Curvature in a Universe with No Cosmological Constant and Zero Density

[MeJennifer]

nrqed:
If k = 0 spacetime is flat. if k != 0 spacetime is curved.

Note that a conformally flat spacetime is not necessarily a flat spacetime.

 

[Wallace]

Nrqed, I don't have a lot of time right now, but GJ I think gave you the transformation needed for the Milne model. If you want to learn more about this and conformal flatness for more general FRW models there are a couple of papers I can recommend. The first is http://arxiv.org/abs/astro-ph/0610590" paper that describes how to consider the same physical situation in Minkowski and FRW co-ordinates for an empty universe and then generalises this to changing general FRW metrics to conformally Minkowski metrics.

The second paper is http://arxiv.org/abs/0707.2106" one that discusses the same issue and hopefully explains this in more detail. I'd probably be just paraphrasing that second paper if I wrote a longer post anyway!

Just qucikly @MJ, I don't think you've got it right. You can't say anything about the curvature of space-time from knowledge of k alone. As discussed in this thread, the Ricci scalar is non-zero for all values of k and is only zero in an empty universe, which can be described by k=0 or k=-1, depending on how the co-ordinates are defined.

The value of k tells you about the curvature of 3D spatial slices of surfaces of constant cosmic time. Since cosmic time in the FRW metric is just one arbitrary way of defining time, there is nothing universal about this. As discussed in the two papers linked to, if you change to a different time co-ordinate, then the curvature of spatial slices of constant time in that co-ordinate changes.

To sensibly talk about curvature of spacetime (as opposed to just spatial curvature) you need to consider things that do not change with a change of co-ordinates, such as the Ricci scalar.

 

[MeJennifer]

You are correct.

Interesting argument against the notion of "expansion of space" in the first referenced document by the way.

 

[nrqed]

Nrqed, I don't have a lot of time right now, but GJ I think gave you the transformation needed for the Milne model. If you want to learn more about this and conformal flatness for more general FRW models there are a couple of papers I can recommend. The first is http://arxiv.org/abs/astro-ph/0610590" paper that describes how to consider the same physical situation in Minkowski and FRW co-ordinates for an empty universe and then generalises this to changing general FRW metrics to conformally Minkowski metrics.

The second paper is http://arxiv.org/abs/0707.2106" one that discusses the same issue and hopefully explains this in more detail. I'd probably be just paraphrasing that second paper if I wrote a longer post anyway!

Just qucikly @MJ, I don't think you've got it right. You can't say anything about the curvature of space-time from knowledge of k alone. As discussed in this thread, the Ricci scalar is non-zero for all values of k and is only zero in an empty universe, which can be described by k=0 or k=-1, depending on how the co-ordinates are defined.

The value of k tells you about the curvature of 3D spatial slices of surfaces of constant cosmic time. Since cosmic time in the FRW metric is just one arbitrary way of defining time, there is nothing universal about this. As discussed in the two papers linked to, if you change to a different time co-ordinate, then the curvature of spatial slices of constant time in that co-ordinate changes.

To sensibly talk about curvature of spacetime (as opposed to just spatial curvature) you need to consider things that do not change with a change of co-ordinates, such as the Ricci scalar.

Thank you to Geaorges for the explicit change of coordinates and to Wallace for the above informative post and for the very good references. I have to ponder a bit and might get with a final question or two. This is very instructive. Thanks to all (Wallace, Hellfire and Georges)

 

[JDoolin]

You are correct.

Interesting argument against the notion of "expansion of space" in the first referenced document by the way.

Nrqed, I don't have a lot of time right now, but GJ I think gave you the transformation needed for the Milne model. If you want to learn more about this and conformal flatness for more general FRW models there are a couple of papers I can recommend. The first is http://arxiv.org/abs/astro-ph/0610590" paper that describes how to consider the same physical situation in Minkowski and FRW co-ordinates for an empty universe and then generalises this to changing general FRW metrics to conformally Minkowski metrics.

http://arxiv.org/abs/astro-ph/0610590][/PLAIN]
Milne (1933) specified what he meant as ‘the space commonly used in physics’: “flat, infinite, static Euclidean space”. He also wrote: “Moving particles in a static space will give the same observable phenomena as stationary particles in ‘expanding’ space” (Milne 1934). These statements are wrong in general.

I think, actually, Milne was quite right.

https://www.physicsforums.com/showthread.php?t=427340&page=5"

Except that when you treat the Milne Model within the Friedmann model, you apparently get an empty universe. Has anyone tried treating the Milne Model within its original context?