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Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes

Yannis Angelopoulos, Stefanos Aretakis, Dejan Gajic

TL;DR

This work rigorously establishes precise late-time tails for solutions of the wave equation on four-dimensional, spherically symmetric, stationary and asymptotically flat spacetimes, including Schwarzschild and sub-extremal Reissner--Nordström. The leading tail is governed by Newman--Penrose constants and, when these vanish, by time-inverted Newman--Penrose constants, with explicit formulas for the leading terms and higher-order derivatives. The authors introduce a time-inverted NP constant hierarchy via the time integral $\\psi^{(1)}$ and prove global decay, asymptotics at null infinity, and interior behavior, all derived through a purely physical-space, vector-field framework without conformal compactification. These results connect tail behavior to conserved quantities and have direct implications for black hole exterior stability and cosmic censorship, while providing sharp, quantitative bounds useful for nonlinear applications. The method spans both the nonvanishing and vanishing NP-constant regimes and yields explicit asymptotics for the radiation field and higher-order time derivatives, offering a complete description of spherical tails and their dependence on initial data.

Abstract

We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner-Nordstrom families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman-Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's polynomial law as a lower bound. Our analysis relies on physical space techniques and uses the vector field approach for almost-sharp decay estimates introduced in our companion paper. In the black hole case, our estimates hold in the domain of outer communications up to and including the event horizon. Our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors.

Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes

TL;DR

This work rigorously establishes precise late-time tails for solutions of the wave equation on four-dimensional, spherically symmetric, stationary and asymptotically flat spacetimes, including Schwarzschild and sub-extremal Reissner--Nordström. The leading tail is governed by Newman--Penrose constants and, when these vanish, by time-inverted Newman--Penrose constants, with explicit formulas for the leading terms and higher-order derivatives. The authors introduce a time-inverted NP constant hierarchy via the time integral and prove global decay, asymptotics at null infinity, and interior behavior, all derived through a purely physical-space, vector-field framework without conformal compactification. These results connect tail behavior to conserved quantities and have direct implications for black hole exterior stability and cosmic censorship, while providing sharp, quantitative bounds useful for nonlinear applications. The method spans both the nonvanishing and vanishing NP-constant regimes and yields explicit asymptotics for the radiation field and higher-order time derivatives, offering a complete description of spherical tails and their dependence on initial data.

Abstract

We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner-Nordstrom families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman-Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's polynomial law as a lower bound. Our analysis relies on physical space techniques and uses the vector field approach for almost-sharp decay estimates introduced in our companion paper. In the black hole case, our estimates hold in the domain of outer communications up to and including the event horizon. Our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors.

Paper Structure

This paper contains 53 sections, 43 theorems, 418 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Introduction and background
  3. Summary of main results
  4. Applications
  5. Overview of techniques
  6. The main theorems
  7. Notation
  8. Late-time asymptotics for solutions to the wave equation with vanishing Newman--Penrose constant
  9. Late-time asymptotics for solutions to the wave equation with non-vanishing Newman--Penrose constant
  10. Relation with the work of Luk--Oh
  11. Outline
  12. Acknowledgments
  13. The geometric setting
  14. The spacetime manifold
  15. Case I: $r_{\rm min}=r_+>0$
  16. ...and 38 more sections

Key Result

Theorem 1

(Asymptotics for solutions with compactly supported data) Let $\psi$ be a solution to the wave equation waveequation on the Lorentzian manifolds $(\mathcal{M},g)$, defined in Section thespacemanif, with smooth compactly supported initial data. Then $\psi$ satisfies the following asymptotic estimate as $\tau\rightarrow \infty$. where the constant $I_{0}^{(1)}[\psi]$ is given by inftinp1. Furthermo

Figures (3)

  • Figure 1: The curve $\gamma_{\alpha}$ and the region $\mathcal{B}_{\alpha}$
  • Figure 2: A Penrose diagram of $\mathcal{M}$ in the case $r_{\rm min}=0$.
  • Figure 3: Penrose diagram of $\mathcal{M}$ in the case $r_{\rm min}=r_+$.

Theorems & Definitions (84)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 3
  • Theorem 4
  • Theorem 1.1
  • Theorem 1.2
  • ...and 74 more