Critical Behavior of Black Hole Formation in a Scalar Wave Collapse

Abstract

Gravitational collapse is one of interesting phenomena in general relativity. In recent years, a large number of studies have been made on the dynamical behavior of a massless scalar field, in the context of spherically symmetric gravitational collapse. It has been shown that for a weak-field implosion scalar waves bounce and disperse to infinity, while a strong-field implosion leads to black hole formation.!) Furthermore, in numerical calculations, ChoptuikJ found an important behavior of a family of solutions which contain the parameter p characterizing the strength of the scalar field: For black hole formation there exists the critical limit p-4 p* to give the gravitational mass satisfying a power law MsHcx: IP-p*l 7 with a universal critical exponent r~0.37. Such a critical behavior may be a general property of gravitational collapse, since imploding axisymmetric gravitational waves also show a similar behavior.l In this paper our purpose is to present an analytical model of the spherically symmetric collapse of scalar waves. We solve the spherically symmetric Einstein equations coupled to a massless scalar field ¢ to find the self-similar solution which has a critical parameter p. This model allows us to see explicitly the black hole formation and the scalar field dispersion. Our main concern is the field evolution in the critical limit p-41, which should be compared with the numerical results. Let us impose a self-similarity on the massless scalar field such that ¢(u, v)= ¢(u/v), where u and v are retarded and advanced times respectively. The line element in the spherical symmetric double-null coordinates is given by