The Rest Mass of an Asymptotically Anti-de Sitter Spacetime

Abstract

Abstract. We study the space of Killing fields on the four dimensional AdS spacetime AdS 3,1 . Two subsets S and O are identified: S (the spinor Killing fields) is constructed from imaginary Killing spinors, and O (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of O is properly contained in that of S. This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. In [10], Chruściel, Maerten and Tod proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the "rest mass" of an asymptotically AdS spacetime is then defined by minimizing the observer energy, and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime.

Small sphere limit of the quasi-local energy with anti de-Sitter space reference

“…Remark 1. Proposition 3.1 of [8] states the above result for κ = 1. It is straightforward to recover the result for general κ from the proof.…”

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. Except for [2], the methods used to prove (3.1) have relied on the spinor approach generalizing that of Witten [51] in the asymptotically flat setting.…”