Computing black hole partition functions from quasinormal modes

TL;DR

This paper introduces a numerical method to compute one-loop determinants in black hole spacetimes using quasinormal modes, with a regularization scheme to handle divergences, demonstrated on the BTZ black hole.

Contribution

It presents a novel numerical approach leveraging quasinormal frequencies and a refined heat kernel technique for calculating black hole partition functions.

Findings

Successfully reproduces scalar one-loop determinant for BTZ black hole

Provides a regularization scheme for divergent quasinormal mode sums

Discusses extension to more complex spacetimes

Abstract

We propose a method of computing one-loop determinants in black hole spacetimes (with emphasis on asymptotically anti-de Sitter black holes) that may be used for numerics when completely-analytic results are unattainable. The method utilizes the expression for one-loop determinants in terms of quasinormal frequencies determined by Denef, Hartnoll and Sachdev in \cite{Denef:2009kn}. A numerical evaluation must face the fact that the sum over the quasinormal modes, indexed by momentum and overtone numbers, is divergent. A necessary ingredient is then a regularization scheme to handle the divergent contributions of individual fixed-momentum sectors to the partition function. To this end, we formulate an effective two-dimensional problem in which a natural refinement of standard heat kernel techniques can be used to account for contributions to the partition function at fixed momentum. We test our method in a concrete case by reproducing the scalar one-loop determinant in the BTZ black hole background. We then discuss the application of such techniques to more complicated spacetimes.