General relativistic collapse of homothetic ideal gas spheres and planes

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Abstract
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≡ξεR. 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. Solutions which develop singularities from regular initial conditions are moreover shown to be necessarily in motion at spacelike infinity on every hypersurface tt=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.