The fundamental theoretical challenge is to explain why so many matter models admit a critical solution, that is, an attractor of codimension one at the black hole threshold. If the existence of a critical solution is really a generic feature, then there should be at least an intuitive argument, and perhaps a mathematical proof, for this important fact. On the other hand, the spherical Einstein-Vlasov system may already be providing a counter-example. A more thorough mathematical and numerical investigation of this system is therefore particularly urgent.

The critical spacetimes and their perturbations are well known only in the past light cone of the singularity. The Cauchy horizon and the naked singularity itself, as well as the possible continuations beyond the Cauchy horizon, of the critical spacetimes have not yet been investigated thoroughly. It is unknown if all possible continuations have a timelike naked singularity, and in what manner this singularity is avoided when one perturbs away from the black hole threshold.

An important mathematical challenge is to make the intuitive dynamical systems picture of critical collapse more rigorous, by providing a distance measure on the phase space, and a prescription for a flow on the phase space (equivalent to a prescription for the lapse and shift). The latter problem is intimately related to the problem of finding good coordinate systems for the binary black hole problem.

On the phenomenological side, it is likely that the scope of critical collapse will be expanded to take into account new phenomena, such as multicritical solutions (with several growing perturbation modes), or critical solutions that are neither static, periodic, CSS or DSS. More complicated phase diagrams than the simple black hole-dispersion transition are already being examined, and the intersections of phase boundaries are of particular interest.

Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
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