Black Hole Perturbations

Abstract

The study of black hole perturbations is an indispensable means of understanding the processes that give rise to gravitational and electromagnetic radiation. The observation of sources of gravitational radiation is an ongoing endeavour worldwide.

Perturbations of the most general black hole states have been first investigated by Chrzanowski and Misner [1, 2], Detweiler [3] and Chandrasekhar [5]. Among these pioneering studies, Chandrasekhar’s is most detailed. He uses a Newman-Penrose (NP) approach both to the unperturbed Kerr metric and to the perturbed space-time.

In this talk we present the full set of black-hole perturbations in the ingoing radiation gauge. Space-time is treated here as a Kerr-metric background on which NP formalism is featured plus a perturbative part h_ab in the metric. In the spirit of the weak-field approximation, we take all perturbations as fields being defined on the background manifold of the Kerr metric.

In treatments of black-hole perturbations, it is a common practice to seek a suitable gauge fixing. A convenient choice, promoted by Chrzanowski, is the incoming (or outgoing) radiation gauge for the normal modes. This choice does not uniquely fix the coordinate gauge, however. In fact, the present work has been launched with the intent to take a second look at the remaining gauge freedom.

In this paper, the following picture emerges for the classical electrovacuum perturbations. There exists a subset of the field equations, consisting of two gravitational equations and two Maxwell equations not containing any mode mixing. (This is because the complex conjugate electromagnetic stresses are absent). From these relations alone, using the definition of the Weyl tensor perturbation ψ₀, it is possible to obtain a pair of coupled equations for ψ₀ and the electromagnetic perturbation φ₀. Thus a normal mode expansion for this doublet of fields is available. Mode mixing occurs only in the further gravitational and electromagnetic perturbation components derived from the fundamental doublet. This picture fully agrees with the results obtained in an entirely different gauge [10].