Self-similar focusing of rarefaction waves in relativistic collapsing clouds

The development of an inhomogeneous collapse of an initially homogeneous cloud in pressure equilibrium with an external medium is considered in a general-relativistic treatment. It is shown that a rarefaction wave propagating from the outer boundary toward the center of the cloud forms in the initial stage of the collapse. The rarefaction front moves through the collapsing gas at the sound speed, and separates the cloud material into an inner, uniform core and an outer, inhomogeneous envelope. Our analysis distinguishes two possible collapse regimes, depending on the ratio of the focusing time for the rarefaction wave and the free-fall time for the cloud. In massive clouds, the focusing time for the rarefaction wave exceeds the free-fall time, and the collapse of such clouds inevitably leads to the formation of an event horizon and black hole. In low-mass clouds, the focusing time is less than the free-fall time. After the focusing, the collapse of such clouds is fully inhomogeneous, and can be appreciably slowed by the pressure gradient. The compression of a cloud in a transitional regime separating these two scenarios is studied. In this case, a self-similar collapse regime is realized in the central part of the cloud near the focusing time for the rarefaction wave. The constructed self-similar solution describes both early stages in the compression up to the focusing of the rarefaction wave and later stages with the accretion of gas onto the black hole that is formed. Asymptotic distributions of quantities in the inhomogeneous region at large distances from the rarefaction front and in the accreting envelope are found. The structure of space-time in the vicinity of the black hole that is formed at the center of the cloud as a result of the focusing of the rarefaction wave is discussed.