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The black hole stability problem for linear scalar perturbations

Mihalis Dafermos, Igor Rodnianski

TL;DR

This work establishes robust linear stability for scalar perturbations on Kerr spacetimes, proving boundedness and decay for solutions of $\Box_g\psi=0$ across the full subextremal range $|a|<M$. It synthesizes Schwarzschild insights with a sophisticated Kerr analysis that isolates superradiance, red-shift, and trapping through Carter’s separation and a spectrum-localized (microlocal) energy-currents framework. The approach combines Noetherian energy identities, virial-type currents, horizon red-shift, and null-infinity estimates, then extends from small $|a|$ to all subextremal Kerr by a continuity argument in $a$ and a detailed frequency decomposition. The results provide quantitative boundedness, integrated local energy decay, and polynomial decay of energy flux and pointwise decay, representing decisive progress toward nonlinear stability of the Kerr family for scalar fields and informing future work on gravitational perturbations. The techniques—especially Carter separation as a geometric microlocalisation and frequency-localized currents—offer a robust framework potentially applicable to nearby axisymmetric spacetimes and nonlinear stability analyses.

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.

The black hole stability problem for linear scalar perturbations

TL;DR

This work establishes robust linear stability for scalar perturbations on Kerr spacetimes, proving boundedness and decay for solutions of across the full subextremal range . It synthesizes Schwarzschild insights with a sophisticated Kerr analysis that isolates superradiance, red-shift, and trapping through Carter’s separation and a spectrum-localized (microlocal) energy-currents framework. The approach combines Noetherian energy identities, virial-type currents, horizon red-shift, and null-infinity estimates, then extends from small to all subextremal Kerr by a continuity argument in and a detailed frequency decomposition. The results provide quantitative boundedness, integrated local energy decay, and polynomial decay of energy flux and pointwise decay, representing decisive progress toward nonlinear stability of the Kerr family for scalar fields and informing future work on gravitational perturbations. The techniques—especially Carter separation as a geometric microlocalisation and frequency-localized currents—offer a robust framework potentially applicable to nearby axisymmetric spacetimes and nonlinear stability analyses.

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.

Paper Structure

This paper contains 76 sections, 9 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. The classical mode analysis
  3. Linear stability?
  4. Schwarzschild
  5. The Kerr family
  6. The interaction of the red-shift, superradiance, and trapped null orbits
  7. Outlook
  8. The Kerr space-time
  9. Boyer-Lindquist coordinates
  10. Kerr-star coordinates
  11. Geometric phenomena of Schwarzschild and Kerr
  12. The stationary Killing field and the ergoregion
  13. Separability of geodesic flow and trapped null geodesics
  14. Red-shift effect
  15. The energy method
  16. ...and 61 more sections

Key Result

Theorem 1

Let $\psi$ be a solution of the wave equation on the Schwarzschild background $(\mathcal{M},g=g_M)$ arising from sufficiently regular initial data on a Cauchy hyperusrface $\Sigma$. Then in the exterior region up to and including the horizon, where $D$ is a suitable higher order energy quantity evaluated on the initial hypersurface $\Sigma$.Otherwise, simply excluding $(nothap)$ without providing

Theorems & Definitions (14)

  • Theorem : Kay--Wald kw
  • Theorem 1
  • Corollary 1
  • Proposition 5.3.1
  • Proposition 5.3.2
  • Lemma 11.1.1
  • proof
  • Lemma 11.1.2
  • proof
  • Lemma 11.1.3
  • ...and 4 more