Perturbations of space-times in general relativity

Abstract

A definition of perturbations of space-times in general relativity is proposed. The definition leads in a natural way to a concept of gauge invariance, and to an extension of a lemma of Sachs (1964). Coupled equations governing linearized perturbations of certain tetrad components of scalar, electromagnetic, and gravitational fields are derived by the use of Geroch, Held & Penrose’s (1973) version of the tetrad formalism of Newman & Penrose (1962). It is shown that these perturbations are gauge invariant if and only if the unperturbed space-time is vacuum of algebraic type {22} or, equivalently, if and only if the perturbation equations decouple. Finally the maximal subclass of type {22} space-times for which the decoupled perturbation equations can be solved by separation of variables is found. This class comprises all the nonaccelerating type {22} space-times, including that of Kerr, thus elucidating earlier results of Bardeen & Press (1972) and Teukolsky (1973).