Homogeneity: does V(X) depend on X?
What forms can
have if it is independent of X?
E.g. a sphere has
independent of X
Condition for homogeneity: JG claim: conformally flat
If independent of X,
is just a function of r. JG claim: first two terms in expansion in “powers” of r are undetermined.
Scale transformation: change the value of r; what happens to
Is the structure after many steps scale invariant? I.e is it a pure power dependence in
? Is it spatially homogeneous? Is it scale free (i.e. r^d)?
Structure of metric
Metric is torsion free if there is symmetry in the geodesics
There could be light cones from A to B and B to A which are not symmetric
To derive a spatial surface, assuming staticness of evolution....
JG claim: a torsion free metric is characterized by
Cotton tensor: 3rd derivative of metric
Weyl tensor etc.
JG : any manifold with constant sectional curvature is conformally flat
time shift between spatial hypersurfaces
spatial shift between hypersurfaces
Independence of X implies R
Can still have R.
Can V depend on X without T
How to measure dimension of space from cosmology?