On Conserved Charges of Asymptotically Flat and Anti-de Sitter Spacetimes in Arbitrary Dimensions

Abstract

Energy, and more generally conserved charges, is a subtle issue in general relativity and it is subject to controversy. Indeed, there aremany different definitions in the literature and in some cases they do not all give the same charges. For example, as emphasized byGibbons, et al. (2005), not even in four dimensions do all authors obtain the same expression for the energy of Kerr-AdS black holes and some of these expressions are in disagreement with the first law of black hole thermodynamics. I adopted in [arXiv:0705.0484] the method formulated by Regge & Teitelboim (1974). Based on the Hamiltonian formulation of gravitation, it associates to each asymptotic Killing vector a conserved charge without having to make any arbitrary choice. The charge is then expressed as a surface integral over a 2-sphere at spatial infinity. I investigate here conserved charges and asymptotic symmetries in both asymptotically flat and AdS spaces in arbitrary spacetime dimensions. The study is presented in general coordinates, including the ellipsoidal coordinates that are well adapted to the Kerr solution.