Black Hole Thermodynamics and Riemann Surfaces

TL;DR

This paper explores the thermodynamics of 2+1 dimensional black holes using Riemann surfaces and hyperbolic 3-manifolds, linking black hole characteristics to conformal moduli and proposing a new entropy bound.

Contribution

It introduces a novel approach connecting black hole thermodynamics with the geometry of Riemann surfaces and Schottky space, and proposes a new entropy bound based on the Kaehler potential.

Findings

Thermodynamic partition function relates to the Weil-Peterson metric's Kaehler potential.

Black hole parameters are encoded in the conformal moduli of boundary Riemann surfaces.

A conjectured strong bound on the Kaehler potential from Bekenstein entropy bound.

Abstract

We use the analytic continuation procedure proposed in our earlier works to study the thermodynamics of black holes in 2+1 dimensions. A general black hole in 2+1 dimensions has g handles hidden behind h horizons. The result of the analytic continuation is a hyperbolic 3-manifold having the topology of a handlebody. The boundary of this handlebody is a compact Riemann surface of genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the physical characteristics of the black hole. The moduli space of black holes of a given type (g,h) is then the Schottky space at genus G. The (logarithm of the) thermodynamic partition function of the hole is the Kaehler potential for the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black hole entropy leads us to conjecture a new strong bound on this Kaehler potential.