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Decay Rates for Spherical Scalar Waves in the Schwarzschild Geometry

Johann Kronthaler

TL;DR

The paper provides a rigorous analysis of late-time decay for a scalar field in Schwarzschild geometry, focusing on spherically symmetric initial data. It constructs an integral spectral representation using Jost solutions of the Regge-Wheeler Schrödinger equation and derives precise small-frequency expansions to analyze the Green's function kernel. The main result proves exact decay rates of $| ext{φ}(t)| \,\le rac{c}{t^3}$ in general and $| ext{φ}(t)| \,\\le rac{c}{t^4}$ for momentarily static data, showing these are optimal, and discusses extension to $l eq 0$ where Price's law predicts $t^{-2l-3}$ and $t^{-2l-4}$. These findings solidify Price's heuristic decay predictions in a rigorous framework and lay groundwork for broader stability analyses in black hole spacetimes.

Abstract

The Cauchy problem is considered for the scalar wave equation in the Schwarzschild geometry. Using an integral spectral representation we derive the exact decay rate for solutions of the Cauchy problem with spherical symmetric initial data, which is smooth and compactly supported outside the event horizon.

Decay Rates for Spherical Scalar Waves in the Schwarzschild Geometry

TL;DR

The paper provides a rigorous analysis of late-time decay for a scalar field in Schwarzschild geometry, focusing on spherically symmetric initial data. It constructs an integral spectral representation using Jost solutions of the Regge-Wheeler Schrödinger equation and derives precise small-frequency expansions to analyze the Green's function kernel. The main result proves exact decay rates of in general and for momentarily static data, showing these are optimal, and discusses extension to where Price's law predicts and . These findings solidify Price's heuristic decay predictions in a rigorous framework and lay groundwork for broader stability analyses in black hole spacetimes.

Abstract

The Cauchy problem is considered for the scalar wave equation in the Schwarzschild geometry. Using an integral spectral representation we derive the exact decay rate for solutions of the Cauchy problem with spherical symmetric initial data, which is smooth and compactly supported outside the event horizon.

Paper Structure

This paper contains 5 sections, 13 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Expansion of the Jost solutions $\grave{\phi}_\omega$
  4. The decay rate for spherical symmetric initial data
  5. Discussion on the case $l \neq0$

Key Result

Lemma 3.1

For all $u>0$, $\omega \in \mathbb{R} \setminus \{0\}$, $\varepsilon >0$, $q \in \mathbb{N}_0$ and $p \in \mathbb{N}$, where the coefficients $c_k$ involve the coefficients $a_0,...,a_q$ of the series expansion of the $\Gamma$-function at $1-p$.

Theorems & Definitions (26)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • proof : Proof of Lemma \ref{['lemma: Entwicklung von acute phi,l=0']}:
  • Lemma 4.1
  • ...and 16 more