Gauge invariant perturbations of black holes. I. Schwarzschild space–time

Abstract

We cast the perturbed Bianchi identities, in the Schwarzschild background, into a form involving only tetrad and coordinate gauge invariant Newman–Penrose field quantities. These quantities, which arise naturally in our approach, are gauge invariant quantities of spin-weight ±2, ±1, and 0. Some of the integrability conditions for the Bianchi identities then provide a system of six gauge invariant perturbation wave equations for the spin-weighted quantities. These wave equations are, respectively, the (spin-weight ±2) Bardeen–Press equations, two new (spin-weight ±1) gravitational wave equations, and two (spin-weight 0) Regge–Wheeler equations. Other integrability conditions provide the transformation identities that relate the field quantities to each other, and hence relate the various perturbation wave equations to one another. In particular, this method provides an alternative derivation of the transformations between the Bardeen–Press and Regge–Wheeler equations. The integrability conditions also allow us to relate the Bardeen–Press quantities of opposite spin-weight, and we investigate how this relationship compares with the Teukolsky–Starobinsky identities. Finally, we give a derivation of the gauge invariant Zerilli equation, and show how it is related to the fundamental equations mentioned above.