Nonspherical Perturbations of Relativistic Gravitational Collapse. I. Scalar and Gravitational Perturbations

Abstract

When a nearly spherical star gravitationally collapses through its event horizon, it cannot leave behind a static gravitational field with nonspherical perturbations. The dynamics of these perturbations during collapse is studied with a scalar-field analog. Computations in comoving coordinates show that the field neither vanishes nor becomes singular as the star falls inside its gravitational radius. The scalar field on the surface of the star must vary as ${a}_{1}+{a}_{2}\mathrm{exp}(\ensuremath{-}\frac{t}{2M})$ due to time dilation. An analysis is presented of the evolution of the exterior scalar field, based on a simple wave equation containing a space-time-curvature-induced potential barrier. This barrier is shown to be impenetrable to zero-frequency waves and thus ${a}_{1}$, the final value of the field on the surface of the star, is not manifested in the exterior; the exterior field vanishes. The detailed nature of the falloff of the field depends on backscattering off the potential. It is shown that an initially static $l$ pole dies out as ${t}^{\ensuremath{-}(2l+2)}$. If there is no initial $l$ pole but one develops during the collapse it must fall off as ${t}^{\ensuremath{-}(2l+3)}$. Wave equations with curvature-induced potential barriers have been derived by Regge and Wheeler and by Zerilli for gravitational perturbations. With these equations the analysis of gravitational perturbations is precisely the same as for the scalar ones. In particular, gravitational multipole perturbations (with $l\ensuremath{\ge}2$) fall off at large $t$ as ${t}^{\ensuremath{-}(2l+2)}$ or ${t}^{\ensuremath{-}(2l+3)}$, depending on initial conditions. (In an accompanying paper it is shown that this result applies as well to the radiatable multipoles of a zero-rest-mass field of any integer spin.)