Linear perturbations of self-gravitating spherically symmetric configurations

Abstract

We present a new covariant, gauge-invariant formalism describing linear metric perturbation fields on any spherically symmetric background in general relativity. The advantage of this formalism relies in the fact that it does not require a decomposition of the perturbations into spherical tensor harmonics. Furthermore, it does not assume the background to be vacuum, nor does it require its staticity. In the particular case of vacuum perturbations, we derive two master equations describing the propagation of arbitrary linear gravitational waves on a Schwarzschild black hole. When decomposed into spherical harmonics, they reduce to covariant generalizations of the well-known Regge-Wheeler and Zerilli equations. Next, we discuss the general case where the metric perturbations are coupled to matter fields and derive a new constrained wave system describing the propagation of three gauge-invariant scalars from which the complete metric perturbations can be reconstructed. We apply our formalism to the Einstein-Euler system, dividing the fluid perturbations into two parts. The first part, which decouples from the metric perturbations, obeys simple advection equations along the background flow and describes the propagation of the entropy and the vorticity. The second part describes a perturbed potential flow, and together with the metric perturbations it forms a closed wave system.

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

“…Introducing J = S 2 /r 2 , G = S 1 /r 2 and k = L/r 2 we arrive at (c.f., [5], Section IV.A) Lemma 1. A generic smooth metric perturbation admits the following decomposition:…”

“…The metric perturbation h αβ may be subjected to gauge transformations of the form (5). This is why we are interested in the decomposition on a spherically symmetric spacetime of covector fields (such as ζ α ) and symmetric rank two tensor fields such as h αβ .…”

“…Q − can be obtained from the set of equations (3) and (29) in [5]. The calculations are tedious and not particularly illuminating.…”

“…In this section we generalize to the case Λ = 0 the derivation in [5] of the Zerilli equation. Equation (140) implies that there is a scalar field ζ : M → R, defined up to an additive function z :…”

“…Among the many current treatments of linear perturbations of spherically symmetric spacetimes, we have made heavy use of the excellent paper [5], which we found particularly well suited to our approach.…”