Overview of the subject

1.1 Overview of the subject

Take generic initial data in general relativity, adjust any one parameter p of the initial data to the threshold of black hole formation, and compare the resulting spacetimes as a function of p. In many situations, the following critical phenomena are then observed:

Near the threshold, black holes with arbitrarily small masses can be created, and the black hole mass scales as

where p parameterises the initial data and black holes form for p > p_*.
The critical exponent $γ$ is universal with respect to initial data, that is, independent of the particular 1-parameter family, although it depends on the type of collapsing matter.
In the region of large curvature before black hole formation, the spacetime approaches a self-similar solution which is also universal with respect to initial data, the critical solution.

Critical phenomena were discovered by Choptuik [48 ] in numerical simulations of a spherical scalar field. They have been found in numerous other numerical and analytical studies in spherical symmetry, and a few in axisymmetry; in particular critical phenomena have been seen in the collapse of axisymmetric gravitational waves in vacuum [1 ]. Scaling laws similar to the one for black hole mass have been discovered for black hole charge and conjectured for black hole angular momentum.

It is still unclear how universal critical phenomena in collapse are with respect to matter types and beyond spherical symmetry, in particular for vacuum collapse. Progress is likely to be made over the next few years with better numerical simulations.

Critical phenomena can usefully be described in dynamical systems terms. A critical solution is then characterised as an attracting fixed point within a surface that divides two basins of attraction, a critical surface in phase space. Such a fixed point can be either a stationary spacetime, or one that is scale-invariant and self-similar. The latter is relevant for the (“type II”) critical phenomena sketched above. We shall also see how the dynamical systems approach establishes a connection with critical phenomena in statistical mechanics.

Therefore, we could define the field of critical phenomena in gravitational collapse as the study of the boundaries among the basins of attraction of different end states of self-gravitating systems, such as black hole formation or dispersion. In our view, the main physical motivation for this study is that those critical solutions which are self-similar provide a way of achieving arbitrarily large spacetime curvature outside a black hole, and in the limit a naked singularity, by fine-tuning generic initial data for generic matter to the black hole threshold. Those solutions are therefore likely to be important for quantum gravity and cosmic censorship.