Positive Mass Theorems for Asymptotically Ads Spacetimes With Arbitrary Cosmological Constant

Abstract

We formulate and prove the Lorentzian version of the positive mass theorems with arbitrary negative cosmological constant for asymptotically AdS spacetimes. This work is the continuation of the second author's recent work on the positive mass theorem on asymptotically hyperbolic 3-manifolds.

Hyperbolic positive mass theorem under modified energy condition

“…4λ 2 e j · e i · φ α + 1 4λ 2 e i · e j · φ α (14) follows from the similar argument of the second theorem in [14]. The result that M is hyperbolic and maximal becomes a consequence that there is a basis {φ α } parallel to the imaginary Killing connection.…”

“…Thanks to the Lax-Milgram theorem [18] , by similar argument in [11,13,14,17,19,20], there exists a unique spinor Φ 1 in H 1 (S) such that…”

Transformations of asymptotically AdS hyperbolic initial data and associated geometric inequalities

“…Initial investigations were carried out in [1,3,27,30], the time symmetric case (k = 0) was treated in [2,18,19,49,53], and the most general versions were given in [20,38,50,52]. See also [15] for a discussion concerning the rigidity statement.…”

Positive mass theorems for asymptotically de Sitter spacetimes

“…In this section, we use hyperbolic coordinates and the positive mass theorem for asymptotically hyperbolic manifolds proved in [31,28] to derive a positive mass theorem for asymptotically half de Sitter spacetime. Let…”

“…Anti-de Sitter spacetimes are characterized by Killing spinors with imaginary Killing constants, which have certain nice properties. Witten's method was extended successfully to asymptotically Antide Sitter spacetimes and the positive mass theorem was proved completely and rigorously in this case [25,10,31,17,28]. As recent cosmological observations indicated that our universe possibly has a positive cosmological constant, the positive mass theorem for asymptotically de Sitter spacetimes gains much more importance.…”