Thermodynamic and gravitational instability on hyperbolic spaces

TL;DR

This work analyzes anti–de Sitter black holes with Gauss–Bonnet corrections across horizon topologies, focusing on the hyperbolic case k = −1 and the role of extremal states as ground states. It establishes that extremal hyperbolic AdS black holes are local energy minima and examines thermodynamic stability via background subtraction, specific heat, and free energy, revealing bounds on the GB coupling α for stability. The gravitational stability analysis extends to D > 4 with tensor perturbations, showing extremal GB backgrounds can be stable for small α, and that negative-mass extremal states may be stabilized by GB corrections. The results indicate that ground-state choice, GB corrections, and horizon topology together shape the thermodynamic phase structure and dynamic stability of AdS black holes, with implications for AdS/CFT and holographic interpretations.

Abstract

We study the properties of anti--de Sitter black holes with a Gauss-Bonnet term for various horizon topologies (k=0, \pm 1) and for various dimensions, with emphasis on the less well understood k=-1 solution. We find that the zero temperature (and zero energy density) extremal states are the local minima of the energy for AdS black holes with hyperbolic event horizons. The hyperbolic AdS black hole may be stable thermodynamically if the background is defined by an extremal solution and the extremal entropy is non-negative. We also investigate the gravitational stability of AdS spacetimes of dimensions D>4 against linear perturbations and find that the extremal states are still the local minima of the energy. For a spherically symmetric AdS black hole solution, the gravitational potential is positive and bounded, with or without the Gauss-Bonnet type corrections, while, when k=-1, a small Gauss-Bonnet coupling, namely, α<< {l}^2 (where l is the curvature radius of AdS space), is found useful to keep the potential bounded from below, as required for stability of the extremal background.

Figures (12)

• Figure 1: The ADM mass $M$ (curved lines) and extremal mass $M_{extr}$ (horizontal lines) as functions of the horizon. The values are fixed at $l=1$, $16\pi G=1$, and (upper plot) $n=4$, $V_3=2\pi^2$, $k=-1$, and $\alpha=1/4,~1/12,~0$ (top to bottom); (lower plot) $n=6$, $V_3=\pi^3$, $k=-1$, and $\alpha=1/4,\,17/100,\,0$ (top to bottom).
• Figure 2: The energy $E$ as a function of horizon ($r_+$). The values are fixed at $l=1$, $16\pi G=1$, and $n=4$ (upper plot), $V_3=2\pi^2$, $k=+1$ (curves that grow with $r_+$) at $\alpha=1/4, 1/12$ (upper to lower), and $k=-1$ (curve that has minimum at $r_+=r_{extr}$); $n=6$ (lower plot), $V_3=\pi^3$, $k=-1$, and $\alpha=1/4, 17/100, 0$ (top to bottom along the $E$-axis).
• Figure 3: The Euclidean period $\beta$ (curves that asymptote to the $C$ or/and $r_+$ axes) and specific heat (curves with one or more cusps) vs horizon position $r_+$. The values are fixed at $l=1$, $n=4$, $V_3/16\pi G=2\pi^2$, $k=-1$, and $\alpha=1/4$, $\alpha=1/8$ and $\alpha=1/12$ (top to bottom plots).
• Figure 4: The Euclidean period (curves that asymptote to the $C$ or/and $r_+$ axes) and specific heat (curves with two or more cusps) vs horizon position $r_+$. The values are fixed at $l=1$, $n=6$, $V_5/16\pi G=\pi^3$, $k=-1$, and $\alpha=0.25$, $\alpha=0.23$, and $\alpha=17/100$ (top to bottom plots).
• Figure 5: The Euclidean period (curves that asymptote to the $F$ or $r_+$ axis) and free energy (curves that are bounded from above or take finite values at $r_+=0$) vs horizon position $r_+$. The values are fixed at $k=-1$, $n=4$, $l=1$, $16\pi G=1$, $V_3=2\pi^2$, and, $\alpha=1/6,\,0.1305,\,1/8$ (top to bottom plots).