[Paper Review] Gravitational Perturbations of Higher Dimensional Rotating Black Holes

Hari K. Kunduri, James Lucietti+1 more|arXiv (Cornell University)|Jun 8, 2006

Black Holes and Theoretical Physics22 citations

TL;DR

This paper investigates linearized gravitational perturbations of higher-dimensional rotating Myers-Perry black holes with equal angular momenta in odd dimensions (d > 5), including cosmological constant effects. It identifies a class of perturbations reducible to a single radial equation and finds no instability in asymptotically flat spacetimes, but reveals a superradiant instability in asymptotically anti-de Sitter spacetimes when the black hole's angular velocity exceeds the speed of light as seen at the conformal boundary, suggesting a possible endpoint of nonaxisymmetric, stationary black holes.

ABSTRACT

Assessing the stability of higher-dimensional rotating black holes requires a study of linearized gravitational perturbations around such backgrounds. We study perturbations of Myers-Perry black holes with equal angular momenta in an odd number of dimensions (greater than five), allowing for a cosmological constant. We find a class of perturbations for which the equations of motion reduce to a single radial equation. In the asymptotically flat case we find no evidence of any instability. In the asymptotically anti-de Sitter case, we demonstrate the existence of a superradiant instability that sets in precisely when the angular velocity of the black hole exceeds the speed of light from the point of view of the conformal boundary. We suggest that the endpoint of the instability may be a stationary, nonaxisymmetric black hole.

Motivation & Objective

• To assess the stability of higher-dimensional rotating black holes under linearized gravitational perturbations.
• To analyze Myers-Perry black holes with equal angular momenta in odd dimensions greater than five.
• To include the effects of a cosmological constant in the analysis of black hole stability.
• To determine whether superradiant instabilities arise in asymptotically anti-de Sitter spacetimes.

Proposed method

• Focus on a specific class of perturbations for which the equations of motion reduce to a single radial equation.
• Use linearized gravity to study small deviations from the Myers-Perry black hole background.
• Apply analytical techniques to solve the radial equation in both asymptotically flat and anti-de Sitter spacetimes.
• Analyze the behavior of the radial equation to detect unstable modes, particularly in the context of superradiance.
• Compare the angular velocity of the black hole to the speed of light as measured at the conformal boundary to identify instability thresholds.
• Use the structure of the effective potential and boundary conditions to determine the existence of growing modes.

Experimental results

Research questions

• RQ1 Does the presence of a cosmological constant alter the stability of higher-dimensional rotating black holes with equal angular momenta?
• RQ2 Are there unstable modes in the linearized gravitational perturbations of Myers-Perry black holes in odd dimensions greater than five?
• RQ3 Under what conditions does a superradiant instability emerge in asymptotically anti-de Sitter spacetimes?
• RQ4 What is the physical significance of the angular velocity exceeding the speed of light at the conformal boundary in relation to instability?
• RQ5 Could the endpoint of the superradiant instability be a stationary, nonaxisymmetric black hole?

Key findings

• In the asymptotically flat case, no evidence of instability is found in the studied class of perturbations.
• In asymptotically anti-de Sitter spacetimes, a superradiant instability is identified when the black hole's angular velocity exceeds the speed of light as observed at the conformal boundary.
• The instability arises precisely at the threshold where the angular velocity surpasses the light speed at infinity, indicating a critical transition.
• The existence of the instability is linked to the presence of a negative effective potential well in the radial equation, allowing for growing modes.
• The instability is confined to a specific class of nonaxisymmetric perturbations that reduce to a single radial equation.
• The results suggest that the endpoint of the instability may be a stationary, nonaxisymmetric black hole, though this remains a conjecture.