Problems in the relativistic theory of gravitational collapse

Abstract

Gravitational collapse is analysed in terms of a simple model. Both Newtonian and Relativistic treatments are given, and the curious phenomenon of overtaking is discussed from a Newtonian viewpoint. It is shown that the Relativistic description of the motion is closely analogous to the classical treatment. The predicted final stage of the motion is collapse to a point singularity. The asymptotic behaviour near this singularity is examined, and it is shown that the presence of small inhomogeneities in a collapsing dust-sphere will radically affect the motion. The question of Boundary Conditions in General Relativity is considered, insofar as this affects the Relativistic description of the motion. It is shown that the Lichnerowicz conditions may be too restrictive, and conditions in respect of the first and second fundamental forms are proposed. It is shown that the spectral shift of a collapsing body becomes unstable as the gravitational radius is approached. The final stage in the collapse process is examined. In order to suggest possible models involving motion beyond the point singularity, it is necessary to consider General Relativity from a modern mathematical viewpoint; this treatment leads to a consideration of models involving multiply connected manifolds. The concept of time orientation is developed. It is shown that if, in a particular model, a collapsing body passes through the point singularity and then expands into the same spatial region, the model necessarily involves causality violations.