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

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 explicitly proved. It is also shown that the charges for higher dimensional Kerr-AdS black holes can be correctly computed from the standard Hamiltonian or Lagrangian surface integrals.

Figure 1

In the example of the four-dimensional Kerr black holes, the solution space is parametrized 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 $Q_{\xi }$.