The Structure of Time and Space

1. Overview

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.

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

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. No physical explanation of what causes the necessary space and time transformations in our Universe, that are implied by the Lorentz transformations, has yet been put forward.

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

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 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, 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.

Maxwell’s equations didn’t specify any velocities for the emission and reception of a beam of light. According to the equations, the speed of light seemed to be a fixed constant. 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.

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.

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 geomentry when time was included.

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 Section 2 and Section 8 below.

Minkowski’s formula, referred to here as the Minkowski metric, or Minkowski spacetime, describing this interval is a particular case of a generalised Rimannian differential geometry. It has some surprising properties.

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

The Lorentz equations showed that length contracted in the direction of a moving observer’s motion relative to another, stationary, observer, and time dilated, (slowed down). As the speed of light was approached, space contracted to zero, and time dilated to infinity.

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.

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.

This leads to a famous problem known as the “Twins Paradox”. Each of the two observers will see themselves as the stationary observer, and the other as the moving observer, with slower time and a slower speed of light. Which twin gets older than the other?

The model of spacetime proposed in this account resolves this paradox.

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. See Section 17, The Light Cone.

It might be uncomfortable for some physicists 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.

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, in the Minkowski spacetime, appears to travel at a fixed speed 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 arriving at an observer, 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 carefully 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. See also Section 17 below for more on the light cone.

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.

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.

In fact, no explanation, of why this necessary length contraction, and time dilation, occurs in an empty Minkowski metric space, has been suggested either. 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.

We do know, however, that Einstein’s General Relativity predicts that mass density 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. Yes, contrary to popular opinion, this can be done. 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. It may be a surprise to some students to learn that this has not been done before now.

No one has yet demonstrated why the quantity we refer to as “mass” is the same thing, within the limits of measurement, when it is used to measure the resistance to acceleration, referred to as “inertial mass”, and the source of the gravitational field, known as “gravitational mass”. These two concepts are quite different, but the same quantity, “mass”, is used to quantify both of them.

It is generally accepted that Einstein’s theory of General Relativity requires this equivalence, but it is an assumption of the theory; it is not derived by the theory.

It will be seen below why inertial and gravitational mass are equivalent.

A space containing a single massive body is considered first in section 2. We imagine a small, insignificantly massive, test body in free fall, and moving with escape velocity with respect to the original mass. This is modeled using the Schwarzschild spacetime. An introduction to differential geomentry is given to demonstrate how the Schwarzschild spacetime metric describes the shape of space and time round a mass.

It is shown that there are length and time variations, for different observers in the Schwarzschild spacetime. Depending on how close to the central mass in a Schwarzschild spacetime an observer is, their lengths will contract in the radial direction, and their times will dilate, relative to an observer an infinite distance away. At a certain distance, close enough to the mass, space contracts to zero, and time dilates to infinity. This is what is known as the event horizon of a black hole.

Next, in section’s 3, 4, 5, and 6, a model Universe is obtained, using Schwarzschild spacetime, together with Birkoff’s theorems¹ , that has a uniform density, finite volume, and finite total mass, at any particular time of observation. This model has an infinite amount of matter within a finite spherical volume. Its mass and volume are increasing, and its density is decreasing, as the Universe evolves in time.

A definition of potential energy in the Schwarzschild spacetime is proposed. The decrease of potential energy of a massive body, as it free falls in a in a gravitational field, is equated to a decrease in the body’s mass. This is shown to match the increase in kinetic energy as the body falls at escape velocity in the gravitational field. It is shown that all the rest mass of a body, in a gravitational field, can be attributed to it’s potential energy.

We see that a body, moving on a geodesic, has its total mass remaining constant as it exchanges kinetic energy and potential energy. As a consequence a body moving on a geodesic is not exchanging energy with its surroundings as observed.

In section 7 we see that this model obeys Hubble’s law. It appears the same for all observers co-moving with the expansion. All these observers see themselves at the centre of a spherical Universe. They all see the constituent matter of this Universe moving away from them, with a velocity proportional to their distance, up to the velocity of light.

As with the single mass described by Schwarzschild’s spacetime, the model has all points moving with escape velocity having the same length and time variations as those specified by the Lorentz factors for frames with a relative velocity to the observer. It is seen that these length and time variations may be equally attributed to the mass distribution of matter in the model.

In section’s 8, and 9, the Schwarzschild spacetime, and the Minkowski spacetime together with the Lorentz transformations, are used to show that every observer co-moving with the expansion will see itself at the centre of a spherical expanding Universe with a uniform speed of light.

In sections 10, 11, and 12 some of the implications arising from the model regarding mass, and the models evolution in time, are explored. The kinetic energy of bodies moving with the overall expansion is quantified, and found to be equal to their potential energy with respect to the mass of the entire Universe. This is identified with a massive body’s rest mass. A bodies rest mass is found to be increasing in time as the Universe expands, and the mass of the Universe as a whole increases in time.

In section 13, Maxwell’s electromagnetism equations are used to show that charge is increasing, along with mass, in the proposed model. Corresponding equations of gravito-magnetism are suggested in section 14. These equations are seen to fit the proposed model.

In section 15, mass increase with velocity is identified with the increase in potential energy of a moving mass with respect to the mass of the Universe as a whole.

The reason for the equivalence of inertial mass and gravitational mass is demonstrated.

In sections 16, 17, and 18, the part played by time in the model is explored, and how this affects the measurement of the speed of light by all observers.

The perfect cosmological principle that the Universe be equivalent at all points in space and time will be seen to hold.

Aspects of this idea that have relevance to Quantum Mechanics will also be explored in section 19. The theories of Quantum Mechanics, and General Relativity, are reconciled. The model proposed has implications for the perennial problem known as “the collapse of the wave function”. The model contains a defined moment of here and now.

This moment of here and now is where General Relativity ends, and Quantum Mechanics takes over. The model developed here shows how space and time are structured around the point of view of an observer. We will see that all observers, in the spacetime described, will see the same structure from a different moment.

At this point, identified with a central co-moving observer, time has proceeded further than any other point. Time, as described by the model, doesn’t continue forever into the future. Instead we see that it is being created at that observer’s present moment. For that observer the future doesn’t yet exist.

We can identify the present moment as the furthest point time has reached. Prior to this time the constituent particles of the universe have a definite position in space and time. Rather than supposing that the wave function “collapses”, it is suggested that all future possibilities are actually realised in the manner first proposed by Hugh Everett 3rd in his “many worlds” interpretation of this aspect of Quantum Mechanics.

Possible futures are described by the evolving wave function, as described by Erwin Schroedinger and Paul Dirac, given the constraints imposed by the present distribution of matter and energy in the Universe. The particular example of a photon emitted by an electron in all possible directions is used to demonstrate how an infinite number of slightly different Universes will result. The photon’s wavefunction spreads out in a sphere, but its momentum and energy are always received at a particular point in spacetime.

An interesting and important consequence is that the Universe, as this model describes it, must be observed. Both the General Relativistic Schwarzschild model proposed, and the theory of Quantum Mechanics, require an observation in their descriptions of the physical world.

This in turn suggests that consciousness is a fundamental part of Physics. It seems that there must be at least one consciousness, and perhaps only one, for a full description of reality. That consciousness is an intrinsic part of every moment. It seems that it is in the process of experiencing every moment there can be, one moment at a time.

Where a new, unusual, or unconventional, equation is used, it is indicated in red, for example

\color{red}{m_{inertia}=m_{gravity}}

2. Differential geometry simplified

The Pythagoras metric

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. This idea can be extended to three coordinate axes of length. Hermann Minkowski further extends it to include time as a fourth component.

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. Here c is the hypotenuse of a triangle with sides a and b.

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 speed of light was a limiting velocity.

Later 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.

The theorem of Pythagoras may be used to define the distance between two points, (x,y) and (x+dx,y+dy) , in a plane.

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.

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}

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.

Three space dimensions

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

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 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

Rdθ, and Rsinθdϕ are infinitesimal orthoganol 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 as

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

Or equivalently as

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

Here c is the velocity of light.

All observers see the same velocity of light

This is the form spacetime must have if all observers are to see the same velocity of light. A fact determined by experiments like the Michaelson Morely experiment, and established as a necessary part of physics by Einstein’s theory of Special Relativity.

Here c²dt² has the dimensions of length, and three dimensional Pythagorean lengths are 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.

When ds =0 in 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}

So if

ds^2\geq0

then

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

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

The intervals, ds, make up the entire space time

It is the intervals, ds, that collectively make up the entire space time. Two points can only coexist in Minkowski space if they are separated by an interval in space and time that is greater than or equal to zero. 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.

A major consequence is that there can be no such thing as a pure length in Minkowski spacetime. The interval ds has units of length, and so it cannot have an imaginary value; imaginary lengths don’t exist. This implies that the quantity ds² cannot have a negative value, and so 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. The spacetime we live in must have this property.

There is only one observer in the present moment

A particularly disturbing, or perhaps enlightening, consequence, is that, for an observer at a point in the present moment of time, there can be no other points, or observers, at the same moment, and at a different place in space. “At the same moment” would imply dt equals zero, and an inspection of the Minkowski metric above will show that, if dt = 0, then so must the length (dx² +dx²+dx²)^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.

The light cone

The “light cone” refers to a graphical representation of Minkowski spacetime. The three space dimensions can be represented as a two dimensional horizontal plane, with the vertical axis used to represent 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.

This “cone” is actually all the points in three dimensional space, continuing to an infinite distance in the r direction, together with a single time coordinate attached to each one. Points in the Minkowski spacetime with the same space coordinates, and a different time coordinate will not be on the light cone. 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 curved space-time, and we should not base our understanding on it.

The Minkowski spacetime is not representing Euclidian 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.

This point at the apex may move into the future light cone, and can send a light signal to points on the future light cone, but it is not possible, in that particular Minkowski metric, for any material object to proceed through time from the past light cone to the future light cone.

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.

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 at the apex of the light cone that could be called the present moment. It is the place where an observer receives light that has been emitted from a different point on the light cone.

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. The emitting electron would have imaginary space time coordinates.

Points in space and time that can’t exist

There are no spacetime coordinates in the future of the emitting electron from the viewpoint of the first observer. There is no Minkowski spacetime ahead of the emitting electron for it to exist in. 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 could 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. The present moment did not feature in Newtonian physics.

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.

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

The collapse of the wavefunction on the light cone

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

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. The present moment, and all points on the “surface” of the light cone, have ds equal to zero.

It is not possible to make sense of this of terms of matter proceeding through time from the past to the future, along with us as observers. It would appear that the future “timelike” region of the Minkowski spacetime can have no matter in it because there is no spacetime in the “spacelike” regions for matter to exist in as it proceeds from the past to the future.

Only the point at the apex of the light cone has a point in its immediate future. If there are no points in the immediate future of points on the past light cone, the world line of a material particle cannot continue in the way that we imagine material objects proceed through time.

It appears instead that all particles are Quantum Mechanical probability waves before the “collapse of the wavefunction” associated with a measurement. This measurement can be the observer’s observation of the entire Universe on the past light cone. It seems that this “collapse of the wavefunction” has just happened on the past light cone as we make an observation of the entire Universe. Then 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 the Universe, 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.

The past light cone is 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.

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 it is 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. It is a complete description. There are no other points.

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 implication is that they cannot be in the same Minkowski metric. The first observer cannot have proceeded through a region of spacetime in the second observers Minkowski metric for which ds < 0, to arrive in the second observer’s future.

Nothing can get through the region of spacetime for which ds < 0 because there are no points of space and time ahead of it in time for it to exist in while it does so. 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.

No material object in a specific Minkowski spacetime 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. The light arrives as a photon of energy where we are. It still is where, and when, it was, when it was emitted, and where and when we see it. Emission and reception happen simultaneously.

The speed of light is actually zero. 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.

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. The length that is the value of ds that the Minkowski metric calculates cannot be negative because lengths cannot be negative.

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 ds 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 is all the spacetime points on the past light cone.

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, known as the Copenhagen interpretation of Quantum Mechanics, after being voted the most likely interpretation at a conference in Copenhagen, the future exists as a predetermined certainty, but we can only make probabalistic predictions about it.

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 Universe as if they haven’t.

An example of this is the popular use of the term “spacelike” to describe the regions where the interval in the Minkowski metric is imaginary. 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.

Each moment has a separate Minkowski spacetime

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.

It also appears that for each 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 each observer as they proceed forward in time. That future is different for each observer, and will be represented by a different spacetime metric. 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 ever existing. Our future does not yet exist.

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.

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 thepast 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.

It does appear that it is being created in the new light cone, as the observer at the apex moves forward in time by each infiitesimal amount, dt.

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

In the description of Pythgorean space 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 Pythgorean plane to be made up of all the possible points for which ds = 0, we could define an entire Minkowski metric as all possible light cones for which ds = 0. Such a definition must include an observer.

With this view, all observers would be at the apex of a light cone. For each of them all that exists, at that moment, is their past lightcone. All their past selves, and all the other observers on their light cone, together with their past selves, are a point where ds = 0, of an extended Minkowski spacetime.

What was considered to be Minkowski spacetime is actually a subset of this extended spacetime. We can view it as a single “point” of this extended Minowski spacetime.

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 space and 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.

This idea will be investigated further in Sections 8 to 19 of this account , and we will see how it can be applied to our Universe.

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. For more on Minkowski spacetime see Section 8 below.

Another big problem remained which means that Minkowski spacetime cannot be a good description of the space and time in our Universe. Minkowski spacetime is devoid of mass and energy. It only applies to inertial, or unaccelerated, reference frames. In the presence of mass, all frames are effectively accelerated. This problem led Einstein to develop his field equations of General Relativity.

The Schwarzschild spacetime metric

Einstein’s field equations specify the way the presence of mass and energy determine the shape of space and time. The Schwarzschild spacetime metric is a solution of Einstein’s equations for a curved space time geometry surrounding a single, spherically symmetric, mass distribution.

Outside a spherical mass distribution, in otherwise empty space and time, the Schwarzschild spacetime metric is 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

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 round a spherical body with mass, M.

As with the Minkowski metric, we still have the subtraction of space from time multiplied by the speed of light, 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 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.

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 factors

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.

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

Proper time

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 between points infinitely close in time, and stationary in the spacetime at r = ∞.

 dt_{(r=∞)} = dτ

Proper time 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.

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 imperceptable. 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. They can all claim that they weren’t protesting at all, they were just going about their usual routine.

Time dilation

The first observer remaining with their clock at infinity will measure a greater 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 tells us 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 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, is greater than the measured time dτ, 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 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 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, 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 will all be different amounts compared to the reference measurement, dt.

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.

If you were asked to choose which of two cats you would like to play with; Fluffykins, or Mauler the Mighty, 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 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.

It would be better 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 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 rate. 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.

An observer a great distance away from a mass might measure ten hours with their clock next to them, while an observer closer to the mass might measure one hour on their clock next to them. The clocks will show different amounts of elapsed time. They will both call their measurements “proper time”.

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 time dilation was not taken into account, the geo-positioning satellite system would not work. If Time dilation 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.

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 this experimentally proven truth. 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.

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.

There is no 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.

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 that given by the Schwarzschild metric, measured at an infinite distance. The definitive time is dt, not dτ. Calling dτ the “proper time”, and then using it to claim that it is the time taken to reach, and pass through, the event horizon of a black hole, is wishful thinking. It is wrong.

At the event horizon time is passing infinitely slowly. Infinitely slowly means stopped. To go through the event horizon would take time. If time stops, you can’t go through. Your proper time seems to pass normally, but, as you get close, every second, dτ, is actually taking thousands of years, 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

Everyone in academic institutions has got there by winning the approval of people who were already there. They didn’t get that approval by telling them that they were wrong. They got there by believing that those already in the institutions were the legitimate custodians of the truth. They wanted to join them, not challenge them.

They proved to them in exams and dissertations, that they knew, and agreed with, their truth. It is an arduous process taking many years. If anyone has any doubts about the system, or those in it, they are unlikely to make it.

Those that do make it have an entrenched interest in fitting in with the status quo. If they think, in the privacy of their own mind, that everyone around them has adopted a belief that is wrong, it still takes enormous courage to voice any doubts. They would need to be very sure of their own status, and the respect of those around them; as well as being sure of their own reasons for believing that they were right about something, and everyone else was wrong. There would be the real danger of being ostracised and not listened to, even if they were right.

Being the first person to be right about a new idea in physics is hard enough, but it is much harder to tell everyone else that they have got something wrong.

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.

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, 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. Eventually a small child, who didn’t know any better, cried out, “The Emperor has no clothes on!”.

There is another story about a small child who wondered what the world would look like if he was travelling along at the same speed as light.

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.

Length contraction

For an infinitesimal length, dr, at a distance, r, 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.

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, and 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, at all radial distances, r.

As with Minkowski spacetime, (see section 8), 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.

If dσ is the 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, then 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, given by

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

The Schwarzschild radius

The Schwarzschild radius, R, 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, so it would be equally true to say that for observers in the real Universe, the beginning of time is always infinitely long ago.

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 Section 12, The nature of mass.

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

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

The observer also contracts

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 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 with their slower clocks.

Time dilation, and length contraction both mean that the event horizon of a black hole cannot be reached. Space and time end at the event horizon. The event horizon itself is not in the space and time of our Universe. It is not something that exists in its own right. It is where space ends, and time begins.

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.

The actual length of a rigid body will change depending on where it is in the gravitational field of the Schwarzschild metric. Its measured length, as measured by an observer at an infinite distance, will change. Its measured length as measured by an observer stationary next to it will stay the same, however, wherever it is in the Schwarzschild metric. This is called its rest length, but that doesn’t mean that it is fixed and constant.

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. 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.

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, t. In practice it is extremely doubtful if it could be done 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 independant. 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 possbility, 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 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 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 edge of a piece of paper is not something that exists in its own right. It is where paper ends, and not paper begins. It is, nevertheless, quite real. If you don’t believe it, try continuing to write when your pen reaches it.

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

Moreover it is assumed that the observer with the metre rule can pass through the event horizon, and carry on to the “central singularity where the laws of physics break down“, whatever that means. 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.

How can the laws of physics break down? Do we set out to explain the Universe except for the parts that can’t be explained? Why bother? Once upon a time nothing was explained. We could have just left it at that. Instead of saying “The laws of physics break down”, perhaps we should try saying “This result means that we need to rethink everything.”

The problem is that everyone seems to be assuming that the observer at an infinite distance, measuring rest lengths next to themselves, can make measurements of a finite distance for the falling observer to reach the event horizon, using the rest length of the metre rule they have next to them.

This finite distance to the event horizon, measured with the uncontracted rest lengths of the observer at infinity, is to be travelled in the falling observers proper time that hasn’t been dilated. In effect this is ignoring time dilation, and length contraction.

It is arbitrarily selecting the rest lengths of the observer at infinity to measure the distance to the black hole, and arbitrarily selecting the proper time of the observer falling in towards it to mesure how long it takes.

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 rest length, and there is contracted length. There are spacelike regions and there are timelike regions. The fundamental understanding of what space and time are is left unaltered.

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 noticable at extreme velocities, and at extremely close distances to a mass. They can be ignored.

The story we can tell ourselves goes like this: In a small enough region of a Schwarzschild metric the curvature of space and time is not noticable; in that small region spacetime is 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. 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, and their detection in our Universe, forces us to reconsider this convenient story.

The distance to the event horizon is infinitely great

The truth is that the distance to the event horizon of a black hole is actually infinitely greater than the distance to a galaxy far beyond it. If that statement doesn’t make sense to you, then we need to think more carefully about length contraction and time dilation.

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 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.

There seems to be two different outcomes depending on whether the situation is observed from the observer at infinity, or 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 at infinity.

One outcome is that the observer falling in towards the event horizon gets frozen onto the horizon forever, and the other outcome is that the falling observer is totally unaffected by the event horizon, and goes on through it, getting stretched out by the tidal forces into spaghetti before disappearing into the central singularity.

Their solution? Both different, and incompatible outcomes, actually happen. https://www.youtube.com/watch?v=uzMUYpemgog

They suggest that one observer, watching from some distance, sees the infalling observer “freezing” on the event horizon where it is burnt up by Hawking radiation in slow motion. 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 Quantum Mechanics requires that no information about the infalling observers can be lost in the singularity. Instead it is supposed to appear as a sort of hologram of themselves back outside the event horizon. After sending this hologram back out in the form of Hawking radiation, they can disappear into the central singularity where their ultimate fate can’t be known.

Then Brian and Jeff suppose that everything outside the event horizon is this hologram.

The rabbit is pulled out of the hat, and it isn’t.

Confused? You should be, and so should Brian and Jeff. This doesn’t make sense.

One reason that so many physicists are so determined to imagine that it is possible to go through the event horizon is that 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 of physics, so that even protons and neutrons would collapse. No other force in physics can overcome gravity when it is strong enough. It is imagined that black holes grow larger as a consequence of matter falling into them. This is all imagined.

There is a major problem with the proposition that blackholes can be formed by gravitational collapse. If matter of any density is considered, and it is imagined that we have a spherical volume of this matter, we can calculate the size of sphere that it would need to be for it all to collapse into a black hole. The less dense the matter, the greater the sphere that would form a black hole. A sphere of water would form a black hole about the size of our solar system.

The matter doesn’t have to fall in. It will already be inside the event horizon of a black hole.

If our Universe has an overall uniform density, and it goes to infinity, there is a spherical volume in it that will form a black hole. All spherical volumes of this size should immediately form black holes. We can’t have a Universe, of any overall uniform density, that won’t immediately turn into black holes.

This density doesn’t have to be uniform on all scales. It could be made up of stars and galaxies. So long as sufficient mass is present, with overall spherical symmetry, within a specific volume, it will form a black hole. Moreover, all this mass would be inside the black hole. According to current popular ideas, all mass should be in the central singularities of black holes. If black holes can be formed by collapsing matter, there could not be a Universe as we see it.

There is another possibility, however, that is suggested here. That is that all mass is in the form of a Schwarzschild metric. It is suggested in Section 12 below that this could be the case.

If this is the case then small black holes can be imagined coalescing into bigger ones without event horizons being crossed. Two small black holes of mass, M, and radius, R, could become one bigger black hole of mass, 2M, and radius, 2R.

This account will show that we can use the Schwarzschild metric, together with Birkhof’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 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 like our Universe does.

In any event it remains true that it is not possible to reach the event horizon of a Schwarzschild black hole by moving through space and time. Those physicists who believe it is, need to think again about crossing the event horizon.

Let’s think about a slightly different thought experiment with the metre rules.

Imagine the observer at, r, with a great many metre rules. 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 near 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. The end furthest down the line of joined metre rules does not move down one metre when a metre rule is added at the top of the stack. The bottom of the stack moves down by the contracted length of the lowest metre rule. All the metre rules get a little bit shorter as they get closer to the event horizon.

The metre rule closest to the event horizon has contracted the most. If that metre rule is used to measure the remaining distance, more metre rules like itself will be needed than if the metre rule being added at the top of the stack was used to measure the distance with its uncontracted length.

The distance left to go will be greater, when measured by the lowest rule, than when measured by the top rule. As the lowest rule is pushed closer, the distance left to go that it measures gets ever greater. This distance will get infintly great as the length of the lowest metre rule tends to infinitely small.

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, the 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 Section 8 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 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.

These changes in length and time due to the relative velocity between observers are also quite real, contrary to popular opinion. The apparent symmetry between two observers in relative motion makes it tempting to suppose that these changes are only apparent, and not real. How do we decide which one of two observers is at rest, and which one is moving?

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. It does seem that a lot of physicists take care not to look at the problems that might lead them to their next insight.

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 Section 15 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.

We can now see that the event horizon of a black hole is the edge of the Universe. 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 an infinite amount of dilated time to fall in.

3. Applying the Schwarzschild spacetime to the Universe

The Universe is spherically symmetric, uncharged, and not rotating

To apply Schwarzschild spacetime to the Universe we need to assume that the Universe is spherically symmetric, and uncharged. We also assume that it is not rotating. 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 a 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, in Section 15.

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, (see also section 7 below), 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 co-moving observer as one with a relative velocity to another co-moving observer that is solely due to the expansion of the Universe.

The velocity of co-moving 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.

Length contraction and time dilation in our Universe

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 co-moving 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 just such 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.

Figure 1. Building a Universe

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 Sections 4 and 5 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. See section 8 for a summary of Minkowskian, or “flat”, spacetime.

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, and this will be seen to be justified in Section 6 below.

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 {8Gπr^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 Schwarzschild spacetime above. The radial coordinate, r, is now the radius of the sphere we are considering. Birkhoff’s theorems only apply to the empty spacetime outside a spherical mass distribution. We cannot investigate this spacetime by simply extending the radius, r, out to infinity as we can with the regular Schwarzschild spacetime.

If we make the radius of the sphere R, where R = 2GM/c² and M=4πρR³/3 then the Schwarzschild factor becomes

(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.

The modified Schwarzschild metric above only applies to the surface of a sphere with a radius, r. At this surface there will be empty spacetime as required by Birkhoff’s theorems. As it stands, however, the modified Schwarzschild spacetime above tells us nothing about the region beyond the surface of the sphere, or about the shape of spacetime within the sphere of matter.

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.

Adding shells of matter at the surface of a sphere

At the surface of the sphere, the radial coordinate 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 infintiely thin shell of thickness dr, and negligible mass, will not affect the spacetime at the surface of the sphere at r.

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. So at r = 0 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

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 Minkowsian 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

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 Section 2, 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 dr 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.

4. 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 .

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.

Motion on a geodesic

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=\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.

Why doesn’t the mass of a falling body increase?

So why doesn’t the mass of a falling body increase?

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}

Relativistic escape velocity

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, 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}

We can see that Hartle’s analysis gives the same result. 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 massive body with 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 equations above as m_rest.

The equation above is Einstein’s equation for total energy. It is telling is that the mass with the observer at r is greater than the rest mass by the factor in brackets, and this excess mass represents the kinetic energy necessary to lift the massive body to infinity.

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.

Massive bodies in free fall from infinity

We can turn this round, and imagine a test body that starts with rest mass, m_rest, at infinity. We assume 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 .

The observer at r would see the body falling past them with escape velocity, v_escape, directed down the r axis.

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, now states that the kinetic energy of the body is increasing as its velocity downwards increases.

As we have noted, the massive body is not exchanging energy with its surroundings as it falls, so its energy cannot increase. How then, can it gain kinetic energy, and therefore gain mass?

A straightforward interpretation of Hartle’s equation is implying that the law of conservation of mass and energy is being violated.

The grandfather clock rewound

If, instead of allowing the massive body to drop in free fall, we lower it slowly, we will be extracting energy from it. This is illustrated by the example of the grandfather clock.

The total energy of the weight dropping to power the clockwork, and therefore its mass, must get less by the amount of energy extracted.

Consequently we would expect a decrease from the rest mass of a massive body at infinity, m_rest, as the body is lowered slowly from infinity, due to the loss of potential energy. Its rest mass will be less than its rest mass at infinity, when it is at rest next to the observer at r.

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.

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^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

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 into 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}

5. Adding spherical shells of uniform density

Bringing matter in from infinity

Starting at a point in empty spacetime, we can imagine an infinitely small sphere, of negligible mass, with a density, ρ₀, and a radius, r=0 , together with an observer at the same point

Now consider increasing the size of the sphere by adding a shell of matter of the same density, and having a thickness, dσ, measured at the shell. We will imagine bringing this shell of matter in from infinity to the surface of the sphere.

Birkhoff’s theorem applies in the empty space outside a spherical shell of mass distribution, so we can apply it on the surface of our imaginary sphere.

The thickness of each shell is reduced

We can imagine adding successive shells, 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 Section 3.

Each shell will start with an infinitesimal thickness, dσ, and a negligible mass. This 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 this shell will make no difference to the local space time, as it increases 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, as described above.

The mass of each shell is reduced

We saw, in Section 4 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, as in Section 4 above, where m_rest-at-r is the mass of a body stationary at a distance, r.

\color{red} m_{rest-at-r} =(1- \frac{v^2}{{c} ^{2}})^{\frac{1}{2}}{m_{rest}}

For a sphere with a uniform density, ρ_, we can write this as

\color{red}{m_{rest-at-r} =  {(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}{m_{rest}}}

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 infnity

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 negligible rest mass m_n. Here we are 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 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

So

\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, made in section 3 above, is justified.

6. 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.

Figure 2. Successive shells of decreasing thickness

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, as we assumed in Section 3. 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, (see Section 7. below). 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 co-moving 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 co-moving 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 co-moving observer seeing itself as being at the centre of its Universe in its own spacetime metric .

The escape velocity 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}}

7. Hubble’s Law and the Lorentz factors

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. 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 co-moving 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}

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 co-moving matter in this model Universe is traveling at escape velocity, it will obey Hubble’s law.

Figure 3. The Universe and Hubble’s law

Consider a co-moving 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 co-moving 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}

Replacing M with 4ρπr³ ⁄ 3 as we did above gives the modified Schwarzschild spacetime for a point on the surface of a sphere of radius r.

\color{red}{ds^2=(1- \frac{8G\rho \pi r^2}{3{c} ^{2}})c^2dt^2-(1- \frac{8G\rho \pi r^2}{3{c} ^{2}})^{-1}dr^2
-r^2d\theta^2-r^2sin^2\theta d\phi^2}

We know that spherical shells of a greater radius than r will not affect this metric space, so this will apply to all points in this model Universe.

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

where

H=(\frac{8\pi G \rho_{critical}}{3})^\frac{1}{2}

We could then write the Schwarzschild spacetime as

\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^2d\theta^2-r^2sin^2\theta d\phi^2}

At the centre of the sphere, where r=0, and v_escape=0, this 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)}}

This is equivalent to the Minkowski spacetime.

Lorentz factors in our Universe

We can write

\color{red}{c^2dt_{(r=0)}^2=(1-\frac{v^2_{escape}}{c^2})c^2dt^2}

and we can write

\color{red}{dr^2_{(r=0)}=(1-\frac{v^2_{escape}}{c^2})^{-1}dr^2}

We can then write

\color{red}{dτ =  dt_{(r=0)}}

and

 \color{red}{dt=(1-\frac{v^2_{escape}}{c^2})^{-\frac{1}{2}}d\tau}

If a co-moving body is located on the surface of the sphere, and is moving away from the center at v_escape, this is the Lorentz time dilation for the body as seen from the centre of the sphere.

We can also write

\color{red}{dσ =  dr_{(r=0)}}

and

\color{red}{dr=(1-\frac{v^2_{escape}}{c^2})^{\frac{1}{2}}d\sigma}

which is the Lorentz length contraction for a body moving with relative velocity, v_escape, away from a central observer.

The co-moving matter in this Universe, and the space it is in, as seen from a central co-moving observer, will have its length contracted, and its time dilated. Moreover this length contraction, and time dilation, will be just what is required, in the curved space and time of this Schwarzschild metric, to allow an observer moving with the co-moving matter to measure the speed of light as constant in all directions relative to themselves.

We see that the edge of this Universe is moving away from a central co-moving 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. We can also now see that the Lorentz length contraction, and time dilation, necessary for co-moving observers to see the velocity of light as being the same in all directions, is due to the gravitational effect of the mass of this model Universe.

8. Moving frames in the Minkowski spacetime

The Lorentz transformations

We saw in Section 2 that the Minkowski spacetime metric defines a three dimensional space together with time. It is empty of mass and energy. This spacetime metric, in rectangular coordinates, is

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

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 the speed of light. We will see that all observers in such frames see themselves in the same type of Minkowski metric.

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, t,x,y,z to represent a stationary frame, and t^/,x^/,y^/,z^/, to represent coordinates of a frame in relative motion to the stationary 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 frame at rest, 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 rest 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 a stationary frame and a moving frame. This is a confirmed experimental result.

If the speed of light is the same in a moving frame, and a rest frame, 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

The wavefront of an expanding sphere of light in a frame moving at a velocity, v, with respect to a rest frame, and starting from the origin, can be shown 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

and

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

We see that an observer in the rest frame also sees a spherical expanding wavefront starting at the origin of the rest frame. The Lorentz transformation was designed to achieve this result.

The interval for a moving frame

In Section 2 above we saw that Minkowski had 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 rest frame coordinate intervals measured by an observer stationary in the rest frame.

The possibe 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 the Lorentz transformations above shows that

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

so

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

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

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 rest frame coordinates are viewed by an observer stationary in the rest frame.

The interval has units of length. If two points in the metric are separated in space and time so that a beam of light can travel between them, the interval, ds, will be zero.

The interval defines the extent of the Minkowski spacetime. We saw in Section 2 above that if we try to calculate an interval between points for which the separation in distance squared is greater than the separation in time squared multiplied by the speed of light squared, we get

ds^2<0

There are no real solutions to this equation. This means that there are no such pairs of points in Minkowski spacetime.

The speed of light cannot be exceeded

Another way to state this is that the speed of light cannot be exceeded. It is an absolute speed limit. The structure of Minkowski spacetime is what causes this speed limit.

A consequence of all observers seeing a uniform speed of light is that the measurements of lengths, and durations of time, must be different for observers stationary in frames with a relative velocity to each other.

An observer stationary in the moving frame will measure a length 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, when measured by an observer and a clock stationary in the rest frame, will measure a dilated time duration, dt.

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 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?

Various explanations have been suggested, but none have gained universal acceptance. It is safe to say that this is because none of them are satisfactory. We will see below that the model Universe proposed does make it clear which twin is moving, and which is stationary. We will see that it depends on their respective velocities with respect to the Universe as a whole.

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 moving body contracts, by the Lorentz factor, in the direction it is moving relative to a 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 length contraction and time dilation from the point of view of an observer stationary in the rest frame.

Proper time and rest length

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 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

dx{^/}=dσ

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

dt{^/}=dτ

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

Time dilation in our Universe

This has implications for all comoving observers in an expanding universe. If our expanding Universe is to be represented by the equivalent of the Minkowski metric, all co-moving points will have time passing more slowly than for 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.

We have seen that in a Minkowski metric all observers will see themselves as being at the apex of their own past light cone. Now we can see that, in our expanding Universe, points on the light cone will be further back in time by the amount of time it would take light to travel to the central co-moving observer. That is as far as they have reached in time. We know this because they cannot have proceeded forward in time into the spacelike region.

Time is passing more slowly for points moving away from the central co-moving observer, and less time has passed in total by the amount of time it would take light to travel to the central co-moving observer.

This is not a radical departure from what Minkowski proposed, and Einstein accepted; it is a logical consequence. It hasn’t been generally recognised in over a century, but it might have been at any time since 1908 when Minkowski proposed his spacetime metric.

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. Now we can see that it is the point where time has proceeded the furthest, with all other points in the Minkowski spacetime in its past.

In our expanding Universe, we can imagine light being emitted from a point, stationary in a moving frame, on the past light cone, with coordinates x, y, z, and t, 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.

The central co-moving observers sees points where where ds = 0 between themselves and the other points. It is the surface of the past light cone. It is the place and time we refer to as here and now. It contains the entire Universe we are presently aware of. It is the moment when the past is about to become the future. It is the moment when, and where, all light is received.

The speed of light

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. If an observer with a clock was approaching the speed of light on the same path as light on the light cone, the observer at the apex of the light cone would see that clock as showing time passing slower and slower as the speed of light was approached. In the limit, as the speed of light is approached, time slows to a stop.

The apparent speed of light is because we are thinking of it moving in Euclidian space, not Minkowski spacetime. The apparent speed of light is due to 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.

The complete set of intervals, for which ds = 0, between the observer, and another point, is the “surface” of the light cone. An observer sees all the other points on the light cone, but they are all in that observers past if light has arrived where they are at the apex of the light cone. With the definition of simultaneous above, all points on the past light cone for which ds = 0 are simultaneous.

The moving observer

The Minkowski metric for the four moving frame coordinates 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 . 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 spacetime in which there is an observer at the apex of a light cone. Points on, and in, that observers past light cone exist, but points in that observers future do not exist.

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 is the set of all points on the light cone with a single point, (x₁, y₁, z₁, t₁ ) at its apex. If a second point 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 point is in its own Minkowski metric. It will see itself at the apex of its own light cone.

The first observer at (x₁, y₁, z₁, t₁ ) is in the future of the observer at (x₂^/, y₂^/, z₂^/, t₂^/ ), but the observer at (x₂^/, y₂^/, z₂^/, t₂^/ ) is at the apex of its own light cone. At this point in time, 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 metric space. The second Minkowski space is contained within the first, and has the same calculated values for ds, but it is not the same Minkowski metric. It doesn’t contain the same set of points.

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.

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 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 electro magentic 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, with a possible observer, in their present moment, at the apex of a light cone in each one. 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 the light cone.

We saw above that the metric with moving frame coordinates has the same values for ds as the original metric expressed with stationary frame coordinates after the Lorentz transformation. The metric experienced by the observer stationary in the moving frame is contained within the original metric using stationary frame coordinates. All the points in the moving frames metric can be obtained by the Lorentz transformation on points in the original stationary frames metric.

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

The reverse is not true, however. The entire collection of points in the original spacetime metric is not contained within the entire collection of points in the moving frames metric. We saw this because the apex of the light cone in the stationary frame is ahead in time of all the other points on its light cone. Other points on its light cone were seen to be at the apex of their own light cone.

There will be points in the spacetime of the original observer in the stationary frame that do not exist in the spacetime of the observer stationary in the moving frame. 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.

The Lorentz transformation will transform stationary frame points to moving frame points. It cannot turn moving frame points back into stationary frame points. If the Lorentz transformation is performed on the moving frame points as if they are stationary, it can find a new set of points for a frame moving with the opposite velocity.

It finds a frame that is time dilated, and length contracted, compared to itself. That is not the original rest frame.

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.

We have seen above that the Lorentz transformation is not a two way transformation. The Minkowski metric that is created by applying the Lorentz transformations to a stationary Minkowski metric’s consituent length and time intervals is not the same as the Minkowski metric that uses the original length and time intervals.

Assuming that it is the same has been the source of much confusion that this account seeks to dispel.

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.

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 them 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. 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.

We can write

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

and

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

We can write

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

dx{^/}=dσ

dt{^/}=dτ

ds^2=c^2dτ^2-dσ^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 see all other lengths and times contracted and dilated according to that frame’s relative velocity.

Substituting the equations above we get, for all infinitesimal lengths and infinitesimal durations stationary in a moving frame, 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, equivalent to the one expressed in moving coordinates above, for the range of possible velocities, v, 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 stationary frame.

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 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 stationary frame, however, these moving frames will have length contraction, and time dilation, determined by their relative velocity to the original rest frame. When v = c, 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.

Observers at rest in all moving frames will see the same rest lengths, and proper times, for all bodies at rest with them in the moving frames. It is only from the point of view of the observer in the original rest frame that the moving frames will show contraction in the length coordinates, and dilation in the time coordinates.

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.

We will see in Section 9, and Section 15, below how this can happen in an expanding Universe full of matter.

Minkowski spacetime 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

Das, Mukul & Misra, Rampada ⁵ have shown that the Minkowski metric transforms 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.

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. If an observer with a clock was approaching the speed of light on the same path as light on the light cone, the observer at the apex of the light cone would see that clock as showing time passing slower and slower as the speed of light was approached. In the limit, as the speed of light is approached, time slows to a stop.

The apparent speed of light is because we are thinking of it moving in Euclidian space, not Minkowski spacetime. The apparent speed of light is due to 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.

The complete set of intervals, for which ds = 0, between the observer, and another point, is the “surface” of the light cone. An observer sees all the other points on the light cone, but they are all in that observers past if light has arrived where they are at the apex of the light cone. With the definition of simultaneous above, all points on the past light cone for which ds = 0 are simultaneous.

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.

9. The Minkowski spacetime and our model Universe

In our Universe, a co-moving observer will see themselves as being at the apex of their own light cone. The expansion of the universe means that this observer sees all other co-moving points moving away from itself. They will see themselves as the central co-moving observer in a rest frame, and they can view all other co-moving 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 co-moving observer’s past. All co-moving observers will see themselves as the central co-moving observer in a rest frame at the furthest point time has reached.

Everything that central co-moving observer sees will be on the “surface” of its past light cone, moving with escape velocity away from them.

We saw, in section 7, that a suitably modified Schwarzschild spacetime will describe a spherical Universe obeying Hubble’s law.

If the density is the critical density, and all co-moving parts of the Universe are moving away from a central observer at escape velocity, the Schwarzschild spacetime becomes

\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 can see that it has the same form as the Minkowski metric at the end of the last section, so it will give co-moving observers with all velocities, v_escape, a uniform speed of light.

It is important to note, however, that this equation was not developed from the Lorentz transformations. It has been developed from the Schwarzschild spacetime, and the theorems of David Birkhoff.

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. 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.

At the center of the 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 then 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}

We 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}

From the equations above, we see that an observer at a distance r in the Schwarzschild spacetime, moving with the magnitude of escape velocity, v_escape, along the r axis, in an infinitely small region of space and time at that distance , will have lengths contracted, and times dilated, as viewed by an observer at the centre.

We now consider a rest frame, in a Minkowski spacetime, written in Cartesian coordinates, and compare it with the Schwarzschild spacetime of our model Universe. This is shown in the diagram at the beginning of this section. We consider that is stationary with respect to the centre of the Schwarzschild spacetime of the model Universe.

We align the x axis of a moving frame in the Minkowski spacetime with the r axis of the Schwarzschild spacetime, so that dx and dx’ in the Minkowski spacetime are in the same direction as dr in the Schwarzschild spacetime. We make the the relative velocity of the moving frame, v, to the rest frame, the the same as the velocity, v_escape , at the distance r in the Schwarzschild spacetime of our model Universe.

This is a comparision only. The Minkowski spacetime we are imagining is not the space time of our model Universe. Minkowski spacetime has no mass in it, so we can’t apply it to our Universe, (except in an infinitely small volume that is accelerating in such a way as to cancel the gravitational acceleration at that point).

We know that, in a Minkowski spacetime, a frame moving with velocity, v_escape , with respect to a rest frame with a uniform speed of light, will have rest lengths, dσ, contracted, and proper time intervals, dτ, dilated, by the Lorentz factor, when observed by an observer stationary in the rest frame. An observer stationary in this moving frame will also see themselves in a Minkowski spacetime, and they will measure a uniform speed of light. These are the hypothetical physical results of using the Lorentz transformation. We need to say “hypothetical” because we can’t have massive bodies, or light, in a Minkowski metric.

Writing

dx{^/}=dσ

we have, for the length contraction of the moving frame from the point of view of an observer in the rest frame

dx=(1-\frac{v^2_{escape}}{c^2})^{\frac{1}{2}}dσ

For the time dilation of the moving frame from the point of view of an observer in the rest frame

and writing

dt{^/}=dτ

we have

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

These equations above will be true for any velocity, v, up to the speed of light, but we are only considering the particular velocity, v_escape . This is not meant to imply that the Minkowski spacetime has an escape velocity. It has no massive bodies in it, so to talk of it having an escape velocity would be meaningless. We are here only saying that the magnitude of escape velocity used in in the equations above, is found by considering the situation of a co-moving observer in our model of an expanding Universe.

Comparing with the results for the Schwarzschild space time above, this means that we are taking the relative velocity between the central co-moving observer, and another co-moving observer at a distance r, in the Schwarzschild spacetime of our model Universe, to be equal to the relative velocity between the rest frame, and the moving frame in the Minkowski spacetime that we are considering above.

We may imagine a body with a rest length, dσ. This body will have the same length, when measured by an observer stationary next to it, when it is stationary at the central co-moving point in the Schwarzschild spacetime of our model Universe, or stationary in the moving frame of the Minkowski spacetime we are comparing it with.

So we will have

\color{red}{dx{^/}=dr_{(r=0)}=dσ}

The rest length, dσ, of an infinitely small real body, when it is stationary next to an observer, is measured by that observer to be the same wherever it is, and however fast the observer and the body are moving.

So, wherever we are in the gravitational field of a Schwarzschild metric, or whatever the relative velocity we have, less than the speed of light, in a Minkowski metric, the rest length, dσ, and proper time, dτ, of the same real body, measured next to us, and moving with the same velocity as us, will be measured to be the same.

They won’t be the same. They will be measured to be the same.

They will be measured to be the same because our measuring rods and clocks will be affected by the gravitational field of a Schwarzschild metric, or, equivalently, their relative velocity in a Minkowski metric, in exactly the same way as the rest length, dσ, and proper time, dτ, of the same real body, are affected.

We imagine this moving frame will be moving with the same velocity as the co-moving bodies, at a distance, r, in the diagram. Bodies at this distance will have a velocity, v_escape , in the model Universe.

So viewing the moving frame of the Minkowski metric from the rest frame, corresponds to viewing the co-moving observers at the distance, r, from the central co-moving observer in the model Universe built with the Schwarzschild metric. The velocity, v_escape , is taken to be the same in both cases.

We can see that dx calculated for the Minkowski spacetime above will be equal to dr calculated for the Schwarzschild spacetime.

\color{red}{dx_{ (Minkowski)}=dr _{(Schwarzschild)}}

Here dx_(Minkowski) is the length of an infinitesimal rest length, dσ, in the moving frame in the Minkowski metric as viewed from the rest frame, and dr_{(Schwarzschild)} is the length of the same infinitesimal rest length, dσ , as viewed from the central co-moving observer, next to a co-moving observer at a distance, r, from the central co-moving observer.

and if we stipulate that

\color{red}{dt{^/}=dt_{(r=0)}=dτ}

We can see that dt calculated for the Minkowski spacetime above will be equal to dt calculated for the Schwarzschild spacetime.

\color{red}{dt _{(Minkowski)}=dt_{(Schwarzschild)}}

Here dt_(Minkowski) is the duration of an infinitesimal proper time dτ in the moving frame in the Minkowski metric as viewed from the rest frame, and dt_{(Schwarzschild)} is the duration of the same infinitesimal proper time dτ, as viewed from the central co-moving observer, next to a co-moving observer at a distance r from the central co-moving observer.

The same infinitely small body has its rest length contracted, and its proper time dilated, by exactly the same amount in the model Universe built from the Swarzschild metric, as it does in the Minkowski metric when the same velocity, v_escape , is used.

This is shown in the diagram at the beginning of this section.

So we can say that the physical conditions, for a co-moving observer, moving with a velocity, v_escape , and at a distance, r, in the Schwarzschild spacetime of our model Universe, are 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. This will be true in an infintesimaly small region of space and time at the distance, r, from the centre of the the frame of the Schwarzschild spacetime.

When the x axis of the Minkowski metric space is aligned with the r axis of the Schwarzschild spacetime, and if there are no massive bodies near enough to significantly change the shape of space and time, these are the same physical conditions as for the situation of a moving frame in a Minkowski spacetime. This means that the observers at r in the Schwarzschild spacetime, should also measure a uniform speed of light in an infinitely small volume of space at the distance r . Since this will be true in all such infinitely small volumes of space, if we are careful to align the r and x axes for each infinitely small volume, it will be true everywhere in this Schwarzschild spacetime, except where a significantly massive body is too close, and changes space and time on its own account.

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 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.

This means that 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.

We now see that the model Universe proposed explains the speed of light being constant 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.

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.

This corresponds to the Many Worlds theory of Hugh Everett 3rd⁶, who suggests that the “collapse of the wave function”, 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 imagining 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.

We have also seen that any co-moving observer will see all other co-moving observers as traveling with exactly escape velocity from their point of view. They will see the edge of their Universe, at a distance, R, traveling away from them at the speed of light. This edge is an event horizon. An infinite Universe will fit inside its boundary.

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. 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.

10. 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.

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, it will depend on the escape velocity for that radius.

At the edge of our model Universe, at a distance, R , escape velocity is equal to the speed of light, but the velocity of light itself will be zero there from the point of view of a central observer.

Any co-moving observer at a distance, r, approaching, R, will see nothing different, however. They will measure the same velocity 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 section 16, Velocity clocks.

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.

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 Section 5. that mass, m_(r=r), at a radius, r, is as follows

\color{red}{m_{(r=r)} =  {(1- \frac{8Gρπr^2}{3c^2})}^{\frac{1}{2}}{m_{(r=0)}}}

Where m_(r=0)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 the same 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.

11. 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 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}}

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 present radius of the Universe, H is Hubble’s constant, and ln2/H is the doubling time of the universe.

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, 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.

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 in section 5 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.

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.

In Section 5. above we saw 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 section 7 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

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 Section 15 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 as potential energy relative to the mass of the Universe as a whole.

12. 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 sections 3. Applying the Schwarzschild spacetime to the Universe, 4. Potential energy and escape velocity, 5. Adding spherical shells of uniform density and 6. 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.

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. An electron’s event horizon would have a 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, as I am suggesting, we have a simple, and complete, explanation for what we observe. William of Ockham would approve.

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.

13. 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.

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.

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.

Figure 4. Electric and magnetic forces on two electrons

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.

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.”

14. 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.

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}}

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}}

Figure 5. Gravitational and gravito-magnetic forces between two massive bodies

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}}}

15. 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.

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.

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.

Figure 6. Moving relative to co-moving observers

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.

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.

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}

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.

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.

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 Section 11 that, 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. 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 Section 2 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.

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.

16. 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.

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.

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.

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.

17.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.

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.

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 Universe. 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.

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.

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 Section 19 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.

18. 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?

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.

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.

19. 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.

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.

We saw in Section 17 above that we could model Maxwell’s spherical electromagnetic waves as an infinite number of plane waves using Fourier transforms, so we should be able to model photon emission, and absorption, by plane waves.

Figure 7 below shows how the interactions as an electron emits a photon, which is then absorbed by another electron, can be modeled by plane waves.

Figure 7. The emission and absorption of a photon by two electrons

The triangle formed by the wave fronts, a, b, and c, in figure 8 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, in figure 8, 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. This means that the directions formed by the interference of wave fronts are identical to the directions obtained by the vector addition of momentum

Figure 8. Wave fronts, wavelengths, and momentum vectors for an electron emitting a photon

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.

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 consequense 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.

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.

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.

20. 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. 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.

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