Thermodynamic instability of rotating black holes

TL;DR

This work shows that quasi-Euclidean sections of rotating black holes possess gauge-invariant negative modes whenever thermodynamic instability is expected in the grand-canonical ensemble. By constructing a TT probe perturbation from Killing vectors and evaluating a Rayleigh-Ritz functional, the authors establish negative modes for Kerr, Myers-Perry (5D and 6D), and singly-spinning black rings, while Kerr-AdS exhibits a nuanced pattern near the critical stability line. The thermodynamic interpretation ties these modes to the Weinhold metric, requiring positive specific heat at constant angular momentum $C_J$ and positive eigenvalues of the isothermal inertia tensor $\epsilon^{ij}$, with subtle violations indicating limits of the thermodynamic description and possible quantum corrections. The results validate the use of quasi-Euclidean instantons for gravitational partition functions and illuminate the link between perturbative gravitational stability and macroscopic thermodynamics, with implications for AdS/CFT and black-hole phase structure.

Abstract

We show that the quasi-Euclidean sections of various rotating black holes in different dimensions possess at least one non-conformal negative mode when thermodynamic instabilities are expected. The boundary conditions of fixed induced metric correspond to the partition function of the grand-canonical ensemble. Indeed, in the asymptotically flat cases, we find that a negative mode persists even if the specific heat at constant angular momenta is positive, since the stability in this ensemble also requires the positivity of the isothermal momentum of inertia. We focus in particular on Kerr black holes, on Myers-Perry black holes in five and six dimensions, and on the Emparan-Reall black ring solution. We go on further to consider the richer case of the asymptotically AdS Kerr black hole in four dimensions, where thermodynamic stability is expected for a large enough cosmological constant. The results are consistent with previous findings in the non-rotation limit and support the use of quasi-Euclidean instantons to construct gravitational partition functions.

Figures (11)

• Figure 1: For the Kerr instanton, $\mathcal{I}$ is negative, decreasing monotonically away from $a=0$ and evaluating to $-12$$r_0^{-2}$ at extremality $|a| = r_0/2$.
• Figure 2: For the five dimensional Myers-Perry instanton, $\mathcal{I}$ is always negative. The dashed lines represent contours of constant $\mathcal{I}r_0^2$.
• Figure 3: $\mathcal{I}$ is negative across the entire range.
• Figure 4: For the six dimensional Myers-Perry instanton, $\mathcal{I}$ is always negative. The dashed lines represent contours of constant $\mathcal{I}r_0^2$.
• Figure 5: Phase diagram of the Kerr-AdS black hole.