Dynamic and Thermodynamic Stability and Negative Modes in Schwarzschild-Anti-de Sitter

TL;DR

This work demonstrates a precise, gauge-invariant link between dynamic stability (presence/absence of a nonconformal negative mode) and local thermodynamic stability (sign of the heat capacity) for Schwarzschild–anti-de Sitter black holes in finite isothermal cavities, and extends the analysis to higher dimensions. By solving the eigenvalue problem for the gauge-invariant spin-2 operator $G$ on transverse-trace-free perturbations, the authors show a single negative mode in 4D that vanishes at the critical horizon radius, and they establish the same correspondence in higher dimensions. In the infinite-cavity AdS limit, this framework yields results consistent with Hawking–Page transitions and supports the Horowitz–Myers positive-energy conjecture, strengthening the link between black hole thermodynamics and Euclidean quantum gravity in AdS. The findings have implications for stability analyses in AdS/CFT contexts and for understanding tachyonic instabilities in periodically identified spacetimes.

Abstract

The thermodynamic properties of Schwarzschild-anti-de Sitter black holes confined within finite isothermal cavities are examined. In contrast to the Schwarzschild case, the infinite cavity limit may be taken which, if suitably stated, remains double valued. This allows the correspondence between non-existence of negative modes for classical solutions and local thermodynamic stability of the equilibrium configuration of such solutions to be shown in a well defined manner. This is not possible in the asymptotically flat case. Furthermore, the non-existence of negative modes for the larger black hole solution in Schwarzschild-anti-de Sitter provides strong evidence in favour of the recent positive energy conjecture by Horowitz and Myers.

Figures (5)

• Figure 1: A constant temperature slice through the $(T_B,r_B,r_+)$ surface defined by (\ref{['sch_surface']}) for a Schwarzschild black hole confined within an ideal isothermal cavity. Also shown are the lines $r_+ = r_B$ and $2 r_B / 3$, between which the heat capacity $C_A$ is positive. The broken line is $r_+ = 8 r_B / 9$, above which the free energy is negative.
• Figure 2: A constant temperature slice through the $(bT_B,\rho_B,\rho_+)$ surface defined by (\ref{['sch_ads_surface']}) for a Schwarzschild black hole in anti-de Sitter space confined within an ideal isothermal cavity. Also shown are the lines $\rho_+ = \rho_B$ and $C_0(\rho_B)$, between which the heat capacity $C_A$ is positive. Note that $C_0(\rho_B) \rightarrow 1$ in the limit of large $\rho_B$. The broken line is $\rho_+ = F_0(\rho_B)$, above which the free energy is negative.
• Figure 3: The black hole solution curve in the $(\rho_+,\beta_{\star})$ plane, in the limit of infinite isothermal cavity radius. The turning point $\beta_{\mathrm{max}} = 1/T_0$ occurs at $\rho_+ = 1 \Leftrightarrow r_+ = b / \sqrt{3}.$
• Figure 4: Numerically generated results for $\tilde{\lambda}_{\mathrm{neg}}$ (vertical) against $\rho_+$ in $n=4$ dimensions Schwarzschild-anti-de Sitter.
• Figure 5: Numerically generated results for $\tilde{\lambda}_{\mathrm{neg}}$ (vertical) against $\rho_+$ in $n=5$ and $n=7$ dimensions Schwarzschild-anti-de Sitter. The spectrum for $n=7$ gives the 'upper' curve (tending to 5), and for $n=5$ the 'lower' curve (tending to 3).