Black hole non-modal linear stability: the Schwarzschild (A)dS cases

Abstract

The nonmodal linear stability of the Schwarzschild black hole established in Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative cosmological constant Λ. Two gauge invariant combinations G± of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map [) with domain the set of equivalent classes [h αβ ] under gauge transformations of solutions of the linearized Einstein's equation, is invertible. The way to reconstruct a representative of [h αβ ] in terms of (G−, G+) is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, G+ and G− are bounded in the the outer static region. At large times, the perturbation decays leaving a linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter background solution. For negative cosmological constant it is shown that there are choices of boundary conditions at the time-like boundary under which the Schwarzschild anti de Sitter black hole is unstable. The root of Chandrasekhar's duality relating odd and even modes is exhibited, and some technicalities related to this duality and omitted in the original proof of the Λ = 0 case are explained in detail. CONTENTS

Theoretical cosmology

“…For example, generic spacelike singularities, traditionally regarded as being cosmological singularities, have been studied in detail [22]. It is also of interest to extend mathematical stability results to the case of a non-zero cosmological constant [9].…”

Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

“…Consider now the zero boost weight case. Fields of this type turned out to be relevant recently for proving the nonmodal linear stability of the 4dimensional Schwarzschild (-de Sitter) black hole in [11,12]. The general case for the class of 4-dimensional Einstein type D spacetimes was analyzed in [4], where it was shown that the decoupling and field reconstruction results are only valid for the case of real Ψ 2 , static black holes being a particular case of this situation.…”

On non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics

“…In this section we analyze the l = 1 modes when ( n N, γ ij ) is a twodimensional round unit sphere. We follow the treatment of the Lorentzian case in [15], which requires only trivial modifications when addressing the Riemannian setting. We present the argument here because of the need of establishing estimates for the gauge vector fields.…”

“…According to [15,Section IV.A.2], it is convenient to define (X, Y a ) by solving the following system of equations:…”