Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric

Abstract

We derive an energy conservation law for the system of gravitational perturbations on the Schwarzschild spacetime expressed in a double null gauge. The resulting identity involves only first derivatives of the metric perturbation. Exploiting the gauge invariance up to boundary terms of the fluxes that appear, we are able to establish positivity of the flux on any outgoing null hypersurface to the future of the initial data. This allows us to bound the total energy flux through any such hypersurface, including the event horizon, in terms of initial data. We similarly bound the total energy radiated to null infinity. Our estimates provide a direct approach to a weak form of stability, thereby complementing the proof of the full linear stability of the Schwarzschild solution recently obtained in [M. Dafermos, G. Holzegel and I. Rodnianski \emph{The linear stability of the Schwarzschild solution to gravitational perturbations}, arXiv:1601.06467].

21 pages, 2 figures

References in corpus (6)

• The linear stability of the Schwarzschild solution to gravitational perturbations
• Superradiant instabilities of asymptotically anti-de Sitter black holes
• Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem
• Black Hole Instabilities and Exponential Growth
• Unstable mode solutions to the Klein-Gordon equation in Kerr-anti-de Sitter spacetimes
• The Penrose inequality on perturbations of the Schwarzschild exterior

Cited by in corpus (12)

• The linear stability of the Schwarzschild solution to gravitational perturbations
• The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge
• A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations
• Mathematical General Relativity
• The non-linear stability of the Schwarzschild family of black holes
• The Case Against Smooth Null Infinity IV: Linearised Gravity Around Schwarzschild -- An Overview
• Open problems in mathematical physics
• A Conserved Energy for Axially Symmetric Newman-Penrose-Maxwell Scalars on Kerr Black Holes
• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• Axially Symmetric Perturbations of Kerr Black Holes I: A gauge-invariant construction of ADM Energy
• Axially Symmetric Perturbations of Kerr Black Holes II: Boundary Behaviour of the Dynamics in the Orbit Space
• Energy Extraction, or Lack Thereof