Towards the classification of static vacuum spacetimes with negative cosmological constant

TL;DR

This paper systematically studies static vacuum solutions with negative cosmological constant, establishing mass and horizon area inequalities, and discusses implications for the uniqueness of generalized Kottler black holes.

Contribution

It provides explicit mass formulas, connectedness of conformal infinity, and inequalities relating mass and horizon area for static solutions with negative cosmological constant.

Findings

Connectedness of conformal infinity for regular solutions.

Explicit Hamiltonian and Hawking mass formulas.

Inequalities relating mass and horizon area, opposing the generalized Penrose conjecture.

Abstract

We present a systematic study of static solutions of the vacuum Einstein equations with negative cosmological constant which asymptotically approach the generalized Kottler (``Schwarzschild--anti-de Sitter'') solution, within (mainly) a conformal framework. We show connectedness of conformal infinity for appropriately regular such space-times. We give an explicit expression for the Hamiltonian mass of the (not necessarily static) metrics within the class considered; in the static case we show that they have a finite and well defined Hawking mass. We prove inequalities relating the mass and the horizon area of the (static) metrics considered to those of appropriate reference generalized Kottler metrics. Those inequalities yield an inequality which is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler black holes if the generalized Penrose inequality can be established.