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A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

Mihalis Dafermos, Igor Rodnianski

TL;DR

The paper proves uniform boundedness for solutions of the wave equation on slowly rotating Kerr spacetimes and more generally on stationary axisymmetric perturbations close to Schwarzschild. It develops a robust vector-field method that avoids Kerr separability by constructing positive-definite energy currents and a horizon-regular redshift current, then employs a high-low frequency decomposition to separate superradiant and dispersive components. The main contributions are quantitative energy bounds up to the horizon, higher-order energy control, and pointwise decay-type bounds that hold under near-Schwarzschild perturbations, providing a robust step toward nonlinear stability analyses of Kerr-like spacetimes. This advances the understanding of wave dynamics in rotating black hole backgrounds and yields tools potentially applicable to the full Einstein equations near Kerr.

Abstract

We consider Kerr spacetimes with parameters a and M such that |a|<< M, Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M>0, with appropriate geometric assumptions on the plane spanned by the Killing fields. We show uniform boundedness on the exterior for sufficiently regular solutions to the scalar homogeneous wave equation. In particular, the bound holds up to and including the event horizon. No unphysical restrictions are imposed on the behaviour of the solution near the bifurcation surface of the event horizon. The pointwise estimate derives in fact from the uniform boundedness of a positive definite energy flux. Note that in view of the very general assumptions, the separability properties of the wave equation on the Kerr background are not used.

A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

TL;DR

The paper proves uniform boundedness for solutions of the wave equation on slowly rotating Kerr spacetimes and more generally on stationary axisymmetric perturbations close to Schwarzschild. It develops a robust vector-field method that avoids Kerr separability by constructing positive-definite energy currents and a horizon-regular redshift current, then employs a high-low frequency decomposition to separate superradiant and dispersive components. The main contributions are quantitative energy bounds up to the horizon, higher-order energy control, and pointwise decay-type bounds that hold under near-Schwarzschild perturbations, providing a robust step toward nonlinear stability analyses of Kerr-like spacetimes. This advances the understanding of wave dynamics in rotating black hole backgrounds and yields tools potentially applicable to the full Einstein equations near Kerr.

Abstract

We consider Kerr spacetimes with parameters a and M such that |a|<< M, Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M>0, with appropriate geometric assumptions on the plane spanned by the Killing fields. We show uniform boundedness on the exterior for sufficiently regular solutions to the scalar homogeneous wave equation. In particular, the bound holds up to and including the event horizon. No unphysical restrictions are imposed on the behaviour of the solution near the bifurcation surface of the event horizon. The pointwise estimate derives in fact from the uniform boundedness of a positive definite energy flux. Note that in view of the very general assumptions, the separability properties of the wave equation on the Kerr background are not used.

Paper Structure

This paper contains 53 sections, 41 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. Statement of the theorem for Kerr and Kerr-Newman
  3. Statement for general stationary axisymmetric perturbations of Schwarzschild
  4. Dispersion and the redshift vs. superradiance
  5. Previous results
  6. Schwarzschild
  7. Kerr
  8. Klein-Gordon
  9. Dirac on Kerr
  10. Heuristic work
  11. Acknowledgements
  12. Constants and parameters
  13. The class of spacetimes
  14. Schwarzschild
  15. The general class
  16. ...and 38 more sections

Key Result

Theorem 1.1

Let $(\mathcal{M},g)$, $\mathcal{D}$, $\Sigma_\tau$ be as above. There exists a universal positive constant $\epsilon>0$, and a constant $C$ depending on $M$ and the choice of $\Sigma_0$ such that if then the following statement holds. Let $\psi$ be a solution of $(waveee)$ on $(\mathcal{M},g)$ such that $\int_{\Sigma_0} J^{n_\Sigma}_\mu(\psi) n^\mu_\Sigma <\infty$. Then Here the integrals are w

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • Theorem 5.1
  • Lemma 9.1
  • proof
  • Lemma 9.2
  • proof
  • Proposition 9.1
  • proof
  • ...and 58 more