Thermodynamics of black holes in finite boxes

TL;DR

This paper investigates the thermodynamics of black holes in finite boxes, analyzing mass evolution, equilibrium states, and entropy, revealing counter-intuitive relationships between black hole mass and box volume, with implications for thermodynamics education.

Contribution

It provides a comprehensive analysis of black hole thermodynamics in finite boxes, including multiple black holes, radiation interactions, and the effects of radiation reflection, with novel insights into equilibrium behavior.

Findings

Black holes can lose all mass above a minimal box volume.

Equilibrium masses are independent of initial black hole masses.

Equilibrium mass scales as a positive power of the box volume.

Abstract

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to $N$ black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a {\it positive} power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.