The linear stability of the Schwarzschild spacetime in the harmonic gauge: even part

TL;DR

This work advances the linear stability analysis of Schwarzschild spacetime by focusing on the even part of metric perturbations in the harmonic gauge. It develops a robust vector-field framework, including red-shift, Morawetz, and rp-energy currents, to prove decay rates of the perturbations after removing the spherically symmetric part, with an explicit τ^{-2+} rate for the non-symmetric components. The analysis reveals precise asymptotics: the spherically symmetric sector converges to linear combinations of a gauge-fixed mass perturbation K^* and a stationary deformation tensor {}^{W^*}{igl()}π, while non-symmetric parts decay, and the ℓ=0 mode is captured by a stationary W^* plus decaying corrections. Collectively, the results establish quantitative decay and asymptotic structure for even perturbations, laying groundwork toward full linear and nonlinear stability questions in the Schwarzschild and Kerr spacetimes.

Abstract

In this paper we study the even part of the linear stability of the Schwarzschild spacetime as a continuation of [22]. By taking the harmonic gauge, we prove that the energy decays at a rate $τ^{-2+}$ for the solution of the linearized Einstein equation after subtracting its spherically symmetric part. We further show that the spherically symmetric part converges to a linear combination of two special solutions. One is the gauge-fixed mass change solution [19]. The other is the deformation tensor of a stationary one form, which solves the tensorial wave equation. As a key ingredient, we prove that the solutions of the tensorial wave equation converge to this stationary one form up to a scalar multiplication.

Figures (1)

• Figure 1: Penrose diagram

Theorems & Definitions (78)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1