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The red-shift effect and radiation decay on black hole spacetimes

Mihalis Dafermos, Igor Rodnianski

TL;DR

The paper develops a robust vector-field method to study linear wave propagation on the Schwarzschild exterior, proving uniform exterior decay, horizon flux decay, and pointwise decay under mild infinity-decay assumptions. By deploying a Killing-field energy, Morawetz-type X_ℓ energies, and a horizon-penetrating redshift vector Y, it yields both global energy decay and precise local energy decay near the event horizon, culminating in explicit pointwise decay rates. The results include a coordinate-form statement of decay rates (F(S) ≤ C((v_+(S))^{-2}+(u_+(S))^{-2}) and |φ| ≤ C v_+^{-1} in the exterior), as well as an independent proof of Kay–Wald boundedness and decay that are robust with respect to non-linear stability considerations. Together, these findings advance the understanding of linear dynamics in black hole spacetimes and provide a foundation for nonlinear stability analyses that avoid reliance on discrete spacetime symmetries.

Abstract

We consider solutions to the linear wave equation on a (maximally extended) Schwarzschild spacetime, assuming only that the solution decays suitably at spatial infinity on a complete Cauchy hypersurface. (In particular, we allow the support of the solution to contain the bifurcate event horizon.) We prove uniform decay bounds for the solution in the exterior regions, including the uniform bound Cv_+^{-1}, where v_+ denotes max{v,1} and v denotes Eddington-Finkelstein advanced time. We also prove uniform decay bounds for the flux of energy through the event horizon and null infinity. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red-shift effect. The results in particular give an independent proof of the classical uniform boundedness theorem of Kay and Wald, without recourse to the discrete isometries of spacetime.

The red-shift effect and radiation decay on black hole spacetimes

TL;DR

The paper develops a robust vector-field method to study linear wave propagation on the Schwarzschild exterior, proving uniform exterior decay, horizon flux decay, and pointwise decay under mild infinity-decay assumptions. By deploying a Killing-field energy, Morawetz-type X_ℓ energies, and a horizon-penetrating redshift vector Y, it yields both global energy decay and precise local energy decay near the event horizon, culminating in explicit pointwise decay rates. The results include a coordinate-form statement of decay rates (F(S) ≤ C((v_+(S))^{-2}+(u_+(S))^{-2}) and |φ| ≤ C v_+^{-1} in the exterior), as well as an independent proof of Kay–Wald boundedness and decay that are robust with respect to non-linear stability considerations. Together, these findings advance the understanding of linear dynamics in black hole spacetimes and provide a foundation for nonlinear stability analyses that avoid reliance on discrete spacetime symmetries.

Abstract

We consider solutions to the linear wave equation on a (maximally extended) Schwarzschild spacetime, assuming only that the solution decays suitably at spatial infinity on a complete Cauchy hypersurface. (In particular, we allow the support of the solution to contain the bifurcate event horizon.) We prove uniform decay bounds for the solution in the exterior regions, including the uniform bound Cv_+^{-1}, where v_+ denotes max{v,1} and v denotes Eddington-Finkelstein advanced time. We also prove uniform decay bounds for the flux of energy through the event horizon and null infinity. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red-shift effect. The results in particular give an independent proof of the classical uniform boundedness theorem of Kay and Wald, without recourse to the discrete isometries of spacetime.

Paper Structure

This paper contains 24 sections, 15 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Schwarzschild
  3. Eddington-Finkelstein and Regge-Wheeler coordinates
  4. The class of solutions
  5. Conservation laws
  6. Conservation of energy
  7. Coordinate version of the main theorem
  8. The vector fields $X_{\ell}$
  9. Identities
  10. The choice of $f_{\ell}$ for $\ell\ge1$
  11. The case $\ell=0$
  12. The identity, summed
  13. The vector field $Y$
  14. The vector field $K$
  15. Comparison estimates
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\phi$ be a sufficiently regular solution of the wave equation on the (maximally extended) Schwarzschild spacetime $(\mathcal{M},g)$, decaying suitably at spatial infinity on an arbitrary complete asymptotically flat Cauchy surface $\Sigma$. Fix retarded and advanced Eddington-Finkelstein coordinates $u$ and $v$ on one of the exterior regions. For any achronal (We also allow $\mathcal{S}$ to

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 7.1
  • Remark 7.1
  • Proposition 11.1
  • proof
  • Proposition 11.2
  • proof
  • Proposition 11.3
  • proof
  • Proposition 11.4
  • ...and 18 more