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The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times

Piotr T. Chrusciel, Gabriel Nagy

TL;DR

The paper develops a Hamiltonian framework to define the total mass of spacelike hypersurfaces in spacetimes asymptotically anti-de Sitter. It proves convergence and background-independence of the mass integrals under precise fall-off conditions, and establishes their covariance under admissible coordinate changes and asymptotic isometries. The authors show that, for AdS backgrounds, the mass reduces to Abbott–Deser-type charges and construct a spectrum of global geometric invariants (including tensorial and angular-momentum-type quantities) from Killing vectors of the background. This yields a robust, coordinate- and background-independent notion of total mass and an organized set of conserved charges for a wide class of asymptotically AdS spacetimes, with concrete connections to Kottler metrics and invariance questions in the Riemannian problem.

Abstract

We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time.

The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times

TL;DR

The paper develops a Hamiltonian framework to define the total mass of spacelike hypersurfaces in spacetimes asymptotically anti-de Sitter. It proves convergence and background-independence of the mass integrals under precise fall-off conditions, and establishes their covariance under admissible coordinate changes and asymptotic isometries. The authors show that, for AdS backgrounds, the mass reduces to Abbott–Deser-type charges and construct a spectrum of global geometric invariants (including tensorial and angular-momentum-type quantities) from Killing vectors of the background. This yields a robust, coordinate- and background-independent notion of total mass and an organized set of conserved charges for a wide class of asymptotically AdS spacetimes, with concrete connections to Kottler metrics and invariance questions in the Riemannian problem.

Abstract

We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time.

Paper Structure

This paper contains 9 sections, 12 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Convergence, covariance under well behaved coordinate transformations
  3. Asymptotic isometries - the Riemannian problem
  4. Global charges
  5. The phase space and the Hamiltonians
  6. Isometries and Killing vectors of the background
  7. $(n+1)$-dimensional anti-de Sitter metrics
  8. $h$'s with a non-positive Ricci tensor
  9. Equality of the Hamiltonian mass with the Abbott-Deser one

Key Result

Theorem 2.1

Let $X$ be a Killing vector of an Einstein metric $b$, set where $\mathring{\nabla}$ is the covariant derivative of $b$; the indices here refer to a $b$-orthonormal frame such that $e_0$ is normal to the hypersurface $t=0$. Suppose that $\lim_{r\to\infty} e^{ab}=0$ and that where $d\mu_b\equiv \sqrt{\det b_{ij}}dr\; dv^2\ldots dv^{n}$ is the Riemannian measure induced on $\{t=0\}$ by $b$. Then t

Theorems & Definitions (12)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3: Asymptotic symmetries
  • Lemma 3.4
  • Corollary 3.5
  • Proposition B.1
  • ...and 2 more