Solution of the Local Mass Problem in General Relativity

Abstract

The local mass problem is solved. That is, in suitable function spaces, it is shown that for any vacuum space-time near flat space, its mass m is strictly positive. The relationship to other work in the field and some discussion of the global problem is given. Our proof is, in effect, a version of critical point analysis in infinite dimensions, but detailed L^p and Sobolev-type estimates are needed for the precise proof, as well as careful attention to the coordinate invariance group. For the latter, we prove a suitable slice theorem based on the use of harmonic coordinates.