# Stability of Black Holes and Black Branes

**Authors:** Stefan Hollands, Robert M. Wald  
**arXiv:** [1201.0463](https://arxiv.org/abs/1201.0463)  
**Date:** 2012-01-02  
**Summary page:** https://www.summarizepaper.com/en/paper/1201.0463v1/

## Key points

- Authors investigate dynamical stability of black holes in spacetime dimensions ≥ 4
- Propose new criterion for dynamical stability based on positive canonical energy on subspace called T
- T consists of linearized solutions with vanishing linearized ADM mass, momentum, and angular momentum at infinity, and satisfying certain gauge conditions at the horizon
- Canonical energy E is nondegenerate on T except for pure gauge perturbations and perturbations towards other stationary black holes
- E exhibits positive flux properties both at infinity and the horizon for axisymmetric perturbations
- Establish close connection between dynamical stability and thermodynamic stability through properties of E
- Canonical energy E can be expressed as δ^2M - ∑A Ω_A δ^2J_A - κ/8π δ^2A, where M represents mass, J_A represents angular momentum associated with different rotational symmetries A, Ω_A represents corresponding rotation rates, and κ represents surface gravity
- For thermodynamically unstable black branes, sufficiently long wavelength perturbations can be found with E < 0 and vanishing linearized ADM quantities
- Result confirms conjecture that all black branes corresponding to thermodynamically unstable black holes are dynamically unstable
- Fulfillment of local Penrose inequality is necessary and sufficient for dynamical stability

## Summary

In their paper "Stability of Black Holes and Black Branes," authors Stefan Hollands and Robert M. Wald investigate the dynamical stability of black holes in spacetime dimensions greater than or equal to four in general relativity, specifically with respect to axisymmetric perturbations. They propose a new criterion for dynamical stability, which states that a black hole is dynamically stable if the canonical energy, denoted as $\E$, is positive on a subspace called $\mathcal T$. This subspace consists of linearized solutions that have vanishing linearized ADM (Arnowitt-Deser-Misner) mass, momentum, and angular momentum at infinity, as well as satisfying certain gauge conditions at the horizon. The authors demonstrate that apart from pure gauge perturbations and perturbations towards other stationary black holes, the canonical energy $\E$ is nondegenerate on $\mathcal T$. Furthermore, they show that for axisymmetric perturbations, $\E$ exhibits positive flux properties both at infinity and the horizon. By establishing these properties of $\E$, they establish a close connection between dynamical stability and thermodynamic stability. The paper also addresses the relationship between the canonical energy $\E$ and second-order variations of mass, angular momentum, and horizon area. The authors find that $\E$ can be expressed as $\delta^2 M - \sum_A \Omega_A \delta^2 J_A - \frac{\kappa}{8\pi} \delta^2 A$, where $M$ represents mass, $J_A$ represents angular momentum associated with different rotational symmetries (denoted by $A$), $\Omega_A$ represents corresponding rotation rates, and $\kappa$ represents surface gravity. This expression establishes a direct link between dynamical stability and thermodynamic stability. While it is known that thermodynamic instability does not necessarily imply dynamical instability for black holes, the authors prove that for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with $\E < 0$ and vanishing linearized ADM quantities. This result confirms a conjecture by Gubser and Mitra stating that all black branes corresponding to thermodynamically unstable black holes are dynamically unstable. This finding establishes that fulfillment of this local Penrose inequality is necessary and sufficient for dynamical stability. In summary, Hollands and Wald present a new criterion for determining the dynamical stability of black holes in higher-dimensional spacetimes. Their analysis reveals important connections between dynamical stability, thermodynamic stability, and variations in mass angular momentum ,and horizon area. The paper provides valuable insights into the fundamental properties and behavior of black holes in general relativity by demonstrating how these concepts are related to one another.