GENERAL-RELATIVISTIC COLLAPSE OF HOMOTHETIC IDEAL-GAS SPHERES AND PLANES

This paper presents in a succinct but self-contained style of our understanding of the gravitational collapse of homothetic, ideal gas spheres and planes. The physical problem is reduced to a study of a nonlinear autonomous system of differential equations. It is first shown that this system is a Cauchy system everywhere in the projective space t/r = xi epsilonR. The concept of sonic Cauchy and apparent horizons is introduced, and it is shown that the set of globally analytic naked solutions is discrete as mentioned by Ori and Piran but is finite and even empty for very strong equations of state. Even when singularities may be ''seen,'' we are able to show that they cannot be ''heard.'' Solutions which develop singularities from regular initial conditions are moreover shown to be necessarily in motion at spacelike infinity on every hypersurface t < 0, and are likely to require inwardly directed radial trajectories at spatial infinity. We give also a parallel analysis of the case of planar homothetic collapse. We find that in this case the singularity is never ''naked.'' It appears then that intersecting particle trajectories are necessary to form visible singularities. We offer in passing the t = const hypersurfaces in the case of spherical collapse as another example of surfaces that come arbitrarily close to a singularity, but which neither contain trapped surfaces nor have any in their past histories. Finally, graphical illustrations, both computed and schematic, are provided.