High and Low Dimensions in The Black Hole Negative Mode

TL;DR

The paper analyzes how the Schwarzschild black-hole negative mode eigenvalue $\lambda(d)$ depends on spacetime dimension, a problem central to Euclidean quantum gravity and the Gregory-Laflamme instability. It employs two perturbative expansions around $d\to\infty$ and $d\to3^+$, connected via matched asymptotic expansions in the near-horizon region, to derive analytic control of $\lambda(d)$. The leading high-$d$ and low-$d$ behaviors are $\lambda(d) \sim d-4 + 2/d + \dots$ and $\lambda(d) \sim c_1(d-3) + c_2(d-3)^2 + \dots$ with $c_1\approx0.71515$, $c_2\approx0.0627$, respectively. An interpolating rational function, constructed to match both limits, reproduces numerical data with better than 2% accuracy across the full range of dimensions, providing practical formulas for 4d and beyond.

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.

Figures (2)

• Figure 1: The regions along the radial direction at high dimension.
• Figure 2: The regions along the radial direction at low dimension.