The spacetime positive mass theorem in dimensions less than eight

Abstract

We prove the spacetime positive mass theorem in dimensions less than eight. This theorem asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition, the inequality E ≥ |P | holds, where (E, P ) is the ADM energy-momentum vector. Previously, this theorem was only known for spin manifolds [38]. Our approach is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case of this theorem [30,27]. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author [14]. An important part of our proof is to introduce an appropriate substitute for the area functional that is used in the time-symmetric case to single out certain minimal hypersurfaces. We also establish a density theorem of independent interest and use it to reduce the general case of the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.

General relativity and gravitation: a centennial perspective

“…Chapter 8 covers these topics extensively, including an outline of a very nice MOTS-based simplification of the proof of the Positive Mass Theorem [80].…”

From sine-Gordon to vacuumless systems in flat and curved spacetimes

“…At λ = 1, we find a flat space, so the stability analysis as presented here fails. In this case, the stability of this space is guaranteed by the positive mass theorems on asymptotically flat spacetimes, which are valid for dimensions ≤ 7 [36,37].…”

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

“…Notice that, from now onwards, we will systematically omit the + sign while referring to these quantities. It is well-know (see, for instance, Section 2 of [EHLS11]) that the first pointwise variation of the null mean curvature is given by…”

“…Boosted harmonic asymptotics. For our purposes, it is appropriate to enlarge the class of data under consideration from those in harmonic asymptotics (see [EHLS11]) to its closure under the operation of relativistic boost (inside a given spacetime), namely when a trasformation of the form…”

“…• data in harmonic asymptotics (as defined in [EHLS11]), which were proven to be a dense class in general asymptotically flat initial data sets with respect to the topology of weighted Sobolev spaces (see Section 6 of [EHLS11]); • the t = constant slices, in isotropic coordinates, of the Kerr-Newman spacetime (thus including, as special cases, Schwarzschild, Kerr and Reissner-Nordström data) as well as boosts thereof.…”