DIMENSIONALITY, SELF-SIMILARITY, AND CRITICAL COLLAPSE

Abstract

Critical phenomena in gravitational collapse have attracted much attention since the pioneering work of Choptuik . From the known results obtained so far, the following emerges : Critical collapse in general can be divided into two different types according to whether or not the black hole mass near the threshold of the black hole formation takes the form MBH OC (P—P*)', where P characterizes the strength of the initial data in such a way that when P > P* a black hole will be formed, and when P < P* no black hole will be formed. When the mass of black holes does not take the above form, the collapse is referred to as Type I collapse, while it does, it is referred to as Type II collapse. In Type I collapse, the critical solutions found so far have neither continuous self-similarity (CSS) nor discrete self-similarity (DSS), while in Type II collapse the critical solutions have either CSS or DSS. By virtue of this, the exponent y is usually also different. Whether the critical solution has CSS, DSS, or none of them, depending on both the matter field and the regions of the initial data space . The co-existence of Type I and Type II collapse was first found in the SU(2) Einstein-Yang-Mills case , and later extended to both the Einstein-scalar case 4 and the Einstein-Skyrme case , while the co-existence of CSS and DSS critical solutions was found in the Brans-Dicke theory . The uniqueness of the exponent 7 in Type II collapse is well understood in terms of perturbations , and is closely related to the fact that the critical solution has precisely one unstable mode. This property now is considered as the main criterion for a solution to be critical. While the uniqueness of the exponent 7 crucially depends on the numbers of the unstable modes of the critical solution, that whether or not the formation of black holes starts with a mass gap seemingly only depends on whether the spacetime has self-similarity or not. Thus, even the collapse is not critical, if a spacetime has CSS or DSS, the formation of black holes may still turn on with zero mass. We studied it for gravitational collapse of massless scalar field and radiation fluid 8 and