Nonspherical perturbations of relativistic gravitational collapse

Abstract

It is known that there can be no gravitational, electromagnetic, or scalar field perturbations (except angular momentum) of a Schwarzschild black hole. A gravitationally collapsing star with nonspherical perturbations must therefore radiate away its perturbations or halt its collapse. The results of computations in comoving coordinates are presented to show that the scalar field in a collapsing star neither disappears nor halts the collapse, as the star passes inside its gravitational radius. On the star's surface, near the event horizon, the scalar field varies as a_1 + a_2 exp (-t/2M) due to time dilation. The dynamics of the field outside the star can be analyzed with a simple wave equation containing a spacetime-curvature induced potential. This potential is impenetrable to zero-frequency waves and thus a_1, the final value of the field on the stellar surface, is not manifested in the exterior; the field vanishes. The monopole perturbation falls off as t^(-2); higher l-poles fall off as ln t/t^(2l+3). The analysis of scalar-field perturbations works as well for electromagnetic and gravitational perturbations and also for zero-restmass perturbation fields of arbitrary integer spin. All these perturbation fields obey wave equations with curvature potentials that differ little from one field to another. For all fields, radiatable multipoles (l ≥ spin of the field) fall off as lnt/t^(2l+3).