Proof of the positive mass theorem. II

Abstract

The positive mass theorem states that for a nontriviat isolated physical system, the total energy, which includes contributions from both matter and gravitation is positive. This assertion was demonstrated in our previous paper in the important case when the space-time admits a maximal slice. Here this assumption is removed and the general theorem is demonstrated. Abstracts of the results of this paper appeared in [113 and [131. #> (~i JiJ) j/2An initial data set will be said to be asymptotically flat if for some compact set C, N\C consists of a finite number of components N 1 ..... Np such that each N i is diffeomorphic to the complement of a compact set in R 3. Under such diffeomorphisms, the metric tensor will be required to be written in the form gij = 6~5 + O(r-i)and the scalar curvature of N will be assumed to be O(r-4).With each N k we associate a total mass M k defined by the flux integral 1 Mk= t-~ ~ .~. (giJ'j-gjj'i)dai LJwhich is the limit of surface integrals taken over large two spheres in N k.This number M k is called the ADM mass of N k (see Arnowitt, Deser, and Misner [1]). Classically it was assumed that the first term in the asymptotic expansion of gii is spherical. It was pointed out by York [11] that physically it is more desirable to relax this assumption to the one mentioned above. The method in this paper will work assuming only this general asymptotic condition of York.In order for the total mass to be a conserved quantity, one assumes Pij = O(r-2) and ~Pu = O(r-3).

Transformations of asymptotically AdS hyperbolic initial data and associated geometric inequalities

“…In this section we propose a different strategy based on deformations of the initial data which result in an asymptotically flat structure. The deformations also preserve the mass and yield 'sufficiently' nonnegative scalar curvature, so that (3.1) follows from the proofs of positivity of the ADM mass given by Schoen and Yau [45,46]. Thus the positive mass theorem in the asymptotically AdS hyperbolic setting is reduced to the positive mass theorem in the ADM context, whenever a canonical system of elliptic equations admits a solution.…”

“…then the scalar curvature of the Jang graph (M 3 , g 3 ) enjoys weak nonnegativity through the formula [46] (3.14)…”

“…Nevertheless, the scalar curvature is 'sufficiently' nonnegative to allow the basic strategy in [46] to be carried out. Namely if ψ > 0 solves…”

“…This could be accomplished by allowing for blow-up of the Jang type equations (3.2), (3.13) at apparent horizons as in [28,46]. We expect that the cylindrical ends which arise from such blow-up behavior can be conformally closed, so that the remaining arguments of the proof remain valid as in [46].…”

“…In [14] we proposed deformations of asymptotically hyperboloidal initial data which transformed the asymptotically hyperbolic structure into an asymptotically flat structure, while preserving the mass up to scaling by a positive constant. These deformations are based on solutions of the so called Jang-type equations [7,8,12,13,33,46], which impart positivity properties to the scalar curvature of the deformed data. In effect, the full round of geometric inequalities in the asymptotically hyperboloidal setting is reduced to their counterparts in the asymptotically flat setting modulo the solution of canonical system of partial differential equations.…”