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Stability of Five-dimensional Myers-Perry Black Holes with Equal Angular Momenta

Keiju Murata, Jiro Soda

TL;DR

The paper investigates the dynamical stability of five-dimensional Myers-Perry black holes with equal angular momenta by exploiting the enhanced $U(2)$ symmetry to construct master variables and decouple perturbations into independent sectors. For the $(J=0,M=0,K=0)$ mode, a Schrödinger-type master equation with a positive potential proves stability, while for $K\neq0$ modes, the authors formulate master equations, apply quasi-normal-mode and WKB analyses, and perform numerical checks. Their results show no unstable modes within the considered parameter range, providing strong evidence for stability of these black holes in the analyzed sector. The approach offers a systematic framework for stability analyses in higher-dimensional rotating spacetimes and suggests paths to extend the analysis to Kerr-AdS and other dimensions.

Abstract

We study the stability of five-dimensional Myers-Perry black holes with equal angular momenta which have an enlarged symmetry, U(2). Using this symmetry, we derive master equations for a part of metric perturbations which are relevant to the stability. Based on the master equations, we prove the stability of Myers-Perry black holes under these perturbations. Our result gives a strong evidence for the stability of Myers-Perry black holes with equal angular momenta.

Stability of Five-dimensional Myers-Perry Black Holes with Equal Angular Momenta

TL;DR

The paper investigates the dynamical stability of five-dimensional Myers-Perry black holes with equal angular momenta by exploiting the enhanced symmetry to construct master variables and decouple perturbations into independent sectors. For the mode, a Schrödinger-type master equation with a positive potential proves stability, while for modes, the authors formulate master equations, apply quasi-normal-mode and WKB analyses, and perform numerical checks. Their results show no unstable modes within the considered parameter range, providing strong evidence for stability of these black holes in the analyzed sector. The approach offers a systematic framework for stability analyses in higher-dimensional rotating spacetimes and suggests paths to extend the analysis to Kerr-AdS and other dimensions.

Abstract

We study the stability of five-dimensional Myers-Perry black holes with equal angular momenta which have an enlarged symmetry, U(2). Using this symmetry, we derive master equations for a part of metric perturbations which are relevant to the stability. Based on the master equations, we prove the stability of Myers-Perry black holes under these perturbations. Our result gives a strong evidence for the stability of Myers-Perry black holes with equal angular momenta.

Paper Structure

This paper contains 13 sections, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Myers-Perry black holes with equal angular momenta and their symmetry
  3. A way to find master variables
  4. Stability analysis for $(J=0,M=0,K=0)$ mode
  5. Stability analysis for $K\neq 0$ modes
  6. Master equation for $(J=0,M=0,K=1)$ mode
  7. Master equations for $(J,M,K=J+2)$ modes
  8. Method to study the stability of $K\neq 0$ modes
  9. WKB analysis
  10. Numerical analysis
  11. Conclusions
  12. Gravitational perturbation equations for $J=M=K=0$ mode
  13. Derivation of master equation for $(J=0,M=0,K=1)$ mode

Figures (3)

  • Figure 1: Typical profiles for the potential $V_0$ are depicted. From top to bottom, each curve represents the potential for $\Omega_H/\Omega_H^\text{max}=0.1,0.7,0.9$ and $0.99$. We see the positivity of these potentials.
  • Figure 2: $\tilde{V}_1$ and $\tilde{V}_2$ for $\Omega_H/\Omega_H^\text{max}=0.99$. We see that $\tilde{V}_2$ tends to be negative for large $\omega$. However, for $\tilde{V}_1$, a positive region remains even for sufficiently large $\omega$.
  • Figure 3: Function $Z_\text{max}(\Omega_H)$ for $(J=0,M=0,K=1)$ mode and $(J,M,K=J+2)$ modes. We see that $Z_\text{max}$ is finite for $\Omega_H<\Omega_H^\text{max}$.