# Robertson-Walker Models Conformally Flat

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
Staff Emeritus
Gold Member
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
Staff Emeritus
Gold Member
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.

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.

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.

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