Generalized Smarr relation for Kerr AdS black holes from improved surface integrals

Abstract

By using suitably improved surface integrals, we give a unified geometric derivation of the generalized Smarr relation for higher dimensional Kerr black holes which is valid both in flat and in anti-de Sitter backgrounds. The improvement of the surface integrals, which allows one to use them simultaneously at infinity and on the horizon, consists in integrating them along a path in solution space. Path independence of the improved charges is discussed and explicitly proved for the higher dimensional Kerr AdS black holes. It is also shown that the charges for these black holes can be correctly computed from the standard Hamiltonian or Lagrangian surface integrals.

Figures (1)

• Figure 1: In the example of the four-dimensional Kerr black holes, the solution space is parameterized by the mass $M$ and rotation parameter $a$. One can for instance use the diagonal path $sM,sa$, $s\in[0,1]$ for the evaluation of the charges $Q_\xi$.