Robertson-Walker Models Conformally Flat

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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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  • #2
bcrowell
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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.
 
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Bill_K
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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.
 
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bcrowell
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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.
 
  • #5
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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.
???
 
  • #6
Bill_K
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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.
 
  • #7
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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:eek:ops, I realize this seems to be precisely your point.
 
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