Semiclassical instabilities of Kerr-AdS black holes

TL;DR

The paper studies semiclassical stability of Kerr-AdS black holes by examining one-loop gravitational corrections to the grand-canonical partition function. It constructs and solves the TT perturbation eigenvalue problem $G h^{TT} = \lambda h^{TT}$ on the Kerr-AdS instanton, yielding three coupled PDEs for perturbation functions in axisymmetric form; a single negative mode appears for $|a| \le r_0/2$, with magnitude increasing with rotation, indicating a Gregory-Laflamme-type instability of the related Kerr string. The authors demonstrate that negative modes appear precisely where the thermodynamic susceptibility $C_\Omega$ is negative, with $\beta C_\Omega = - \frac{8 \pi^2 r_+ (r_+^2 + a^2)}{\Xi (1 + a^2 r_+^{-2} + a^2 \ell^{-2} - 3 r_+^2 \ell^{-2})}$, confirming the agreement between quantum-stability analysis and grand-canonical thermodynamics beyond the instanton approximation. The work thus supports the physical relevance of gravitational partition functions in rotating spacetimes and showcases a spectral-method approach to a long-standing instability problem.

Abstract

We study the thermodynamic stability of the Kerr-AdS black hole from the perturbative corrections to the gravitational partition function. The line of critical stability is identified by the appearance of a negative mode of the Euclidean action that renders the partition function ill-defined. The eigenvalue problem, consisting of a system of three coupled partial differential equations for the metric perturbations, is solved numerically. The agreement with the standard condition of thermodynamic stability in the grand canonical ensemble is remarkable. The results illustrate the physical significance of gravitational partition functions for rotating spacetimes beyond the instanton approximation.

Figures (2)

• Figure 1: For the Kerr instanton, $\lambda_*$ is negative, decreasing monotonically away from $a=0$ and evaluating to a finite value at extremality $|a| = r_0/2$.
• Figure 2: Phase diagram of the Kerr-AdS black hole. The points represent the parameter region with a negative mode and the line represents the change of sign of $C_\Omega$, which is negative in the Kerr limit $\ell \to \infty$. The diagram is limited above by the extremality bound and by the singular limit $a=l$.