The black hole stability problem for linear scalar perturbations

Abstract

We review our recent work on linear stability for scalar perturbations of Kerr spacetimes, that is to say, boundedness and decay properties for solutions of the scalar wave equation \Box_gψ = 0 on Kerr exterior backgrounds. We begin with the very slowly rotating case |a| \ll M, where first boundedness and then decay has been shown in rapid developments over the last two years, following earlier progress in the Schwarzschild case a = 0. We then turn to the general subextremal range |a| < M, where we give here for the first time the essential elements of a proof of definitive decay bounds for solutions ψ. These developments give hope that the problem of the non-linear stability of the Kerr family of black holes might soon be addressed. This paper accompanies a talk by one of the authors (I.R.) at the 12th Marcel Grossmann Meeting, Paris, June 2009.

48 pages, 5 figures, to appear in Proceedings of the 12 Marcel Grossmann Meeting

References in corpus (3)

• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• The wave equation on Schwarzschild-de Sitter spacetimes
• Local energy estimate on Kerr black hole backgrounds

Cited by in corpus (64)

• Testing General Relativity with Present and Future Astrophysical Observations
• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I
• The linear stability of the Schwarzschild solution to gravitational perturbations
• Slowly Decaying Waves on Spherically Symmetric Spacetimes and an Instability of Ultracompact Neutron Stars
• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II
• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• A scattering theory for the wave equation on Kerr black hole exteriors
• Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations
• Stability for linearized gravity on the Kerr spacetime
• Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds
• A scattering theory construction of dynamical vacuum black holes
• On the quasilinear wave equations in time dependent inhomogeneous media
• Nonlinear instability of scalar fields on extremal black holes
• Decay of linear waves on higher dimensional Schwarzschild black holes
• The Stability of the Minkowski space for the Einstein-Vlasov system
• Quasinormal modes in extremal Reissner-Nordström spacetimes
• The linear stability of Reissner-Nordström spacetime: the full subextremal range
• Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem
• A model problem for quasinormal ringdown on asymptotically flat or extremal black holes
• The weak null condition and global existence using the p-weighted energy method
• The linear stability of Reissner-Nordström spacetime for small charge
• A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations
• Sharp decay estimates for massless Dirac fields on a Schwarzschild background
• The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves
• Price's law and precise late-time asymptotics for subextremal Reissner-Nordström black holes
• A non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom
• The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Stability and Instability Results
• A Vector Field Method for Radiating Black Hole Spacetimes
• The $r^p$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications
• Existence of Zero-damped Quasinormal Frequencies for Nearly Extremal Black Holes
• The Case Against Smooth Null Infinity II: A Logarithmically Modified Price's Law
• Price's law for spin fields on a Schwarzschild background
• Boundedness and decay for the Teukolsky equation of spin $\pm 1$ on Reissner-Nordström spacetime: the $\ell =1$ spherical mode
• Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry
• The Scattering Map on Oppenheimer--Snyder Spacetime
• Pointwise decay for the wave equation on nonstationary spacetimes
• Small data global existence and decay for two dimensional wave maps
• Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes
• On Late Time Tails in an Extreme Reissner-Nordström Black Hole: Frequency Domain Analysis
• Pointwise decay for semilinear wave equations in ${\mathbb{R}}^{!+3}$
• Evanescent ergosurface instability
• Boundedness and decay for the Teukolsky system of spin $\pm 2$ on Reissner-Nordström spacetime: the case $|Q|\ll M$
• Asymptotic blow-up for a class of semilinear wave equations on extremal Reissner-Nordström spacetimes
• Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere
• A Vector Field Method for Non-Trapping, Radiating Spacetimes
• Price's Law on Nonstationary Spacetimes
• Explicit formulas and decay rates for the solution of the wave equation in cosmological spacetimes
• Global solutions of nonlinear wave equations with large energy
• Soliton resolution for equivariant wave maps on a wormhole: I
• Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes
• Pointwise decay for semilinear wave equations on Kerr spacetimes
• Uniform bound for solutions of semilinear wave equations in ${\mathbb{R}}^{1+3}$
• Global solutions of nonlinear wave equations in time dependent inhomogeneous media
• Decay of Maxwell Fields on Reissner-Nordstrøm-de Sitter Black Holes
• A vector field method on the distorted Fourier side and decay for wave equations with potentials
• Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities
• Instability results for the wave equation in the interior of Kerr black holes
• On the global behaviors for defocusing semilinear wave equations in ${\mathbb{R}}^{1+2}$
• Rough Solutions of the Relativistic Euler Equations
• Localized energy estimates on Myers-Perry space-times
• Multi-center Vector Field Methods for Wave Equations
• On global behavior of solutions of the Maxwell-Klein-Gordon equations
• On the link between the Maxwell and linearised Einstein equations on Schwarzschild
• Small data solutions of the Vlasov-Poisson system and the vector field method