Discrete Calculus

In 1960, Regge formulated a discrete calculus as a numerical scheme for relativity. We can formulate it as a theory on a weighted graph . where is a distance which is a discrete q-manifold. The graph defines a q-dimensional pure simplicial complex generated by the maximal complete subgraphs as facets. We can in parallel look also look at completely combinatorial settings, where no weights are given.

To model classical notions like Ricci curvature or scalar curvature in the discrete, one has to get a good notion of sectional curvature. I myself worked on such notions for a year now and suggested a notion of sectional curvature that is based on geodesic flow on the frame bundle and which is purely combinatorial without looking at a geometric realization of the complex. Regge suggested to look just at the excess angle of the dual of a bone and weight this with the area of the bone. A bone is a q-2 simplex in the simplicial complex which is in q=4 dimensions a triangle. Every bone b in a discrete manifold defines a circle which is the intersection of all unit spheres in b. The dual of a simplex x is always a sphere: examples:

• if x is a q-simplex, a facet, then its dual is the empty set which is a (-1)-sphere.
• If x is a q-1 simplex, a wall, then its dual is a two point set, which is a 0-sphere.
• If x is a q-2 simplex, a bone, then its dual is a circular graph, a 1-sphere.
• If x is a 0-simplex, a vertex, then its dual is a q-1 sphere, by assumption of having a manifold.

What I had suggested during the last year is to use the geodesic flow to extend the circle for a bone to get so the dual of a 2-manifold and use the curvature of this 2-manifold (the analog of the image of the exponential map of a 2-dimensional plane in the tangent plane) as sectional curvature.

In any case, whether one has assigned a curvature K(b) to the bone in a combinatorial manner or by using a geometric realization, one can then define the Ricci curvature of an edge as the average over all sectional curvature of bones containing it. The scalar curvature is the average of all sectional curvatures of all the bones containing it.

A nice idea in general is to read the Einstein equations R-g S/2 = T as an equation defining T. S.T. Yau once told me almost 10 years ago, that this is an approach that has been considered by geometries. So, instead of giving a mass tensor T and then painfully try to find a metric g that solves the Einstein equations, one takes a geometry g and looks what “matter” it produces. This is a way how “geometry” can induce “matter”. This is especially handy, in a situation, where one has not yet any notion of “matter”. The nice thing about this approach is that we do not have to work at all. In the discrete this means that once we have defined the Ricci R and scalar S, and a notion of g, then we the geometry has a mass distribution.