Euler numbers of four-dimensional rotating black holes with the Euclidean signature

Abstract

For a black hole's spacetime manifold in the Euclidean signature, its metric is positive definite and therefore a Riemannian manifold. It can be regarded as a gravitational instanton and a topological characteristic which is the Euler number to which it is associated. In this paper we derive a formula for the Euler numbers of four-dimensional rotating black holes by the integral of the Euler density on the spacetime manifolds of black holes. Using this formula, we obtain that the Euler numbers of Kerr and Kerr-Newman black holes are $2.$ We also obtain that the Euler number of the Kerr-Sen metric in the heterotic string theory with one boost angle nonzero is $2,$ which is in accordance with its topology.