The rigorous investigation of the spherically symmetric collapse of collisionless matter in general relativity was initiated by Rein and the author , who showed that the evolution of small initial data leads to geodesically complete spacetimes where the density and curvature fall off at large times. Later, it was shown  that independent of the size of the initial data the first singularity, if there is one at all, must occur at the centre of symmetry. This result uses a time coordinate of Schwarzschild type; an analogous result for a maximal time coordinate was proved in . The question of what happens for general large initial data could not yet be answered by analytical techniques. In , numerical methods were applied to try to make some progress in this direction. The results are discussed in the next paragraph.
Despite the range and diversity of the results obtained by Christodoulou on the spherical collapse of a scalar field, they do not encompass some of the most interesting phenomena that have been observed numerically. These are related to the issue of critical collapse. For sufficiently small data the field disperses. For sufficiently large data a black hole is formed. The question is what happens in between. This can be investigated by examining a one-parameter family of initial data interpolating between the two cases. It was found by Choptuik  that there is a critical value of the parameter below which dispersion takes place and above which a black hole is formed, and that the mass of the black hole approaches zero as the critical parameter value is approached. This gave rise to a large literature in which the spherical collapse of different kinds of matter was computed numerically and various qualitative features were determined. For reviews of this see [120, 119]. In the calculations of  for collisionless matter, it was found that in the situations considered the black hole mass tended to a strictly positive limit as the critical parameter was approached from above. These results were confirmed and extended by Olabarrieta and Choptuik . There are no rigorous mathematical results available on the issue of a mass gap for either a scalar field or collisionless matter and it is an outstanding challenge for mathematical relativists to change this situation.
Another aspect of Choptuik's results is the occurrence of a discretely self-similar solution. It would seem hard to prove the existence of a solution of this kind analytically. For other types of matter, such as a perfect fluid with linear equation of state, the critical solution is continuously self-similar and this looks more tractable. The problem reduces to solving a system of singular ordinary differential equations subject to certain boundary conditions. This problem was solved in  for the case where the matter model is given by a massless scalar field, but the solutions produced there, which are continuously self-similar, cannot include the Choptuik critical solution. Bizon and Wasserman  studied the corresponding problem for the Einstein equations coupled to a wave map with target SU (2). They proved the existence of continuously self-similar solutions including one which, according the results of numerical calculations, appears to play the role of critical solution in collapse. Another case where the question of the existence of the critical solution seems to be a problem that could possibly be solved in the near future is that of a perfect fluid. A good starting point for this is the work of Goliath, Nilsson, and Uggla [115, 116]. These authors gave a formulation of the problem in terms of dynamical systems and were able to determine certain qualitative features of the solutions. See also [53, 54].
A possible strategy for learning more about critical collapse, pursued by Bizon and collaborators, is to study model problems in flat space that exhibit features similar to those observed numerically in the case of the Einstein equations. Until now, only models showing continuous self-similarity have been found. These include wave maps in various dimensions and the Yang-Mills equations in spacetimes of dimension greater than four. As mentioned in Section 2.3, it is known that in four dimensions there exist global smooth solutions of the Yang-Mills equations corresponding to rather general initial data [97, 155]. In dimensions greater than five it is known that there exist solutions that develop singularities in finite time. This follows from the existence of continuously self-similar solutions . Numerical evidence indicates that this type of blow-up is stable, i.e. occurs for an open set of initial data. The numerical work also indicates that there is a critical self-similar solution separating this kind of blow-up from dispersion. The spacetime dimension five is critical for Yang-Mills theory. Apparently singularities form, but in a different way from what happens in dimension six. There is as yet no rigorous proof of blow-up in five dimensions.
The various features of Yang-Mills theory just mentioned are mirrored in two dimensions less by wave maps with values in spheres . In four dimensions, blow-up is known while in three dimensions there appears (numerically) to be a kind of blow-up similar to that found for Yang-Mills in dimension five. There is no rigorous proof of blow-up. What is seen numerically is that the collapse takes place by scaling within a one-parameter family of static solutions. The case of wave maps is the most favourable known model problem for proving theorems about critical phenomena associated to singularity formation. The existence of a solution having the properties expected of the critical solution for wave maps in four dimensions has been proved in . Some rigorous support for the numerical findings in three dimensions has been given by work of Struwe (see the preprints available from ). He showed, among other things, that if there is blow-up in finite time it must take place in a way resembling that observed in the numerical calculations.
Self-similar solutions are characteristic of what is called Type II critical collapse. In Type I collapse an analogous role is played by static solutions and quite a bit is known about the existence of these. For instance, in the case of the Einstein-Yang-Mills equations, it is one of the Bartnik-McKinnon solutions mentioned in Section 3.1 which does this. In the case of collisionless matter the results of  show that at least in some cases critical collapse is mediated by a static solution in the form of a shell. There are existence results for shells of this kind  although no connection has yet been made between those shells whose existence has been proved and those which have been observed numerically in critical collapse calculations. Note that Martín-García and Gundlach  have presented a (partially numerical) construction of self-similar solutions of the Einstein-Vlasov system.
|Theorems on Existence and Global Dynamics for the
Alan D. Rendall
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