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Local decay of waves on asymptotically flat stationary space-times

Daniel Tataru

TL;DR

The paper addresses the pointwise decay of solutions to the wave equation on stationary, asymptotically flat 3+1 dimensional spacetimes, motivated by Price’s conjecture on Schwarzschild and Kerr backgrounds. It develops a resolvent-based framework with a coordinate normalization to Regge-Wheeler-like coordinates, decomposes the operator into long- and short-range pieces, and leverages forward energy bounds together with weak local energy decay to obtain uniform resolvent estimates near the real axis and a detailed zero-frequency expansion. By translating these resolvent bounds into pointwise decay via vector-field Sobolev methods and separating large- and small-frequency analyses, it proves a sharp local decay rate $|u(t,x)| \lesssim \frac{1}{\langle t+|x|\rangle \langle t-|x| \rangle^2}$ and $|\partial_t u(t,x)| \lesssim \frac{1}{\langle t+|x|\rangle \langle t-|x| \rangle^3}$, confirming Price’s Law in Schwarzschild and small-angular-momentum Kerr spacetimes within the forward cone. The results hinge on low-frequency behavior and trapped set geometry, and provide a robust framework for decay in a broad class of stationary asymptotically flat geometries.

Abstract

In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary 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 for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds, where such a decay rate has been conjectured by R. Price. Our results apply to both of these cases.

Local decay of waves on asymptotically flat stationary space-times

TL;DR

The paper addresses the pointwise decay of solutions to the wave equation on stationary, asymptotically flat 3+1 dimensional spacetimes, motivated by Price’s conjecture on Schwarzschild and Kerr backgrounds. It develops a resolvent-based framework with a coordinate normalization to Regge-Wheeler-like coordinates, decomposes the operator into long- and short-range pieces, and leverages forward energy bounds together with weak local energy decay to obtain uniform resolvent estimates near the real axis and a detailed zero-frequency expansion. By translating these resolvent bounds into pointwise decay via vector-field Sobolev methods and separating large- and small-frequency analyses, it proves a sharp local decay rate and , confirming Price’s Law in Schwarzschild and small-angular-momentum Kerr spacetimes within the forward cone. The results hinge on low-frequency behavior and trapped set geometry, and provide a robust framework for decay in a broad class of stationary asymptotically flat geometries.

Abstract

In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary 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 for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds, where such a decay rate has been conjectured by R. Price. Our results apply to both of these cases.

Paper Structure

This paper contains 12 sections, 15 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. The coordinate normalization
  3. Normalized coordinates
  4. Left and right multiplication by scalar functions
  5. The resolvent and energy estimates
  6. The resolvent and local energy decay
  7. Vector field bounds and the resolvent limit on the real axis
  8. The resolvent analysis near zero frequency
  9. From $L^2$ to pointwise bounds
  10. Conclusion
  11. The case of large $\tau$, $|\tau| \gtrsim 1$.
  12. The case of small $\tau$, $|\tau| \ll 1$.

Key Result

Theorem 4

Let $m$ be a large enough integer. Let $g$ be a metric which satisfies the conditions (i), (ii), (iii) in ${\mathbb R} \times {\mathbb R}^3$, or (i), (ii), (iii), (iv) in ${\mathbb R} \times {\mathbb R}^3\setminus B(0,R_0)$. Let $V$ be a potential as in vassume. Assume that the evolution hom is forw respectively

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Corollary 5: Price's Law
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8
  • proof
  • ...and 19 more