Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

Abstract

The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in L p , 1 ≤ p < 2. In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.

Properties of the Null Distance and Spacetime Convergence

“…In the second part of the proof, where a null curve β is constructed from an (almost) -minimizing α, (20) which shows that…”

“…The question of geometric stability has already been studied extensively for spacelike slices of spacetimes by Allen [2,3,5], Bray and Finster [19], Bryden, Khuri, and Sormani [20], Cabrera Pacheco [26], Finster [37], Finster and Kath [38], Huang and Lee [51], Huang, Lee, and Sormani [52], Jauregui and Lee [53], Lee [61], Lee and Sormani [62], LeFloch and Sormani [64], Sakovich and Sormani [76], Sormani and Stavrov Allen [82]. Lee and Sormani [62] conjecture that SWIF convergence is the correct notion of convergence for stability.…”

“…Singular static metrics appear, for instance, when the Einstein equations are coupled to matter fields [12,23]. Using spacetime convergence results, it may be possible to study their spacetime stability using techniques analogous to [20,52,62,64].…”

Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space

“…Stability of the positive mass theorem has also been studied for asymptotically hyperbolic manifolds, by Allen [4]; Dahl, Gicquad and Sakovich [17]; Sakovich and Sormani [39]; and by the second-named author [13]. Further related stability results can be found in [5,10,12,16,21,22].…”

The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting

“…[BK11]). Other important works where the Jang equation plays a prominent role include (but do not restrict to) [ADGP18] of Andersson, Dahl, Galloway and Pollack on topological censorship, [BM19] of Bourni and Moore on the null mean curvature flow, of Wang and Yau [WY09] on the notion of quasilocal mass, as well as the recent work of Bryden, Khuri, and Sormani [BKS19] on the stability of the spacetime positive mass theorem. In the view of these results, we hope that our study of the Jang equation in the asymptotically hyperbolic setting will be useful in other contexts that are out of the scope of the current paper.…”