Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability and the negative mode for Schwarzschild in a finite cavity

James P. Gregory, Simon F. Ross

TL;DR

Addresses the Gubser–Mitra conjecture that classical stability of black branes with translational symmetry coincides with local thermodynamic stability, by studying an uncharged Schwarzschild brane in a finite cavity. The authors relate the Gregory–Laflamme threshold to the Euclidean negative mode and extend York’s canonical-ensemble thermodynamics to $d$ dimensions, showing the onset of instability occurs precisely when the specific heat $C_A$ becomes negative. In 4D they solve the Euclidean negative-mode problem with an isothermal boundary, finding a critical wall radius $\rho_b \approx 3M$ under the true boundary condition, and they generalize to $d>4$ where onset is governed by a scaled parameter $\tilde{\rho}_b>1$, consistent with $C_A<0$ across dimensions. Overall, the work provides explicit, finite-volume validation of the conjecture and clarifies how boundary conditions influence classical instability via the Euclidean negative mode.

Abstract

It has been proposed that translationally-invariant black branes are classically stable if and only if they are locally thermodynamically stable. Reall has outlined a general argument to demonstrate this, and studied in detail the case of charged black p-branes in type II supergravity. We consider the application of his argument in the simplest non-trivial case, an uncharged asymptotically flat brane enclosed in a finite cylindrical cavity. In this simple context, it is possible to give a more complete argument than in the cases considered earlier, and it is therefore a particularly attractive example of the general approach.

Stability and the negative mode for Schwarzschild in a finite cavity

TL;DR

Addresses the Gubser–Mitra conjecture that classical stability of black branes with translational symmetry coincides with local thermodynamic stability, by studying an uncharged Schwarzschild brane in a finite cavity. The authors relate the Gregory–Laflamme threshold to the Euclidean negative mode and extend York’s canonical-ensemble thermodynamics to dimensions, showing the onset of instability occurs precisely when the specific heat becomes negative. In 4D they solve the Euclidean negative-mode problem with an isothermal boundary, finding a critical wall radius under the true boundary condition, and they generalize to where onset is governed by a scaled parameter , consistent with across dimensions. Overall, the work provides explicit, finite-volume validation of the conjecture and clarifies how boundary conditions influence classical instability via the Euclidean negative mode.

Abstract

It has been proposed that translationally-invariant black branes are classically stable if and only if they are locally thermodynamically stable. Reall has outlined a general argument to demonstrate this, and studied in detail the case of charged black p-branes in type II supergravity. We consider the application of his argument in the simplest non-trivial case, an uncharged asymptotically flat brane enclosed in a finite cylindrical cavity. In this simple context, it is possible to give a more complete argument than in the cases considered earlier, and it is therefore a particularly attractive example of the general approach.

Paper Structure

This paper contains 6 sections, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Schwarzschild in a finite cavity
  3. Finding the Negative Mode
  4. Four dimensions
  5. Higher dimensions
  6. Conclusions

Figures (3)

  • Figure 1: $\lambda M^2$ vs $\rho_b / M$ for the boundary condition (\ref{['eqn:Allenbc']})
  • Figure 2: $\lambda M^2$ vs $\rho_b / M$ for the boundary condition (\ref{['eqn:truebc']})
  • Figure 3: $\lambda (\omega M)^{\frac{2}{d-3}}$ vs $\tilde{\rho}_b$ for $d=5$ (closest to axis) to $d=10$ (furthest from axis) dimensions