That's a **great** question ! What you are asking about is one of the missing links between classical and quantum gravity.

On their own, the Einstein equations are local field equations:

$$ G_{\mu\nu} = 8 \pi G T_{\mu\nu} $$

and do not contain any topological information. At the level of the action principle:

$$ S_{eh} = \int_\mathcal{M} d^4 x \sqrt{-g} \mathbf{R} $$

the term we generally include is the Ricci scalar $ \mathbf{R} = Tr[ R_{\mu\nu} ] $, which depends only on the first and second derivatives of the metric and is, again, a local quantity. So the action does not tell us about topology either, unless you're in two dimensions, where the Euler characteristic is given by the integral of the ricci scalar:

$$ \int d^2 x \mathcal{R} = \chi $$

(modulo some numerical factors). So gravity in 2 dimensions is **entirely** topological. This is in contrast to the 4D case where the Einstein-Hilbert action appears to contain no topological information.

This should cover your first question.

All is not lost, however. One can **add** topological degrees of freedom to 4D gravity by the addition of terms corresponding to various topological invariants (Chern-Simons, Nieh-Yan and Pontryagin). For instance, the Chern-Simons contribution to the action looks like:

$$ S_{cs} = \int d^4 x {}^\star R \, R $$

where $ R \equiv R_{abcd} $ is the Riemann tensor and $ {}^\star R_{abcd} = 1/2 \epsilon_{ab}{}^{ij} R_{cd\,ij} $ is its dual. Here is a very nice paper by Jackiw and Pi for the details of this construction.

There's plenty more to be said about topology and general relativity. Your question only scratches the surface. But there's a goldmine underneath ! I'll let someone else tackle your second question. Short answer is "yes".

This post imported from StackExchange Physics at 2014-04-01 16:47 (UCT), posted by SE-user user346