Is Homogeneous Line Universe Possible in Higher Dimensions?

[122520253025]

I have been trying to work out the solutions to a homogenous line universe using special relativity, and have found that, as per special relativity, one of the solutions is $$v = tanh(d)$$, where $v$ is the velocity and $d$ is the distance of recession of galaxies in this one dimensional case. Now, by homogenous, i mean that between any two points in my line universe, the velocity of recession of the two points can be given by $$v = g|d|$$ where $g$ is some function to be found [irrespective of the location of points]. One can then observe that due to velocity addition, considering three points on the stretching line: $$\frac{g|b-a| + g|c-b|}{1+g|b-a|g|c-b| = g|c - a|.$$ [The fact that this right hand side is $$g|c-a|$$ is due to the assumption of homogeneity]. Now, when i try to extend this to two dimensions or higher. i encounter difficulties and impossibilities, and this is hinting that homogeneous space, as i have defined it at least, is not possible. Does anyone know of any solution to this problem?

 

[pervect]

You might try reading about the Milne universe, for instance http://en.wikipedia.org/wiki/Milne_model

However, to understand fully what you read, you'll probably need to know a bit about GR. Minamally, you'll need to know how SR works with arbitrary coordinates. The needed mathematical techniques are usually taught in a GR course, and much of the language used to describe the Milne universe is borrowed from what's taught in GR courses as well.

In the Milne paramaterization, the spacelike hypersurfaces have a negative spatial curvature, though the underlying space-time itself is flat. This is most likely your difficulty in going above the 1 dimensional model, handling the curvature of the space-like surfaces.

Sorry I can't make this post more SR like, I don't see a way at the moment to avoid using a lot of GR concepts.

 

[122520253025]

Thank you very much. I shall definitely look closely at the Milne universe, and have been wanting to learn general relativity for a while, so this is a great beginning point. You reply seems to make sense in that if space time was curves then of course the curvature can be adjusted to imply homogeneity. But this is profound!: because this means that for a homogenous universe you can mathematically identify the specific class of spacetime curves such that homogeneity is satisfied! Thank you once again.