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Price's Law on Nonstationary Spacetimes

Jason Metcalfe, Daniel Tataru, Mihai Tohaneanu

TL;DR

The paper proves Price's Law decay for linear waves on a broad class of nonstationary, asymptotically flat 3+1 dimensional spacetimes by establishing robust local energy decay bounds that do not require stationarity. Using a vector-field framework, commutator estimates, a one-dimensional reduction, and cone-Sobolev embeddings, the authors derive a sequence of improving decay bounds that culminate in a sharp $t^{-3}$ local decay inside the forward light cone, plus precise gradient decay. The results accommodate small perturbations of Kerr spacetimes and rely on stationary (and, when needed, weak) local energy decay bounds, thereby providing a robust pathway toward nonlinear stability analyses in general relativity. The work combines geometric microlocal analysis with energy methods to achieve Price’s Law in nonstationary backgrounds, with explicit decay rates and quantitative norms controlling the forcing and initial data.

Abstract

In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate (Price's law \cite{MR0376103}) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric. This result was previously established by the second author in \cite{Tat} on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.

Price's Law on Nonstationary Spacetimes

TL;DR

The paper proves Price's Law decay for linear waves on a broad class of nonstationary, asymptotically flat 3+1 dimensional spacetimes by establishing robust local energy decay bounds that do not require stationarity. Using a vector-field framework, commutator estimates, a one-dimensional reduction, and cone-Sobolev embeddings, the authors derive a sequence of improving decay bounds that culminate in a sharp local decay inside the forward light cone, plus precise gradient decay. The results accommodate small perturbations of Kerr spacetimes and rely on stationary (and, when needed, weak) local energy decay bounds, thereby providing a robust pathway toward nonlinear stability analyses in general relativity. The work combines geometric microlocal analysis with energy methods to achieve Price’s Law in nonstationary backgrounds, with explicit decay rates and quantitative norms controlling the forcing and initial data.

Abstract

In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a local uniform decay rate (Price's law \cite{MR0376103}) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric. This result was previously established by the second author in \cite{Tat} on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.

Paper Structure

This paper contains 27 sections, 23 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Notations and the regularity of the metric
  3. Normalized coordinates
  4. Uniform energy bounds
  5. Local energy decay
  6. The main result
  7. Vector fields and local energy decay
  8. Vector fields: notations and definitions
  9. The weak local energy decay
  10. The stationary local energy decay
  11. The pointwise decay
  12. The cone decomposition and Sobolev embeddings
  13. The one dimensional reduction
  14. An initial decay bound
  15. Improved $L^2$ gradient bounds
  16. ...and 12 more sections

Key Result

Theorem 5

Let $g$ be a metric which satisfies the conditions (i), (ii) in ${\mathbb R} \times {\mathbb R}^3$, or (i), (ii), (iii) in ${\mathbb R} \times {\mathbb R}^3\setminus B(0,R_0)$, and $V$ belonging to $S(r^{-3})$. Assume that the evolution box satisfies the stationary local energy bounds from Definitio where If in addition the weak local energy bounds lew hold then the same result is valid for all f

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 33 more