Regge-Wheeler equation, linear stability, and greybody factors for dirty black holes

Abstract

So-called "dirty" black holes are those surrounded by non-zero stress-energy, rather than vacuum. The presence of the non-zero stress-energy modifies key features of the black hole, such as the surface gravity, Regge-Wheeler equation, linear stability, and greybody factors in a rather nontrivial way. Working within the inverse-Cowling approximation, (effectively the test-field limit), we shall present general forms for the Regge-Wheeler equation for linearized spin 0, spin 1, and axial spin 2 perturbations on an arbitrary static spherically symmetric background spacetime. Using very general features of the background spacetime, (in particular the classical energy conditions for the stress-energy surrounding the black hole), we extract several interesting and robust bounds on the behaviour of such systems, including rigorous bounds on the greybody factors for dirty black holes.

V1: 4 pages. V2: two minor typos fixed. V3: Slight change in title to emphasise linear stability; now 5 pages; additional discussion to make clearer the role of the inverse-Cowling approximation; references updated. This version accepted for publication in Physical Review D as a Rapid Communication. V4: minor typos fixed. Published version