Robertson-Walker Models Conformally Flat

[Airsteve0]

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.

 

[bcrowell]

By homogeneity, tidal forces vanish, i.e., the Weyl tensor is zero. Since the Weyl tensor is zero, the spacetime is conformally flat.

 

[Bill_K]

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.

 

[bcrowell]

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.

 

[Austin0]

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.
?

 

[Bill_K]

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.

 

[TrickyDicky]

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?

Editops, I realize this seems to be precisely your point.