Thermodynamical Characteristics of "Crystal Lattice" of Many Interacting Kerr Black Holes in Touching Limit

TL;DR

This paper uses approximate methods to analyze the thermodynamics of many interacting Kerr black holes in the touching limit, revealing discrete and continuous spectra of entropy and horizon surface depending on the number of black holes.

Contribution

It generalizes previous results on two black holes to many black holes using a simple, quasi-classical approach based on horizon circumference quantization.

Findings

Entropy and horizon surface become discrete spectra as the number of black holes increases.

For a single black hole, entropy and horizon surface tend toward continuous spectra.

The approach reproduces and extends known thermodynamical characteristics of interacting Kerr black holes.

Abstract

In this work, starting by simple, approximate (quasi-classical) methods presented in our previous works, we reproduce effectively and generalize final results of Herdeiro and Rebelo on the basic thermodynamical characteristics (entropy and temperature) of two interacting Kerr black holes (in touching limit) obtained recently by accurate analysis. Like as it has been done in our previous works, we simply suppose that circumference of the horizon of total black hole (that includes two or, generally, a "crystal lattice" of many interacting Kerr black holes in touching limit, without angular momentum) holds integer number of reduced Compton wave lengths corresponding to mass spectrum of a small quantum system captured at horizon. (Obviously it is conceptually analogous to Bohr quantization postulate interpreted by de Broglie relation in Old, Bohr-Sommerfeld, quantum theory.) It, by simple mathematical methods, first neighbour approximation of the black holes interaction and first thermodynamical law, implies mentioned basic thermodinamical characteristic of the total black hole as well as any its part, i.e. single black hole. Especially, it is shown that, in limit of increasing number of the black holes, entropy and horizon surface of the total black hole stand observables of the discrete spectrum while entropy and horizon surface of the single black hole tends toward observables of the continuous spectrum.