3.4 Cylindrically symmetric solutions3 Global symmetric solutions3.2 Spatially homogeneous solutions

3.3 Spherically symmetric solutions

  The most extensive results on global inhomogeneous solutions of the Einstein equations obtained up to now concern spherically symmetric solutions of the Einstein equations coupled to a massless scalar field with asymptotically flat initial data. In a series of papers Christodoulou [43, 42, 45, 44, 46, 47, 48Jump To The Next Citation Point In The Article, 52Jump To The Next Citation Point In The Article] has proved a variety of deep results on the global structure of these solutions. Particularly notable are his proofs that naked singularities can develop from regular initial data [48Jump To The Next Citation Point In The Article] and that this phenomenon is unstable with respect to perturbations of the data  [52]. In related work Christodoulou  [49, 50, 51] has studied global spherically symmetric solutions of the Einstein equations coupled to a fluid with a special equation of state (the so-called two-phase model).

The rigorous investigation of the spherically symmetric collapse of collisionless matter in general relativity was initiated by Rein and the author  [134], 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 [137] 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 [148]. The question of what happens for general large initial data could not yet be answered by analytical techniques. In  [138Jump To The Next Citation Point In The Article] numerical methods were applied in order 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 which 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 [38] 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 where the spherical collapse of different kinds of matter was computed numerically and various qualitative features were determined. For reviews of this see [86] and [85]. In the calculations of  [138] 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. 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. A problem of this type was solved in [48], but the solutions produced there, which are continuously self-similar, cannot include the Choptuik critical solution. In the case of a perfect fluid the existence of the critical solution seems to be a problem which could possibly be solved in the near future. A good starting point for this is the work of Goliath, Nilsson and Uggla [82], [83]. These authors gave a formulation of the problem in terms of dynamical systems and were able to determine certain qualitative features of the solutions.

3.4 Cylindrically symmetric solutions3 Global symmetric solutions3.2 Spatially homogeneous solutions

image Local and Global Existence Theorems for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de