The Power of Action: "The" Derivation of the Black Hole Negative Mode

TL;DR

The paper tackles the negative mode of the Schwarzschild black hole, a central feature in Euclidean quantum gravity around hot flat space and in the Gregory-Laflamme instability, by deriving it from a fully action-based, gauge-invariant framework. It postpones gauge fixing and uses a maximally general ansatz to obtain a canonical, gauge-invariant master field formulation with a Schrödinger-type potential $V(r)$ and a master field $\widetilde{c}$, yielding the equation $-\triangle \widetilde{c} + V(r)\widetilde{c} = -k^2\widetilde{c}$. The analysis reveals a single negative eigenvalue $\lambda_{GPY} \\equiv k_{GL}^2$, aligning with GPY/GL results but in a clearer, gauge-free framework. The work further generalizes to perturbations of arbitrary co-homogeneity-1 geometries, outlining a canonical action structure with gauge functions and dynamic gauge-invariant fields, and suggesting extensions beyond quadratic order.

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. Numerous gauges were employed in the past to analyze it. Here _the_ analytic derivation is found, based on postponing the gauge fixing, on the power of the action and on decoupling of non-dynamic fields. A broad-range generalization to perturbations around arbitrary co-homogeneity 1 geometries is discussed.

Figures (1)

• Figure 1: A graph of the perturbation potential $V$ as a function of the radial coordinate $r$ (\ref{['Vschw']}) in $r_0=1$ units. Drawn at $d=4$ -- in higher dimensions $V$ becomes even more concentrated around $r/r_0=1$.