The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge

Abstract

We prove in this paper that the Schwarzschild famiily of black holes are linearly stable as a family of solutions to the system of equations that result from expressing the Einstein vacuum equations in a generalised wave gauge. In particular we improve on our recent work \cite{Johnsonlinstabschwarzold} by modifying the generalised wave gauge employed therein so as to establish asymptotic flatness of the associated linearised system. The result thus complements the seminal work \cite{DHRlinstabschwarz} of Dafermos-Holzegel-Rodnianski in a similar vein as to how the work \cite{LRstabmink} of Lindblad-Rodnianski complemented that of Christodoulou-Klainerman \cite{CKstabmink} in establishing the nonlinear stability of the Minkowski space. This paper is the content of the authors PhD thesis.

arXiv commentRemoved the statement of the conjecture regarding the nonlinear stability of Schwarzschild due to Dafermos--Holzegel--Rodnianski so as to remove a text overlap

References in corpus (13)

• Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism
• Strichartz estimates on Schwarzschild black hole backgrounds
• Gauge invariant perturbations of Schwarzschild black holes in horizon-penetrating coordinates
• The wave equation on the Schwarzschild metric II: Local decay for the spin 2 Regge Wheeler equation
• On the asymptotic behavior of solutions to Einstein's vacuum equations in wave coordinates
• Characterisation of the Energy of Gaussian Beams on Lorentzian Manifolds - with Applications to Black Hole Spacetimes
• Energy and angular momentum of the weak gravitational waves on the Schwarzschild background -- quasilocal gauge-invariant formulation
• Non-modal linear stability of the Schwarzschild black hole
• Linear perturbations of self-gravitating spherically symmetric configurations
• The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge
• The Global Stability of the Minkowski Spacetime Solution to the Einstein-Nonlinear Electromagnetic System in Wave Coordinates
• Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric
• Decay of Solutions of the Teukolsky Equation for Higher Spin in the Schwarzschild Geometry

Cited by in corpus (12)

• The linear stability of Reissner-Nordström spacetime: the full subextremal range
• The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge
• A sharp version of Price's law for wave decay on asymptotically flat spacetimes
• The linear stability of Reissner-Nordström spacetime for small charge
• Sharp decay estimates for massless Dirac fields on a Schwarzschild background
• The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves
• Price's law for spin fields on a Schwarzschild background
• Boundedness and decay for the Teukolsky system of spin $\pm 2$ on Reissner-Nordström spacetime: the case $|Q|\ll M$
• Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes
• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• On the link between the Maxwell and linearised Einstein equations on Schwarzschild
• Physical-space estimates for axisymmetric waves on extremal Kerr spacetime