Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields

Abstract

A nearly spherical star collapses through its gravitational radius. Nonspherical perturbations exist in its density, pressure, electromagnetic field, and gravitational field, and in other (hypothetical) zero-rest-mass, integer-spin fields coupled to sources in the stars. Paper I analyzed the evolution of scalar-field and gravitational-field perturbations. This paper treats fields of arbitrary integer spin and zero rest mass, using the Newman-Penrose tetrad formalism. The analysis of each multipole ($\mathrm{order}=l$) of each field ($\mathrm{spin}=s$) is reduced to the study of a two-dimensional wave equation, with a curvature potential that differs little from one field to another. The analysis of this wave equation for the scalar case ($s=0$) carries over completely to fields of arbitrary spin $s$. In particular, any radiatable multipole ($l\ensuremath{\ge}s$) gets radiated away completely in the late stages of collapse; if the multipole is static prior to the onset of collapse, it will die out as ${t}^{\ensuremath{-}(2l+2)}$ at late times. Nonradiatable multipoles ($lls$) are conserved. This paper also treats gravitational perturbations in the Newman-Penrose framework, and supplies some technical details missing in the gravitational-perturbation analysis of Paper I.