[Paper Review] Linear stability of slowly rotating Kerr black holes

Dietrich Häfner, Peter Hintz+1 more|arXiv (Cornell University)|Jun 3, 2019

Black Holes and Theoretical Physics166 references59 citations

TL;DR

This paper establishes the linear stability of slowly rotating Kerr black holes under the Einstein vacuum equation by proving that linearized perturbations decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. Using a wave map/DeTurck gauge with constraint damping, the authors analyze the low-energy resolvent of the linearized gauge-fixed Einstein operator via microlocal and Fredholm theory, showing that the asymptotic behavior is controlled by a 7-dimensional space of pure gauge modes, with no growing modes or pathological zero-energy states.

ABSTRACT

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equation: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the linearization of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equation have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications.

Motivation & Objective

• To establish the linear stability of slowly rotating Kerr black holes as solutions to the Einstein vacuum equation.
• To show that linearized perturbations decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term.
• To eliminate pathological generalized zero-energy modes using constraint damping in a hyperbolic formulation of the linearized Einstein equation.
• To provide a robust, perturbative framework for stability analysis that does not rely on special algebraic properties of the Kerr metric.
• To characterize the asymptotic behavior of solutions for general initial data and forcing terms, enabling applications in nonlinear and numerical relativity.

Proposed method

• Formulates the linearized Einstein equation in a wave map/DeTurck gauge to break diffeomorphism invariance and simplify the analysis.
• Introduces a modified gauge-fixed Einstein operator with constraint damping to eliminate spurious zero-energy modes and ensure correct asymptotics.
• Applies a robust microlocal and non-elliptic Fredholm framework for analyzing resolvents on asymptotically flat spacetimes.
• Uses spherical harmonic decomposition to reduce the problem to radial operators on the Schwarzschild background.
• Relies on mode stability results for scalar and 1-form wave operators on Schwarzschild spacetime as input to the perturbative argument.
• Analyzes the low-energy resolvent of the linearized operator in detail, showing its precise structure near zero frequency, including the role of the 7-dimensional pure gauge space.

Experimental results

Research questions

• RQ1 Do linearized perturbations of slowly rotating Kerr black holes decay over time, and at what rate?
• RQ2 Can the asymptotic behavior of solutions to the linearized Einstein equation be described as a linearized Kerr metric plus a pure gauge term?
• RQ3 What is the role of constraint damping in eliminating pathological zero-energy modes and ensuring correct decay and asymptotics?
• RQ4 How can the resolvent of the linearized gauge-fixed Einstein operator be analyzed at low energies using microlocal and Fredholm theory?
• RQ5 Is the linear stability of slowly rotating Kerr black holes provable via perturbative methods without relying on special algebraic symmetries of the Kerr metric?

Key findings

• Linearized perturbations of slowly rotating Kerr black holes decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term.
• The pure gauge term lies in a fixed 7-dimensional space with a clear geometric interpretation, corresponding to diffeomorphisms generated by Killing vector fields.
• Constraint damping successfully eliminates generalized zero-energy modes and ensures that solutions have the correct asymptotic behavior for general initial data.
• The resolvent of the linearized modified gauge-fixed Einstein operator is shown to be well-behaved at low energies, with a precise structure near zero frequency.
• The solution to the initial value problem for the linearized Einstein equation satisfies the gauge condition and constraint equations, with decay rates matching those of the initial data.
• The proof is perturbative and does not require special algebraic properties of the Kerr metric, relying instead on mode stability of the Schwarzschild case and microlocal analysis.