#### Homogeneity: does V(X) depend on X?

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

V

r

E.g. a sphere has

V

r

Condition for homogeneity: JG claim: conformally flat

If independent of X, (X) is just a function of r. JG claim: first two terms in expansion in “powers” of r are undetermined.

V

r

Scale transformation: change the value of r; what happens to ?

V

r

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)?

V

r

#### Structure of metric

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 Γ

Cotton tensor: 3rd derivative of metric

Weyl tensor etc.

JG : any manifold with constant sectional curvature is conformally flat

Weyl tensor etc.

JG : any manifold with constant sectional curvature is conformally flat

####

#### Covariant derivative

Covariant derivative

time shift between spatial hypersurfaces

spatial shift between hypersurfaces

spatial shift between hypersurfaces

Independence of X implies Rμ⋁=0 ?????

Can still have R.

Can V depend on X without Tμ⋁?

#### Vacuum solutions

Vacuum solutions

How to measure dimension of space from cosmology?