Topological deformation of isolated horizons

We show that the Gauss-Bonnet term can have physical effects in four dimensions. Specifically, the entropy of a black hole acquires a correction term that is proportional to the Euler characteristic of the cross sections of the horizon. While this term is constant for a single black hole, it will be a nontrivial function for a system with dynamical topologies such as black-hole mergers: it is shown that for certain values of the Gauss-Bonnet parameter, the second law of black-hole mechanics can be violated.

Figure 1

The region of the four-dimensional spacetime $\mathcal{M}$ being considered has an internal boundary $\Delta$ representing the event horizon, and is bounded by two (partial) Cauchy surfaces $M^{\pm }$ which intersect $\Delta$ in two-surfaces ${\mathfrak{S}}^{\pm }$ and extend to the boundary at infinity $\mathfrak{B}$.