Black hole nonmodal linear stability under odd perturbations: The Reissner-Nordström case

Abstract

Following a program on black hole nonmodal linear stability initiated in Phys. Rev. Lett. 112 (2014) 191101, we study odd linear perturbations of the Einstein-Maxwell equations around a Reissner-Nordström (A)dS black hole. We show that all the gauge invariant information in the metric and Maxwell field perturbations is encoded in the spacetime scalars F = δ(F * αβ F αβ ) and Q = δ( 1 48 C * αβγδ C αβγδ ), where C αβγδ is the Weyl tensor, F αβ the Maxwell field, a star denotes Hodge dual and δ means first order variation, and that the linearized Einstein-Maxwell equations are equivalent to a coupled system of wave equations for F and Q. For nonnegative cosmological constant we prove that F and Q are pointwise bounded on the outer static region. The fields are shown to diverge as the Cauchy horizon is approached from the inner dynamical region, providing evidence supporting strong cosmic censorship. In the asymptotically AdS case the dynamics depends on the boundary condition at the conformal timelike boundary and there are instabilities if Robin boundary conditions are chosen.

Black hole nonmodal linear stability: Even perturbations in the Reissner-Nordström case

“…For odd perturbations i) and ii) were proved in the companion paper [1]. In this paper we complete the proof of nonmodal linear stability of the charged black hole by considering the even sector of the linear perturbation fields.…”

“…These two papers are devoted to the nonmodal linear stability of the Schwarzschild and Schwarzschild de Sitter black hole. The nonmodal linear stability of the Reissner-Nordström black hole with Λ ≥ 0, under odd perturbations, was established in [1]. In the following Sections we complete the proof of nonmodal linear stability of this black hole by proving its stability under even perturbations.…”

“…Similarly, a symmetric tensor field S αβ = S (αβ) , such as h αβ , G αβ := dG αβ /dε| 0 and T αβ := dT αβ /dε| 0 , decomposes as [1,3,15]…”

“…Even and odd fields are not mixed by the LEME. The restriction of the LEME to the odd sector was the subject of [1], even perturbations are studied in the following sections.…”

Scattering and absorption of a scalar field impinging on a charged black hole in the Einstein-Maxwell-dilaton theory

“…Moreover, the linear stability of RN black hole was fully examined Refs. [86]- [87] in the last couple of years.…”