TITLE: Is there such a thing as discrete Riemannian geometry?
QUESTION [5 upvotes]: General relativity expresses gravity as a curvature in space time, created by the stress energy tensor.
$$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu}$$
Given I put the fact that energy is quantizied and can only change within discrete levels into the stress energy tensor, then the Ricci tensor and as a result, the Christoffel symbols, as well as the metric would have to change into some finite difference form.
For large space, this finite difference would approximate the equation above, but for small areas, it would behave differently. Is there any kind of discrete Riemannian geometry that satisfies this criteria? Any sources?

REPLY [4 votes]: Here's a fuzzy argument I've heard since my undergraduate days in physics: If the universe is bounded then quantum mechanics would seem to indicate that the momentum operator does not assume a continuous spectrum of eigenvalues. It seems a reasonable extension to suppose that spacetime itself would be some sort of discrete object. The continuum which we frame classical physics would just be a macroscopic approximation of some quantum plank-length geometry. 
One way to implement such an idea is to replace the commutative coordinates of classical manifold theory with noncommutative coordinates where the departure from commutivity is introduced by in physics literature by the introduction of star-products. Look up the Moyal bracket, or the paper Deformation quantization of Poisson manifolds by M. Kontsevic http://arxiv.org/abs/q-alg/9709040. The literature that follows this is voluminous, see the work of Julius Wess and his collaborators in particular. The features of GR that have been recovered in this formalism are surprising. I'll leave it to someone more adept than myself to comment further on the connections with algebraic geometry and Alain Connes program of noncommutative geometry.