This note is about 97 years late. Had Hubbel noted the recession of the galaxies after Einstein had formulated special relativity but before GR, then this would have been a good paper.

Presume Special Relativity. Consider one spatial dimension and time: <t, x>. Suppose that we have for each integer, j, a particle whose worldline is <τ*cosh(j), τ*sinh(j)>. τ is the proper time as measured by a clock with the particle. For each j, particle j is at <0, 0> for τ = 0. In our laboratory frame, at time t, particle j is:

• at <t, t*tanh(j)>
• moving with velocity tanh(j) .
Note that −1 < tanh(x) < 1. Consider another frame:
t' = t cosh 1 + x sinh 1
x' = t sinh 1 + x cosh 1
(Illustrated here) This Lorentz transformation merely renumbers particles by one since <τ*cosh(j), τ*sinh(j)> is the same event as <τ*cosh(j+1), τ*sinh(j+1)> in the primed coordinates.

The group of transformations generated by this transformation makes the particles equivalent to each other. The view is the same from each particle.

The laboratory frame where we started favors particle 0. We introduce a new curved and democratic coordinate system:
t'' = √(t2 − x2)
x'' = tanh−1 (x/t)
In this new system every observer thinks his age, since the big bang, is t'' and x'' = j. Each particle observes his nearest neighbor receding at velocity tanh 1. The n’th particle away is receding with velocity tanh n, or with rapidity n. A photon launched from a particle will eventually overtake all of the particles in the direction that it travels. Furthermore each particle can see each other particle; there is no event horizon. Early ionized matter would limit the view, of course. The covariant metric tensor for this system is
 1 0 0 t''2

From 1D to 3D

It is the 3D version of this that is germane. There are even regular arrangements of particles with asymptotically uniform densities as in the 1D case, but these arrangements are obscure because the 3D space that they tile is negatively curved! It is somewhat analogous to the curved S3 embedded in the flat R4.

Lorentz transformations that leave <0, 0, 0, 0> unmoved leave t'' = √(t2 − x2 − y2 − z2) unchanged.

We denote an event in the lab coordinates: <t, x, y, z>. We transform to more democratic coordinates:
t'' = √(t2 − x2 − y2 − z2)
x'' = x; y'' = y; z'' = z
These coordinates have the advantage that t'' is the interval from <0, 0, 0, 0>. The coordinates are used only for positive t and where the expression for t'' is real—inside the light cone with vertex at the origin. A time slice with t'' = const is a 3 space with constant negative intrinsic curvature and thus no convenient set of coordinates. They are homogeneous and isotropic.

That the curvature is constant can most easily be seen from a geometric argument. Such a slice is invariant under Lorentz transformations that leave the origin fixed, for the slice is defined in terms of a Lorentz property: interval to the origin. Lorentz transformations of the 4D space time become the congruence transformations of each such time-slice. These time-slices are thus self congruent which means exactly that they have constant intrinsic curvature. We go into the math in more detail here.

To model cosmology we presume dust at some constant density throughout a time slice. That density will be proportional to t''−1/3. The total dust and volume of a time-slice is infinite.

This is not a solution to the GR equations near time zero for the large density of the dust is incompatible with a flat 4 space time. As the density become negligible, the solution fits GR, even Special Relativity.

Red Shift

Lets do the red shift calculation. Observer at j=0 observes the light of frequency 1 emitted by the source at j ≠ 0. Since the frequency is 1 the emitter will emit one cycle as τ increases by 1. When does the photon which emitter j at its τ = 1 reach observer at j = 0? In the observer’s frame the particle is emitted at location sinh(j) at time cosh(j). It reaches the observer at his τ = cosh(j) + sinh(j) = ej. He computes a red shift factor of e−j. He computes that the emitter is on a sphere whose area was 4πsinh(j)2 as the light was emitted.