Topological constrains for the solutions of EFE

[cianfa72]

TL;DR

About the constrains for the topology of spacetime given by Robertson - Walker metric

I'm keep watching the lectures on GR from F.P. Shuller and D. Giulini -- International Winter School on Gravity and Light.

As far as I can tell, the spacetime metric $g_{a b}$ in the Robertson-Walker (RW) form relies on the hypothesis of spatial homogeneity & isotropy (which provide constraints for the spacelike hypersurfaces of the relevant spacetime foliations). This implies a constrain on the topology of the spacetime itself. Namely its topology must be the product topology of a maximally symmetric 3D Riemman manifold $\Sigma$ times $\mathbb R$ -- see Lecture 18.

In the definition of product topology enter the topology of $\Sigma$ as topological 3D manifold plus the standard topology of $\mathbb R$. So far so good.

I was thinking that assumption might be too restrictive, putting a too strong constrain on the possible solutions of Einstein Field Equations (EFEs).

What do you think about ?

 

[PeterDonis]

I was thinking that assumption might be too restrictive, putting a too strong constrain on the possible solutions of Einstein Field Equations (EFEs).

The topology constraint you refer to is global, not local. But the EFE is local. A global topological constraint cannot restrict the solutions of a local equation.

To put this another way: the EFE is a tensor equation, and tensor equations are equations in the tangent space at a point in spacetime. A "solution" of the EFE is a solution of the tensor equation that is valid in the tangent space at every point in some open region of spacetime. Every possible global topology has such open regions. So the global topology can't restrict what EFE solutions you can get.