A Note on Conserved Charges of Asymptotically Flat and Anti-de Sitter Spaces in Arbitrary Dimensions

TL;DR

This paper systematically derives conserved charges like energy and angular momentum for asymptotically flat and AdS black holes in any dimension, clarifying boundary conditions and symmetry algebras.

Contribution

It provides a Hamiltonian framework for defining conserved charges in arbitrary dimensions, resolving previous ambiguities especially in AdS spaces.

Findings

Explicit formulas for conserved charges of Kerr and Kerr-AdS black holes in any dimension.

Asymptotic symmetry algebra is isomorphic to Poincare or so(D-1,2).

Boundary conditions generalize Regge-Teitelboim parity conditions.

Abstract

The calculation of conserved charges of black holes is a rich problem, for which many methods are known. Until recently, there was some controversy on the proper definition of conserved charges in asymptotically anti-de Sitter (AdS) spaces in arbitrary dimensions. This paper provides a systematic and explicit Hamiltonian derivation of the energy and the angular momenta of both asymptotically flat and asymptotically AdS spacetimes in any dimension D bigger or equal to 4. This requires as a first step a precise determination of the asymptotic conditions of the metric and of its conjugate momentum. These conditions happen to be achieved in ellipsoidal coordinates adapted to the rotating solutions.The asymptotic symmetry algebra is found to be isomorphic either to the Poincare algebra or to the so(D-1, 2) algebra, as expected. In the asymptotically flat case, the boundary conditions involve a generalization of the parity conditions, introduced by Regge and Teitelboim, which are necessary to make the angular momenta finite. The charges are explicitly computed for Kerr and Kerr-AdS black holes for arbitrary D and they are shown to be in agreement with thermodynamical arguments.