The Proof of the Positive Mass Theorem using Minimal Surfaces

The positive mass theorem of general relativity to be addressed here states that the total mass of an isolated gravitational system (viewed from spatial infinity, known as the ADM mass) must be non-negative if its local mass density is non-negative everywhere, and in fact the mass is strictly positive unless the space-time is flat. In this essay we are only concerned with the first part of this theorem. Equivalently, the theorem asserts that the total energy-momentum vector must be non space-like. Mathematically, the basic setting can be described as follows [1]: Let (M,γ) be a space-time whose local mass density is non-negative everywhere. Suppose M admits an oriented, three-dimensional maximal space-like hypersurface N , with induced Riemannian metric g and second fundamental form h. We assume that there exists a compact subset K of N such that N \K consists of a finite number of ends N1, N2, . . . , Nr, with each Nk being diffeomorphic to the exterior of a ball in R. We say that the metric g is asymptotically flat if for each end Nk, under the diffeomorphism described above, in the Euclidean coordinates x = (x, x, x) of R, g and h have the asymptotic behaviour