[Paper Review] Classical and thermodynamic stability of black holes

Ricardo Monteiro|arXiv (Cornell University)|Jun 28, 2010

Black Holes and Theoretical Physics4 citations

TL;DR

This thesis investigates the classical and thermodynamic stability of black holes using linearised perturbations and gravitational partition functions. It establishes a direct link between negative modes in the partition function and classical instabilities, identifying the onset of instability in higher-dimensional rotating black holes (Myers-Perry) as a bifurcation point to new phases, with numerical evidence for exponential growth in D=9 and confirmation of the Emparan-Myers conjecture in the singly-spinning case.

ABSTRACT

We consider the stability of black holes within both classical general relativity and the semiclassical thermodynamic description. In particular, we study linearised perturbations and their contribution to the gravitational partition function, addressing technical issues for charged (Reissner-Nordstrom) and rotating (Kerr-AdS) black holes. Exploring the connection between classical and thermodynamic stability, we find classical instabilities of Myers-Perry black holes and bifurcations to new black hole families.

Motivation & Objective

• To establish a rigorous connection between classical instability and thermodynamic instability in black holes.
• To investigate negative modes in the gravitational path integral as indicators of quantum and classical pathologies.
• To analyze the stability of higher-dimensional rotating black holes (Myers-Perry) in the ultraspinning regime.
• To determine whether classical instabilities in vacuum black holes lead to new families of solutions, as conjectured by Emparan and Myers.
• To extend the analysis of negative modes to charged (Reissner-Nordström) and rotating (Kerr-AdS) black holes using gauge-invariant and numerical methods.

Proposed method

• Utilizes gauge-invariant perturbation theory to analyze the Einstein-Maxwell action for Reissner-Nordström black holes, reducing the problem to a five-dimensional Kaluza-Klein framework.
• Applies spectral methods with Chebyshev grids to solve the eigenvalue problem for perturbations, ensuring exponential convergence for analytic solutions.
• Employs a numerical approach based on discretizing the perturbation equations on a Chebyshev grid to compute eigenvalues of the linearised operator.
• Implements Dirichlet boundary conditions by truncating the first and last rows/columns of the resulting matrices to ensure well-posedness.
• Solves a generalized eigenvalue problem of dimension $ n( ext{N}-1) $, where $ ext{N} $ is the number of grid points, to locate negative modes.
• Extends the method to two-dimensional systems for PDEs by using a tensor product grid in multiple coordinates, maintaining spectral accuracy.

Experimental results

Research questions

• RQ1 Does the presence of a negative mode in the gravitational partition function correspond to a classical instability in vacuum black holes?
• RQ2 Can the onset of classical instability in higher-dimensional rotating black holes be identified with a bifurcation to a new family of solutions?
• RQ3 Is the Gubser-Mitra conjecture on black brane stability applicable to individual black holes, not just their brane analogs?
• RQ4 Do Reissner-Nordström and Kerr-AdS black holes exhibit negative modes precisely in regions where local thermodynamic stability fails?
• RQ5 Can numerical methods confirm the existence of exponentially growing modes in the ultraspinning regime of Myers-Perry black holes in odd dimensions?

Key findings

• The first numerical evidence is found for classical instabilities in vacuum asymptotically flat black holes, specifically in the ultraspinning regime of Myers-Perry black holes.
• In D=9, exponentially growing perturbations are found for the equal-spinning Myers-Perry solution, supporting the existence of a classical instability.
• The threshold for instability in the singly-spinning case coincides with the appearance of a stationary negative mode, confirming the Emparan-Myers conjecture.
• New black hole solutions bifurcate at the instability threshold, and these solutions are expected to have a single rotational symmetry, saturating the rigidity theorem.
• For Reissner-Nordström and Kerr-AdS black holes, negative modes are found precisely in the region where local thermodynamic stability fails, confirming the expected correspondence.
• The use of Chebyshev spectral methods enables high-accuracy resolution of the perturbation equations, allowing reliable detection of negative modes and unstable eigenvalues.