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The wave equation on Schwarzschild-de Sitter spacetimes

Mihalis Dafermos, Igor Rodnianski

TL;DR

This work establishes quantitative decay for solutions of the linear wave equation on Schwarzschild–de Sitter spacetimes within the region bounded by the black hole and cosmological horizons. It advances the vector-field method by constructing a hierarchy of currents (X, Y, and Θ) that control the photon sphere and sector-specific angular modes, while exploiting the red-shift near horizons to obtain uniform energy decay up to the horizons. The main contributions are (i) a robust energy decay bound tied to initial data through a high-regularity norm E_s, (ii) a decomposition into spherical harmonics yielding exponential decay per mode with rates depending on the harmonic number, and (iii) a detailed framework relating bulk, flux, and horizon terms that can inform nonlinear stability analyses. Together, these results clarify the decay landscape for linear waves in SdS spacetimes and lay groundwork for non-linear stability studies in the presence of a positive cosmological constant.

Abstract

We consider solutions to the linear wave equation $\Box_gφ=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may contain the sphere of bifurcation of the black/white hole horizons and the cosmological horizons.) We prove that in the region bounded by a set of black/white hole horizons and cosmological horizons, solutions $φ$ converge pointwise to a constant faster than any given polynomial rate, where the decay is measured with respect to natural future-directed advanced and retarded time coordinates. We also give such uniform decay bounds for the energy associated to the Killing field as well as for the energy measured by local observers crossing the event horizon. The results in particular include decay rates along the horizons themselves. Finally, we discuss the relation of these results to previous heuristic analysis of Price and Brady et al.

The wave equation on Schwarzschild-de Sitter spacetimes

TL;DR

This work establishes quantitative decay for solutions of the linear wave equation on Schwarzschild–de Sitter spacetimes within the region bounded by the black hole and cosmological horizons. It advances the vector-field method by constructing a hierarchy of currents (X, Y, and Θ) that control the photon sphere and sector-specific angular modes, while exploiting the red-shift near horizons to obtain uniform energy decay up to the horizons. The main contributions are (i) a robust energy decay bound tied to initial data through a high-regularity norm E_s, (ii) a decomposition into spherical harmonics yielding exponential decay per mode with rates depending on the harmonic number, and (iii) a detailed framework relating bulk, flux, and horizon terms that can inform nonlinear stability analyses. Together, these results clarify the decay landscape for linear waves in SdS spacetimes and lay groundwork for non-linear stability studies in the presence of a positive cosmological constant.

Abstract

We consider solutions to the linear wave equation on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may contain the sphere of bifurcation of the black/white hole horizons and the cosmological horizons.) We prove that in the region bounded by a set of black/white hole horizons and cosmological horizons, solutions converge pointwise to a constant faster than any given polynomial rate, where the decay is measured with respect to natural future-directed advanced and retarded time coordinates. We also give such uniform decay bounds for the energy associated to the Killing field as well as for the energy measured by local observers crossing the event horizon. The results in particular include decay rates along the horizons themselves. Finally, we discuss the relation of these results to previous heuristic analysis of Price and Brady et al.

Paper Structure

This paper contains 66 sections, 26 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. The initial value problem for the wave equation
  3. The main theorem
  4. Norms on initial data
  5. First statement of the theorem
  6. Comparison with the Schwarzschild case
  7. Second statement of the theorem
  8. Overview of the proof
  9. Vector fields and compatible currents
  10. The photon sphere and the currents $J^X_\mu$
  11. The red-shift effect and the currents $J^Y_\mu$ and $J^{\overline{Y}}_\mu$
  12. Comparison with the Schwarzschild case
  13. Discussion
  14. Note added
  15. Acknowledgements
  16. ...and 51 more sections

Key Result

Theorem 1.1

Let $(\mathcal{M},g)$ denote the Schwarzschild-de Sitter spacetime with parameter $M$ and cosmological constant $\Lambda$ satisfying $(NEBH)$ and let $\Sigma$ be a Cauchy surface for $\mathcal{M}$. Let $\mathcal{D}\subset \mathcal{M}$ denote a region as defined in $(Ddef)$ and let $s\ge 0$. Then, th holds, where $u$ and $v$ denote fixed Eddington-Finkelstein advanced and retarded coordinatesSee Se

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 6.1
  • Lemma 7.3.1
  • Lemma 7.3.2
  • proof
  • Proposition 8.3.1
  • proof
  • Proposition 10.2.1
  • proof
  • ...and 32 more