Rigid upper bounds for the angular momentum and centre of mass of non-singular asymptotically anti-de Sitter space-times

TL;DR

This paper establishes upper bounds on angular momentum and center of mass for non-singular asymptotically anti-de Sitter space-times, relating them to the Hamiltonian mass and cosmological constant across various dimensions and boundary conditions.

Contribution

It provides new geometric inequalities for asymptotically anti-de Sitter space-times, including a comprehensive analysis of the borderline cases and examples saturating the bounds.

Findings

Spherical cross-sections in 4D AdS space only saturate bounds in pure AdS.

Toroidal cases admit regular initial data saturating the bounds.

Inequalities hold under broad conditions, including various asymptotic backgrounds.

Abstract

We prove upper bounds on angular momentum and centre of mass in terms of the Hamiltonian mass and cosmological constant for non-singular asymptotically anti-de Sitter initial data sets satisfying the dominant energy condition. We work in all space-dimensions larger than or equal to three, and allow a large class of asymptotic backgrounds, with spherical and non-spherical conformal infinities; in the latter case, a spin-structure compatibility condition is imposed. We give a large class of non-trivial examples saturating the inequality. We analyse exhaustively the borderline case in space-time dimension four: for spherical cross-sections of Scri, equality together with completeness occurs only in anti-de Sitter space-time. On the other hand, in the toroidal case, regular non-trivial initial data sets saturating the bound exist.