need to let anyone know that you haven’t made sure that you understand it for yourself. It is important that they don’t think that you are a fool.

If a great many people publicly agree about something, it doesn’t make it true. This is beautifully illustrated by Hans Christian Andersen’s story “The Emperor’s new clothes.”

The Emperor was told that clothes made of an amazing magical cloth had a wonderful way of becoming invisible to anyone who was unfit for his office, or who was unusually stupid. His most trusted advisors told him that the clothes were wonderful, so the King proudly wore them around town. The townsfolk, who had heard about the clothes, praised the fabulous new suit of clothes, and their handsome Emperor. Eventually, however, a small child, who was unconcerned about the opinions of his fellow citizens, cried out, “The Emperor has no clothes on!”

To see, and understand, the truth in physics, we need to think about it with a child’s mind. We need a mind that is oblivious to the loudly proclaimed opinions of others. It is very hard to think independently, in the presence of statements about the truth made by people whose authority we respect, but, to understand physics, it is imperative that we do. We need to find the truth for ourselves.

No one reading this is expected to believe the statement that it is not possible to pass through an event horizon. They will have to satisfy themselves about the truth of that statement by engaging with the physics themselves in the solitude of their own mind.

Length contraction in the Schwarzschild metric

For an infinitesimal length, dr, at a distance, r, when dθ, and dφ equal zero,we can write

ds^2=c^2dτ^2-(1- \frac{2GM}{{rc} ^{2}})^{-1}dr^2

It is important to include the proper time term in the equation above because pure lengths cannot exist in the Schwarzschild metric. A pure length, with dτ = 0, implies a negative value for ds², so there can be no real number for ds. There is no spacetime if there is no real value for ds between any pairs of infinitely close points being considered.

As with the Minkowski metric, there are no pure lengths in the Schwarzschild metric. That is hard for the mind to accept. We are so used to the idea of a “length” having a clear meaning independant of time that it is just about impossible to truly believe that it doesn’t. We don’t live in a Universe with pure lengths in it. That is a fact of life. Eventually most of us may fully accept it, but it is likely that there will be many who simply won’t be able to.

At an infinite distance from the central mass in the Schwarzschild metric, when dθ, and dφ equal zero, we can write

ds^2=c^2dτ^2-dr^2_{(r=∞)}

If an infinitesimal length, dr, at an infinite distance, is measured by an observer next to it, it will be a rest length, dσ. This rest length is the length of an object that is next to, and stationary with respect to, an observer that measures it with a ruler, or measuring rod. So, we can write, when dθ, and dφ equal zero.

ds^2=c^2dτ^2-dσ^2

This equation is for rest lengths and proper times in the Schwarzschild metric, so it will be true for all pairs of infinitely close points in the Schwarzschild spacetime we are considering, when dθ, and dφ equal zero, and at all radial distances, r.

If dr equals zero as well as dθ, and dφ, then we are considering an infinitesimal change in dt for a body that stays at the same radial distance, r. Such a body will have coordinates ( t₁, r₁, θ₁, φ₁), and then ( t₂, r₂, θ₂, φ₂). An observer with a clock moving between these points will measure a proper time, dτ.

dt =t_2-t_1=dτ

As with Minkowski spacetime, if a point exists in Schwarzschild spacetime, then a second point can only exist if it is possible to travel between the points at the speed of light or less. This will take some proper time, dτ, and we must have

c^2dτ^2\geq dσ^2

dσ is a radial length component measured by an observer at r, and the same radial length component at r, as measured by an observer at infinity, is dr. The two proper times in the equations below are the same, so for all r > 2GM/c²

c^2dτ^2-dσ^2=c^2dτ^2-(1- \frac{2GM}{{rc} ^{2}})^{-1}dr^2

so

dr=(1- \frac{2GM}{{rc} ^{2}})^\frac{1}{2}dσ

This is telling us that length contracts in the r direction, from the point of view of an observer at infinity, as we approach the mass, M. It will be infinitely small at a distance r, when

r=\frac{2GM}{c^2}

The Schwarzschild radius

The Schwarzschild radius, R, when r = 2GM/c², defines what is commonly known as the event horizon of a black hole.

The diagram above shows how the radial co-ordinate, r, of the Schwarzschild spacetime behaves. It doesn’t show the time co-ordinate, t.

We have seen above that as the radial distance, r, approaches the event horizon, time intervals, dt, are dilated to infinity. Time stops at r = R. At R no time has passed.

We could say that the event horizon is where time begins and space ends. We will see below that it can’t actually be reached by moving in space.

Measuring the distance to the event horizon

It is useful to see how the radial distance, as measured by an observer at an infinite distance from r = 0, behaves as the event horizon is approached. The diagram above shows how metre rules laid end to end should appear to an observer at infinity. The metre rules are assumed to be of negligible mass.

No, this couldn’t actually be done. It is a thought, or “gedanken”, experiment, as Einstein called them. Connected metre rules like this form the r axis. This idea shows how the r axis itself is contracted in physical reality. As he describes the effects of Special and General Relativity on space and time, Einstein is careful to talk in terms of real measuring rods, and real clocks. The effects on space and time that his equations describe are not apparent, they are real; they have real effects on real measuring rods, and real clocks.

The mass, M, that would cause a black hole of the size in the diagram above would be about ten times the mass of the Earth. This couldn’t be formed by gravitational collapse, according to accepted current ideas in physics, but, if all mass has the structure of Schwarzschild spacetime, there is no reason why there should not be Schwarzschild spacetime of any size. That possibility is not ruled out by the Schwarzschild spacetime solution above. See “The nature of mass” below.

Everyone seems to have assumed that black holes can only be formed by gravitational collapse. That is an assumption.

The observer also contracts

What happens to an observer as they approach the radial distance, R?

If an observer at infinity has an item that has a rest length of one metre, a metre rule for example, and takes it to a distance, r, its length will contract in the r direction. The length of the observer next to the item will contract in the r direction along with all their surroundings. All distances in the r direction will be contracted. The r axis itself will be contracted.

When the observer next to the metre rule measures its length with another metre rule, they find that it measures one metre. This doesn’t surprise them. It seems the same length to them because they themselves have shrunk in length to exactly the same degree. An observer stationary next to a rigid body will always measure its length as a rest, or proper, length.

From the point of view of a second observer at infinity, the metre rule, and the first observer, have both shrunk in length.

Time dilation means that an observer traveling towards the event horizon, and laying out metre rules as they go, would take longer and longer to lay down each metre rule. It would take an infinite amount of time to lay an infinite number of metre rules to reach the event horizon, so it can’t be reached.

The observer laying down the metre rules in this way would be measuring proper length and proper time as far as they were concerned. Their metre rules would measure one metre, and their seconds would measure one second on their clocks. An observer at infinity, however, would see them covering less and less distance with their contracted metre rules, and taking more and more time to do it.

If the observer laying the metre rules is laying them at a rate of one per second of their local proper time, they will say that their velocity is one metre per second. One of their seconds is actually much longer than one second, and one of their metres is actually much shorter than one metre, when measured by the reference length and time at an infinite distance. It is at this infinite distance that observers measure a finite distance to the event horizon. They see a velocity of much less than one metre covered in much more than one second. They see the observer approaching the event horizon going slower and slower the closer they get.

If the observer laying the contracted metre rules uses one of them as a standard metre to measure the distance they still have to go, they will find that they need more of them than the observer at infinity says that they will need of the standard metre rules that they have with them at an infinite distance.

As they get closer their clocks run slower and slower. Each second on their clocks takes longer to pass. Eventually it is taking millions of years to travel each contracted metre. Their time dilates to infinity and their lengths contract to zero. At any point on their way, they continue to say that they are traveling at one metre per second, but they are not.

Their time has dilated, and their lengths have contracted. They just don’t notice it because all their local measurments of time have dilated, and all their local measurements of length have contracted. They have no local way to tell that their time and space are not the same.

Time dilation, and length contraction both mean that the event horizon of a black hole cannot be reached. There is an infinite number of contracted metres to travel in your contracted spaceship, and your seconds take longer and longer to pass the closer you get. Space ends, and time stops, at the event horizon. The event horizon itself is not in the space and time of our Universe.

Both the observer at infinity, and the observer moving towards the event horizon measure proper times and rest lengths with the clocks and rulers that they have next to them. These proper times and rest lengths remain the same measurements as the observer moving towards the event horizon moves closer, but they are not the same actual lengths and times. Proper time is not absolute, and neither is rest length.

Aligned opinions are not safe

The reader is invited to carefully think this through for themselves. It doesn’t matter that nearly all physicists, who commit their opinions to print, seem to think that it is possible for an observer to travel right up to an event horizon, and even pass through it in their proper time. The physics is what tells us the truth, if we can interpret it correctly. Aligning one’s opinion with the aligned opinions of others might feel safe, but it really isn’t.

If we examine human history it is clear that what the majority of any group of people, and this includes physicists, believe is the truth, depends on popular opinion in any age we look at. We all look back on the fanciful notions of bygone times with a feeling of superiority that is quite unjustified. The earth is flat. Thunder and lightning are caused by angry gods fighting each other. Draining people of blood is a good medical procedure. Sacrificing children helps crops to grow. There is an invisible, undetectable, substance that light waves vibrate in. It is possible for an observer to travel through the event horizon of a black hole.

If enough people agree about something, it is treated as true, and their opinions are respected. That seems to be just as true today. Instead of assuming that at last we have got it right, perhaps we should expect that the opinions of the majority are suspect. They always have been. Why should that be any different nowadays.

There is plenty of room in the interpretation of physics for popular opinion to hold all the unjustified power it has ever done in the past.

Some physicists seem to think that, because the first observer next to the metre rule still measures one metre for its length, it hasn’t actually shrunk. From the point of view of the observer at an infinite distance there will remain a finite number of their rest length metres from the distance, r, where it is, to the event horizon. It is then argued that, with a finite number of metres to go, the event horizon could be reached in a finite amount of the first observers proper time, as the observer, and their metre rule, move towards it.

This is ignoring the Schwarzschild metric, and what it is telling us. It is continuing to think in terms of the absolute space and time of Newton.

The event horizon is not a coordinate singularity

It is frequently stated that the event horizon of a black hole is not a true singularity of the metric space. It is claimed that it is a coordinate singularity. If it was only a coordinate singularity, and not real, it is argued that it should have no physical significance.

If we describe a flat plane with coordinate axes x and y, the origin at x = 0 and y = 0 does not have any physical significance. It is a coordinate singularity. It could be arbitrarily placed anywhere on the plane. The point at r = 0 in the description of three dimensional space using spherical coordinates is another example of a coordinate singularity with no physical significance.

The North and South poles on a globe of the earth are examples of a coordinate singularity with a physical significance. The physical axis of the rotation of the earth is taken to pass through the two poles where the lines of longitude converge. If two different points were chosen, navigation would be impossible. The axis of rotation would stay the same, however.

To argue that the event horizon is a coordinate singularity that can be “removed” by a change of coordinates is to suppose that it has no physical significance in the same way that the point r = 0 can be moved without changing the physical three dimensions of Euclidean space, or any objects in it.

The event horizon does have a physical significance. It is where the escape velocity in the Schwarzschild spacetime is the speed of light. It is where length contraction, and time dilation, physically stops real bodies from reaching it. It is not a definite place in space and time; it is not in space and time. It is a limit that cannot be reached by moving through real space in a real amount of time. It is where the time and space of the real Universe cease to exist.

The Schwarzschild metric does present us with a physically real event horizon. The fact that this is disturbing should not rule out its acceptance. The fact that ships arrived back home after sailing steadily westwards was, no doubt, disturbing to those who advocated the theory of a flat earth. Being disturbed by the evidence is not a good reason for rejecting it.

The Schwarzschild metric depends on the mass, M, the radial distance r, the gravitational constant, G, and the speed of light, c. It doesn’t depend on the coordinate system used.

It would be theoretically possible to describe the Schwarzschild spacetime using x, y, z, and t. The mass, M, could be placed somewhere in the three Cartesian coordinate axes x, y, z, together with the time measurement, t. In practice it would extremely difficult to do as a mathematical description of Schwarzschild spacetime, but if it could, there would still be an event horizon exactly where it is using the t, r, θ, and φ coordinates. That is what is meant when the interval ds is said to be coordinate independent. The event horizon is real; it doesn’t depend on the coordinates used.

Just as with the longitude lines being chosen to pass through the axis of rotation of the Earth to make navigation a practical possibility, the t, r, θ, and φ coordinates, with the mass, M, at r = 0, are chosen for the enormous simplification they provide in describing the Schwarzschild metric. They are not necessary, they are convenient.

The whole point, and the utility, of differential geometry is that it describes a geometrical space and time independently of a change of coordinates. It is strange, then, for some to claim that a change of coordinates can change the physical conditions created by the event horizon in the Schwarzschild spacetime.

The laws of physics break down

It is sometimes assumed that the observer with the metre rule can not only pass through the event horizon, but can also carry on to the central singularity where the laws of physics break down, whatever that means. Does it mean that the laws of physics don’t apply any more? Since the laws of physics have broken down, physicists presumably don’t need to investigate further. They can let magicians take over. After deciding to name something a place where “the laws of physics break down”, the possibility of using physics to understand what is going on is ruled out.

The problem is that everyone seems to be assuming that the observer at an infinite distance, measuring rest lengths and proper times next to themselves, can make measurements of a finite time for the falling observer to reach the event horizon. They measure the distance using the rest length of the metre rule they have next to them, and calculate the time it would take at free fall velocity if it was measured with a clock they have next to them.

Then it is assumed that these rest lengths and proper times are the same as the rest lengths and proper times measured by the free falling observer.

Length contraction and time dilation are relegated to just being an apparent effect when the situation is viewed from an infinite distance. Physicists talk about an “image”, as viewed from an infinte distance being “frozen” onto the event horizon, as the real body falls straight through it.

The problem is that most physicists just can’t seem to shake the deep rooted belief that space and time are absolute, even as we all happily agree with Minkowski, Schwarzschild, and Einstein, that they are not.

The mathematics is learned. New names are given. Space and time are now spacetime. There is coordinate time, and there is proper time. There is real rest length, and there is apparent contracted length. There are spacelike regions and there are timelike regions. These names are used to indicate that the user is aware about something new and esoteric, but the fundamental beliefs about what space and time actually are is left unaltered.

The new physics of Einstein, Schwarzschild and Minkowski is praised and celebrated, but no one seems to really want to take on the implications. Firstly, the effects described by the Minkowski, and the Schwarzschild spacetime metrics are considered to be apparent, and therefore not real. Secondly, they are only noticeable at extreme velocities, and at extremely close distances to a mass. They can be ignored.

The story that is told goes like this: In a small enough region of a Schwarzschild metric the curvature of space and time is not noticable; if that small region is in free fall, spacetime in it can be taken to be Minkowskian. If the velocity is low compared to light, Minkowskian spacetime is Euclidian. For most situations then, we can go on thinking of space and time as the old Euclidian three dimensions of space, with the same properties everywhere, and with time flowing independently, and without reference to anything.

We can reassure ourselves that proper time, and rest length, are what are real, simply by calling them “proper”. Then the effects described by Special and General Relativity are only apparent, and only need to be considered in extreme circumstances.

The prediction of the existence of black holes, that the Schwarzschild metric makes, forces us to reconsider this convenient story. Trying to think in the old way, and believing that we can somehow plunge through the event horizon, leads to all sorts of peculiar explanations of what might happen if we do.

The truth is that the amount of space between an observer and the event horizon of a black hole, is actually infinitely greater than the amount of space between them and a galaxy far beyond the black hole. If that statement doesn’t make sense to you, then we need to think more carefully about length contraction and time dilation.

Two incompatible outcomes for an infalling body

The Schwarzschild metric is defined as seen, and measured, by an observer at an infinite distance from a single mass, M.

It is acknowledged that the observer at infinity sees time slow down for the observer moving towards the event horizon with their metre rule, and it is supposed that they will see this motion eventually stop at the event horizon after an infinite amount of time. It is suggested, in the popular opinion of many physicists, that they will see the observer, that we are imagining moving in towards the event horizon, becoming a sort of image frozen at the event horizon for ever.

Meanwhile the observer moving towards the event horizon somehow splits off from this Image, at some unspecified time, and in some unspecified way, and carries on, through the event horizon, into the interior of the black hole; eventually reaching the magical singularity at the centre, where physicists don’t need to ask what happens to them because physics doesn’t apply anymore.

There seems to be two different outcomes depending on whether the situation is observed from the observer at a distance, or by the observer falling into the black hole.

See, for example Professor Brian Cox and Professor Jeff Forshaw discussing their book on Black Holes. They both happily talk about two quite different physical outcomes depending on whether space and time are measured by an infalling observer, or by an observer remaining at a distance away.

One outcome is that an observer falling in towards the event horizon gets frozen onto the horizon forever, after which they get burnt up by Hawking radiation, and are vaporised; and the other outcome is that the falling observer is totally unaffected by the event horizon, doesn’t notice it, or a holographic image of themselves stuck on its surface, and goes on through it into space and time that have swapped over with one another, getting stretched out by the tidal forces into spaghetti, before disappearing into the central singularity where magic takes over from physics.

Their solution? Both different, and incompatible outcomes, actually happen. Look at https://www.youtube.com/watch?v=uzMUYpemgog 27 minutes and 30 seconds in.

They suggest that one observer, watching from some distance, sees the infalling observers “freezing” on the event horizon where they are burnt up by Hawking radiation. The infalling observers themselves don’t feel anything. They don’t notice the event horizon at all, and they continue through the event horizon, getting spaghettified on the way.

Meanwhile they say that Quantum Mechanics requires that no information about the infalling observers can be lost in the singularity. It is worth noticing at this point that this condition is satisfied if it isn’t possible for matter to fall through the event horizon. Instead Brian and Jeff say that the information is supposed to appear as a sort of hologram of the infalling observers back outside the event horizon. After sending this hologram back outside, they can carry on, and disappear into the central singularity where their ultimate fate can’t be known.

It is all very strange and wonderful. Popularisers of physics seem to delight in declaring how very unintuitive physics is. The mathematics is done, and interpreted as implying something strange and unexpected. Its weirdness is presented almost as a justification for its truth.

The impossibility of using physics to understand how the central singularity can exist should prompt a search for the reason why it can’t exist. Instead it is accepted as being too strange to explain, but it is still allowed to exist at the centre of the current idea of a black hole. It is believed that it can be reached by travelling through an event horizon that surrounds it, but doesn’t really exist, through an infinte amount of contracted space, and in time that has stopped passing.

Instead of being a search for understanding, physics is presented in many popular science shows as being necessarily impossible to understand. It is arcane and mysterious knowledge that only the initiates into the cult have access to. Understanding has been replaced by a sort of mystical reverence, and worship.

A thought experiment with metre rules

Let’s think about a slightly different thought experiment with the metre rules.

Imagine an observer at, r, with a great many metre rules. They all have the same rest length. The observer remains where they are, but they lower metre rules towards the event horizon, sticking them together end to end, as shown in the diagram above. They have negligible mass. They have a special gedanken experiment sauce on them that stops them getting spaghettified.

As they get nearer the event horizon, their lengths get contracted. As a metre rule is added, the whole line of metre rules will move down, each one contracting a little bit more on the way, as it is pushed further down. All the metre rules get a little bit shorter as they get pushed closer to the event horizon.

The metre rule added at the top, at a distance, r, moves the stack down very slightly less than one metre because length contraction is happening at all distances less than infinity, but, at the bottom, the stack only moves down by the more contracted length of the lowest metre rule. The amount of the contraction gets greater the closer the metre rules get to the event horizon.

As the whole stack gets closer to the event horizon, the lowest metre rule gets more and more contracted. If an infinite number of metre rules could be added to the top of the stack, the lowest metre rule would reach the event horizon. Its length would then be contracted to infinitely small. In other words its length would be zero. This cannot happen. We cannot add an infinite number of metre rules to the stack.

It would take an infinite number of metre rules to reach the event horizon, so there is an infinite amount of space to travel through to get there. Adding an infinite number of metre rules would take an infinite amount of time at the top of the stack, so the lowest metre rule can never actually reach the event horizon however many metre rules are added at the top.

An important point to notice is that all the metre rules, in this gedanken experiment, started out with the same length at the top of the stack. They are all equivalent to each other. They all have the same rest length. If they were laid end to end in empty space, they would make an infinitely long line. So an infinitely long length of regular metre rules would be needed to reach the event horizon of a black hole.

This means that there is an infinite amount of space between an observer at a distance, r, and the event horizon of a black hole. We can see now that it is true that the amount of space between an observer and the event horizon of a black hole, is actually infinitely greater than the amount of space between them, and a galaxy far beyond it.

This statement still feels strange, and there is a resistance to accepting it, but the logic of interpreting the mathematical description of the shape of Schwarzschild’s space time means that it must be true.

All of us, except the flat earthers, will accept that the straight line distance to a place on the other side of the earth is different to the actual distance that would need to be travelled to get there. We have a picture in our minds of the globe, and we know that we live on a spherical surface. It seems flat because of the scale. The earth is large and the curvature of the surface is not apparent locally.

It may still feel strange to a person new to air travel, that their flight from London to Los Angeles takes them over Greenland. They might have to stop and think of a great circle on the globe of the earth to overcome their inclination to think of their travel as a straight line on a flat map.

The surface of the Earth is not flat, and space is not flat. Length is contracted in the gravitational field of a mass. Time is not flowing continuously at the same rate everywhere. Even if we know the truth of this, it is still difficult not to assume that we live in flat space with uniform time in our everyday life, and for most practical purposes this assumption is quite workable.

The existence of blackholes, and attempting to understand them, makes that assumption untenable. The event horizon of a black hole is the edge of the Universe. Our spacetime actually ends at the event horizon of a black hole, but there is an infinite amount of contracted space available to fall through before it is reached, and there is not enough time passing for it to fall in. This means that an observer in our spacetime cannot reach an event horizon, let alone pass through it.

A physicist who believes that matter can pass through an event horizon has unconsciously invented a spacetime separate from the Schwarzschild spacetime, and imagined that these two separate spacetimes can actually exist together side by side.

This is why Brian, Jeff and many others believe that there are two outcomes for a massive body falling towards an event horizon. They have effectively invented a separate spacetime. One outcome happens in the Schwarzschild spacetime, and the other outcome happens in their separate invented spacetime where space hasn’t contracted, and time hasn’t dilated.

There is no separate spacetime. The Schwarzschild spacetime is the one and only definitive solution to Einstein’s equation for a mass with zero charge and zero angular momentum.

If an observer at r starts to move down the length of the joined metre rules, the observer themself will contract in the r direction. Each metre rule they pass will look the same as the one before. If they stop to measure the one they are passing, with a metre rule that they are carrying with them, it will measure one metre. They will say that the rest lengths of the metre rules they are measuring, with the metre rule they are carrying with them, are the same.

The rest length they measure will always be one metre, but the actual length is getting contracted. This isn’t an illusion due to the point of view of an observer at infinity. It is real length contraction. There is an infinite amount of rest length to move down before any event horizon is reached.

Rest length is not a fixed length; it is a fixed measurement of length. Similarly proper time is not a fixed duration of time; it is a fixed measurement of a duration of time. Length and time, contrary to what Newton must have thought, are not absolute; they are relative. A duration of time can only be compared to another duration of time. There is no absolute time to compare it to. Calling something “proper time” does not make it an absolute duration. Similarly, calling something a “rest length” does not make it an absolute length.

Because the rulers and clocks of the observers change by exactly the same factor that the lengths and times being measured next to them change, these changes of lengths and times can only be measured by observers somewhere else. In the case of the Schwarzschild metric being discussed here, the length and time measurements are stipulated to be made by an observer at infinity. This observer sees lengths and times next to them as rest lengths and proper times, and they see the lengths and times of other observers, closer to the central mass, as being contracted and dilated. These length contractions and time dilations are quite real.

The rest length of a body measured by an observer next to it at infinity, will be the same measurement as measured by an observer next to the same body at a distance, r. The actual lengths are not the same. Time really does slow down, and lengths really do contract, as a real physical body approaches a mass.

Relative velocity between observers also contracts lengths, and dilates time. This must be true in the Schwarzschild metric space are well as in the Minkowski metric space. See “Moving frames in the Minkowski spacetime” below. It must also be true in our Universe where experiments show that all observers measure the same speed of light. Changes in space and time must occur in order that two observers in relative motion see the same speed of light.

The measurements of proper time, and rest length, made by moving observers, will be the same, but the actual length of a body, and the rate at which time is passing for it, will vary depending on its velocity relative to an observer at rest. Proper time, and rest length, will always be measured to be the same by an observer at rest next to the body. That doesn’t mean that their lengths and times don’t change with their relative velocity.

Which of two observers in relative motion to each other has their time passing more slowly than the other? Which one do we choose to be at rest and which one to be moving? Does that choice affect which one’s time passes more slowly?

If we choose to consider that the length contraction, and time dilation, affecting observers with a relative velocity, do not actually happen, but are only apparent effects due to a point of view, we don’t need to figure out what the solution to this problem is.

We will see, in this account, that changes in length and time due to relative velocity are quite real, and are, in fact, caused by the gravitational effect of mass on space and time in the same way that mass causes changes in space and time in the Schwarzschild metric. If we consider motion through the Universe, we can say that one observer is at rest with respect to the Universe as a whole, and the other is moving with respect to the Universe. See the section “The equivalence of inertial and gravtational mass” below.

We have seen above that it is clearly not possible to reach the event horizon of a black hole. This will potentially upset the entire physics community, but they will just have to get used to it. They need to think through this carefully for themselves, and not trust the opinion of everyone else.

The formation of black holes

It has been imagined that black holes can only be formed by supermassive stars collapsing in on themselves. They would have to be massive enough to overcome all the other forces known to physics, so that even protons and neutrons would collapse. Physicists know of no other force that can overcome gravity when it is strong enough.

One reason that so many physicists are so determined to imagine that it is possible to go through the event horizon may be because it is imagined that black holes can only grow larger as a consequence of matter falling into them. Supermassive black holes, with masses on the same scale as galaxies themselves, have been detected at the centre of many galaxies. It is supposed that these supermassive black holes can only be formed by stars falling through the event horizon of a collapsed star.

The story goes that if the nuclear processes, that provide the outward radiation pressure that balances the inward pull of gravity, end, then gravity pulls the material of the star inwards. The density of this material increases, and the stars radius decreases.

At any particular density there is a radius that corresponds to the radius of an event horizon. If the collapse proceeds, and it is not halted by the pressure created by the constituent protons, neutrons, and electrons that make up the material of the star, the collapse continues until its density is great enough to cause an event horizon at its radius.

Stars below a certain mass will have this collapse halted by pressure exerted by their constituent atoms forced together. At greater masses, when protons and electrons have been compressed enough by gravity, it is expected that they will become neutrons. It is thought that the material of the star could consist of a solid mass of neutrons, known as a neutron star. If the gravity is strong enough to overcome the pressure of these neutrons, the neutron star collapses within the event horizon for its mass, and a black hole is formed.

There is another way that a mass distribution could form an event horizon. George Birkhoff’s theorems¹ show that Schwarzschild spacetime will exist outside any spherically symmetric distribution of mass.

If matter of any uniform density is considered, and it is imagined that we can have any sized spherical volume of this matter, we can calculate the size of sphere that it would need to be for it all to form an event horizon. The less dense the matter, the greater the sphere that would form a black hole.

Such a sphere of sufficient size for its density would already be inside an event horizon. Stars don’t have to collapse to form a black hole. Matter doesn’t have to fall in to an existing black hole for us to be able to imagine a black hole of any size.

If our Universe has an overall uniform density, and it extends to an infinite distance, the Schwarzschild metric tells us that there will be spherical volumes in it that will already be inside an event horizon. All spherical volumes of a large enough size, and with a uniform density, should be inside an event horizon.

Birkhoff has shown that it makes no difference if the Universe is expanding. If matter has a density that is the same in all directions, or in other words, it has spherical symmetry, it will have a Schwarzschild metric outside the mass, even if it is in motion. So long as sufficient mass with spherical symmetry is present, within the volume given by R = 2GM/c², it will form an event horizon.

Schwarzschild’s spacetime is a Universe with one mass in it. There has not, as yet, been a solution of Einstein’s equation proposed for two masses, or more than two. The presence of each mass would affect the space time of the other. Schwarzschild found the solution for one mass by making enough simplifying assumptions. It was still an amazing feat. Einstein himself thought that his equation was too difficult to be solved.

To investigate the way a spherical distribution of mass could be made large enough to have an event horizon, we need to examine how Birkhoff’s theorems predict the way a spherical distribution of mass changes the spacetime outside its extent.

The Schwarzschild spacetime in our Universe

It has been said that the Schwarzschild spacetime cannot be applied to our Universe since one of its conditions is that the spacetime it describes cannot contain mass or energy apart from the central point source mass, M.

We are going to construct a model Universe, using the Schwarzschild spacetime, that keeps to this condition.

George Birkhoff’s theorems¹ state that a spherically symmetric distribution of mass will have a Schwarzschild spacetime outside its radius. We are going to imagine starting with an infinitley small spherical volume, and increasing its size by adding matter into this Schwarzschild spacetime, maintaining spherical symmetry as we do so.

As we add to this hypothetical sphere we are adding matter into this spacetime, and we need to take account of how the shape of spacetime affects this process. We cannot simply postulate a sphere of uniform density simply springing into existence. Such a sphere exists in space and time, and its presence will affect the shape of the spacetime it is in.

To take account of the sphere affecting the spacetime outside its extent, we can imagine starting with an infinitely small spherical distribution of mass. We can imagine incrementally adding to this spherical volume, increasing its radius, and its total mass. If we maintain spherical symmetry, we can continue to use Birkhoff’s theorems to model the space time outside its boundary.

We can imagine growing the size of our hypothetical sphere until it is on the point of being large enough to form an event horizon at its surface. At this point we can still use Birkhoff’s theorems, and at this point, Birkhoff’s theorems, and the Schwarzschild metric, predict the formation of an event horizon. This means that if a spherically symmetric sphere of the right density and radius exists, it will have an event horizon at its surface.

Our Universe could be inside an event horizon without falling into a central singularity. This account will show that we can use the Schwarzschild metric, together with Birkhoff’s theorems, to model an infinite Universe, with an average uniform density, that nevertheless has a finite mass, M, and a spherical structure with a radius, R, that a black hole of that mass would have.

The model is a Universe like ours inside the event horizon of a modified Schwarzschild metric. It is based on known, and accepted, laws of Physics. The mass of this Universe does not fall into a central singularity where it is beyond understanding by the laws of physics. It is an expanding spherical distribution of matter that appears to all observers exactly like our Universe does, and it is inside an event horizon.

If the whole Universe we perceive is inside an event horizon, then it is clear that it is not true that anything inside an event horizon must continue to collapse until it is consumed by the central singularity. That hasn’t happened to us.

George Birkhoff¹ has shown that a spherically symmetric distribution of mass, even if it is in motion, can act as the central source, but the spacetime described by the metric must be outside this mass distribution, and empty.

In what follows Birkhoff’s theorem¹ is used to describe the shape of spacetime on the surface of a hypothetical spherical distribution of mass. It is imagined that successive, infinitesimally thin, shells of mass are added to this surface, and a spherical structure is constructed.

Birkhoff has additionally shown that the spacetime inside a spherical shell of mass is Minkowskian. This implies that nothing outside any particular radial point in this spherical structure will affect the spacetime inside the radial distance, r.

In other words the only mass in the structure that will affect the spacetime on the surface of radius, r, is mass inside the radius. We can envisage adding successive shells without changing the structure already built.

To apply Schwarzschild spacetime to the Universe we need to assume that the Universe is spherically symmetric. In building the proposed model spherical symmetry is maintained throughout the process. A spherical structure is imagined being built of successive shells of mass. Each shell that is added is imagined being placed on the surface of the sphere that is already constructed.

In the process of building this hypothetical structure, each shell is the last one placed on the surface of a spherical structure that, as allowed by Birkhoff’s theorem, has Schwarzschild spacetime on its surface.

It is also assumed that it is not rotating, and uncharged, in order to apply the Schwarzschild metric.

We will see that Birkhoff’s theorems¹ allow us to describe it with the four non-zero terms of the Schwarzschild metric, even though it is expanding. The model, that is developed here, is from the point of view of an observer co-moving with the expansion, as we will see below, and later extended to observers with all velocities less than the speed of light, relative to the expansion. See “The equivalence of inertial and gravitational mass” below.

Hubble’s law, and the velocity of light

The recession velocity of the galaxies, v, is observed to be related to their distance away by Hubble’s law. Galaxies at a distance, r, will have a velocity, v, given by

v=Hr

Where H is Hubble’s constant.

Let’s suppose that the distance, R, implied by Hubble’s law, and the velocity of light, c, is a real boundary. We will have

c=HR

Where R is the distance that the recession velocity of the galaxies is equal to the speed of light, c.

We define a comoving observer as one with a relative velocity to another comoving observer that is solely due to the expansion of the Universe.

Length contraction and time dilation in our Universe

The velocity of comoving matter near this distance, R, will be approaching the speed of light from the point of view of such an observer. According to Special Relativity, an observer in spacetime empty of matter and energy, will see lengths at this distance contracted to zero in the direction of the motion, and time dilated infinitely.

If we suppose that this length contraction and time dilation happens in our Universe, which it must if all observers are to measure the same speed of light, the space between the receding galaxies is contracted, along with the galaxies themselves. To undergo a length contraction in whatever direction a comoving observer looks, the contraction would have to be directed radially away from that observer, and it is this that points to gravitation as the cause.

A spherical distribution of mass produces a radially distributed length contraction. Birkoff’s theorem¹ tells us that we can consider any spherically distributed set of masses as if they are a point source at the centre of the distribution, and we can use the Schwarzschild spacetime to describe the shape of space and time outside the distribution of masses. This is true even if they are in motion.

Using Birkoff’s theorem in our Universe

We can use Birkoff’s theorem¹ to build a model Universe as follows.

We will consider a sphere of matter, with a radius, r, containing matter at the same uniform average density of matter throughout the Universe, and centred on a co-moving observer. The assumption of average density will be fully justified in “Potential energy and escape velocity”, and “Adding shells of uniform density” below.

Birkoff’s theorem¹ tells us that, for the exterior solution to be given by the Schwarzschild spacetime, the only requirements for the distribution of mass is that it is spherically symmetric, and that there is zero mass density at the radius considered. Any spherically symmetric solution of the vacuum field equations must be static, and asymptotically Minkowskian. It does not matter if the mass is expanding, or contracting.

Birkoff’s theorem¹ also states that a spherically symmetric shell of mass will have Minkowskian spacetime inside it.

We will assume we can use Birkoff’s theorems¹, including spherical symmetry round the sphere of matter we are considering.

Space and time at the surface of a sphere of matter

At the surface of this sphere, with a radius, r, there will be gravitational effects on space and time, as predicted by the General Theory of Relativity. As we have noted, the shape of space and time outside any such spherically symmetric distribution of mass will obey the Schwarzschild spacetime equation.

ds^2=(1- \frac{2GM}{{rc} ^{2}})c^2dt^2-(1- \frac{2GM}{{rc} ^{2}})^{-1}dr^2-r^2dθ^2-r^2sin^2θdφ^2

Where G is the gravitational constant, M is the mass, c is the velocity of light, and dt, dr, dθ, and dφ are infinitesimal coordinate intervals, in spherical coordinates, as measured by an observer at an infinite distance from the sphere of matter.

Birkhoff’s theorems allow us to replace the mass, M, with a spherical distribution of mass. If we consider the sphere to consist of matter with a uniform density, ρ, it will have a mass given by

M=\frac{4ρπr^3}{3}

A modified Schwarzschild spacetime equation

Replacing M with 4ρπr³ ⁄ 3 gives a modified Schwarzschild spacetime equation

\color{red}{ds^2 = (1-\frac {8Gρπr^2}{3{c} ^{2}})c^2 dt^2- (1-\frac {8Gρπr^2}{3{c} ^2 })^{-1} dr^2 -r^2dθ^2 -r^2sin^2θdφ^2}

We should note that this is not the same as the original Schwarzschild spacetime above. The radial coordinate, r, is now the radius of the sphere we are considering. We cannot investigate this spacetime by simply extending the radius, r, out to infinity as we can with the regular Schwarzschild spacetime.

We have equated the radius, r, or the sphere with the radius, r, defining the Schwarzschild factor. This determines the density, ρ. This density is the critical density that results in a spherical volume with a radius equal to the radius, r, in the original Schwarzschild metric above. This critical density will give the same mass, M, for the sphere as was in the original Schwarzschild factors.

As we increase r ,we are increasing the size, and the mass, of a spherical distribution of mass. The Schwarzschild metric describes the shape of space and time at this distance, in exactly the same way as before, but the mass, M, is now a sphere of mass, M, with that radius. As we increase the radius, r, there will come a point, at a radius R, where the radius corresponds to the radius of an event horizon.

If we make the radius of the sphere R, where R = 2GM/c² and M=4πρR³/3 then the Schwarzschild factor in the modified equation above becomes

(1-\frac {8GρπR^2}{3{c} ^{2}})=(1- \frac{2GM}{{Rc} ^{2}})=(1- 1)=0

All we have done is make a spherical distribution of mass with a density that makes a sphere with a radius equal to the radius of a black hole with that mass. This is a black hole.

As it stands the modified Schwarzschild spacetime above tells us nothing about the region beyond the surface of the sphere at a radius, r, or about the shape of spacetime within this sphere of matter.

We can, however, use Birkhoff’s theorems at the radius, r, given by the modified Schwarzschild spacetime above. At this surface there will be empty spacetime as required by Birkhoff’s theorems. This modified Schwarzschild spacetime will apply to the spacetime immediately outside the radius, r, of the sphere we are considering. Birkhoff’s theorems apply to the empty spacetime outside a spherical mass distribution, and so they will apply at this radius, r, when r<R.

To investigate this situation with more care we need to imagine how such a spherical mass distribution could be created. The process will need to take account of how such a spherical distribution of mass could be assembled, starting from empty Minkowskian spacetime, and determining how the shape of spacetime affects the process at each stage.

An infinitly small sphere of mass in Minkowskian spacetime

What we can do with the modified Schwarzschild spacetime equation above is investigate the shape of spacetime at the surface of a spherical distribution of mass. We can use it to investigate how we can start with an infinitely small spherical distribution of mass, and model the spacetime at its surface as we incrementally add to it in infinitesimal steps while maintaining spherical symmetry.

At the surface of the sphere, the radial coordinate, r, is equal to the radius of the sphere, and we can use the modified Schwarzschild equation to determine what happens to an infinitesimal increase dr to the radial coordinate, r, by adding a shell of the same density, and with a thickness dr.

We will assume that adding an infinitely thin shell of thickness dr, and negligible mass, will not affect the spacetime at the surface of the sphere at r. It will be outside the surface in empty spacetime, as reqired by Birkhoff’s theorems.

To investigate what happens as we increase the size of the sphere we are considering, we will imagine starting with a sphere that has a finite density, but has an infinitely small mass, and an infinitely small volume in otherwise empty Minkowskian spacetime.

When r = 0, the sphere will be infinitely small, and this modified Schwarzschild spacetime is then equivalent to the Minkowski spacetime in spherical coordinates. As we add spherical shells, Birkhoff tells us that the space inside the shells will remain Minkowskian. At r = 0 the Schwarzschild factors are equal to one, and we will have

ds^2=c^2dt_{(r=0)}^2-dr^2_{(r=0)}-r^2dθ^2_{(r=0)}-r^2sin^2θdφ^2_{(r=0)}

Time dilation at the surface of the sphere

When dr, dθ, and dφ equal zero we can write for an infinitesimal time dt on the surface of a sphere of radius, r

ds^2=(1-\frac {8Gρπr^2}{3{c} ^{2}})c^2 dt^2

and at the centre of the sphere we have

ds^2=c^2dt^2_{(r=0)}

We again let dσ and dτ be equivalent to rest length and proper time respectively where dσ is an infinitesimal radial length that is measured next to an observer stationary in the spacetime, and dτ is an infinitesimal time duration measured next to an observer stationary in the spacetime. Birkhoff has shown that the space inside a spherical mass distribution is Minkowskian in form, so, for an observer at the centre of the sphere we are considering, we have

dτ^2 =  dt^2_{(r=0)}

so

ds^2=c^2dτ^2 

and so for all all r²<3c²/8Gπρ

\color{red}{(1-\frac {8Gρπr^2}{3{c} ^{2}})c^2 dt^2=c^2dτ^2}

Looking out to the surface of the sphere from the centre of the sphere, a duration of time, dτ, measured at a distance r, will correspond to a dilated time, dt, measured from the centre, given by

\color{red}{dt=(1- \frac{8Gρπr^2}{3{c} ^{2}})^{-\frac{1}{2}}dτ}

This means that time will be flowing more slowly on the surface of the sphere we are considering than it does for an observer at the centre.

Length contraction at the surface of the sphere

As we increase the radius of the sphere by adding a shell, we see that the thickness of the shell, dσ, as measured at the shell, is contracted to a length dr, as measured from the centre of the sphere.

As above, we can write for all r²<3c²/8Gπρ

(1-\frac {8Gρπr^2}{3{c} ^{2}})c^2dt^2=c^2dτ^2

So, when dθ, and dφ equal zero we can write

\color{red}{ds^2 = c^2dτ^2- (1-\frac {8Gρπr^2}{3{c} ^2 })^{-1} dr^2} 

and

\color{red}{ds^2 = c^2dτ^2-dr^2_{(r=0)}}

As we saw in “The Mikowski spacetime metric” above, it is necessary to include the proper time terms in the two equations above because pure lengths cannot exist in the Schwarzschild metric. A pure length, with dτ = 0, implies a negative value for ds², so there can be no real number for ds. There is no spacetime if there is no real value for ds between any pairs of infinitely close points being considered.

We can however easily show that the term in dr² in the Schwarzchild spacetime metric is equal to dσ²

We have for an observer at r = zero

dr^2_{(r=0)}=dσ^2

The rest length, dσ, will be the same everywhere in the Schwazschild metric, so, for a rest length, dσ, when measured at r, and a proper time at r of dτ

ds^2=c^2dτ^2-dσ^2

so

\color{red}{c^2dτ^2- (1-\frac {8Gρπr^2}{3{c} ^2 })^{-1} dr^2}  = c^2dτ^2-dσ^2

and so for all r²<3c²/8Gπρ

(1-\frac {8Gρπr^2}{3{c} ^2 })^{-1} dr^2= dσ^2

A shell in Minkowskian spacetime at an infinte distance, i.e. initially not on the surface of a spherical mass distribution, that has a thickness dσ, will have that thickness reduced to a thickness dr when it is placed on the surface of the sphere as follows

\color{red}{dr=(1- \frac{8Gρπr^2}{3{c} ^{2}})^{\frac{1}{2}}dσ} 

By adding successive shells to an infinitely small sphere, we can attempt to build a model of our universe obeying the Schwarzschild spacetime and Birkhoff’s theorems, but we have a problem with this analysis.

Constant density is assumed

If a shell of space containing massive bodies has its thickness reduced, as described above, and there are the same number of massive bodies in the shell, its density must increase. We assumed earlier, though, that the density of the model universe we are considering remains uniformly the same as it is at the centre. To continue to add shells, using the equations above, the density of the shells must remain the same as the density of the sphere. For that to be true, the mass of each body in the shell must decrease by the same factor as the thickness of the shell is decreased. We can see how this could be the case by considering the potential energy of the masses in the shell.

Potential energy and escape velocity

Energy in a grandfather clock

Let us consider the following situation. A mass moving straight down in a gravitational field can be used to operate machinery, for example a grandfather clock. A tension force is transmitted up the chain holding the weight, and, as the mass drops, this force moves the machinery of the clockwork. This takes energy, and this energy is eventually dispersed into the environment of the clockwork in the form of heat from friction, and sound. This is kinetic energy on a molecular level.

Where does this energy come from? It can only come from the mass of the dropping weight. Nothing is supplying energy to the dropping weight, so, if energy is supplied by the dropping weight, its mass must decrease according to E = mc² .

It can be shown that rest mass in the Schwarzschild spacetime is dependent on the radial coordinate r. The rest mass at a point r, m_{rest_at_r} , can be written as

\color{red}{m_{rest-at-r} = {m_{rest}} {(1- \frac{2GM}{{rc} ^{2}})^\frac {1} {2} }}

Where m_rest is the rest mass of a body at an infinite distance.

We can identify this decrease of rest mass as a decrease in the bodies potential energy, E_potential , in the Schwarzschild spacetime as follows

\color{red}E_{potential}={m_{rest-at-r}c^2 = {m_{rest}}c^2 {(1- \frac{2GM}{{rc} ^{2}})^\frac {1} {2} }}

A mass lowered slowly from infinity, in the manner of the weight dropping to run the grandfather clock, will lose energy. The energy lost, E_lost, will be the total rest mass energy at infinity minus the potential energy at the radial point r.

E_{lost} ={m_{rest}}c^2- {m_{rest}}c^2 {(1- \frac{2GM}{{rc} ^{2}})^\frac {1} {2} }

At r=2GM/c² all the original rest mass energy the mass had at infinity will be lost. The rest mass of a body stationary at the event horizon is zero.

Motion on a geodesic

If, instead, the weight is dropped the same distance in free fall, we expect that its mass remains the same. This is what is observed.

A mass moving freely in a gravitational field is not exchanging energy with its surroundings. This means that the mass of a body in free fall should remain constant. In General Relativity a mass moving in this way is said to be moving on a geodesic.

It’s velocity, however, will increase, and Special Relativity tells us that its mass should increase with increasing velocity.

Mass increase with velocity

According to Einstein the total energy, E, of a mass, m, moving at velocity, v, should increase according to

E=mc^2

where m is the bodies relativistic mass. Richard Feynman¹⁰ shows in “Energy and mass” above how Einstein’s equation for total energy, E , relates the relativistic mass, m, to the rest mass, m_rest , as follows.

m=\frac {m_{rest}} {(1-\frac{v^2}{c^2})^{\frac 1 2}}

Where c is the velocity of light, and the rest mass, m_rest, is the mass when the relative velocity is zero. The mass, m, giving the total energy, is greater than the rest mass, m_rest, by an amount we refer to as kinetic energy.

We can write this as

E=m_{rest}c^2(1-\frac{v^2}{c^2})^{-\frac{1}{2}}

So why doesn’t the mass of a falling body increase?

Relativistic escape velocity

In Newtonian mechanics the quantity 2GM/r can be identified with the square of the escape velocity, v_escape , at a distance, r, from a mass, M. The potential energy lost by a mass, m, falling from an infinite distance to a distance, r, from M, appears as an equal amount of kinetic energy. Equating kinetic energy, KE, and potential energy, PE, gives

KE = PE

\frac{mv_{escape}^2}{2} = \frac{GMm}{r}

​{v_{escape}^2} = \frac{2GM}{r}

The equations above for escape velocity use the non-relativistic form for kinetic energy.

J B Hartle ⁷, has shown, in a general relativistic analysis, that the escape velocity of a test mass, as measured by a stationary observer at a distance, r, in a gravitational field with a Schwarzschild metric space, is given by

\frac{v_{escape}^2}{c^2}=\frac{2GM}{rc^2}

This test mass is taken to have an insignificant effect on the Schwarzschild spacetime it is moving in.

This is Hartle’s description of the situation he is modelling.

“An observer maintaining a stationary position at Schwarzschild coordinate radius, r, launches a projectile radially outward with velocity, v, as measured in his or her own frame. How large does v have to be for the projectile to reach infinity with zero velocity? This is the escape velocity, v_escape . The outward-bound projectile follows a radial geodesic since there are no forces acting on it.”

We can see that Hartle’s analysis gives the same result as the Newtonian analysis with the addition of the constant c². The velocity is now expressed as a proportion of the speed of light. The concept of an escape velocity that is the same for all bodies, whatever their mass, arises in both the Newtonian regime, and the relativistic regime. This in turn depends on the equivalence of inertial, and gravitational mass.

The mass in the kinetic energy equation is inertial mass, and the mass in the potential energy equation is gravitational mass. The equations above for escape velocity assume the equivalence of inertial mass and gravitational mass. There is no a priori reason for this equivalence. It was an assumption in Newtonian Mechanics, and it remains an assumption in General Relativity.

Hartle shows that the total energy, E, needed to give a test body with negligable mass, m_rest, escape velocity, v_escape , away from a point at a radius, r, from a mass, M, in the Schwarzschild metric, as measured by an observer next to the body at that radius, is

 E ={m_{rest}c^2}(1- \frac{2GM}{{rc} ^{2}})^{-\frac{1}{2}}

and equivalently

E=m_{rest}c^2(1-\frac{v_{escape}^2}{c^2})^{-\frac{1}{2}}

This assumes that the rest mass at at the radius, r, is the same as the rest mass at infinity. It is the mass at rest at r, as measured by an observer at r, that Hartle is using in the first of the two equations above as m_rest.

The second equation above is Einstein’s equation for total energy. It is telling us that the total energy with the observer at r is greater than the rest mass by the factor in brackets, and this excess energy represents the kinetic energy necessary to lift the massive body to infinity.

As the body ascends up the r axis, it will lose velocity, and its energy, given by Hartle’s equations above, will get less. At an infinite distance all the extra energy that it had at r, needed to give the massive body enough energy to escape to infinity, is gone. The massive body now has a total energy equal to its rest mass. At an infinte distance v_escape is zero, and Hartle’s equation above becomes

E=m_{rest}c^2

This is conventionally correct. Rest mass is assumed to be constant, when measured by an observer stationary next to it, wherever the massive body and that observer are. A bodies rest mass is equal to its total energy when it isn’t moving.

Having stated that the body is moving on a geodesic, since there are no forces acting upon it, Hartle does not explain what happens to the extra energy that the body has at the radius, r, and that it no longer has when it reaches an infinite distance.

Massive bodies in free fall from infinity

We can turn this round, and imagine a test body that starts with rest mass, at infinity. We assume, as above, that the mass of this test body is small enough to have a negligible effect on the Schwarzschild spacetime. We can imagine this body in free fall from infinity towards the source of the Schwarzschild metric, M, at r=0.

It will have a velocity straight down, with a magnitude equal to its escape velocity, v_escape, for its position on the r axis. Hartle’s equation above suggests that it would gain kinetic energy, so that its total energy, E, would be as stated above when it reaches an observer at r, and its total mass would increase by the factor in brackets.

The observer at r would see the body falling past them with escape velocity, v_escape, directed down the r axis.

It now has sufficient energy so that, if its velocity were reversed at the point, r, it would have the velocity needed to escape back to an infinite distance.

We may notice that a massive body dropping in this way is in free fall, and is moving on a geodesic . It is not exchanging energy with its surroundings, so we would expect that its mass will remain constant, and equal to m_rest. Hartle’s equation above, however, states that the kinetic energy of the body is increasing as its velocity downwards increases, and its total energy increases according to Einstein’s equation for total energy.

Hartle envisions that this is a Schwarzschild metric, with a test body of insignificant mass, m_rest . The Schwarzschild mass, M, could be inside an event horizon. Hartle’s equation for escape velocity gives v_escape equal to the speed of light, c, when r = 2GM/c². This implies that energy cannot be coming from the mass, M, since no energy can escape from behind the event horizon.

The mass, m_rest, starts with rest mass at an infinite distance, and when it reaches the point, r, its energy is greater by the factor in brackets, according to Hartle’s equations above.

As we have noted, the massive body is not exchanging energy with its surroundings as it falls, so its energy cannot increase. The energy that the body has at an infinite distance when it starts to fall must be the same as the energy it has at a distance r. The only place the extra kinetic energy that the body has at r can come from is the rest mass of the body.

To see what this implies we will label the rest mass at r as m_{rest_at_ r} . Hartle’s equation for the energy needed to escape to infinity becomes

E ={m_{rest-at-r}c^2}(1- \frac{2GM}{{rc} ^{2}})^{-\frac{1}{2}}

and if no energy is gained by a body in free fall from an infinite distance then

E ={m_{rest-at-r}c^2}(1- \frac{2GM}{{rc} ^{2}})^{-\frac{1}{2}}= {m_{rest}c^2}

We can see that if the total energy of the falling body doesn’t change then m_{rest_at_ r} must be less than m_rest .

The kinetic energy that the body has at r must come from the rest mass that the body has at infinity. There is nowhere for the kinetic energy to come from, as the mass, m_rest, descends in free fall, unless the rest mass of the body is less at r than it is at an infinite distance.

The grandfather clock rewound

If, instead of allowing the massive body to drop in free fall, we imagine lowering it slowly, we will be extracting energy from it. This is illustrated by the example of the grandfather clock.

As we noted above the mass, M, that is the source of the Schwarzschild metric could be inside an event horizon.

This gedanken experiment now consists of the mass, M, inside an event horizon, and the test mass that is exerting a force on the clockwork as the test mass moves down the r axis. We are imagining the clockwork receiving energy, not supplying it.

The situation envisaged consists of the central mass, M, and the test mass, together with the clockwork of the grandfather clock. There is nothing else in this gedanken experiment that can supply the energy needed to run the clockwork. We will assume that the clockwork, like the test mass, has an insignificant effect on the Schwarzschild spacetime.

The Schwarzschild metric models the entire space time being considered here. If the mass, M, is inside an event horizon, it cannot be supplying energy to the clockwork. The clockwork is receiving energy, and eventually it is dissipating as heat. This energy must be coming from somewhere. There is only one other place in this system that can supply energy to operate the clockwork, and that is from the rest mass of the dropping weight.

Consequently we would expect a decrease of the test bodies mass from its rest mass at infinity, m_rest, as the body is lowered slowly from infinity, due to the loss of potential energy. There is nothing else that could be the source of the energy supplied to the clockwork. The mass of the test body must be less than its rest mass at infinity, when it is at rest next to the observer at r.

The total energy of the weight dropping to power the clockwork, and therefore its mass, must get less by the amount of energy extracted.

How much should a bodies mass be reduced in this way?

Rest mass and potential energy

To counter the expected increase in mass due to increasing velocity, so that a mass in free fall on a geodesic remains constant , we will postulate that a mass being lowered slowly will lose mass according to

\color{red}{m_{rest-at-r} = {m_{rest}} {(1- \frac{2GM}{{rc} ^{2}})^\frac {1} {2} }}

Here we distinguish between m_rest as the rest mass of a body at an infinite distance from, and stationary with respect to, a gravitating mass, M, in a Schwarzschild metric, and m_rest-at-r as the rest mass of the same body stationary next to an observer at r. The massive body ends up at rest with an observer at r, but its mass, m_rest–at-r, is less than its rest mass at infinity.

This represents the body losing potential energy as it is lowered. This energy is not lost. It appears as the energy used to run the clockwork of the grandfather clock. Eventually it disperses into the envirionment of the clock as heat. Mass and Energy are conserved.

Hartle’s equation is applied by an observer at r. It is this lower mass, m_rest-at-r, that needs to be used with Hartle’s equation in place of m_rest.

If we do this, then Hartle’s equation for the total energy to give a massive body escape velocity becomes

\color{red} E ={m_{rest-at-r}c^2}(1- \frac{v_{escape}^2}{{c} ^{2}})^{-\frac{1}{2}}

We can combine these two factors for a massive body in free fall with escape velocity, and Hartle’s escape velocity equation above becomes

\color{red}{E = \frac{{m_{rest}}c^2 {(1- \frac{2GM}{{rc} ^{2}})^\frac {1} {2} }} {{(1-\frac{v_{escape}^2}{c^2})^{\frac 1 2}}}}

E = m_{rest}c^2

We see that the total mass of a massive body in free fall remains constant, as expected.

A body in free fall exchanges potential energy for kinetic energy

A body in free fall will be exchanging potential energy for kinetic energy, and its total energy will remain constant. It is not exchanging energy with its surroundings, so It is moving on a geodesic.

In both the case of the mass supplying energy to the clock, and in the second case of dropping the mass in free fall, the potential energy of the mass decreases by the same amount. In the case of the clock, the potential energy appears as heat and sound in the mechanism of the clock, and its environment; in the case of free fall, the potential energy of the mass becomes kinetic energy of the mass.

If there were no corresponding decrease in potential energy, and decrease in mass associated with it, then as a bodies velocity increases in free fall, we must suppose that its relativistic mass would go to infinity as it approaches the event horizon of a black hole. This can’t be the case, as it would mean that any object falling onto a black hole would gain infinite mass, and therefore infinite energy.

So, we expect that a body lowered slowly into a gravitational field will lose mass. We used the example of a weight dropping to operate a grandfather clock. Energy is conserved; the energy to run the clock comes from the rest mass of the dropping weight.

A massive body lowered slowly into the gravitational field round a black hole in this way would lose all its mass if it could reach the event horizon, r, where

{r}=\frac {2GM}{c^2}

Lorentz factors in the Schwarzschild spacetime

We saw above that there are length and time variations in the Schwarzschild metric. Time is dilated as follows

dt=(1-  \frac{2GM}{{rc} ^{2}})^{-\frac{1}{2}}dτ

Length is contracted in the radial direction as follows

dr=(1- \frac{2GM}{{rc} ^{2}})^\frac{1}{2}dσ

Hartle’s equations above for the total energy needed for a test mass to escape from a Schwarzschild metric to infinity show us that the escape velocity, v_escape, is given by

\frac{v_{escape}^2}{c^2}=\frac{2GM}{rc^2}

So for a test mass in free fall

(1- \frac{2GM}{{rc} ^{2}})=(1-\frac{v_{escape}^2}{c^2})

A test mass in free fall from an infinite distance will experience a time dilation

dt=(1- \frac{v_{escape}^2}{c^2})^{-\frac{1}{2}}dτ

and a length contraction

dr=(1- \frac{v_{escape}^2}{c^2})^\frac{1}{2}dσ

These are the same factors that appear in the Lorentz time dilation and length contraction equations. They apply here in the special case where the relative velocity is escape velocity in a Schwarzschild spacetime. They will change lengths and times in the Schwarzschild spacetime, for free falling bodies travelling at escape velocity on the r axis, by the same amount that the Lorentz transformations do for inertial frames moving in a Minkowski metric with that velocity.

It should be noted that this is not the Lorentz transfomation. It is the effect on space and time produced by a mass, as described by the Schwarzschild metric. The factors in brackets are the same, however.

We can see that this models Einstein’s assumption that he made to arrive at his field equation. A massive test body in free fall from infinity, and moving with escape velocity for the point r they have reached in the Schwarzschild metric, will experience exactly the degree of length contraction and time dilation needed for observers moving with that test mass to see a uniform velocity of light.

We need to remember that the Lorentz transformation is a mathematical device to explain the constancy of the speed of light in Minkowski metrics in relative motion. Neither Lorentz, nor Minkowski, nor Einstein proposed a physical reason for the Lorentz transformation to operate in Minkowski spacetime.

There is no need to appeal to an abitrary application of the Lorentz transformation. Einstein does propose a physical reason for the shape of spacetime. That reason is the distribution of mass and energy. General relativity proposes that the distribution of mass and energy determines a shape of spacetime in which all laws of physics are to operate in.

The constancy of the speed of light for all observers is one of those laws, and the Schwarzschild metric is a solution of Einstein’s equation.

We can expect then, that the Schwarzschild metric is a solution that provides for a constant speed of light for all observers. It does indeed do so, but only for observers in free fall from, or to, infinity, on a geodesic, with no forces acting on them

It is interesting to note, at this point, that in a Schwarzschild universe containing only one mass affecting the shape of space and time, there is no other velocity possible for a test body. A test body with any other velocity, apart from escape velocity on the radial axis, would have to have been accelerated by a force, and no other force exists in the Schwarzschild metric.

We can say that all observers in a Schwarzschild metric will be in free fall at escape velocity, and therefore we can say that all observers in the Schwarzschild metric must see a uniform velocity of light.

It was clearly Einstein’s intention, in developing his field equations, to show that the uniformity of the velocity of light, that applied in Minkowski spacetime, also applied in our Universe. His equivalence principle states that “the outcome of any local, non-gravitational test experiment is independent of the experimental apparatus’ velocity relative to the gravitational field and is independent of where and when in the gravitational field the experiment is performed.“

We are not quite there yet, but the simplified case of a single mass in the Schwarzschild metric does give a uniform velocity of light for all possible observers.

We can see above that we needed to introduce the way that potential energy affects a mass in a Schwarzschild metric to show how the necessary space and time variations occur to give free falling observers with escape velocity a uniform velocity of light. We are now in a position to show how the mass distribution, in the Universe we inhabit, results in the space and time variations necessary to give all observers a uniform speed of light, as required by Einstein’s equivalence principle.

Adding spherical shells of uniform density

Bringing matter in from infinity

Starting at a point in empty Minkowski spacetime, we can imagine introducing an infinitely small sphere, of infinitely small mass, with a density, ρ₀, and a radius, r=0 , together with an observer at the same point. This sphere will effectively be in Minkowskian spacetime. A Schwarzschild space time with M equal to zero is Minkowskian.

Now consider increasing the size of the sphere by adding a shell of matter of the same density, and having a thickness, (r+dσ)-r, measured at the shell. We will imagine bringing this shell of matter in from infinity to the surface of the sphere. This means that we are imagining the shells initially in Minkowskian spacetime. This is the condition of the Schwarzschild metric at infinity. At an infinite distance the shells are not affected by the presence of the mass as we increase the size of the sphere.

Birkhoff’s theorem applies in the empty space outside a spherical shell of mass distribution, so we can apply it immediately outside the surface of our imaginary sphere.

The thickness of each shell is reduced

We can imagine adding successive shells to the surface of the sphere, and by doing so, we can increase the radius, r, of the sphere. As we do so the thickness of each shell will be reduced by a factor determined by the Schwarzschild metric for the particular radius, r, of the sphere, as we saw in “The Schwarzschild spacetime metric ” above.

Each shell will start with an infinitesimal thickness, dσ, and an infinitetismal mass. This infinitesimal mass will have a negligible effect on the Schwarzschild space time at the surface of the sphere as it grows larger. The infinitesimal thickness, dσ, will be reduced to a thickness, dr, as viewed from the centre of the sphere, when we place the shell on the surface of the sphere. Birkoff¹ tells us that a spherical shell of matter will not affect the configuration of space time inside it; so these shells will individually make no difference to the local space time on the surface of the sphere, as they increase the radius of the sphere. As we add a shell in the vacuum at the surface of the sphere, we can use the Schwarzschild spacetime at the surface, as described above.

The mass of each shell is reduced

We saw, in “Potential energy and escape velocity” above, that a massive body on the surface of the sphere of our model Universe, at a distance, r, will have had its mass reduced by the Schwarzschild factor, corresponding to its loss of potential energy, if it is moved slowly into the gravitational field of the sphere. The greater the radius of the sphere, the smaller the Schwarzschild factor gets.

We expect a bodies rest mass, m_rest, to be reduced to, m_rest-at-r , where m_rest-at-r is the mass of a body stationary at a distance, r.

\color{red}{m_{rest-at-r} =  {(1- \frac{2GM}{{rc} ^{2}})^{\frac{1}{2} }}{m_{rest}}}

For a sphere with a uniform critical density, ρ_, we can write this as

\color{red}{m_{rest-at-r} =  {(1- \frac{8Gρπr^2}{3c^2})^{\frac{1}{2} }}{m_{rest}}}

The critical density is the density that will give a sphere of radius, r, and with a mass, M.

We are postulating that this is due to the body’s loss of potential energy in a gravitational field. If we move massive bodies in from infinity to build the next shell, these bodies will lose potential energy, and therefore mass.

The density of shells at infinity

Now consider the density of a spherical shell, of thickness dσ, and radius, r, on its own at infinity, i.e. not round a sphere exerting a gravitational influence, containing n bodies each with a rest mass m_n. Here we will be assuming that the masses of the shell’s constituent bodies will be negligible in their effect on the Schwarzschild spacetime at the surface of the sphere, but not in their contribution to the density of the shell. We are imagining an infinitely thin shell, so it will have an infinitely small mass. We will suppose that this shell will initially have the same density as a sphere with a uniform density, ρ₀. The density, ρ₀, of the sphere will be given by its total rest mass, m_rest, and its volume, V, as

ρ_0=\frac {m_{rest}}{V}

If the total rest mass, dm_rest, of the shell at an infinite distance is

dm_{rest}=\sum_{n=1}^{n} m_n 

and the volume of the shell at an infinite distance, is, dV _∞ , where

{dV_{∞}=4πr^2dσ}

The mean density, ρ₀, of this shell will be

ρ_0=\frac{dm_{rest}}{dV_∞}

so

ρ_0=\frac{dm_{rest}}{4πr^2 dσ }

Uniform density is maintained for the sphere

If we place this shell outside the sphere of uniform density, ρ₀, there will now be a gravitational influence on length and mass as viewed from the centre of the sphere

The thickness of the shell is reduced to dr from its original thickness, dσ where

\color{red}{dr=(1- \frac{8Gρπr^2}{3{c} ^{2}})^{\frac{1}{2}}dσ} 

The total mass of the bodies in the shell is now dm_rest-at-r where

\color{red}{dm_{rest-at-r} =  {(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}{dm_{rest}}}

The volume, dV, of the shell is now

\color{red}{{dV=4πr^2dr}}

and so

\color{red}{{dV=4πr^2dσ(1- \frac{8Gρπr^2}{3{c} ^{2}})^{\frac{1}{2}}}}

The density, ρ, of the shell will now be

ρ=\frac{dm_{rest-at-r}}{dV}

so

\color{red}{ρ=\frac{dm_{rest}{(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}}{{4πr^2 dσ }{{(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}}}}

and so

ρ=ρ_0

The assumption of uniform density is justified.

The size of the Universe

Adding successive spherical shells

Next we will consider how big we can make this sphere.

We have seen that we can start with an infinitely small sphere of radius, r = 0, and add shells until we have a sphere of radius r. Now we consider continuing to add successive spherical shells. As we do so, the radius, r, of the sphere will increase, and so the thickness, dr, of the shells we are adding will decrease, but their density will remain the same.

The edge of the Universe

Successive shells will have their thickness reduced until a radial distance is reached where their thickness is zero.

This will occur when

\color{red}{{(1- \frac{8Gρπr^2}{3c^2})}=0}

To reach this radial distance, we will have to add an infinite number of shells. At this point the size of the sphere cannot be increased any further, so we will have produced a sphere of finite size. The sphere would have a radius, as viewed from its centre, which would be determined by its density, and this density would be the same throughout the sphere.

Spherical symmetry justifies relying on Birkoff’s theorems¹. In particular the spherical symmetry of the outer shells justifies using Birkoff’s result that the space inside a spherical shell is Minkowskian, whether it is in motion or not. Adding spherical shells, as we build a bigger sphere, will not affect the shells already placed.

The edge to this Universe will be at a distance R, where dr = 0. It will also have dt = ∞ . At this edge time is dilated to infinity, so the edge of this Universe in space is also its beginning in time. Time is more stretched out the closer to the edge we go, so there is an infinite amount of time in the past of any co-moving point where the Universe is being observed.

An infinite Universe with a finite mass

We have added an infinite number of shells of matter, and so we can envisage a Universe containing an infinite amount of matter, but having a finite mass, and volume, and a uniform density.

Crucially this Universe is structured around the central observer. This is, of course, tantamount to scientific heresy since the time of Galileo, but remember that the argument is the same for any comoving observer. All such observers will see the same infinite Universe, and they can all claim that they are at the centre. The distribution of matter will appear as a sphere around every comoving observer.

We will see below that the effect of the Schwarzschild spacetime will have an equivalent effect to the Minkowski spacetime for every co-moving observer, with each comoving observer seeing itself as being at the centre of its Universe in its own spacetime metric .

The event horizon of the Universe

The edge of this spherical volume, when dr = 0, will be at the distance, R, at which the event horizon of a black hole of that mass would be. This is the distance where the equivalence of inertial mass and gravitational mass predicts that the escape velocity is equal to the speed of light. This will be when

\color{red}{{(1- \frac{8GρπR^2}{3c^2})}=0}

\color{red}{R=(\frac{3c^2}{8Gρπ})^\frac{1}{2}}

Hubble’s Law and escape velocity

Mass increase in time

A massive body at a distance, r, from a central observer will have its mass reduced by

\color{red}{m_{rest-at-r}=(1- \frac{8G\rho\pi r^2}{3{c} ^{2}})^{\frac{1}{2}}m_{rest}}

and time will be passing more slowly for it according to

\color{red}{dt=(1- \frac{8G\rho\pi r^2}{3{c} ^{2}})^{-\frac{1}{2}}d\tau}

If its mass is less than it would be at the centre, and less time has passed for it, then mass is less in the past, and mass must be increasing in time, as measured by an observer at the centre. If the mass of the constituent massive bodies in the Universe is increasing, then the mass of the entire Universe must be increasing, and its radius, R, must also be increasing in proportion to it’s mass.

An expanding Universe does not change the analysis above, as Birkoff¹ has shown that the solution of Einstein’s equations is still the Schwarzschild solution outside a spherically symmetric mass distribution even if it is in motion. The sphere we are considering can be expanding, and the space time at the surface will be as modeled above.

We are now imagining a Universe that would have a finite mass, M, and a volume determined by the the radius of the Universe, R, which would both be defined by the local mass density of comoving matter at the place and time it is observed from.

Our model of the structure of the Universe is bounded when the thickness, dr, of the shells we are adding equals zero. As we have seen above, this is when

\color{red}{{(1- \frac{8Gρπr^2}{3c^2})}=0}

Relativistic escape velocity

Hartle’s result ⁷ for general relativistic escape velocity gives

\frac{v_{escape}^2}{c^2}=\frac{2GM}{rc^2}

and

M=\frac{4ρπr^3}{3}

Where ρ is the critical density which makes a sphere that has a mass, M, and an escape velocity of v_escape at the radius, r, in the equation for escape velocity.

so

\color{red}{\frac{v_{escape}^2}{c^2}=\frac{8Gρπr^2}{3c^2}}

When v_escape = c we get

\color{red}{\frac{c^2}{c^2}=\frac{8Gρπr^2}{3c^2}}

so as above we have

\color{red}{(1- \frac{8Gρπr^2}{3c^2} )=0}

We can see that the edge of the Universe where dr = 0 is also where the velocity of light is the escape velocity of the Universe.

Hubble’s law

If all the comoving matter in this model Universe is traveling at escape velocity from the sphere of radius, r, it sees itself at the surface of, it will obey Hubble’s law.

Consider a comoving observer at a distance, r, from the centre, as observed by an observer at the centre of the spherical Universe we have built. We can imagine this observer on the surface of an imaginary sphere of radius, r. That comoving observer will be moving away from the centre with a velocity given by

v=Hr

Where H is Hubble’s constant.

Escape velocity in the Schwarzschild metric

Birkoff¹ has shown that a spherical shell has no gravitational effect inside itself, and so the gravitational effect felt by the moving observer will be due to the inner sphere only. This will be a sphere with a radius, r.

This sphere of matter will have the same density as the rest of the Universe, and so, as we saw above, there will be length and time variations on its surface according to its mass.

Its mass, M, is given by

M=\frac{4\rho \pi r^3}{3}

Hartle’s result above for escape velocity of a massive body in a gravitational field at distance r from a mass M gives

\frac{v^2_{escape}}{c^2}=\frac{2GM}{rc^2}

so on the surface of a sphere of radius r

\color{red}{\frac{v^2_{escape}}{c^2}=\frac{8\pi G \rho_{critical}r^2}{3c^2}}

\color{red}{v_{escape}=(\frac{8\pi G \rho_{critical}}{3})^\frac{1}{2} r}

The critical density, ρ_critical, is the density which makes a sphere that has an escape velocity of v_escape at its surface that is proportional to r. This will be true for spheres of any radius, r, for v_escape up to the velocity of light.

The velocity of its constituent galaxies is related to their distance away by Hubble’s constant, H. Galaxies at a distance, r, will have a velocity given by

v=Hr

So, for an expanding Universe, with all co-moving matter travelling at escape velocity, we could write Hubble’s equation as

v_{escape}=Hr

And when v_escape = c

c=HR

where

H=(\frac{8\pi G \rho_{critical}}{3})^\frac{1}{2}

As the Universe expands, the mass of this expanding spherical Universe will be increasing in proportion to its radius, R. It’s critical density, ρ_critical, will be decreasing as the inverse square of the radius, R, so Hubble’s constant will be decreasing in inverse proportion to the radius, R. The measured speed of light, c, will remain the same at any time this universe is observed.

We see that the edge of this Universe is moving away from a central comoving observer at the speed of light, and this is escape velocity for the model Universe built of shells we have been considering. We see that all the matter obeying Hubble’s law will be travelling at escape velocity.

The Minkowski spacetime and our model Universe compared

Hubble’s law and the observed expansion of the Universe means that a comoving observer will see all other comoving points moving away from themself. They may all view themselves as the central comoving observer in a rest frame, and they can view all other comoving observers as being in moving frames. These other moving frame observer’s time will be dilated, and less time will have passed for them. They will all be in the central comoving observer’s past.

All co-moving observers will see themselves as the central comoving observer in a rest frame at the furthest point time has reached. Everything that comoving observer sees will be on the “surface” of their past light cone, moving with escape velocity away from them.

Lorentz factors in the modified Schwarzschild spacetime

This account is proposing that our Universe can be modeled by a modified Schwarzschild spacetime

\color{red}{ds^2 = (1-\frac {8G\rho_{critical}πr^2}{3{c} ^{2}})c^2 dt^2- (1-\frac {8G\rho_{critical}πr^2}{3{c} ^2 })^{-1} dr^2 -r^2dθ^2 -r^2sin^2θdφ^2}

At the center of the modified Schwarzschild spacetime, when r = 0, the spacetime becomes

\color{red}{ds^2=c^2dt_{(r=0)}^2-dr^2_{(r=0)}-r^2d\theta^2_{(r=0)}-r^2sin^2\theta d\phi^2_{(r=0)}}

We can imagine smaller spheres of radius, r, in a spherical Universe of radius, R. These smaller spheres will be inside a spherical shell of thickness R – r. Birkhoff tells us that such a spherical shell will not affect the spacetime inside it, so we can now say that the modified Schwarzschild metric above will be true for the entire spherical Universe we are modeling.

If the density is the critical density, and all comoving parts of the Universe are moving away from a central observer at escape velocity, Hartle’s result for relativistic escape velocity gives

\color{red}{\frac{v^2_{escape}}{c^2}=\frac{8 G \rho_{critical}\pi r^2}{3c^2}}

Substituting into the modified Schwarzschild metric above gives

\color{red}{ds^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^2-( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^2-r^2dθ^2-r^2sin^2θdφ^2}

This has the same form as the Minkowski metric in the subsection “Moving frames in spherical coordinates” at the end of “Moving frames in the Minkowski spacetime”. It is the metric for a moving frame as seen by an observer stationary in a rest frame.

This is equivalent to the metric for a moving frame, but it is expressed in rest frame coordinates. In this metric dt is a dilated time and dr is a contracted length compared to dt_(r=0) and dr_(r=0) in the modified Schwarzschild spacetime, when r = 0, above.

In this case the moving frame is a frame moving with velocity, v_escape, at a distance, r, and the rest frame is the frame of the central co-moving observer at r = 0 .

We have seen that the interval, ds, must be a real number in all Minkowski metrics, so the interval, ds, must be a real number in the modified Schwarzschild metrics above. This means that all the implications for ds being real that were discussed in “The Minkowski spacetime metric”, and “Moving frames in the Mikowski spacetime” will apply to this modified Schwarzschild metric.

Length contraction in the model Universe

From the modified Schwarzschild metric, and the escape velocity equation can write

\color{red}{dr^2_{(r=0)}= ( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^2}

If dσ is an infinitesimal radial rest length that is measured next to an observer at r = 0, then, for an observer at r = 0, we have

\color{red}{dr^2_{(r=0)}=dσ^2}

and so

\color{red}{dσ^2= ( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^2}

and

\color{red}{dr=(1-\frac{v^2_{escape}}{c^2})^{\frac{1}{2}}dσ}

The rest length, dσ, is the same length measured next to an observer anywhere that observer is. Observers moving with a body will measure its rest length as dσ. At the same time central comoving observers, at r = 0, measure the contracted length of that same body as dr.

If we consider a reference frame moving with a velocity v_escape, with its origin at r, and with coordinates t^/, r^/, θ^/, and ϕ^/ then an observer stationary in this moving frame will have

dr^{/}=dσ

so

\color{red}{dr=(1-\frac{v^2_{escape}}{c^2})^{\frac{1}{2}}dr{^/}}

This is the same as the Lorentz length contraction for frames moving with a relative velocity, v_escape.

Time dilation in the model Universe

From the modified Schwarzschild metric, and the escape velocity equation can write

\color{red}{c^2dt_{(r=0)}^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^2}

If dτ is an infinitesimal proper time that a clock next to an observer at r = 0 shows, then we can write

\color{red}{c^2dt_{(r=0)}^2=c^2dτ^2}

and we can write

\color{red}{c^2dτ^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^2}

and so

\color{red}{dt=(1-\frac{v^2_{escape}}{c^2})^{-\frac{1}{2}}dτ}

A reference frame moving with a velocity v_escape, with its origin at r, and with coordinates t^/, r^/, θ^/, and ϕ^/ will have, for an observer stationary in this moving frame

dt{^/}=dτ

so

\color{red}{dt=(1-\frac{v^2_{escape}}{c^2})^{-\frac{1}{2}}dt{^/}}

This is the same as the Lorentz time dilation for frames moving with a relative velocity, v_escape.

Moving reference frames in the model Universe

From the equations above we can write

\color{red}{dt{^/}^2}=(1-\frac{v^2_{escape}}{c^2})dt^ 2

and

\color{red}{dr{^/}^2=(1-\frac{v^2_{escape}}{c^2})^{-1}dr^2}

also, since dθ and dφ are unchanged

r{^/}^2dθ{^/}^2=r^2dθ^2

and

r{^/}^2sin^2\theta{^/} d\phi{^/}^2=r^2sin^2\theta d\phi^2

Substituting into the metric for the moving frame in rest frame coordinates

\color{red}{ds^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^2-( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^2-r^2dθ^2-r^2sin^2θdφ^2}

We get a metric for the moving frame in moving frame coordinates.

ds^2=c^2dt{^/}^2-dr{^/}^2-r{^/}^2d\theta{^/}^2-r{^/}^2sin^2\theta{^/} d\phi{^/}^2

This is equivalent to the Minkowski metric for a moving frame, but it has been developed by considering the modified Schwarzschild metric, and the relativstic equation for escape velocity. It is telling us that in our Universe, modelled by the modified Schwarzschild metric, an observer at a comoving point at a distance, r, will see themselves stationary in a metric equivalent to the moving Minkowski metric.

We can see above that in this metric dt^/ is a proper time, and dr^/ is a rest length. In the modified Schwarzschild metric above for r = 0, dt_(r=0) and dr_(r=0) are also proper times and rest lengths, but dt^/ is not the same as dt_(r=0), and dr^/ is not the same as dr_(r=0). They are equal measurements, but they are not the same actual time and length intervals.

As we saw in “Moving frames in the Mikowski spacetime” proper times and rest lengths are the same measurements of time and length intervals but they are not the same actual lengths and times. This is the same here. The spacetime in moving coordinates that is derived from the modified Schwarzschild spacetime is length contracted and time dilated. It appears the same, and has the same structure, as the modified Schwarzschild metric at r = 0. They both have the same structure as a Minkowski metric.

They are both written in rest lengths and proper times, so they both appear the same to observers that are stationary in them. They are equivalent, but they are not the same.

To clarify the terms “equivalent” and “the same” as they are being used here, consider two model trains. They are models of the same type of train but one is twice the size of the other.

The model tracks the trains run on have a model sign next to them saying “track guage 2 metres”. In each model the track guage is the same. Each model states that the guage is two metres.

They are equivalent in that they describe the same type of train, but they are not the same.

Now the actual guage of the tracks is measured, and one model’s tracks are five centimetres wide, while the other models tracks are ten centimetres wide. We find a scale factor between the models of one to two. One model is twice the size of the other.

If the models both say that the guage is two metres on a sign in the model, there is no way of telling from within the models that one of them is twice the size of the other.

If there was no external reality in which a real train existed, with real people, and real clocks, and real metre rules, to compare the models with, there would be nothing about the models that could tell someone what their actual, real, size was.

In the same way if the rest length of a body is measured at rest in a rest frame, and the same body has the same rest length when it is measured at rest in the moving frame, there is no way to tell from within the reference frames that one frame is twice the size of the other.

Only if you knew that one frame was moving at 3^1/2/2 of the speed of light relative to the other would you be able to calculate, with the Lorentz length contraction equation, that the moving frame’s lengths were half the other frame’s lengths.

The equations for length contraction and time dilation derived from the Lorentz transformations give us the scale factor between the length and time intervals in the rest frame compared with the length and time intervals in the moving frame.

There is no external, “real”, reality between the moving Minkowski metric and the Minkowski metric at rest. We can only compare one spacetime metric with another. Each is describing an entire reality. The best we can do is describe the moving frame as it looks from the rest frame. We take a rest frame’s measurement, dr, of a length that is measured as dr^/ in the moving frame and say that

dr=(1-\frac{v^2}{c^2})^{\frac 1 2}dr{^/}

We can only tell that the moving frame has contracted by looking at it from the rest frame. The equation above is giving us the scale factor between the moving frame and the rest frame.

In the analogy above the entire model, the tracks, and the sign next to the tracks are analogous to an entire Minkowski spacetime. The entire moving Minkowski spacetime is length contracted, and time dilated, when viewed by an observer stationary in the Minkowski spacetime at rest relative to it.

If a coordinate change is a purely geometrical one, then the Minkowski spacetime is the same after such a coordinate change For example the change from rectangular Cartesian coordinates to spherical coordinates does not change the spacetime. The Lorentz transformation is not a purely geometrical coordinate change; it is a physical coordinate change. It changes the physics of the situation.

In our physical Universe velocity is a physical property of a physical body. It is intimately bound up with energy in Einstein’s equation. The mass of a body is dependent on the relative velocity of an observer no matter how much some physicists don’t want to acknowledge it. The velocity of light appears in the Lorentz equations. It is a physical constant. it is much more than a purely numerical constant; it appears in most of the equations in physics.

The moving frame is in its own distinct spacetime. It has a different size to the rest frame’s spacetime. It is equivalent when it is observed from within it, but it is not the same.

We have seen that the moving frame spacetime does not even contain all the same points as the rest frame spacetime. The observer seeing a moving spacetime is at rest at a point in their spacetime that doesn’t exist in the moving spacetime.

The spacetimes are not entirely separate, they are connected by light. There is a sense in which the moving spacetimes are contained within the observer’s spacetime, but the observer’s spacetime is not contained within the moving spacetimes. To be able to observe a moving spacetime, an observer must be in a separate spacetime of their own.

It is the observer’s own progress through time that determines which spacetime is ahead of all the other possible ones. The spacetime’s own observer is at the furthest time has progressed.

When we consider comoving points in our expanding Universe we see that the expansion may be viewed from any comoving point. That comoving point will see itself as the centre of the expansion. An observer at that point can consider themselves to be the central comoving observer in their Universe.

We can now see that they will see themselves as being in the equivalent of a Mikowski metric in moving coordinates, and so they will see a uniform speed of light. A light wave, as described by Maxwell, originating where they are, will make a spherical wavefront as it travels away from them in every direction.

The speed of light in the moving frame

We can place the origin, r^/ = 0, of this moving frame, at a distance r in the modified Schwarzschild spacetime. We can imagine a sphere of light centred at this origin. This sphere will be the set of coordinates in this Minkowski metric for which ds = 0.

We can write, when ds = 0

0=c^2dt{^/}^2-(dr{^/}^2+r{^/}^2d\theta{^/}^2+r{^/}^2sin^2\theta{^/} d\phi{^/}^2)

c^2dt{^/}^2=(dr{^/}^2+r{^/}^2d\theta{^/}^2+r{^/}^2sin^2\theta{^/} d\phi{^/}^2)

c^2=\frac{(dr{^/}^2+r{^/}^2d\theta{^/}^2+r{^/}^2sin^2\theta{^/} d\phi{^/}^2)}{dt{^/}^2}

c=\frac{(dr{^/}^2+r{^/}^2d\theta{^/}^2+r{^/}^2sin^2\theta{^/} d\phi{^/}^2)^\frac{1}{2}}{dt{^/}}

This is the equation for the spherical coordinates of a sphere expanding at the velocity c.

An observer stationary in this moving frame will see a sphere of light, with its centre at r^/ = 0, in the same way that the original central comoving observer in the rest frame would see a sphere of light with its centre at at r = 0 . The speed of light will be the same in all moving frames.

The Universe as viewed by the co-moving observer

This point at r is surrounded by matter expanding with the expansion of the Universe. All such comoving points will see themselves as the center of the expansion. This expansion is seen to be directed radially away from the point at r.

We saw how the modified Schwarzschild spacetime models the expanding Universe we see ourselves in. This model is in the form of a sphere, with its edge moving away at the speed of light.

An observer at a co-moving point at a distance, r, will also see the matter of the Universe moving away from them. They will see themselves at the centre of a sphere, with the edge moving away at the speed of light, just as the original central comoving observer does in the model discussed above.

Just as the points for which ds = 0 model an expanding sphere of light for all comoving observers, we can also see that it models the sphere of the expanding Universe for all comoving observers. All comoving observers in the modified Schwarzschild spacetime will see themselves at the centre of a sphere of points moving away from them at the speed of light. They will all view themselves as the central comoving observer in their Universe.

Just as a sphere of points moving away at the speed of light from the central comoving observer contains all the matter in the expanding universe, we can see that all the matter in the expanding universe will also be within a sphere of points moving away at the speed of light from the comoving observer at a distance, r. The length contraction and time dilation, due to the modified Schwarzschild metric, will affect all the matter in the Universe within that expanding sphere.

The matter that was distributed in a sphere around the original central comoving observer, and moving away from them at escape velocity, is also distributed in a sphere around the comoving observer at a distance, r.

The two viewpoints are not completely equivalent, however. The original central comoving observer is further ahead in time than the observer at r. We saw in “The Minkowski spacetime metric” that the original central comoving observer cannot exist in the spacetime of the observer at r.

Time is flowing faster for the original central comoving observer so more time has passed. They are in just one of the infinite possible future universes ahead in time from the point of view of the observer at r. The observer at the present moment will always see themselves as the furthest advanced in time.

This sphere of points can also be viewed as the point in spacetime when the expansion began. From this viewpoint all comoving observers are moving away from this original spacetime moment at the speed of light. All such comoving observers are at the furthest point that can be reached, from this initial moment of spacetime, in the total time the Universe has existed for them.

The matter of the Universe will form a spherical shell around this comoving point, and so Birkhoff tells us, as before, that it will not affect the spacetime inside it, and we can have Minkowski spacetime, as shown above, at r^/ = 0.

We can imagine building a modified Schwarzschild spacetime around this point as we did at r = 0 in “Applying the Schwarzschild spacetime to the Universe“, but now this comoving point at r, where r^/ = 0, is acting as the central comoving observer. We can imagine an infinite amount of matter, with a finite mass and volume, in the form of a sphere around this new central comoving observer.

We will get the same shape of spacetime, but now it will be centred on the moving point at r^/ = 0 instead of the original rest point at r = 0, and it will be in moving coordinates. The new central comoving point, stationary in the moving frame, will be at r^/ = 0.

\color{red}{ds^2 = (1-\frac {8G\rho_{critical}πr{^/}^2}{3{c} ^{2}})c^2 dt{^/}^2- (1-\frac {8G\rho_{critical}πr{^/}^2}{3{c} ^2 })^{-1} dr{^/}^2 -r{^/}^2dθ{^/}^2 -r{^/}^2sin^2θ{^/}dφ{^/}^2}

When r^/ = 0 this becomes

ds^2=c^2dt{^/}_{(r{^/}=0)}^2-dr{^/}^2_{(r{^/}=0)}-r{^/}^2dθ{^/}^2_{(r{^/}=0)}-r{^/}^2sin^2θ{^/}dφ{^/}^2_{(r{^/}=0)}

This has the same form as Minkowski spacetime. It is using a frame of reference moving at a velocity, v_escape, and it has its origin at r^/ = 0.

We have seen above that the interval, ds, must be a real number in all Minkowski metrics so ds must be a real number in the two metrics above. All the implications that followed for the Minkowski metric, for the non-existence of points in the “spacelike” regions, must also apply to this modified Schwarzschild spacetime, and the moving spacetimes, described above.

Just as with the Minkowski spacetime, all comoving observers will see themselves as being at the tip of their lightcones, and at the furthest point time has reached.

An observer at this point will see a Universe with exactly the same structure, but the coordinate elements dr^/ will be length contracted in the original r direction, and time coordinate elements, dt^/, will be dilated, when viewed by the original central comoving observer in the original rest frame.

An observer stationary at r^/ = 0 in this moving frame will not see their length differentials as contracted, or their time differentials as dilated. This moving observer will themselves be length contracted and time dilated along with their rulers and clocks.

Just as an observer approaching a Schwarzschild radius does not notice the time dilation and length contraction, comoving observers will not percieve the length contraction and time dilation measured by the central comoving observer at r = 0.

In fact, they will not perceive the central comoving observer as being part of their Universe. As was discussed in “The Minkowski spacetime metric” and “The moving Minkowski spacetime” above, these co-moving observers will be on the surface of the light cone of the central co-moving observer. The original central comoving observer will not be in their Minkowski spacetime. All comoving observers will be at the furthest point time has reached in their own Universe.

They may see an earlier version of the central comoving observer, but they won’t see the version that is looking at them.

In Minkowski metrics that have a relative velocity to each other, we saw, in “The Minkowski spacetime metric” , that the equation of the speed of light was the same for moving frames as it was for the rest frame. It is the equation of a sphere expanding at the speed of light from both the rest frame’s origin, at r = 0, and the moving frame’s origin, at r^/ = 0.

We may note that the edge of the Universe we are modelling will be a sphere expanding at the speed of light from the origin where the central comoving observer is watching it at r = 0, and it will also be a sphere expanding at the speed of light from the origin where the moving comoving observer is watching it at r^/ = 0. The structure of time and space adapts to the observer.

A pair of comoving observers in our Universe each see their entire Universe behaving like a light sphere in a pair of Minkowski metrics with a relative velocity, v_escape, to each other. Each of them will see themselves as central in their own Universe, but one of them will be ahead of the other in time.

The one that is ahead in time will see the other one on their lightcone, but the one that is behind will not see the one that is ahead. The Universe they see will be isotropic and homogenous. Each view of the Universe will look the same in all directions, and it will look the same at all points, but they will not be equivalent.

An observer, at this distance r in our Universe, will see the constituent galaxies of an expanding Universe moving away from them. They will all be in that observer’s past.

We have seen that the speed of light in Minkowski spacetime is the maximum magnitude of velocity possible between two points in the spacetime. There are no points with a separation that implies a velocity greater than light.

We have seen that every comoving point in our expanding spacetime can be considered to be at r^/ = 0 in a stationary spacetime with the same properties as a Minkowski metric. We can conclude that every comoving point will see themselves as being at the apex of a lightcone in a spacetime with the same properties as Minkowski spacetime.

We can conclude that an observer at every comoving point in our Universe will see a uniform speed of light, with a value that they measure to be c, using their local infinitesimal moving coordinates.

This account is suggesting that this will be what every co-moving observer in our Universe will experience.

Every such observer is seeing the galaxies moving away from themselves according to Hubble’s law. They will be in the equivalent of Minkowski spacetime, but it will be full of matter. This was what was required for all observers in our Universe to see a uniform speed of light.

We see that every such observer will see themselves at the centre of a spherical Universe with its edge receding at the speed of light. Every such observer will see length contracted, and time dilated, as this edge is approached. Every such observer will see an infinite amount of matter, and a finite mass and volume. Every such observer will see the edge of their Universe infinitely far back in time. Every such observer will see length contracted to zero, and time stopped, at this edge.

They will see themselves as the central co-moving observer in their Universe.

The comoving observer at r is the new central comoving observer

The new central comoving observer will be at the central point of the following modfied Schwarzschild spacetime at r^/ = 0

\color{red}{ds^2 = (1-\frac {8G\rho_{critical}πr{^/}^2}{3{c} ^{2}})c^2 dt{^/}^2- (1-\frac {8G\rho_{critical}πr{^/}^2}{3{c} ^2 })^{-1} dr{^/}^2 -r{^/}^2dθ{^/}^2 -r{^/}^2sin^2θ{^/}dφ{^/}^2}

This will give the spacetime at the radius r^/, where r^/ is the radius of a new sphere centered on r^/ = 0. We can build this sphere out to a radius R^/ With the same argument as above in “Hubble’s law and escape velocity” we can say that the surface of this sphere at r^/ will be moving at escape velocity.

Hartle’s equation for escape velocity, v_escape , from this sphere will be

\color{red}{\frac{v_{escape}^2}{c^2}=\frac{8Gρπr^{/2}}{3c^2}}

but r is now r^/

Substituting into this new modified Schwarzschild spacetime we can write

\color{red}{ds^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^{/2} -( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^{/2}-r^{/2}dθ^2-r^{/2}sin^2θdφ^2}

The escape velocity, v_escape , is now the escape velocity from a point, r^/, in a spherical mass distribution with its centre at r^/ = 0.

At r^/ = 0 we have

ds^2=c^2dt{^/}_{(r{^/}=0)}^2-dr{^/}^2_{(r{^/}=0)}-r{^/}^2dθ{^/}^2_{(r{^/}=0)}-r{^/}^2sin^2θ{^/}dφ{^/}^2_{(r{^/}=0)}

We can write

\color{red}{c^2dt{^/}_{(r{^/}=0)}^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^{/2}}

An observer stationary at r^/=0 measures proper time next to them so

\color{red}{c^2dt{^{/2}}_{(r{^/}=0)}}=c^2dτ^2

and we can write

\color{red}{c^2dτ^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^{/2}}

We can treat the t^/, r^/, θ^/, and ϕ^/ frame as the new rest frame and we can imagine a frame of reference moving relative to this frame along r^/, with coordinates t^//, r^//, θ^//, and ϕ^// . The origin of the new frame, r^//=0, will be moving along r^/. An observer in this new frame will see dt^// as a proper time so

c^2dt^{//2}= c^2dτ^2

and

\color{red}{c^2dt^{//2}=(1-\frac{v^2_{escape}}{c^2})c^2dt^{/2}}

An observer stationary at r^/=0 measures rest length next to them so

( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^{/2}= dσ^2

An observer in this new frame will see dr^// as a rest length so

\color{red}{dr^{//2}_{(r=0)}=dσ^2}

and

dr^{//2}_{(r=0)}=( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^{/2}

θ^//, and ϕ^// are unchanged so

r{^{//2}} dθ{^{//2}}=r{^{/2}}dθ{^{/2}}

and

r{^{//2}}sin^2\theta{^{//}} d \phi{^{//2}}=r{^{/2}}sin^2\theta{^/} d\phi{^{/2}}

Substituting into

\color{red}{ds^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^{/2} -( 1-\frac{v^2_{escape}}{c^2})^{-1}dr^{/2}-r^{/2}dθ^2-r^{/2}sin^2θdφ^2}

gives

\color{red}{ds^2=c^2dt^{//2} -dr^{//2}-r^{//2}dθ^2-r{^{//2}} dθ{^{//2}}}

The spacetime seen by a comoving observer at r in moving coordinates, t^/, r^/, θ^/, and ϕ^/ looks like the original spacetime with a rest frame at r = 0. These coordinates are then treated as the rest frame of a comoving observer at r where the central point of this moving coordinate frame, r^/=0, is at rest and a point at r^/ is moving away from r^/=0 . A new moving coordinate frame at this point r^/ can be denoted with moving coordinates, t^//, r^//, θ^//, and ϕ^// centred at r^//=0, and will have a new escape velocity v_escape at r^//

This process may be repeated any number of times. Each time a new spacetime in a Minkowski form is created. Each new moving reference frame is in a new spacetime, but it will be in the past of the previous one. This spacetime will be contained within the previous one, but an observer at the origin of each new moving reference frame will see themselves as the observer that has proceeded furthest in time in their own spacetime. They will not, and cannot, be aware of any observers in their future. For them these observers do not exist.

New physics

It is important to note that these equations above were not developed from the Lorentz transformations, as was the equation at the end of “Moving frames in the Minkowski spacetime “. They have been developed from the Schwarzschild spacetime, and the theorems of David Birkhoff and J B Hartle.

The Lorentz transformations assume that space contracts, and time dilates, for the transformations to produce a transformed Minkowski metric in the same form as the original rest frame metric. Lorentz, Einstein, and Minkowski give no physical reason for these transformations to occur. The spacetime metric above, that was developed from the Schwarzschild metric, has the gravitational field of the Universe causing the space contraction, and time dilation. Moreover it applies in a Universe full of matter, whereas the Minkowski metrics only apply in a hypothetical empty Universe.

What is shown here is that our expanding universe full of matter has a gravitational effect on space and time that results in all co-movomg observers seeing the same velocity of light in the same way that the Lorentz transformations do in the empty Minkowski spacetime. There is a physical reason for the necessary length contraction and time dilation due to the velocity of co-moving matter. It is due to the effect of mass on space and time. This has not been shown before. It is new physics.

The result is that all co-moving observers will see themselves as the central comoving observer in a Universe distributed as a sphere around them. They will see the constituent matter in their Universe moving away from them with a velocity according to Hubble’s law. They will see a constant speed of light. They will all see themselves as being at the furthest point that time has reached, with all other comoving bodies existing at an earlier time. They will see the edge of their Universe, at its furthest extent in all directions, as also being its beginning an infinitely long time ago.

Comparing the Minkowski spacetime with the model Universe

Minkowski spacetime is normally written in rectangular, or Cartesian, coordinates. The Minkowski metric in Cartesian coordinates has been used here to show how the moving frame coordinates will transform with the Lorentz transformations. Using Cartesian coordinates makes the algebra much more friendly than using spherical coordinates.

The work of Mukul Das, & Rampada Misra, has been quoted to show that the same result can be found with spherical coordinates.

When a moving frame is considered in the Minkowski metric in spherical coordinates, it is taken to apply to a small region that is moving in the r direction. Mukul Das, & Rampada Misra specify that it is valid only in an infinitesimally small region at r. Length contraction occurs in the r direction, which is also the direction of the relative velocity.

The Schwarzschild metric and the modified Schwarzschild metric developed in this account use spherical coordinates. It would be extremely difficult to write the Schwarzschild metric in Cartesian coordinates. I doubt if it has ever been done. It must be theoretically possible; the physics cannot be dependant on the type of coordinates used, but the spherical shape of the Schwarzschild spacetime makes the choice of spherical coordinates inevitable.

The Minkowski metric in spherical coordinates is used here because it has an identical structure to the modified Schwarzschild metric, and has the same consquenses for length contraction, time dilation, and the speed of light. It has been made clear throughout this account, that the modified Schwarzschld metric that is proposed here as a model for our Universe is not the Minkowski metric. It matches the Minkowski metric in spherical coordinates, but it is full of mass and energy, and the the Minkowski metric is empty.

Strictly speaking we can’t even talk about light in Minkowski spacetime. Light is energy, and energy has mass. If we talk about light in Minkowski spacetime, we are really referring to the mathematical description of a spherically distributed set of points moving away from a central point with a velocity equal to the speed of light. It has been pointed out above that we cannot talk about Minkowski spacetime in our Universe, and particularly not superimposed onto Euclidean space and Newtonian time in the way it is often presented.

The crucial difference is that the modified Schwarzschild metric has length contraction and time dilation caused by gravitational fields, while the Minkowski metric has to invoke the Lorentz transformations, which are arbitrary mathematical assumptions with no physical cause.

Even with the modified Schwarzschild metric presented here it is an assumption that a spherical distribution of points in the spacetime, moving at the speed, c, represents light. Light certainly doesn’t seem to be a spherical wavefront in the future lightcone, and neither does it seem to be a wavepacket.

It is tempting to take refuge in the mathematics, but as physicists we need to be concerned with the interface between mathematics and reality. We make mathematical models, but the reality that they model is our real concern.

Length contraction in the Minkowski metric in Cartesian coordinates occurs at all points throughout the space in the direction of the velocity, when one frame is compared with a frame in relative motion to it. A body in motion in the x direction will contract in the x direction at all its points, according to the Minkowski metric and the Lorentz transformations.

This doesn’t happen in our real Universe. In our Universe the length contraction is directed along the r axis, from the central co-moving observer, in every direction. In our Universe the spherical coordinates are appropriate and necessary. The length contraction and time dilation apply everywhere, not just in an infinitely small region at a point at the distance, r.

The Minkowski metric in spherical coordinates lends itself to a direct comparision with the modified Schwarzschild metric we have been investigating here. The length contraction due to the mass distribution of the whole Universe is directed radially. On a small enough scale it will appear to be all in the same direction, but it will actually be occurring in the r direction which is not the same over any real volume.

The Mikowski metric in spherical coordinates is an exact match for the modified Schwarzschild metric. All the motion of co-moving observers in the modified Schwarzschild metric is directed radially. We don’t have to limit our result with the Minkowski metric in spherical coordinates to an infinitesimal region of spacetime. It matches the modified Schwarzschild metric everywhere.

The distribution of mass causes length contraction and time dilation

So we can say that the physical distribution of mass around a co-moving observer, moving with a velocity, v_escape , and at a distance, r, in the Schwarzschild spacetime of our model Universe, causes length contraction, and time dilation, equal to those caused by the Lorentz factor in the case of stationary frames, and moving frames, in the Minkowski spacetime. This is provided that the moving frames are moving at the same velocity, v_escape , with respect to a rest frame.

Co-moving observers, moving with a velocity, v_escape, along the r axis from the central co-moving observer, will have had their time pass more slowly, and so less time will have passed for them. In other words they will be existing at an earlier time than the central co-moving observer, and so they will be in the central co-moving observer’s past.

Time moving more slowly means that all clocks will be showing time pass more slowly. This will include light clocks, and so light itself will be moving more slowly. Any measurement of the velocity of light will not show this slower velocity since any clock used to measure it will be running slow. The measured velocity of light will be constant, although its actual velocity will be increasing along with the increase in the size of the Universe. See the next section below.

All co-moving observers will see themselves at the centre of their Universe

All co-moving observers will see themselves as being at the centre of their Universe, with all other co-moving matter moving away according to Hubble’s law, and they will see a speed of light that is the same in all directions. They will all see themselves as being the furthest evolved point in time.

All co-moving observers will see a uniform speed of light in our model Universe, and this is an effect of the mass density as seen by co-moving observers in the Schwarzschild spacetime we have used to model our Universe.

The Minkowski spacetime was fine to show, hypothetically, that observers in a frame moving with a relative velocity to a frame with a uniform speed of light, should also see a uniform speed of light, but this was for a Universe empty of all massive bodies, and devoid of light energy.

We might say that an empty universe, with space and time obeying the Lorentz transformations, will have a uniform speed of light, but that still doesn’t explain why, in a Universe full of matter and energy, all observers in it, moving with a constant velocity, measure a uniform speed of light.

The Universe we are modeling has a constant density, on a sufficiently large scale, and so we can’t apply the Minkowski spacetime to it.

Minkowski started with the assumption that light moved with a constant speed. The idea that light moved with a uniform speed for all inertial observers, came originally from experiment, and was backed up by Maxwell’s equations.

Einstein had considered that the uniformity of the speed of light for all observers was more fundamental than the Newtonian mechanics that was current at the time, and proceeded to reform Newtonian mechanics.

Minkowski then used Einsteins idea’s to show that, if light had a uniform speed for all observers, the shape of space and time that was implied by that fact must be different to the shape of space and time which had been assumed up until then. He wasn’t quite at a working model of our Universe, though, because his new metric of space-time had no mass, or energy in it. That means it had no light either.

All co-moving observers will see a uniform speed of light

We now see that the model Universe proposed explains the speed of light being uniform for all co-moving observers. It shows that the necessary length contraction and time dilation is due to the distribution of mass round the observer. It shows that the speed of light itself is equal to the escape velocity of the entire Universe for that particular observer.

This must be the case since, when we build a Universe from successive shells of thickness dr, any co-moving point could be selected as the starting point. This means that all observers at co-moving points at a distance, r, with r < R and v_escape < c, can be considered to be at the centre of their own modified Schwarzschild spacetime, and with their own uniform speed of light.

There has not, until now, been a physical reason proposed for the speed of light to be uniform in our Universe full of massive bodies. Length contraction and time dilation were consquences of the postulates, made by Lorentz and his proposed transformation, for two inertial frames with a relative velocity, to explain the results of the Michaelson Morely experiment. We can now see that the required length contraction and time dilation are due to the gravitational effect of the distribution of mass in our Universe.

The future hasn’t happened yet

We have seen that all co-moving observers will see all other co-moving observers as being in their past. So what has happened to the central co-moving observer that is observing another co-moving observer in a moving frame from the point of view of the observer in the moving frame?

A straightforward interpretation for the observer in the moving frame, is that the original central co-moving observer is in a future that hasn’t yet happened. This future doesn’t actually exist yet, and so it is not part of the Schwarzschild spacetime of the co-moving observer. For all co-moving observers, the whole of space and time is on the past light cone, and the point where it is being observed from is a single point at the apex of the past light cone.

Every co-moving observer is the central co-moving observer in its own Schwarzschild spacetime. There is nothing in its future. We will see that it makes sense to consider the original central co-moving observer to be in just one of an infinite number of possible futures that have not yet happened.

The Many Worlds theory of Hugh Everett 3rd

This corresponds to the Many Worlds theory of Hugh Everett 3rd⁶, who suggests that the “collapse of the wave function” when a measurement is made, as postulated in the Copenhagen interpretation of Quantum Mechanics, does not actually happen; rather the universe splits into all the possible outcomes predicted by the wave function, each one existing in its own right.

For example we can imagine a split of a photon’s wave function into infinitely many futures, as each photon goes in every way it can, interfering with itself on the way, and each possible way it goes creating a new Universe. These new Universes would differ from each other by where the energy and momentum of that photon ended up.

This interaction isn’t observed by an observer at the point in space and time where it occurs. In practice it will already be in the past, as the observer becomes aware of it. The observation of an event and the original event are not the same thing. We don’t directly observe a photon interacting with an electron. We observe a scintillation on a screen, or a record on a measuring instrument. A measurement is ultimately an observation of something.

In our description of spacetime there is a central comoving observer. This observer corresponds to the observer at the apex of the light cone in Minkowski spacetime. We have seen that the proposed modified Schwarzschild spacetime has the same structure as Minkowski’s spacetime. We have seen that all co-moving observers will see themselves as the central comoving observer in their own Universe.

This central comoving observer sees their Universe on the surface of a light cone. Light arriving from points on that light cone has come from events where the wavefunction has just collapsed. These events have a value of ds equal to zero between them and the observer. They are effectively simultaneous since time, for this central comoving observer, has not progressed from the point of emission, or from the point of arrival of the light. Everything the observer perceives on the light cone is in that observer’s present moment.

We could say that the present moment is the collapse of the wavefunction.

In the next moment the central co-moving observer, and every point on the light cone, will proceed into a possible future. In Hugh Everett’s “Many Worlds” model this will be one of an infinite number of possible futures. Every possibility is realised in an infinite number of possible new moments that the central co-moving observer now percieves as the moment they are in.

What happens to that observer’s conscious awareness? There must be a conscious awareness of every new moment in an observers body that differs by an infinitesimally small degree to an infinite number of almost identical bodies. If there are an an infinite number of possible new moments, does our consciousness split and multiply as it goes forward into an infinity of possible futures? Do we select a future out of all the possibilities?

If there are an infinite number of other consciousnesses, what makes us the one consciousness we are rather than one of the infinite number of other consciousnesses? Put more succinctly; why are you the conscious consciousness?

We have the experience of being aware of one moment at a time, but the only one we are actually aware of is the present one. We also have a memory of time passing. It appears that the present moment is continually becoming a future moment, but it only ever seems to have been one future moment. What has happened to all the others?

Is the present moment an attribute of time independent of an observer? Is it just chance that you, who are reading this, happen to be alive and conscious in this particular moment? The probability that your lifetime should coincide with the present moment is zero in a Universe in which time has no beginning and no end.

How many consiousnesses do there need to be to be aware of all possible past and future moments?

We can talk about photons entering the eye, forming an image on our retina, getting converted to an electrical signal in our optic nerves, and then this signal getting passed on for “processing” in the center of the brain responsible for seeing. That is the conventional model.

Then, somehow, this information gets converted back into a image for our consciousness to become aware of. At this stage the model starts getting a bit uncertain. We leave the realm of physics, and even biology. We just don’t know how this biological process becomes an experience that a consciousness is aware of.

When we experience the Universe we live in, it does not seem as if we are looking at an image in our heads; it seems as if we are experiencing reality directly. We don’t actually know if there is a separate Universe “out there” that we only see a representation of, or whether the experience is all there is.

A separate universe “out there” is an idea in our minds. The experience of it in the moment actually has more reality. We take the experience, and make a mental model of the Universe in which we try to exclude the experiencer. Both Quantum Mechanics and Minkowski spacetime are strongly implying that we can’t do this.

Professional physicists don’t seem to talk much about consciousness, possibly because it isn’t a sensible career move. Roger Penrose, and his book “The Emperor’s New Mind” is a notable exception. Even for biologists, for whom the study of life is their legitimate business, consciousness is rarely part of their experimental or theoretical investigations.

This is, perhaps, because it is very hard to say what consciousness is. We only know of one; our own. We don’t actually know if there are any other consciousnesses. We assume that other people have a consciousness, but that is an assumption. We can’t detect consciousness.

Most people would say that a dog is conscious, but is an ant conscious? Is a bacterium conscious? In trying to decide, we are actually not detecting their consciousness, we are judging based on their reactions to what we assume they are experiencing.

We can’t even detect our own consciousness. We can only infer its existence by being aware of what it is aware of.

This doesn’t mean that awareness is not part of the process of determining the structure of the Universe. It is a pure, and likely unjustified, assumption that it isn’t.

This is illustrated whimsically by this limerick by Ronald Knox.¹¹

There was a young man who said “God
Must find it exceedingly odd
To think that the tree
Should continue to be
When there’s no one about in the quad.”

And its anonymous reply:

Dear Sir: Your astonishment’s odd;
I am always about in the quad.
And that’s why the tree
Will continue to be
Since observed by, Yours faithfully, God.

One Universe complete with past and future all mapped out in advance does not fit with our personal experience. We all have the idea that our choices determine the future.

If we consider consciousness to be an emergent property of living organisms, with each organism of sufficient complexity possessing consciousness, we have to wonder why it evolved at all in a Universe in which all events were predetermined.

It seems that physics is necessarily starting to come to terms with consciousness, and including it directly in our description of reality. It is beginning to look as if there are a lot of possible realities in one consciousness rather than a lot of consciousnesses inside one reality. That idea isn’t new. The Bagavad Gita has been saying that for a few thousand years, as have the Buddhists

What did Jesus mean when he said “The kingdom of God is within you”? Is what Jesus called the kingdom of God what we call the Universe today? Is he actually saying that the Universe exists within one consciousness?

Are the infinity of possible moments unfolding right now in this present moment? Are all possible present moments being experienced one moment at a time by one conscious being?

The mass of the Universe and the speed of light are increasing in time

If the rate of flow of time is less in the past, and the mass of massive bodies is less in the past, then the rate of flow of time is increasing as the Universe expands, and the mass of massive bodies must be increasing. The speed of light must also be increasing with the rate of flow of time, but its measured speed will always be the same. The radius of the Universe must be increasing in keeping with Hubble’s law, as the mass of the entire Universe increases.

So now we have a picture of a spherical Universe centred on any co-moving observer, with a uniform density, a finite mass that is increasing in time, and a finite radius that also is increasing in time. Because its radius is directly proportional to its mass, its volume will be increasing as the cube of its radius, and its density will be decreasing as the square of its radius.

Since this would be true for all co-moving matter in the Universe, the cosmological principle, that the Universe appears the same from all places, and in all directions, is satisfied; at least for all co-moving observers.

The rate of flow of time

Einstein was once asked by a student to say what time was; he replied “Time is what a clock shows”.

We are never comparing a clock with absolute time; all we can actually do is compare it with another clock. We say that clocks that agree with each other “keep good time”.

We can imagine a hypothetical light clock, where we bounce a beam of light between two parallel mirrors, and count the reflections. We have conventional gravity driven pendulum clocks. We have electromagnetically driven quartz wristwatches, and atomic clocks operating with nuclear forces. For time to pass more quickly, all these clocks must run more quickly.

The speed of light is fundamentally connected to time dilation due to mass, and the rate of flow of time.

We can expect that the speed of light will increase in time as the radius of our model Universe increases with the increase of mass. As we look out to a radius, r, where time is flowing more slowly, the speed of light will depend on the escape velocity for that radius, and so the speed of light will be less.

At the edge of our model Universe, at a distance, R , escape velocity is equal to the speed of light, but the speed of light itself will be zero there from the point of view of a central observer.

Co-moving observers see the same speed of light

Any co-moving observer at a distance, r, approaching, R, will see nothing different, however. They will measure the same speed of light.

If all physical activity proceeds at a faster rate, what would we notice? Would we see objects moving faster? Would we see light moving faster? Would we see our watches and clocks ticking faster? Well, actually, no we wouldn’t.

If all our clocks, and ourselves, started running faster, or slower, we would have no reference time to detect it. If all our time measuring devices showed time flowing faster, would we say that time itself was flowing faster? In fact we wouldn’t notice anything. To say our time was flowing faster could only be from the point of view of an observer somewhere else whose time was passing more slowly according to their clocks

Anything that moves in an inertial reference frame, and in the absence of external forces, is a clock. Any such moving object can be used to measure time flowing by measuring the time it takes to travel equal lengths. Time flowing faster implies that all actual velocities must be increasing, though measured velocities will remain constant. See also “Velocity clocks” below.

Time in the model Universe

What is happening to time in our model? We have seen that

\color{red}{dt=(1- \frac{8Gρπr^2}{3{c} ^{2}})^{-\frac{1}{2}}dτ}

And equivalently, if the density of the Universe is the critical density,

\color{red}{dt=(1-\frac{v^2_{escape}}{c^2})^{-\frac 1 2}dτ}

This is telling us that as we look out to a radius, r, we see time pass more slowly. This means that all clocks will be running more slowly, and this will include light clocks, so light itself must be moving more slowly. At a distance r, where co-moving observers are moving away at escape velocity for that distance, we expect that

\color{red}{c_2=(1-\frac{v^2_{escape}}{c_1^2})^{\frac 1 2}c_1}

Where c₁ is the speed of light measured by the central co-moving observer where they are, and c₂ is the speed of light, measured by the central co-moving observer , where a co-moving observer is moving away at a distance r, and at a velocity, v_escape.

A co-moving observer, in the frame moving with respect to the frame of the central co-moving observer, will equivalently be at the apex of the past light cone in their own Schwarzschild spacetime, with a lower velocity of light. They will still call it the speed of light, but they will be measuring it with slower clocks. According to the central co-moving observer, they will be measuring a slower speed of light with dilated time.

The shape of space and time depends on the observer

Since the mass of the Universe determines the radius of the model being considered, it follows that, if the mass of the Universe is increasing, its radius must be increasing in step with its mass. This was the assumption made earlier.

We saw, in “Adding shells of uniform density”, that mass, m_rest-at-r, at a radius, r, is as follows

\color{red}{m_{rest-at-r} =  {(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}{m_{rest}}}

Where m_rest is the mass a body would have next to the central observer. The mass of all bodies at this distance, r, would be reduced by the same factor as time is dilated. If an observer is at the distance at which time is flowing at half the rate compared to time at the centre, then his or her mass will be halved, and the radius of his or her observed Universe will also be halved. Their velocity of light will also be halved, but they will not see this. Their measured velocity of light will remain constant.

We can now see that, in this model, the velocity of recession of the edge of the Universe, and the velocity of light, are one and the same thing. The edge, where length is contracted to zero, and time is dilated to infinity, will be moving away from the central observer with a velocity equal to the speed of light.

This edge is also the beginning of time an infinitely long time ago, or in other words, our Universe has always been here. The Universe is going from an infinitely small mass, an infinitely small radius, and a critical density; to an infinitely large mass, an infinitely large radius, and a critical density. At any time it is observed, it appears the same as at any other time.

As was suggested at the beginning of this account, the shape of space and time is determined by the distribution of mass in space and time, and this is dependent on the point of view of the observer.

Exponential expansion

We saw in the last Section that if we look out towards the edge of our model Universe, we will come to a distance where time is flowing at half the rate it is where we are. This point will see itself as a central co-moving observer in the Universe it perceives but with half the radius, half the mass, and, because time is passing more slowly, the velocity of light will be halved.

Since time is passing more slowly, all clocks will be running at half the rate of ours, so the observers in this Universe will not say their velocity of light is half of our measured velocity of light. They will measure it to be the same since they will be using clocks running at half the rate of ours.

Conversely they will consider us to be in a possible future Universe where time is flowing twice as fast, and the speed of light will be doubled, but we will be measuring it to be the same because all our clocks will be running faster.

All co-moving observers see themselves as central

All co-moving observers will see themselves as central in the Universe they perceive. The edge of their Universe is moving away at the velocity of light, and that velocity will increase, in proportion to the increase of mass and radius, r, as the rate of flow of time increases.

This implies that the expansion of the Universe is exponential. As its radius, r, doubles, the velocity of light, which is the velocity of the edge away from the centre, must also double. Here we are using R for the radius of the entire Universe as it evolves in time, with R_(t=0) denoting the radius at the present time.

We can model R as a function of t as

\color{red}{R=R_{(t=0)}e^{Ht}}

\color{red}{\frac{dR}{dt}=HR_{(t=0)}e^{Ht}}

\color{red}{\frac{dR}{dt}=HR}

All observers will measure the same velocity of light, c, which will be escape velocity at R, as viewed by a central co-moving observer.

\color{red}{\frac{dR}{dt}=c}

The actual velocity of the edge will be inreasing exponentially in proportion to the radius, R. The current rate of flow of time will be that of the central co-moving observer. As the radius doubles, and the speed of light doubles, the rate of flow of time will double for the central co-moving observer. The result will be that the relationship below will hold for all central co-moving observers as the Universe evolves in time.

\color{red}{c=HR}

Where t = 0 is the present moment, R is the radius of the Universe, H is Hubble’s constant, and ln2/H is the doubling time of the universe, as measured with our present rate of flow of time.

We saw in “Hubbles law and escape velocity” that the critical density of the Universe is inversely proportional to the square of the radius, R, so Hubbles constant, H, is inversely proportional to the radius, R. The measured speed of light will remain the same at any time the Universe is observed.

The present Universe looks just like it did in its past

This view is consistent. The present Universe looks just like it did in its past. A co-moving observer in the present sees themselves as being at the centre of a spherical Universe, with its edge moving away from it at the velocity of light, and so does a co-moving observer in the past.

They both measure the speed of light to be a constant, c, though the actual velocity of light is greater for the observer in the present. Both of them see themselves as being at a point in space and time that has had more time pass than all other points. They both see themselves as being the furthest advanced in time, and at the apex of their light cone.

All points in space and time are equivalent

All points in space and time, for all co-moving observers, are equivalent. The exponential function asymptotically approaches the time axis in the past, and continues doubling for ever into the future. This means there was no “big bang”, or perhaps we could say that the big bang has been happening for all time, and will continue to do so.

We saw above that the edge of this model Universe is also its beginning in time. The universe started as an infinitely small, infinitely dense, sphere with infinitely small mass an infinitely long time ago.

According to this model the Universe has zero mass an infinitely long time ago, and its volume was zero, so the mass of its constituent bodies was zero an infinitely long time ago, and their distance apart was zero.

In the model we are considering a central comoving observer sees the other matter in the Universe moving away from it. If mass increases with velocity, according to Einstein’s mass energy relationship, we would expect that the matter moving away from that observer would increase accordingly. Instead, in this model, mass seems to decrease with velocity away from the central co-moving observer.

The centre is moving away from the edge

We see that it makes more sense to consider that the centre of this model is moving away from its edge. The edge is in fact a singular point at the beginning of space and time. It is a definite distance away, but it is an infinitely long time ago. At any real point on the time axis there will be the entire Universe in the process of exponential expansion. The exponential function has the interesting property of looking the same at any point on the time axis.

It is the mass of a body at the centre that is greater by the Lorentz factor than the mass that same body would have near the edge. Velocity is relative. We need to make the appropriate choice as to which of any pair of bodies, that are co-moving with the general expansion, has their mass increase.

We saw above that a co-moving massive body has less potential energy at a distance, r, in the model, and that this corresponds to an earlier time in the evolution of the Universe. If m_rest-at-r is the mass of a body at a radius, r , in the model, this mass will be less relative to the mass, m_rest, that the same body would have at the centre of the model.

Every co-moving observer is the central co-moving observer

We have seen that every co-moving observer, next to one of these bodies, will see themselves as being the central co-moving observer in their Universe. They will be moving, relative to the edge they perceive, at the escape velocity of their perceived Universe. They will measure the edge of the Universe as moving away from themselves at the speed of light.

The velocity, v, due to the expansion, is a relative velocity, so we may consider the central co-moving observer at r = 0, to be moving away from the edge, where r = R, at the speed of light, c. When we consider the expansion of the Universe like this, every central co-moving observer is moving faster than all the other bodies in the expanding Universe it is observing, and we expect that the mass of a massive body next to this co-moving observer will have increased according to Einstein’s equation for the total energy of a moving body.

Einstein’s equation for total energy in an expanding Universe

We saw above that the potential energy of a mass at a distance r is given by

\color{red}{m_{rest-at-r} =  {(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}{m_{rest}}}

When we developed this model we saw that every co-moving observer will see themseves as being the central comoving observer in their Universe. They see all other comoving observers as having a reduced mass. We saw in “Hubble’s law and escape velocity” that we can write this as

\color{red}{m_{rest-at-r} =  {{(1-\frac{v_{escape}^2}{c^2})}^{\frac{1}{2}}}{m_{rest}}}

This velocity, v_escape, is relative. The model Universe we have developed has its beginning at a radius, R, an infinitely long time ago, where the Universe was infinitely small. All comoving observers in this model see themselves as the central co-moving observer. So all co-moving observers will consider themselves to have rest mass. They see themselves as moving away from all other comoving observers. Their mass is greater than all other comoving observers in their Universe. Expressing m_rest In terms of m_rest-at-r we get

\color{red}{m_{rest} =  {(1-\frac{v_{escape}^2}{c^2})}^{-\frac{1}{2}}{m_{rest-at-r}}}

Where v_escape is the relative velocity between the point at a radius, r, and the centre of the model at r = 0. In this case we are considering m_rest-at-r to be the rest mass of the body, and m_rest is the mass of the moving body. We can see that m_rest corresponds to the mass of the moving body, m, and m_rest-at-r corresponds to m_rest in Einstein’s equation for the total energy, E, given by

E=mc^2

where

m=\frac {m_{rest}} {(1-\frac{v^2}{c^2})^{\frac 1 2}}

and where m is the moving mass, and m_rest, is the rest mass.

We saw above that when v_escape = c, and r = R, we have m_(r=R) = 0. We saw that the radius, R, corresponds to the beginning of the Universe in this model, so the rest mass of the body, at the beginning of the Universe, is zero

Rest mass is potential energy

This view is consistent with exponential expansion. At the beginning of the Universe, an infinitely long time ago, the total mass of the Universe is zero, so the kinetic energy of its constituent bodies, like their potential energy, is equal to zero.

As the Universe expands its mass increases. The mass of all its constituent co-moving bodies increase, as they move apart, and their relative velocity increases.

This means that we can consider all the mass of its constituent co-moving bodies bodies at , m_(r=0), as being due to their gain in potential energy, or equivalently, to their gain in kinetic energy.

A bodies mass at m_(r=0) is what is commonly referred to as its rest mass.

We see that the total energy of a co-moving body is equal to its kinetic energy due to the expansion, or, equivalently, it is due to its potential energy relative to the mass of the entire Universe.

In fact, with this model of the Universe, we don’t need to consider that a co-moving body has kinetic energy. It’s mass will all be due to it’s potential energy. In “The equivalence of inertial and gravitational mass” below we see that this is also true for bodies moving with all velocities up to the speed of light moving relative to, or through, the expansion.

We can always consider what we thought of as kinetic energy to be potential energy relative to the mass of the Universe as a whole.

The nature of mass

What is the nature of mass? What is the mechanism that causes a gravitational field, and the corresponding shape of space and time? We are looking for something that increases its mass in step with the increase of mass of the entire Universe.

Consider a structure exactly like the structure of the entire model Universe we have been describing, with a mass, and a corresponding radius, as described. If we took this structure, and placed it in an otherwise empty Universe, it would produce, outside its radius, a shape of space and time described by the Schwarzschild metric space. In this new Universe it would appear as a black hole. We have, in effect, built a black hole from the inside out. This black hole would also have a mass that would be increasing in time.

We can imagine moving such massive bodies in from infinity in the manner described above in the sections “Applying the Schwarzschild spacetime to the Universe”, “Potential energy and escape velocity”, “Adding spherical shells of uniform density” and “The size of the Universe”. An infinitely large collection of these objects, moving with escape velocity from a central observer, and obeying Hubble’s law, would form a Universe just like the one we have been describing.

The model Universe can have any mass

We can go further, and notice that the model Universe we have been developing can have any mass, and therefore any size. Consider a black hole with the mass of an electron, or the mass of a quark. We can describe its mass completely in terms of the shape of space and time. A mass the size of an electron’s would have an event horizon radius of approximately 10 ^-57 m.

What is there inside an electron? Another Universe? This is not really as outrageous as it might appear at first sight. Any alternative would need another, credible, explanation for the nature of mass, together with an explanation for why it is increasing. We would also need to find an explanation for how mass bends space-time. If mass is bent space-time, we have a simple, and complete, explanation for what we observe. William of Ockham would approve.

The Universe is self similar on all scales.

We could then imagine a structure that was self similar on all scales. What we perceive as our Universe would be a tiny part of an infinite assembly of such Universes, both larger and smaller.

This conjecture raises the possibility that the cosmological principle, the notion that the spatial distribution of matter in the Universe is both homogeneous and isotropic on a large enough scale, could have a new addition; namely that the Universe looks the same at all scales.

Increase of charge with the evolution of the universe

We have seen that, in this model of the Universe, mass is increasing along with the radius of the Universe, the rate of flow of time, and the velocity of light.

As co-moving observers looking out along a cosmological radius, we are also looking back into the past. We see time proceeding more slowly, length contracted, and mass reduced, all by the same factor. This factor is equivalently the Lorentz factor determined by the velocity of recession, or the Schwarzschild factor determined by the size and density of the sphere of matter that radius describes.

Charge is increasing along with mass

What about electric charge? For the perfect cosmological principle to hold, the Universe should appear the same at all times, as well as at all places, and in all directions. We would expect that charge is reduced by the same factor as mass when we look out along a radius from the centre.

We would expect that lengths and times, as measured by electromagnetic clocks and measuring rods, would give the same measurements as gravitational time and space measuring devices like solar systems. For time to pass more slowly, all clocks must operate more slowly

For this to be true, charge must be increasing along with mass for a central co-moving observer. We should see charge reduced by the same Lorentz and Schwarzschild factors as mass when we look out along a cosmological radius .

It turns out that charge is increasing in time along with mass in the proposed model. This can be shown by applying the standard Maxwell equations to this model of the Universe.

The force on one moving charge due to another moving charge

Consider first the force on one moving charge due to another moving charge if both charges are moving side by side with a velocity, v, away from the observer. This force is a combination of electric, F_elec, and magnetic, F_mag, forces. For mathematical simplicity consider two equal charges,q₁ and q₂ moving away from the co-moving observer with velocity, v, along a cosmological radius, r, such that their distance relative to each other, x, is perpendicular to r.

One such moving charge, q₁, will produce an electric field, E, at a distance, x, given by

E=\frac{q_1}{4πεx^2}

and a magnetic field, B, at a distance, x, that will depend on their velocity, v, along the radius, r, where

B=\frac{μq_1v}{4πx^2}

We have the relationship between μ and ε given by

c^2=\frac{1}{με} 

μ=\frac{1}{εc^2}

The resultant force on a second charge, q₂ a distance, r, away is given by

F_{elec}-F_{mag}=q_2(E-vB)

The subtraction indicates that the force due to the magnetic field, F_mag is in the opposite direction to the force due to the electric field, F_elec Like charges will produce a repulsive electric force and an attractive magnetic force.

F_{elec}-F_{mag}=q_2(\frac{q_1}{4πε x^2}-\frac{q_1v^2}{4πε x^2c^2})

\color{red}{F_{elec}-F_{mag}=\frac{q_1q_2}{4πεx^2}({1}-\frac{v^2}{c^2})}

\color{red}{F_{elec}-F_{mag}=\frac{({1}-\frac{v^2}{c^2})^\frac{1}{2}q_1({1}-\frac{v^2}{c^2})^\frac{1}{2}q_2}{4πεx^2}}

The last equation above suggests that we can consider each charge, q, to be reduced by the factor

\color{red}{({1}-\frac{v^2}{c^2})^\frac{1}{2}}

as required by the proposed cosmological model.

This simplified analysis was for a pair of charges orientated perpendicular to their velocity away from an observer due to the expansion of the Universe, i.e. perpendicular to r. The symmetry of the situation, with respect to a sphere of observers at the same distance, r, demands that this must be true for all orientations.

The magnetic force is the relativistic manifestation of the electric force.

Einstein was of the opinion that the magnetic force was simply the relativistic manifestation of the electric force. In 1953 Albert Einstein wrote to the Cleveland Physics Society:

“What led me more or less directly to the Special Theory of Relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field.”

Gravito-magnetism

It has been suggested by several people that there may be the equivalent of the Maxwell equations for electromagnetic fields that describe the operation of gravitational fields. One of the latest contributors to this idea is Arbab I. Arbab⁴. Arbab has shown, using quaternions, that an exact corollary to Maxwell’s equations can be made for gravity. This includes the idea that there is a gravito-magnetic field, corresponding to Maxwell’s magnetic field, that is attributed to moving mass.

Corresponding equations for masses to Maxwell’s electromagetism equations

We can rewrite Maxwell’s electromagetism equations for electrical charges, to make corresponding equations for masses. If we denote the gravitational field round a mass, m₁ as, E_grav we would have

\color{red}{E_{grav}=\frac{m_1}{4πε_{grav}x^2}}

Where we rewrite the gravitational constant, G, as

\color{red}{G=\frac{1}{4πε_{grav}}}

We would also have a gravito-magnetic field, B_grav, at a distance, r, that will depend on their velocity, v, along the radius, R, from the central co-moving observer, and a gravito-magnetic constant, µ_grav, corresponding to Maxwell’s magnetic constant.

\color{red}{B_{grav}=\frac{µ_{grav}m_1v}{4πx^2}}

We would have the relationship between μ_grav and ε_grav given by

\color{red}{c^2=\frac{1}{μ_{grav}ε_{grav}}}

\color{red}{μ_{grav}=\frac{1}{ε_{grav}c^2}}

The resultant force on a second mass, m₂ a distance, r, away is given by

\color{red}{F_{grav}-F_{gravmag}=m_2(E_{grav}-vB_{grav})}

Where F_grav is the usual force of gravity, and F_gravmag is the gravito-magnetic force. The subtraction indicates that the force due to the gravito-magnetic field is in the opposite direction to the force due to the gravitational field. Masses will produce an attractive gravitational force, and a repulsive gravito-magnetic force.

\color{red}{F_{grav}-F_{gravmag}=m_2(\frac{m_1}{4πε_{grav}x^2}-\frac{m_1v^2}{4πε_{grav}x^2c^2})}

\color{red}{F_{grav}-F_{gravmag}=\frac{m_2m_1}{4πε_{grav}x^2}(1-\frac{v^2}{c^2})}

\color{red}{F_{grav}-F_{gravmag}=\frac{(1-\frac{v^2}{c^2})^\frac{1}{2}m_2(1-\frac{v^2}{c^2})^\frac{1}{2}m_1}{4πε_{grav}x^2}}

Mass is reduced by the Lorentz factor

The last equation above suggests that we can consider each mass, m, to be reduced by the Lorentz factor below, in exactly the same way as we saw with charge.

\color{red}{({1}-\frac{v^2}{c^2})^\frac{1}{2}}

If a mass has rest mass, m_rest at the central observers position, it will have a mass, m, when it is at a distance, r, and co-moving with the expansion.

\color{red}{m=({1}-\frac{v^2}{c^2})^\frac{1}{2}m_{rest}}

This corresponds exactly with the decrease in mass we assumed to model a universe of uniform density.

\color{red}{m =  {(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}{m_{rest}}}

By analogy we would expect that, for an electric charge q

\color{red}{q=({1}-\frac{v^2}{c^2})^\frac{1}{2}q_{rest}}

and

\color{red}{q =  {(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}{q_{rest}}}

The equivalence of inertial and gravitational mass

We can now see that the length and time transformations, attributed to the relative velocity of co-moving observers, can be attributed equally well to the effect of the mass distribution of the entire Universe. This mass distribution will be dependent on the observer. Mass will be distributed spherically around every co-moving observer. This will give the correct length and time transformations for every other co-moving observer, so that they too will see themselves at the centre of a spherical Universe.

In the model we are proposing, the central co-moving observer sees the mass distributed around him or her reduced by the Lorentz factor as it moves away with the expansion of the Universe. We have seen that this can be attributed to a loss of potential energy. Here we are treating the masses of co-moving bodies as gravitational mass.

So the mass of a body moving away from from a central observer, as part of the general expansion of the Universe, is less than it would be next to the observer. This appears to be contrary to what relativity theory says happens when we observe a body moving with relative velocity to us.

Special relativity tells us that that the total energy of a massive body is

E=mc^2

where

m=\frac {m_{rest}} {(1-\frac{v^2}{c^2})^{\frac 1 2}}

This tells us that mass increases with velocity. It gains kinetic energy. The velocity, v, is taken to be the relative velocity between two observers, one moving with the massive body, and the other moving with velocity, v, relative to the body.

Which mass increases with velocity?

There is something unsettling about the current theory of special relativity when it comes to mass increase with velocity. The mass increase is real, and has been observed in the laboratory, but there is a paradox here. Why should we look at the situation from the point of view of someone in the laboratory? From the point of view of an observer moving with a particle moving at speeds where relativistic increase in mass becomes noticeable, it is the laboratory that is moving with relativistic speed. Why isn’t it the laboratory that increases in mass? We can now resolve this.

For a co-moving object with rest mass, m_rest to be accelerated we must change its momentum by application of a force. If this is done by a purely external method, by bouncing light off it for example, it will also absorb energy, and it’s mass will consequently increase. It has been suggested that spacecraft could be accelerated in this way by using a light sail.

Moving through the expansion

Now consider a body that is initially co-moving, and seeing itself at the centre of our model Universe. If it is accelerated, it starts to move past the co-moving bodies next to it with a velocity, v. It will see all the matter in its universe have a velocity, –v, added. In particular, a point ahead of it, that had been moving away with velocity, v, due to the expansion, will now be stationary relative to itself. This point up ahead will now be where a new central co-moving observer is observing the Universe.

The centre of the sphere of the Universe has moved away ahead of the body, and the edge has come up closer behind it. To the new central co-moving observer, the moving body will now appear to be in the past of a more evolved Universe.

This is counter intuitive. As observers moving past co-moving matter accelerate, they will perceive the sphere of the Universe shift farther ahead of them. As they accelerate towards the centre of the Universe they see themselves in, their distance to the centre will actually increase. Their position in the sphere of the Universe depends on their velocity past nearby co-moving matter.

The moving body, and the co-moving matter it is moving past, will both experience the the same rate of flow of time, but the moving body will have gained mass.

For the central co-moving observer, the body moving through the expansion, past co-moving observers, will be on the edge of a sphere of matter with radius, r, centred where we are observing it at the new centre of the Universe ahead of them.

The central co-moving observer will also see a spherical shell, made up of the rest of their Universe, with a radius greater than r. According to Birkoff, as we have seen, this spherical shell will have no effect on the space and time inside the shell.

Increase in mass is increase in potential energy

Observers with the moving body also see themselves on the edge of the same sphere of matter with radius, r. they will associate their increase in mass with their increase in potential energy with respect to this sphere of matter.

The central co-moving observer measures the same velocity of light coming from all directions. Observers with the body we are considering, moving through the expansion past nearby co-moving matter, are stationary relative to the central co-moving observer. They will also see the same velocity of light coming from all directions. The light will actually be slower, but time is passing more slowly, and so their measured velocity of light will be the same.

Inertial mass increases with velocity

We can say that the accelerated body we are considering will have its inertial mass, m_inertia , increase with increasing velocity, relative to the co-moving matter it is passing, in the manner prescribed by the equations of special relativity.

E=m_{inertia}c^2

where

m_{inertia}=({1}-\frac{v^2}{c^2})^{-\frac{1}{2}}m_{rest}

Velocity is proportional to radius

As the moving body increases its velocity, it will increase the radius of that sphere ahead of it in proportion to its velocity. In effect it will be lifting itself above a sphere of increasing mass and radius, and so we would expect that its potential energy, and therefore its gravitational mass, m_gravity , will be increasing by the factor

\color{red}{{(1- \frac{8Gρ_{critical}πr^2}{3c^2})}^{-\frac{1}{2}}=(1-\frac{v_{escape}^2}{c^2})^{-\frac{1}{2}}}

So

\color{red}{E=m_{gravity}c^2}

where

\color{red}{m_{gravity}={(1- \frac{8Gρ_{critical}πr^2}{3c^2})}^{-\frac{1}{2}}m_{(r=0)}}

and where m_(r=0) is the mass the body would have at the centre of the sphere.

The mass at the centre is the rest mass

What about the mass at r=0 ? This is the rest mass of a body. We saw above that the mass of the entire Universe is increasing exponentially, and so the mass of its constituent bodies is increasing exponentially. We can identify this increase with the increase in potential energy of its constituent bodies as the distance between them increases with the expansion of the Universe . We can now see that the rest mass of a body is its potential energy at the present time. Einstein’s equation, E = mc² , applies to potential energy, as well as kinetic energy, and all mass, including rest mass, is equivalent to gravitational potential energy.

At r = 0

\color{red}{m_{rest}=m_{(r=0)}}

So

\color{red}{m_{inertia}=m_{gravity}}

It is the massive body moving through the expansion that will have its mass increase compared to the co-moving matter it is moving past. It is clear now which way round this mass increase is. The co-moving matter sees the body moving past it have its mass increase. They are not equivalent.

We can see now that there are two distinct ways an object can have a relative velocity to another. One is velocity between co-moving objects due to the overall expansion of the universe, and the other is relative velocity of an object moving through the expansion, past co-moving objects.

If a body is moving through the expansion, an observer moving with it will experience the same dilation of time, and the same contraction of space, with respect to the central co-moving observer, as the co-moving matter it is moving past, but the moving body will have had its mass increase with its velocity, as predicted by special relativity.

The body moving through the expansion experiences a length contraction, and time dilation, due to the gravitational effect of the mass of the sphere of matter ahead of them. Observers with this body will know that this is exactly the length contraction, and time dilation, necessary for observers co-moving with the matter they are passing to continue to perceive themselves at the centre of their universe, with a speed of light that is the same in all directions.

Observers with the body, moving past co-moving matter, are stationary relative to the centre of their spherical Universe which is now ahead of them, and observers at that centre confirm that they measure a light speed that is equal to the same value of c in all directions. Light is actually moving faster, but time is also flowing faster by exactly the same factor.

A body moving through the expansion will be in free fall towards this sphere, and it’s velocity relative to the co-moving matter it is passing will be equal to escape velocity from the sphere.

The moving body will gain in mass compared to the co-moving bodies it is passing. This is observed experimentally. As it accelerates, the moving body will have its potential energy increase. It will still be stationary relative to the centre of the sphere, but its distance to the centre will be increasing. It will attribute its increase of mass as being due to its gain in potential energy.

Co-moving observers, observing the moving body passing them, will associate its increase in mass as being due to its velocity relative to the matter they are moving past. They may wonder, though, if they believe that they are in an inertial frame with the body moving past them, why mass increase is associated with the moving body, and not with the them, since they have the same relative velocity.

Kinetic energy is potential energy

We can see now that a body’s kinetic energy, and its potential energy, are one and the same thing. What we refer to as kinetic energy is really the potential energy a massive body possesses with respect to the mass of the Universe as a whole.

This is why inertial mass is the same as gravitational mass. As a massive body is accelerated by the application of a force, increasing its velocity relative to the co-moving matter it is passing, the process is said to be happening to it’s inertial mass. We can now see that the applied force is actually lifting the mass in the gravitational field of the universe, increasing its potential energy. In this case we say that the force is acting on the bodies gravitational mass. Inertial mass, and gravitational mass, are the same thing.

We saw in “Exponential expansion” above that, we don’t need to consider that a co-moving body has kinetic energy. We can consider its mass to be due to it’s potential energy. We now see that this is also true for bodies moving with all velocities up to the speed of light moving relative to, or through, the expansion.

We saw in “The Schwarzschild spacetime metric” above that a massive body, falling in free fall near a Schwarzschild spacetime round a mass, loses mass as it falls due to its loss of potential energy with respect to the mass it is free falling towards, and gaining an equivalent amount of kinetic energy. The gain in kinetic energy is actually due to the massive body gaining an equivalent amount of potential energy “above” the central co-moving observer in the way discussed in this section.

A geodesic is simply a massive bodies path, through the Universe, that maintains the same amount of potential energy due to both its proximity to a gravitating mass, and its position relative to the sphere of the Universe. It is the “natural path” that a body follows when it is unaffected by external forces, and moving under the influence of gravity alone.

We can see this as the body exchanging potential energy for kinetic energy as it “free falls” in a local gravitational field.

The potential energy it has due to its position near a mass is being exchanged for potential energy with respect to the sphere of the entire Universe in a way that keeps its total energy constant for that particular time in the overall expansion. We saw above that its mass will actually be increasing as it gains potential energy due to the overall expansion of the Universe.

The above is all speculating that the proposed model is valid. The prediction of the equivalence of gravitational, and inertial, mass above suggests that it is.

Velocity clocks

Because an observer moving through the expansion sees a sphere of matter ahead of them, they will be accelerated in the direction they are moving in by that sphere of matter.

As a body accelerates, it’s velocity will increase, so the radius, r, of the sphere it sees ahead of it is also increasing. It is always travelling at escape velocity, but instead of decreasing its radial distance as it falls, its radial distance will increase.

The entire Universe is increasing in mass as its radius, R, increases at its escape velocity, c, so we expect the body’s mass, and radial distance from the centre, will increase in step with the increasing mass and radius of the Universe.

This means that the body’s velocity would be expected to increase along with the velocity of light as the Universe expands, but a measurement of its velocity with a light clock will give a constant value. The acceleration of a body moving past co-moving matter, due to the mass of the sphere of matter ahead of it, will exactly match the increasing rate of flow of time.

In practice this means that a bodies measured velocity will remain constant, as a fraction of the speed of light, just as the measured speed of light itself will remain constant.

A massive body moving with a constant velocity is a clock

Any massive body moving with a constant velocity can be used as a clock. It will cover equal distances in equal times. We could call it a velocity clock.

We can envisage other types of clock. Our solar system is a gravitational clock.

We have seen that, as mass doubles with the expanding Universe, velocities must also double. Viewed as a gravitational clock, our solar system, and all other gravitationally bound systems, should show time passing twice as fast, just as a light clock, or a velocity clock would.

Consider a mass, m, moving in a circular orbit round another mass M. It will have a force on it due to the gravitational field resulting in a constant centripetal acceleration.

F=ma

\frac{GMm}{r^2}=\frac{mv^2}{r}

r=\frac{GM}{v^2}

We saw that we could write

\color{red}{G=\frac{1}{4πε_{grav}}}

and, supposing that the idea of gravito-magnetism is valid, we have

\color{red}{c^2=\frac{1}{μ_{grav}ε_{grav}}}

\color{red}{c^2=\frac{4πG}{μ_{grav}}}

If the velocity of light increases, we would expect that the gravitational constant, G, would increase, and the gravito-magnetic constant, μ_grav would decrease.

From which we can see that, r, the orbital radius given by

r=\frac{GM}{v^2}

will remain the same as G, M, and v, all increase in step with the expanding Universe.

If orbital radii are constant, and orbital velocities double as the universe doubles its size, gravitational clocks, like solar systems and galaxies, will show a doubling of their periods in step with all other clocks.

We may also note that length in our model, if measured by gravitational orbits, stays constant. What about length measured electromagnetically? We have for the velocity of light, c.

c=fλ

where f = frequency and λ = wavelength.

Time flows faster and lengths stay the same

If the rate of flow of time doubles, we can expect the velocity of light to double, and its frequency to double, so its wavelength will stay the same.

The second is presently defined as the duration of 9,192,631,770 periods of the radiation from a specified state of the caesium-133 atom, so the second, as defined, will be decreasing in step with the increase of the velocity of light.

So we can expect gravitationally measured lengths, and electromagnetically measured lengths to stay the same. Using these measures of length we see that the Universe is expanding.

Faster time can’t be detected

We can see that all clocks are measuring an increasing rate of flow of time for the observer at the centre. If we time the velocity of light with any other type of clock, it shows a constant velocity. The velocity of light has increased, but the increase cannot be detected. The speed of light, as measured, will be constant for any co-moving observer.

The light cone in our model Universe

The Schwarzschild spacetime is similar to the Minkowski spacetime with the effect of the presence of a single mass, M. Factors appear in front of the terms in dt and dr.

The Schwarzschild factors introduce a curvature to the time coordinates, dt, and the radial coordinates, dr.

This is a better depiction of our model Universe than the standard “‘light cone” shown above.

We saw above, in our model, that this curvature causes a time difference between the central co-moving observer, and other co-moving points some distance away. These points appear at an earlier time than the central co-moving observer. This time difference is interpreted as the time it would take light to cover this distance.

Emission and absorption are simultaneous events

In fact, in our proposed model, the time of emission is at the same moment as the time of absorption. These two events are simultaneous, from the point of view of the central co-moving observer, even though they record different times. We see now that the time difference between emission and absorption is entirely due to the curvature of space and time in our model.

The proposed modified Schwarzschild metric defines the “shape” of space and time. The modified Schwarzschild metric tells us that points stationary in frames moving with a relative velocity greater than the speed of light can’t actually exist because we can’t have an imaginary value for ds in the metric.

This implies that there cannot be pure lengths in the modified Schwarzschild metric spacetime. If the modified Schwarzschild metric is a good representation of our actual Universe, we cannot have such a thing as a pure length in our Universe. When we measure a “length” in practice, we must include a measurement of time.

What is the present moment in this model spacetime with a modified Schwarzschild metric? It cannot be a horizontal line through the crossover point on the spacetime diagram. Such a horizontal line would have only one real point; the crossover point. All other points on this line would have pure lengths measured from the central co-moving observer at the crossover point, and so they cannot exist in this spacetime.

In the model Universe being considered here there are only points further in the past than the central co-moving observer.

There is only one observer at the present moment

Consider a central co-moving observer at a point ( x₁, y₁, z_1, t₁) and another central co-moving observer a point ( x₂, y₂, z_2, t₂) and further stipulate that dt = t₂ – t₁ = 0 . In that case we must also have x₂ – x₁ = y₂ – y₁ = z₂ – z₁ = 0. Since we can’t have two observers at exactly the same place and time, we must conclude that only one central co-moving observer can exist at any one time.

This implies that there is only one observer in the modified Schwarzschild spacetime. If we take a snapshot of the Universe, we will find only one observer in it.

How many central co-moving observers do there need to be for this model to fully describe the Universe up to the present moment? The answer would seem to be; just one.

This central observer sees a universe surrounding him or her with points further away having time passing more slowly. Although less time has passed for points further away, all the points that the central observer sees are in that observers present moment. When we built this Universe, it was all in the central observers present; a moment in his or her stream of time.

The central observer of this universe is where time is flowing fastest. It is where time has progressed the furthest. At the edge of this universe time is passing infinitely slowly. The closer to the edge we look, the further back in time we are seeing.

There is no future for this central observer. He or she is progressing into the future as fast as his or her present is becoming the future for the central observer to exist in.

All other co-moving points are at the central point of their Universe, but at an earlier time from the point of view of the central observer. They are, like all co-moving observers, in their present moment, and at the furthest point time has reached for them.

So we can see that all co-moving points exist on the past light cone. This is the present moment of the central observer. That same moment, in the view of each co-moving observer, is their own present moment since they are the central observers in their Universe.

What do we mean by “the speed of light”

The past light cone is the term used for the part of a space time diagram that corresponds to points that could be connected by signals travelling at “the speed of light”. What do we mean by “the speed of light” in this Universe described by the modified Schwarzschild metric?

Consider an observer watching their Universe where a photon is emitted by an electron, and absorbed by another electron where another observer is watching their Universe.

In our model the observer where the electron is absorbed is a co-moving central observer. When they look out towards the place where the photon has apparently come from, they see a point where time is progressing at a slower rate than it is for them. This point appears to be an earlier time, and they interpret the time difference as the time it has taken light to reach them.

But we have seen that the co-moving observer where the photon has been emitted is actually in the same present moment as the central observer.

Time has not proceeded beyond the time of emission at the place of emission, and it has not proceeded beyond the time of absorption at the place of absorption, because both these observers are co-moving central observers at the apex of their light cone.

This means that these events are simultaneous. The appearance of speed is because a different total amount of time has passed for the emitting electron, and the absorbing electron. This difference in time does not imply that the events cannot be simultaneous. It just means that time has been progressing more slowly at the point of emission than at the point of absorption.

The present moment is not a set of points in space time with the same time coordinate, as Newton must have thought. We can now appreciate that the past light cone is describing all points at the same moment of time. The past light cone is the set of all points at the present moment of the central observer.

The speed of light defines the shape of space and time

We can now see that the speed of light, c, in the modified Schwarzschild metric, is a constant that defines the shape of space and time. It doesn’t mean that light signals travel at this speed. In fact it is the shape of space and time that results in a light signal appearing to travel with a speed c.

Since all points on the past light cone are at the same moment in time, a photon exists on the past light cone for an instant only; it will have a zero duration in time.

The point where it is emitted is at the tip of a past light cone. Where is it absorbed?

The Quantum Mechanical description of a photon emitted by an electron has its wave function giving the probability of delivering its energy and momentum at an absorbing particle in a possible future.

This wave function can be identified with the electromagnetic wave predicted by Faraday and Maxwell. This describes a spherical wavefront of varying electric and magnetic fields spreading out in all directions.

A spherical wave can be Fourier transformed into an infinite number of plane waves, travelling outwards from a source in all directions. We can imagine that Maxwell’s spherical light wave is actually composed of an infinite number of plane waves.

We will use this idea in “Quantum mechanics” below where we see that these plane waves can transfer momentum, and energy in the manner required by Quantum Mechanics.

If the moment of emission is the same as the moment of absorption, we can envisage an infinite number of emission and absorption events corresponding to every possible momentum direction the photon could be emitted in. Each one will be on the apex of a new light cone in a slightly different Universe. Readers may recognise once again the “many worlds” interpretation of Quantum Mechanics, as suggested by Hugh Everett 3rd⁶.

The Nature of Time

What do we mean by time? If Einstein is right, and time is literally what a clock shows, and not, by implication, something that exists continuously in its own right, then it is just the changes we see. If nothing changes, if no clocks tick, then no time is passing. So what causes change? What, in particular, is the smallest amount of change there can be?

A photon is the smallest tick of a clock

If we call the smallest amount of change there can be an instant of time, then we can view time as a succession of these instants. In the proposed model, a single photon is the smallest tick of a clock there can be. It is instantaneous, and so cannot be deconstructed into smaller events, nor can it be measured against a continuous time flow. In fact, the existence of instantaneous events implies that there cannot be a flow of time in the way that Newton imagined it.

All events in our model, like the ticking of a mechanical clock, will be sequences of these fundamental photon “ticks”

Actually we can suppose that there will also be gravitons, apparently traveling at the speed of light, but actually interacting instantaneously to provide the change of momentum we perceive as forces between masses, and gluons providing the strong nuclear force, also apparently traveling at the speed of light, but actually instantaneous. All these interactions will exist, one at a time, on the past light cone.

The sum of these changes would be time. There wouldn’t be a continuous, smoothly flowing “stuff” against which we notice events occurring, as Newton imagined. Time, as something separate from events, wouldn’t exist. There would only be events.

So, if we say time is flowing faster in one region of space compared to another, we would actually mean that we counted more events in the first region compared to the second. If we counted all the events involved in two ticks of a clock, and we notice the first half of these events, or one tick, happening somewhere else, while they all happened where we are, we would say that time was flowing at half the rate there than it was where we are. In fact we count the ticks, but each tick is just a number of events in a sequence, so by counting ticks of a clock, we are actually counting events.

Only one photon can exist in each moment

Each event, or instant of time, would involve only one photon. Any past light cone of a central co-moving observer would contain only one photon. A single moment of time would consist of two almost identical Universes, and a light cone containing one photon’s worth of energy and momentum.

The standard picture of space-time with time like and space like regions; and showing a future light cone and a past light cone, would not apply to this Universe. There would be no space like regions, and no future light cone. Only the past light cone would exist, with a co-moving observer at the tip of the cone.

The present moment would not be a “flat plane” in the three dimensions of space, and the same point in time. The present moment would be the “surface” of the past light cone. This means that what we perceived as the velocity of light would be entirely due to the shape of space time. Light itself would be instantaneous.

This also means that, at any one moment, only one photon can exist. There is only one photon in the entire universe at any instant of time.

Quantum mechanics

One photon transfers an amount of energy, E, and momentum, p. So, in the proposed model, there will exist two, almost identical, universes that differ from each other by that amount of energy, and that change of momentum. Of course the photon might have any frequency and wavelength corresponding to the velocity of light, c, and therefore any corresponding energy, E, and momentum, p, such that

c=fλ

Where f is the frequency and λ is the wavelength

E=hf

p=\frac{h}{λ}

Where h is Planck’s constant.

In the universe we are modelling, this event, consisting of the exchange of energy and momentum by one photon, is instantaneous. There is no time for it to have a duration in. Another photon exchange would have to happen either before, or after it.

The only solution to Schrodinger’s wave equation that describes a particle that has a momentum with definite magnitude and direction is a plane wave.

A photon is a plane wave

Plane waves satisfying Schrodinger’s equation have the form

Ψ(r,t)=Ae^\frac{2πi(p.r-Et)}{h}

Here p and r are vector quantities, r is the radial distance from a point, the momentum, p, is perpendicular to the wave front, and p.r is the vector dot product. E is the energy, and h is Plancks constant.

Emission and absorption modeled by plane waves

The diagram below on “Electron photon interference” shows how the interactions as an electron emits a photon, which is then absorbed by another electron, can be modeled by plane waves.

The triangle formed by the wave fronts, a, b, and c below, has altitudes λ_a, λ_b, and λ_c. It is a standard geometrical result from Heron’s formula that

a=\frac{\frac{1}{2}((a+b+c)(a+b-c)(c+a-b)(b+c-a))^\frac{1}{2}}{λ_{a}}

The quantity

\frac{1}{2}((a+b+c)(a+b-c)(c+a-b)(b+c-a))^\frac{1}{2}

is constant for any particular triangle, so we can write

a=\frac{k}{λ_{a}}

and

b=\frac{k}{λ_{b}}

c=\frac{k}{λ_{c}}

where

k=\frac{1}{2}((a+b+c)(a+b-c)(c+a-b)(b+c-a))^\frac{1}{2}

The triangle with sides a, b, and c, will be similar to all triangles with sides k/λ_a k/λ_b k/λ_c where k is any constant.

So the triangle, with sides a, b, and c, is similar to the triangle with sides h/λ_a h/λ_b h/λ_c where h is Planck’s constant.

We know that momentum p_a p_b and p_c is related to wavelength by

p_{a}=\frac{h}{λ_{a}}

p_{b}=\frac{h}{λ_{b}}

p_{c}=\frac{h}{λ_{c}}

The vector addition triangle and the wave interference triangle are similar

The vector addition triangle, and the wave interference triangle, are similar. This means that the directions formed by the interference of wave fronts are identical to the directions obtained by the vector addition of momentum

This suggests that the vector addition of momentum is a consequence of wave interference. It suggests that the assumption that we can model the two electrons, and the photon, as plane waves, or pure momentum states, and that the transfer of momentum is by interference between these waves, is justified.

If this is how momentum is transferred then the collapse of the wavefunction doesn’t happen. The electron will be at the apex of the light cone and this will determine the time and place of the event.

We are supposing that the photon’s plane wave exists for an instant, with both emission and absorption are events occurring at the apex of the light cone for their respective electrons. Emission and absorption are simultaneous events.

We have seen that events on the light cone, in our proposed model, are simultaneous from the point of view of a co-moving observer at the apex of the light cone. When viewed by an observer where the photon is absorbed, both emission, and absorption, are events occurring on the light cone, so emission and absorption are simultaneous events for that observer. This suggests that a photon can be thought of as a plane wave that exists for an instant of time.

As was suggested above, an interesting consequence of this idea is that only one photon can exist at any one moment of time in the Universe this model is describing. There is only one photon in the Universe we observe.

Paul Dirac in his book Quantum Mechanics states³:

“Each photon then interferes only with itself. Interference between different photons never occurs.”

Richard Feynman describes the reflection of a photon in detail in his Auckland lectures². He points out that a photon is not reflected from an electron; rather it is absorbed and readmitted. A photon that exists as a plane wave cannot change direction without interacting with an electron since a change of direction implies a momentum with a new direction, and a consequent equal, and opposite, change of momentum for the electron it interacts with. We don’t know, however, which direction the emitted photons momentum will be in.

In the Auckland lectures² Feynman refers to the wave function as “going every way it can”. We are here envisaging the emission of a photon as an infinite number of plane waves, going every way they can from the emitting electron.

A photon is emitted into an infinity of different future Universes

We can make sense of this if we imagine a photon emitted into an infinity of different future Universes.

We can imagine an infinite collection of these Universes, each differing from the next one by one photon’s worth of energy and momentum.

As an infinite number of plane waves are emitted in all directions, instantaneous reflection or refraction allows interference. When the energy and momentum is transferred to an electron, a new Universe is created differing by that amount of energy and momentum from the universe of the emitting electron.

Readers will recognise the “many worlds” interpretation of Quantum Mechanics put forward by Hugh Everett 3rd⁶. Each way a photon goes creates an infinite number of new Universes, and each each of these Universes differs by the direction in which the plane wave of the photon has travelled.

From the point in space and time at which a photon is emitted, there will be an infinite number of possible absorption events at an absorbing electron. Each absorbing electron existing in a slightly different future, with each of these futures differing from each other by the absorption of one photon.

The emission and absorption of a photon is instantaneous.

Since this event cannot be broken down into a sequence of smaller events, it cannot take any time. The emission and absorption of a photon is instantaneous.

Instantaneous reflection from an electron, or other charged particle, for instance by a diffraction grating, will allow interference between this wave and itself.

The implication is that until an absorption event occurs between one of these waves and a charged particle, the transfer of energy and momentum has not yet occurred. As each wave interacts, an infinite number of new universes appear with the emitting and absorbing particles having equal and opposite momentum. Even if there are many light-years, and apparently as many years of time, between emission and absorption, the transmission of momentum and energy is actually instantaneous.

Conclusions

One new idea has been introduced. Mass is now supposed to be a function of the potential energy a body has, as well as its velocity. Massive bodies, moving in free fall in a gravitational field, will have their mass stay constant. Mass increase due to increasing velocity is exactly compensated by mass decrease as the body falls in the gravitational field. This is the reason for the equivalence of inertial and gravitational mass. We can identify this with moving on a “geodesic”.

With this idea, and using Birkoff’s theorems¹ together with the Schwarzschild spacetime, we have constructed a model of the Universe.

This model is spherical in shape. It looks the same from all co-moving points. It looks the same at all places, in all directions, and at all times, so it satisfies the perfect cosmological principle. This states that “’Viewed on a sufficiently large scale, the properties of the Universe are the same for all observers.” or, “The Universe is homogeneous, and isotropic, in both space and time”.

We have introduced a further conjecture that this Universe is the same on all scales when viewed on a sufficiently large breadth of scale.

This Universe is a finite sphere containing an infinite amount of matter. Its mass, and it’s radius, are increasing exponentially in time. The rate of flow of time is, itself, increasing exponentially. This Universe’s density is decreasing as its constituent matter moves apart in a cosmological expansion, just as we observe our Universe doing.

Electric charge is also increasing in time. This is shown using Maxwell’s equations together with the model of the Universe we are presenting here. Corresponding equations of gravito-magnetism are suggested to show the increase of mass expected in this model.

The speed of light in this Universe is increasing exponentially also, but all observers will see it as constant because the rate of flow of time is exponentially increasing.

The length contractions, and time dilations, required by the Lorentz transformations, are due to the distribution of mass as observed by a central co-moving observer.

The changes in length and time required for all co-moving observers to see themselves as being at the center of their own Universe, with a speed of light the same in all directions, can be attributed to their relative position in the universe of a central co-moving observer.

The Schwarzschild spacetime is shown to be equivalent to the Minkowski spacetime for all co-moving points. Each co-moving observer sees himself or herself as being stationary at the centre of their own reference frame in their own Schwarzschild spacetime.

Observers moving relative to the expansion have their mass increase with their increasing relative velocity, and this corresponds to a mass increase due to them lifting themselves in the gravitational field of the Universe.

It is shown why inertial mass is equivalent to gravitational mass.

We can envisage this Universe as beginning as an infinitely small, infinitely dense sphere of infinitely small mass, and expanding exponentially towards a state of zero density, and infinite mass and radius.

At any time in the history of this Universe all co-moving observers see themselves in the same situation. They are at the apex of the past light cone, and moving into the future in every possible way they can. All these ways correspond to Hugh Everett 3rd’s “many worlds” theory of Quantum Mechanics ⁶.

Light is actually instantaneous, and the apparent velocity of light is due entirely to the structure of time and space.

There is a deeper message that hopefully the reader has become aware of. The picture of the Universe presented here is backed by solid Physics over one hundred years old. This physics has developed since Galileo’s great insights, and experimental work. It was Galileo who spotted the key idea that led, first Newton, and then Einstein, Minkowski, and Schwarzschild to their formulations of the laws of gravity and spacetime. This idea was the equivalence of inertial and gravitational mass.

Galileo saw that massive bodies fell with the same acceleration in the Earth’s gravitational field. For him to recognise the significance of this observation four hundred years ago, and well before the development of the mathematical framework we use today, is astounding. Einstein is the physicist who is most celebrated today, but it is Galileo who really deserves to be in my opinion.

Galileo provoked the ire of the Roman Catholic Church because he was suggesting that the the Earth was not the center of the Universe. This was primarily caused by the observation of the moons of Jupiter, so the story goes.

There was a far deeper rift that Galileo started, however. He, and other renaissance scientists, started the separation between Science and the Church. This has become established as the separation between the material and the spiritual worlds.

This separation seems to have become accepted to allow both groups to live in peace with each other. Scientists are no longer kept under house arrest, as Galileo was, or threatened with being burnt at the stake, for suggesting alternatives to the views of the church.

We can imagine that before this, priests, prophets, and other holy men, were one and the same as the scientists who sought to understand the world. They were the people who were searching for the truth.

In the east, in Asia and India, this search was an investigation into the nature of conscious being. Consciousness was central to the ideas developed in the Bhagavad Gita of the Hindu beliefs, and the teachings of the Buddha.

In the West, the division between Science and the Church has resulted in a strange rejection by scientists of anything to do with consciousness. The notion of consciousness has been relegated to an emergent property of central nervous systems. It is something that exists inside our brains. It is not investigated with anything like the determination that “hard” science is.

Rupert Sheldrake, a scientist with a refreshingly independent mind, has given us the following illustration of the absurdity of this viewpoint. He says that the conventional view of perception has light reflected off objects in our inanimate Universe. This light enters our eyes through a lens that forms an image of the reality outside on the retina of the eye. This image is then converted into electrical signals in our optic nerves by special cells in the retina. These signals travel to the region of the brain where they are processed, but then what happens to them?

Rupert suggests that, in this conventional view, these signals are then somehow converted back into an image, that is presented to our consciousness, and which is also somewhere in the brain. All we can actually experience is this image. All our sense perceptions are just perceptions; they are representations of reality, and not reality itself.

Why then, do most of us believe that we are directly experiencing reality?

With the view that consciousness is a non-essential property of biological evolution, it is hard to explain what it is for. Why do organisms of a certain level of complexity need to be conscious? What survival benefits does it confer?

To accurately describe a mechanical world obeying the laws and rules of physics, it has been imagined that this description is possible without any reference to conscious awareness. This present attempt to understand the Minkowski spacetime, and how our Universe can possess the same structure, has come from this purely mechanical set of equations.

We have seen that these impersonal equations have necessitated the role of an observer in a central position of the description of the Universe. The structure of time and space is completely dependent on this observer. This observer is at a defined position in space and time, but it is not a unique position. All points of view are seen as equivalent. The Universe that has been described above is truly universal. there is nothing special about any particular moment as it is being observered.

We can identify this observer as consciousness. The question was proposed above as to whether there is only one observer seeing just one moment, or an infinite number of observers seeing all possible moments.

The “Atman” of the Bhagavad Gita, the “Father within me” that Jesus spoke of, and the “Buddha nature” referred to by the Buddhists, are all referencing a self that is beyond the individual self. It is a “true self” within us. There is a sense in all these teachings that that there is just one “true self”, and that it is our true reality. It is our perception that we are a separate ego self that is an illusion.

The view of spacetime developed here, as a modified Schwarzschild metric, has been shown to have the same structure as the Minkowski metric. We have seen that it is not possible to force a great many observers, each in their own Minkowski metric, into a Euclidian space, and Newtonian time. This would be, in effect, denying the essential property of a Riemannian spacetime metric that it is a whole, entire, and complete description of all the possible points in that spacetime metric.

We cannot put one Riemannian metric inside other. We cannot imagine each observer, at the apex of their light cone, in their own Minkowski metric spacetime in an infinitely small volume of space and time, and imagine that there are an infinite number of such observers inside a Euclidian space and Newtonian time. Neither can we imagine these Minkowski metric spacetimes in infinitely small volumes inside a Schwarzschild spacetime when they are close to a mass.

It appears instead that there are an infinite number of what we are calling “modified Schwarzschild metrics”, each with an observer in the present moment, but that these are not the same spacetimes. Instead each observer in a present moment is about to move into a new spacetime.

Each observer sees the whole of the past light cone, with that moment at its apex. That past lightcone is described in its entirety by a modified Schwarzschild metric. At that present moment there is nothing else.

That present moment, at the apex of the light cone, is where light has just arrived at an absorbing electron, and changed its momentum and energy. This is the smallest change that there can be. It is the smallest tick of a clock. It creates a whole new spacetime with the smallest possible difference to the one that has just been.

It arrives from just one direction after having been emitted in all directions. Each different direction it could have gone creates a new spacetime moment. Each one has the possibility of an observer at the apex of the light cone it arrives at. That observer sees a moment of spacetime, as described by the modified Schwarzschild metric. That observer sees a history particular to the present moment it is in. In this way all possibilities are realised in the way first described by Hugh Everett 3rd in his “many worlds” proposal.

We have to abandon the idea that there is a spacetime in which massive bodies move through space, and over time, realising just one future outcome, with probability waves predicting their paths, in a Universe that operates with, or without, a conscious observer. There is another possibility. That is that there is just one one Atman, or Father that dwells within, or Buddha nature, in the process of seeing all possible moments, one moment at a time.

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