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Linear Stability of Schwarzschild Spacetime: Decay of Metric Coefficients

Pei-Ken Hung, Jordan Keller, Mu-Tao Wang

TL;DR

The paper proves linear stability of the Schwarzschild spacetime by decomposing perturbations into lower (ℓ<2) and higher (ℓ≥2) angular modes. It advances a dual approach: (i) analyze lower harmonics to extract Kerr perturbations and fix gauge, and (ii) study higher harmonics via gauge-invariant master quantities (P, Q^{(-)}, Q^{(+)}, Z^{(+)}) obeying Regge-Wheeler and Zerilli-type equations, obtaining uniform decay through vector-field multipliers (red-shift, Morawetz, quasi-conformal) and energy methods. Decay of the co-closed sector is established in a Regge-Wheeler gauge, while the closed sector uses a Chandrasekhar interpolation gauge to derive decay for the Zerilli-Moncrief function and metric components, culminating in decay of all metric perturbations modulo a linearized Kerr perturbation. The results yield explicit decay rates and norms, showing that linearized solutions converge to a Kerr perturbation in the exterior region, thereby completing a robust linear stability theory for Schwarzschild with a gauge-sensitive reconstruction of the metric. This work provides an accessible alternative to DHR by emphasizing geometric, gauge-invariant quantities and elementary multiplier techniques.

Abstract

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. We employ Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For the co-closed portion, we extend previous results by the first two authors, deriving Regge-Wheeler type equations for two gauge-invariant master quantities without the earlier paper's need of axisymmetry. For the closed portion, we build upon earlier work of Zerilli and Moncrief, wherein the authors derive an equation for a gauge-invariant master quantity in a spherical harmonic decomposition. We work with gauge-invariant quantities at the level of perturbed connection coefficients, with the initial value problem formulated on Cauchy data sets. With the choice of an appropriate gauge in each of the two portions, decay estimates on these decoupled quantities are used to establish decay of the metric coefficients of the solution, completing the proof of linear stability. Our result differs from that of Dafermos-Holzegel-Rodnianski, both in our choice of gauge and in our identification and utilization of lower-level gauge-invariant master quantities.

Linear Stability of Schwarzschild Spacetime: Decay of Metric Coefficients

TL;DR

The paper proves linear stability of the Schwarzschild spacetime by decomposing perturbations into lower (ℓ<2) and higher (ℓ≥2) angular modes. It advances a dual approach: (i) analyze lower harmonics to extract Kerr perturbations and fix gauge, and (ii) study higher harmonics via gauge-invariant master quantities (P, Q^{(-)}, Q^{(+)}, Z^{(+)}) obeying Regge-Wheeler and Zerilli-type equations, obtaining uniform decay through vector-field multipliers (red-shift, Morawetz, quasi-conformal) and energy methods. Decay of the co-closed sector is established in a Regge-Wheeler gauge, while the closed sector uses a Chandrasekhar interpolation gauge to derive decay for the Zerilli-Moncrief function and metric components, culminating in decay of all metric perturbations modulo a linearized Kerr perturbation. The results yield explicit decay rates and norms, showing that linearized solutions converge to a Kerr perturbation in the exterior region, thereby completing a robust linear stability theory for Schwarzschild with a gauge-sensitive reconstruction of the metric. This work provides an accessible alternative to DHR by emphasizing geometric, gauge-invariant quantities and elementary multiplier techniques.

Abstract

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. We employ Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For the co-closed portion, we extend previous results by the first two authors, deriving Regge-Wheeler type equations for two gauge-invariant master quantities without the earlier paper's need of axisymmetry. For the closed portion, we build upon earlier work of Zerilli and Moncrief, wherein the authors derive an equation for a gauge-invariant master quantity in a spherical harmonic decomposition. We work with gauge-invariant quantities at the level of perturbed connection coefficients, with the initial value problem formulated on Cauchy data sets. With the choice of an appropriate gauge in each of the two portions, decay estimates on these decoupled quantities are used to establish decay of the metric coefficients of the solution, completing the proof of linear stability. Our result differs from that of Dafermos-Holzegel-Rodnianski, both in our choice of gauge and in our identification and utilization of lower-level gauge-invariant master quantities.

Paper Structure

This paper contains 47 sections, 23 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. The Schwarzschild Spacetime
  3. Linearized Gravity in a Spherically Symmetric Background
  4. Spherically symmetric background
  5. The Linearized Vacuum Einstein Equations
  6. Hodge Decomposition
  7. Spherical Harmonics
  8. Linearized Kerr Solutions and Pure Gauge Solutions
  9. Linearized Kerr Solutions
  10. Linearized Change in Mass
  11. Linearized Change in Angular Velocity
  12. Pure Gauge Solutions
  13. Analysis of the Lower Harmonics and Proof of Theorem 1
  14. The $\ell = 0$ Case
  15. The $\ell = 1$, Closed Case
  16. ...and 32 more sections

Key Result

Theorem 1

Let $\delta g$ be a smooth, symmetric $(0,2)$ tensor on the Schwarzschild spacetime, satisfying the linearized vacuum Einstein equations linearized_Einstein. For the $\delta g^{\ell < 2}$ component of $\delta g$, there exists a unique smooth co-vector $X^{\ell<2}$ (modulo Killing fields) on the Schw where $K, K_{-1}, K_0, K_1$ are smooth symmetric $(0, 2)$ tensors that correspond to linearized Ker

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Definition 6
  • Lemma 7
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 28 more