The rest mass of an asymptotically Anti-de Sitter spacetime

TL;DR

This paper explores the geometric structure of Killing fields in four-dimensional Anti-de Sitter spacetime, introduces a new concept of rest mass based on observer energy, and establishes its positivity and rigidity properties.

Contribution

It defines the rest mass of asymptotically AdS spacetimes via observer energy minimization and proves its positivity and characterization of AdS spacetime.

Findings

Positivity of observer energy follows from spinor energy positivity.

Rest mass is characterized by the minimal observer energy.

Rest mass has a rigidity property that characterizes AdS spacetime.

Abstract

We study the space of Killing fields on the four dimensional AdS spacetime $AdS^{3,1}$. Two subsets $\mathcal{S}$ and $\mathcal{O}$ are identified: $\mathcal{S}$ (the spinor Killing fields) is constructed from imaginary Killing spinors, and $\mathcal{O}$ (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 $ \mathcal{O}$ is properly contained in that of $\mathcal{S}$. This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. In [10], Chru\'sciel, 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.