Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards the classification of static vacuum spacetimes with negative cosmological constant

Piotr T. Chrusciel, Walter Simon

TL;DR

This work develops a systematic framework for static vacuum spacetimes with negative cosmological constant that are asymptotically generalized Kottler. It unifies three asymptotic formalisms (3D conformal compactification, 4D conformal completion, and coordinate approaches), proves connectedness of conformal infinity, and defines multiple mass notions—coordinate mass $M_c$, Hamiltonian mass $M_{Ham}$, and Hawking mass $M_{Haw}$—with explicit relations among them. It derives mass and area inequalities relative to generalized Kottler references, discusses a generalized Penrose inequality and its implications for uniqueness, and provides a rigorous maximum-principle framework (via Beig–Simon-type lemmas) to compare arbitrary solutions to reference ones. The results illuminate the structure of AdS-ish static spacetimes, connect asymptotic data to horizon properties, and offer a pathway toward a Penrose-type uniqueness theorem in the negative-$\Lambda$ setting.

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.

Towards the classification of static vacuum spacetimes with negative cosmological constant

TL;DR

This work develops a systematic framework for static vacuum spacetimes with negative cosmological constant that are asymptotically generalized Kottler. It unifies three asymptotic formalisms (3D conformal compactification, 4D conformal completion, and coordinate approaches), proves connectedness of conformal infinity, and defines multiple mass notions—coordinate mass , Hamiltonian mass , and Hawking mass —with explicit relations among them. It derives mass and area inequalities relative to generalized Kottler references, discusses a generalized Penrose inequality and its implications for uniqueness, and provides a rigorous maximum-principle framework (via Beig–Simon-type lemmas) to compare arbitrary solutions to reference ones. The results illuminate the structure of AdS-ish static spacetimes, connect asymptotic data to horizon properties, and offer a pathway toward a Penrose-type uniqueness theorem in the negative- setting.

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.

Paper Structure

This paper contains 17 sections, 19 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. The generalized Kottler solutions
  3. $k=-1$
  4. Asymptotics
  5. Three dimensional formalism
  6. Four dimensional conformal approach
  7. A coordinate approach
  8. Connectedness of $\partial_{\infty}\Sigma$
  9. The mass
  10. A coordinate mass $M_c$
  11. The Hamiltonian mass $M_{Ham}$.
  12. A generalized Komar mass
  13. The Hawking mass $M_{Haw}(\psi)$
  14. The generalized Penrose inequality
  15. Mass and area inequalities
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Lambda<0$, consider a set $(\Sigma,g,V)$ which is $C^3$ conformally compactifiable in the sense of Definition Dc below, suppose that (f0)--(f2) hold. Then the conformal boundary at infinity $\partial_{\infty}\Sigma$ of $\Sigma$ is connected.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 10 more