High and Low Dimensions in The Black Hole Negative Mode

TL;DR

This paper investigates the negative mode of Schwarzschild black holes across various dimensions by developing perturbative expansions for large and small dimensions, resulting in an accurate interpolating function.

Contribution

It introduces two perturbative expansions for the negative mode eigenvalue at large and small dimensions and combines them into a precise interpolating function.

Findings

Derived analytical coefficients for eigenvalue expansions

Created an accurate interpolation covering all dimensions including d=4

Enhanced understanding of black hole stability across dimensions

Abstract

The negative mode of the Schwarzschild black hole is central to Euclidean quantum gravity around hot flat space and for the Gregory-Laflamme black string instability. We analyze the eigenvalue as a function of space-time dimension by constructing two perturbative expansions: one for large d and the other for small d-3, and determining as many coefficients as we are able to compute analytically. Joining the two expansions we obtain an interpolating rational function accurate to better than 2% through the whole range of dimensions including d=4.