Exploring a Flat, Expanding Universe with GR & Newtonian Mechanics

[Pony]

Global Newtonian mechanics seems to be compatible with
1) Hubble's law, and
2) the cosmological principle:

take a vector space, set the velocity of a galaxy at x to be v=x, where x is the position vector. Then from any galaxy, the other galaxies seem to go away with velocity v2-v1 = x2-x1. That is, from looking anywhere, everyone observes Hubble's law.

I wonder if this example has a special relativistic analogy? If not, why not?

If yes, then why is wikipedia claiming

In 1927, Georges Lemaître independently reached a similar conclusion to Friedmann on a theoretical basis, and also presented observational evidence for a linear relationship between distance to galaxies and their recessional velocity.[7] Edwin Hubble observationally confirmed Lundmark's and Lemaître's findings in 1929.[8] Assuming the cosmological principle, these findings would imply that all galaxies are moving away from each other.

Based on large quantities of experimental observation and theoretical work, the scientific consensus is that space itself is expanding,

https://en.wikipedia.org/wiki/Expansion_of_the_universe
?
I am wondering if galaxies have speed (and GR only refutes the existence of their speed in the same way as it refutes the existence of a car's speed for example), and that causes the cosmic redshift, and not because "space is expanding" or something like that.
It maybe works in a flat spacetime, in a special relativistic setting. I am not sure if it's consistent, if there is a model for it.

 

[Ibix]

take a vector space, set the velocity of a galaxy at x to be v=x, where x is the position vector.

Well, it would need to be $v=Hx$, where $H$ is a constant of dimension $[T]^{-1}$.

I wonder if this example has a special relativistic analogy? If not, why not?

Above some distance $x=c/H$ the galaxies would have to be superluminal, which isn't possible.

It is possible to create a special relativistic expanding universe, the Milne cosmology. This is only defined in the future light cone of some arbitrarily chosen event and its spatial surfaces are hyperbolae centered on that event. It is the zero mass limit of an FLRW spacetime, so is empty of anything except test particles (which is the fundamental problem with SR models of anything).

"space is expanding"

This is an ordinary language description of metric expansion and not wholly accurate, IMO. Anyway, you can change coordinates so that this is not (or not entirely) responsible for galaxy recession. However, doing so hides a lot of the symmetry of the FLRW spacetime and changes nothing physically.

 

[Orodruin]

I am wondering if galaxies have speed (and GR only refutes the existence of their speed in the same way as it refutes the existence of a car's speed for example), and that causes the cosmic redshift, and not because "space is expanding" or something like that.

This is just a question of what coordinates one chooses to use. See https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

I wonder if this example has a special relativistic analogy? If not, why not?

This depends on what exactly you try to reproduce. Hubble’s law on the form $v = Hd$ obviously cannot hold in special relativity because letting $d$ become large enough would result in speeds larger than $c$. What you could do in SR is to consider a family of observers traveling away from a single event at constant velocity. This scenario is homogeneous in the sense that all observers will see the same thing. Being special relativistic, the scenario does not include any gravity and is excluded by observation.

 

[Pony]

It is possible to create a special relativistic expanding universe, the Milne cosmology. This is only defined in the future light cone of some arbitrarily chosen event and its spatial surfaces are hyperbolae centered on that event. It is the zero mass limit of an FLRW spacetime, so is empty of anything except test particles (which is the fundamental problem with SR models of anything).

According to the wikipedia article it seems to be the analogue of my expanding Newtonian cosmology. Thank you for pointing it to me!

Do you know what is the distribution of velocities in this model when the space is parametrized with standard Minkowski coordinates?

This depends on what exactly you try to reproduce. Hubble’s law on the form $v = Hd$ obviously cannot hold in special relativity because letting $d$ become large enough would result in speeds larger than $c$.

Good question! I expect that there is a model in minkowski space where the universe is expanding, and the cosmological principle holds (in space at least).
Hubble's law will hold on a short scale automatically anyway, since continuous functions are approximable with their derivatives.

 

[Orodruin]

Do you know what is the distribution of velocities in this model when the space is parametrized with standard Minkowski coordinates?

A flat distribution in rapidity rather than in velocity.

I expect that there is a model in minkowski space where the universe is expanding, and the cosmological principle holds (in space at least).

This does not hold in normal Minkowski coordinates. It does hold in Milne coordinates but then you could interpret ”space” as expanding in those coordinates. It really is a coordinate dependent interpretation.

 

[Ibix]

Do you know what is the distribution of velocities in this model when the space is parametrized with standard Minkowski coordinates?

As Orodruin says, uniform in rapidity. Another way to look at it is that the velocity at Minkowski distance $x$ is $x/t$, where $t$ is the "cosmological time" since the initial event. Note that the Milne cosmology does not cover things with $x>ct$. Also note that if the mass density (everywhere uniform in Milne coordinates) is even slightly non zero then in Minkowski coordinates it goes to infinity as $x\rightarrow ct$, making this model inconsistent (Milne coordinates are pathological at Minkowski $x=ct$, which is why you can't see the problem in those coordinates). You must throw out either the cosmological principle or the existence of any matter at all.

 

[Orodruin]

You must throw out either the cosmological principle or the existence of any matter at all.

Considering the SR spacetime is a vacuum solution, the existence of matter was doomed from the start …

 

[Ibix]

Considering the SR spacetime is a vacuum solution, the existence of matter was doomed from the start …

Yeah, but introductory SR uses flat spacetime containing rocket ships and even planets all the time. You can get away with that because the curvature is negligible, and therefore one might naively think that Milne would be a decent approximation if the mass density were "low enough". I'm just pointing out one way to see why that isn't correct - any non-zero Milne-uniform mass density is singular on the light cone, so not "low".