Thermodynamics of a Black Hole with Moon

TL;DR

This paper analyzes how a small orbiting particle (moon) affects a Kerr black hole's thermodynamic properties, revealing a cooling effect and changes in angular frequency without altering entropy.

Contribution

It provides the first detailed calculation of surface gravity perturbation and horizon changes for a Kerr black hole perturbed by a co-rotating particle.

Findings

Perturbation in surface gravity is negative, indicating a cooling effect.

Horizon angular frequency increases due to the perturbation.

Surface area and entropy remain unchanged.

Abstract

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.