Thermodynamics of a black hole with moon

For a Kerr black hole perturbed by a particle on the “corotating” circular orbit (angular velocity equal to that of the unperturbed event horizon), the black hole remains in equilibrium in the sense that the perturbed event horizon is a Killing horizon of the helical Killing field. The associated surface gravity is constant over the horizon and should correspond to the physical Hawking temperature. We calculate the perturbation in surface gravity/temperature, finding it negative: the moon has a cooling effect on the black hole. We also compute the change in horizon angular frequency, which is positive, and the change in surface area/entropy, which vanishes.

Figure 1

Spacetime diagram depicting a black hole tidally perturbed by a corotating moon. (One spatial dimension is not shown.)

Figure 2

Carter-Penrose diagram showing the causal structure of our perturbative spacetime. (Two spatial dimensions are not shown.)

Figure 3

Horizon angular velocity ${\overline{\omega }}_H$ of a Kerr black hole of mass $\overline{M}$ and spin $\overline{J}$ (red) and the perturbation $\mathcal{D}{\omega }_H$ induced by a corotating moon of mass $m$ (blue).

Figure 4

Horizon surface gravity $\overline{\kappa }$ of a Kerr black hole of mass $\overline{M}$ and spin $\overline{J}$ (red) and the perturbation $\mathcal{D}\kappa$ induced by a corotating moon of mass $m$ (blue).