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On pointwise decay of linear waves on a Schwarzschild black hole background

Roland Donninger, Wilhelm Schlag, Avy Soffer

TL;DR

The paper establishes sharp pointwise decay for linear waves on a Schwarzschild black hole, proving t^{-3} decay for scalar perturbations and faster t^{-4} and t^{-6} for electromagnetic and gravitational perturbations, respectively. The authors reduce the problem to a family of 1D Regge–Wheeler equations via angular momentum decomposition, then analyze low-energy (near-zero) and high-energy regimes semiclassically. A key dichotomy is revealed: tails are governed by the far-field, low-energy behavior of the Regge–Wheeler potential, while the required angular derivatives are dictated by dynamics near the potential maximum, treated via a Mourre bound and Sigal–Soffer propagation. They also discuss general-data extensions (Kay–Wald) and note that the approach yields arbitrarily fast decay rates for suitably filtered data, aligning with Price’s law in spirit. The work combines WKB, Jost solutions, spectral measures, Mourre theory, and semiclassical propagation to achieve a comprehensive decay analysis with robust angular-spectral control.

Abstract

We prove sharp pointwise $t^{-3}$ decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates $t^{-4}$, and $t^{-6}$, respectively. We proceed by decomposition into angular momentum $\ell$ and summation of the decay estimates on the Regge-Wheeler equation for fixed $\ell$. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in $\ell$ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.

On pointwise decay of linear waves on a Schwarzschild black hole background

TL;DR

The paper establishes sharp pointwise decay for linear waves on a Schwarzschild black hole, proving t^{-3} decay for scalar perturbations and faster t^{-4} and t^{-6} for electromagnetic and gravitational perturbations, respectively. The authors reduce the problem to a family of 1D Regge–Wheeler equations via angular momentum decomposition, then analyze low-energy (near-zero) and high-energy regimes semiclassically. A key dichotomy is revealed: tails are governed by the far-field, low-energy behavior of the Regge–Wheeler potential, while the required angular derivatives are dictated by dynamics near the potential maximum, treated via a Mourre bound and Sigal–Soffer propagation. They also discuss general-data extensions (Kay–Wald) and note that the approach yields arbitrarily fast decay rates for suitably filtered data, aligning with Price’s law in spirit. The work combines WKB, Jost solutions, spectral measures, Mourre theory, and semiclassical propagation to achieve a comprehensive decay analysis with robust angular-spectral control.

Abstract

We prove sharp pointwise decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates , and , respectively. We proceed by decomposition into angular momentum and summation of the decay estimates on the Regge-Wheeler equation for fixed . We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.

Paper Structure

This paper contains 23 sections, 20 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Extension to more general data
  3. Strategy of proof of Theorems \ref{['thm:main']} and \ref{['thm:main2']}
  4. Angular momentum decomposition
  5. Notations
  6. Low lying energies and WKB
  7. The far field
  8. Liouville-Green transform and reduction to a perturbed Airy equation
  9. A basis for the perturbed Airy equation
  10. The outgoing Jost solution for the far field
  11. Approaching the event horizon
  12. Transforming the problem to a compact interval
  13. The outgoing Jost solution towards the event horizon
  14. The Wronskian of the outgoing Jost solutions
  15. The spectral measure $e(E,x,x';\hbar)$ near the maximum of the potential
  16. ...and 8 more sections

Key Result

Theorem 1.1

The following decay estimates hold for solutions $\psi$ of eq:wave with data $\psi[0]=(\psi_0,\psi_1)$: where ${\slash\!\!\! \nabla}$ stands for the angular derivativesThe notation $a\pm$ stands for $a\pm{\varepsilon}$ where ${\varepsilon}>0$ is arbitrary (the choice determines the constants involved). Also, instead of $( {\slash\!\!\! \nabla} ^{10}, {\slash\!\!\! \nabla} ^9)$ in eq:decaywaveL1 o

Figures (1)

  • Figure 1: The Kruskal spacetime. The two shaded regions together comprise the original exterior Schwarzschild manifold, denoted by $\mathcal{S}$. The solution of the wave equation $\Box_g \psi=0$ in the darker shaded region $\mathcal{D}$ is uniquely determined by Cauchy data on $\Sigma_1 \cap \mathcal{S}$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4: CST
  • proof
  • Corollary 2.5
  • ...and 25 more