Robertson-Walker Models Conformally Flat

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Discussion Overview

The discussion revolves around the properties of Robertson-Walker models in the context of general relativity, specifically focusing on whether these models are conformally flat. Participants explore the implications of homogeneity and isotropy in relation to the Weyl tensor and the nature of cosmological expansion.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that all Robertson-Walker models are conformally flat due to the vanishing of tidal forces, indicated by a zero Weyl tensor.
  • Another participant questions whether homogeneity or isotropy is the determining factor, noting that a cosmology can be homogeneous but not isotropic, which may lead to a nonzero Weyl tensor.
  • A different perspective is offered, arguing that anisotropic expansion rates could imply that the space is only homogeneous at a specific moment, potentially leading to inhomogeneity thereafter.
  • It is mentioned that each hypersurface in the spacetime is flat and homogeneous, but the differing expansion rates in various directions could complicate the conformal flatness.
  • One participant expresses doubt about the possibility of achieving a conformal transformation from a non-isotropic to an isotropic cosmology while preserving angles.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the implications of homogeneity and isotropy on conformal flatness and the behavior of the Weyl tensor in different cosmological models. No consensus is reached on whether all Robertson-Walker models can be considered conformally flat under varying conditions.

Contextual Notes

The discussion highlights limitations in understanding the relationship between homogeneity, isotropy, and the properties of the Weyl tensor, as well as the complexities introduced by anisotropic expansion rates.

Airsteve0
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As an understanding exercise in my general relativity course my professor recommended proving to ourselves that all Robertson-Walker models are conformally flat. However, I am unsure of how to approach such a proof. Thanks in advance for any help.
 
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By homogeneity, tidal forces vanish, i.e., the Weyl tensor is zero. Since the Weyl tensor is zero, the spacetime is conformally flat.
 
Is it homogeneity or isotropy? The Robertson-Walker metric is both. But you can have a cosmology with different expansion rates in different directions:

ds2 = dt2 - a(t)dx2 - b(t)dy2 - c(t)dz2

which is homogeneous but not isotropic. I'd guess the Weyl tensor for this type of cosmology is nonzero.
 
Bill_K said:
Is it homogeneity or isotropy? The Robertson-Walker metric is both. But you can have a cosmology with different expansion rates in different directions:

ds2 = dt2 - a(t)dx2 - b(t)dy2 - c(t)dz2

which is homogeneous but not isotropic. I'd guess the Weyl tensor for this type of cosmology is nonzero.

Yep, thanks for the correction.
 
Bill_K said:
Is it homogeneity or isotropy? The Robertson-Walker metric is both. But you can have a cosmology with different expansion rates in different directions:

ds2 = dt2 - a(t)dx2 - b(t)dy2 - c(t)dz2

which is homogeneous but not isotropic. I'd guess the Weyl tensor for this type of cosmology is nonzero.

I don't get this. If the expansion rates are anisotropic then it would seem that the space could only be homogeneous at a particular moment in time. From that point becoming inhomogeneous from then on.
?
 
Each hypersurface t = const in the spacetime is a flat 3-space, hence it is always homogeneous. But the expansion rate is different in different directions. This is an example of a Bianchi cosmology, of which there are nine types, each with a different symmetry. See for example Ken Jacobs' thesis.
 
Bill_K said:
Is it homogeneity or isotropy? The Robertson-Walker metric is both. But you can have a cosmology with different expansion rates in different directions:

ds2 = dt2 - a(t)dx2 - b(t)dy2 - c(t)dz2

which is homogeneous but not isotropic. I'd guess the Weyl tensor for this type of cosmology is nonzero.
I'm doubtful that a non-isotropic cosmology can be made flat thru a conformal transformation. How do you preserve the angles when going from anisotropy to
isotropy?

Edit:oops, I realize this seems to be precisely your point.
 
Last edited:

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