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The evolution of unstable black holes in anti-de Sitter space

Steven S. Gubser, Indrajit Mitra

TL;DR

This work ties local thermodynamic instability to dynamical instability for charged AdS black holes, showing large AdS$_4$-RN holes become dynamically unstable precisely when the entropy Hessian signals instability. It derives and numerically solves a fourth-order ODE for equal-charge perturbations, confirming a dynamical mode that tracks the thermodynamic instability boundary. The analysis argues that adiabatic evolution tends toward charge clumping without violating cosmic censorship, while also highlighting the potential for non-uniform stationary end-states or curvature growth. Overall, the study furnishes a concrete link between Hessian-based stability and Lorentzian dynamics in AdS, with implications for no-hair theorems and cosmic censorship in holographic settings.

Abstract

We examine the thermodynamic stability of large black holes in four-dimensional anti-de Sitter space, and we demonstrate numerically that black holes which lack local thermodynamic stability often also lack stability against small perturbations. This shows that no-hair theorems do not apply in anti-de Sitter space. A heuristic argument, based on thermodynamics only, suggests that if there are any violations of Cosmic Censorship in the evolution of unstable black holes in anti-de Sitter space, they are beyond the reach of a perturbative analysis.

The evolution of unstable black holes in anti-de Sitter space

TL;DR

This work ties local thermodynamic instability to dynamical instability for charged AdS black holes, showing large AdS-RN holes become dynamically unstable precisely when the entropy Hessian signals instability. It derives and numerically solves a fourth-order ODE for equal-charge perturbations, confirming a dynamical mode that tracks the thermodynamic instability boundary. The analysis argues that adiabatic evolution tends toward charge clumping without violating cosmic censorship, while also highlighting the potential for non-uniform stationary end-states or curvature growth. Overall, the study furnishes a concrete link between Hessian-based stability and Lorentzian dynamics in AdS, with implications for no-hair theorems and cosmic censorship in holographic settings.

Abstract

We examine the thermodynamic stability of large black holes in four-dimensional anti-de Sitter space, and we demonstrate numerically that black holes which lack local thermodynamic stability often also lack stability against small perturbations. This shows that no-hair theorems do not apply in anti-de Sitter space. A heuristic argument, based on thermodynamics only, suggests that if there are any violations of Cosmic Censorship in the evolution of unstable black holes in anti-de Sitter space, they are beyond the reach of a perturbative analysis.

Paper Structure

This paper contains 10 sections, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The $AdS_4$-RN solution and its cousins
  3. Thermodynamics
  4. Generalities
  5. Explicit formulas
  6. Adiabatic evolution
  7. Existence of a dynamical instability
  8. Dyadic index derivation of (\ref{['FinalODE']})
  9. Numerical results from the fourth order equation
  10. Conclusions

Figures (4)

  • Figure 1: An example of a mass function whose convex hull is flat. The region we interpret as stable is from $A$ to $B$.
  • Figure 2: Plots of the most unstable eigenvector of the Hessian matrix of $S(M,Q_1,Q_2,Q_3,Q_4)$. The inner curves are boundaries of stability. The outer curves (when they are present) denote the boundary between regular black branes and naked singularities.
  • Figure 3: The Penrose diagram of a regular AdS black hole. We can take $T=R=P_+=P_-=0$ at the center of the diagram. The black hole horizon is the diagonal line going up and right from the origin.
  • Figure 4: (a) A topologically correct representation of dynamical and thermodynamic stability in the whole $\chi$-$\sigma$ plane (but see the text regarding possible overlap of the two shaded regions). (b) A sample normalizable wave-function with negative $\omega^2$: here $\sigma = 0.3$, $\chi = 0.96$, and $\tilde{\omega}^2 = -0.281$.