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Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

Mihalis Dafermos, Igor Rodnianski, Yakov Shlapentokh-Rothman

TL;DR

<3-5 sentences high-level summary> This work completes the program of establishing boundedness and decay for the scalar wave equation on Kerr spacetimes in the full subextremal range $|a|<M$, without symmetry assumptions. It combines Carter separation-based frequency analysis with sophisticated multiplier constructions to prove phase-space integrated local energy decay, then patches these estimates into a global, non-degenerate bound using a continuity argument in the rotation parameter and red-shift results. The approach yields uniform energy and higher-order energy bounds, leading to polynomial decay in time for the energy and pointwise decay of the scalar field, with corollaries applicable to nonlinear problems. This provides a robust linear stability framework for Kerr spacetimes, laying groundwork for nonlinear stability analyses and extending prior axisymmetric or small-$a$ results to the full subextremal regime.

Abstract

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal |a| < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, 2011, pp. 132-189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincare], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notations so that it can be read independently of previous work.

Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

TL;DR

<3-5 sentences high-level summary> This work completes the program of establishing boundedness and decay for the scalar wave equation on Kerr spacetimes in the full subextremal range , without symmetry assumptions. It combines Carter separation-based frequency analysis with sophisticated multiplier constructions to prove phase-space integrated local energy decay, then patches these estimates into a global, non-degenerate bound using a continuity argument in the rotation parameter and red-shift results. The approach yields uniform energy and higher-order energy bounds, leading to polynomial decay in time for the energy and pointwise decay of the scalar field, with corollaries applicable to nonlinear problems. This provides a robust linear stability framework for Kerr spacetimes, laying groundwork for nonlinear stability analyses and extending prior axisymmetric or small- results to the full subextremal regime.

Abstract

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal |a| < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, 2011, pp. 132-189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincare], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notations so that it can be read independently of previous work.

Paper Structure

This paper contains 114 sections, 65 theorems, 500 equations.

Table of Contents

  1. Introduction
  2. Overview of the main difficulties
  3. Review of the very slowly rotating case $|a|\ll M$
  4. Structure of trapping and its disjointness from superradiance
  5. Absence of bounded frequency superradiant obstructions
  6. Higher order estimates
  7. Continuity argument
  8. Non-degenerate boundedness from integrated local energy decay
  9. Outline of the paper
  10. Acknowledgements
  11. Review of the setup
  12. Ambient manifold and coordinate systems
  13. Fixed coordinates $(y^*,t^*,\theta^*,\phi^*)$
  14. Kerr-star coordinates $(r,t^*,\theta^*,\phi^*)$
  15. Boyer-Lindquist coordinates $(r,t, \theta,\phi)$
  16. ...and 99 more sections

Key Result

Theorem 1

1. General solutions $\psi$ of $(WAVE)$ on the exterior of a Kerr black hole background $(\mathcal{M}, g_{a,M})$in the full subextremal range $|a|<M$, arising from bounded initial energy on a suitable Cauchy surface $\Sigma_0$, have bounded energy flux through a global foliation $\Sigma_\tau$ of the

Theorems & Definitions (146)

  • Theorem
  • Corollary
  • Remark 2.3.1
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.1
  • Lemma 4.1.1
  • Proposition 4.5.1
  • Proposition 4.5.2
  • Proposition 4.5.3
  • ...and 136 more