publication . Preprint . 2018

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

Johnson, Thomas;
Open Access English
  • Published: 11 Mar 2018
Abstract
In a recent seminal paper \cite{D-H-R} of Dafermos, Holzegel and Rodnianski the linear stability of the Schwarzschild family of black hole solutions to the Einstein vacuum equations was established by imposing a double null gauge. In this paper we shall prove that the Schwarzschild family is linearly stable as solutions to the Einstein vacuum equations by imposing instead a generalised wave gauge: all sufficiently regular solutions to the system of equations that result from linearising the Einstein vacuum equations, as expressed in a generalised wave gauge, about a fixed Schwarzschild solution remain uniformly bounded on the Schwarzschild exterior region and in...
Subjects
free text keywords: General Relativity and Quantum Cosmology, Mathematical Physics, Mathematics - Analysis of PDEs, Mathematics - Differential Geometry
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7 The Regge-Wheeler and Zerilli equations and the gauge-invariant hierarchy 45 7.1 The Regge-Wheeler and Zerilli equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.2 The connection to the system of gravitational perturbations . . . . . . . . . . . . . . . . . . . . 46

8 Initial data and well-posedness of linearised gravity 49 8.1 Initial data for linearised gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8.2 The well-posedness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.3 Pointwise strong asymptotic flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

9 Gauge-normalised solutions and identification of the Kerr parameters 53 9.1 The Km,a-adapted Regge-Wheeler gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 9.2 Achieving the Km,a-adapted Regge-Wheeler gauge: the initial-data-normalised solution S˚m,a . . 54 9.3 Global properties of the solution S˚m,a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10 Precise statements of the main theorems 57 10.1 Flux and integrated decay norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 10.2 Theorem 1: Boundedness and decay for the initial-data-normalised solution S˚m,a . . . . . . . . 59 10.3 Theorem 2: Boundedness and decay for solutions to the Regge-Wheeler and Zerilli equations on the Schwarzschild exterior spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11 Proof of Theorem 1 61 11.1 Decay for the gauge-invariant quantities Φ(1) and Ψ(1) . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11.2 Completing the proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Initial data for the equations of linearised gravity 62 A.1 Constructing admissible initial data from seed data . . . . . . . . . . . . . . . . . . . . . . . . 62 A.2 Propagation of strong asymptotic flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 [7] S. Aretakis, Horizon instability of extremal black holes, Adv. Theor. Math. Phys., 19 (2015), pp. 507-530.

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