An isolated system in general relativity, such as a star, or ball of radiation fields, or even of pure gravitational waves, typically ends up in one of three kinds of final state. It either collapses to a black hole, forms a stable star, or explodes and disperses, leaving empty flat spacetime behind. The phase space of isolated gravitating systems is therefore divided into basins of attraction. A boundary between two basins of attraction is called a critical surface. All numerical results are consistent with the idea that these boundaries are smooth hypersurfaces of codimension one in the phase space of GR. Inside the dispersion basin, Minkowski spacetime is an attractive fixed point. Inside the black hole basin, the 3-parameter family of Kerr-Newman black holes forms a manifold of attracting fixed points .
A phase space trajectory starting in a critical surface by definition never leaves it. A critical surface is therefore a dynamical system in its own right, with one dimension less. Say it has an attracting fixed point or attracting limit cycle. This is the case for the black hole threshold in all toy models that have been examined (with the possible exception of the Vlasov-Einstein system, see Section 6 below). We shall call these a critical point, or critical solution, or critical spacetime. Within the complete phase space, the critical solution is an attractor of codimension one. It has an infinite number of decaying perturbation modes tangential to the critical surface, and a single growing mode that is not tangential.
Any trajectory beginning near the critical surface, but not necessarily near the critical point, moves almost parallel to the critical surface towards the critical point. As the critical point is approached, the parallel movement slows down, and the phase point spends some time near the critical point. Then the phase space point moves away from the critical point in the direction of the growing mode, and ends up on a fixed point. This is the origin of universality: Any initial data set that is close to the black hole threshold (on either side) evolves to a spacetime that approximates the critical spacetime for some time. When it finally approaches either empty space or a black hole it does so on a trajectory that appears to be coming from the critical point itself. All near-critical solutions are passing through one of these two funnels. All details of the initial data have been forgotten, except for the distance from the black hole threshold. The phase space picture in the presence of a fixed point critical solution is sketched in Fig. 1 . The phase space picture in the presence of a limit cycle critical solution is sketched in Fig. 2 .
All critical points that have been found in black hole thresholds so far have an additional symmetry, either continuous or discrete. They are either time-independent (static) or periodic in time, or scale-independent or scale-periodic (discretely or continuously self-similar). The static or periodic critical points are metastable stars. As we shall see below in Section 4.1, they give rise to a finite mass gap at the black hole threshold. In the remainder of this section we concentrate on the self-similar fixed points. They give rise to power-law scaling of the black hole mass at the threshold. These are the phenomena discovered by Choptuik. They are now referred to as type II critical phenomena, while the type with the mass gap, historically discovered second, is referred to as type I.
Continuously scale-invariant, or self-similar, solutions arise as intermediate attractors in some fluid dynamics problems (without gravity) [5, 6, 7]. Discrete self-similarity does not seem to have played a role in physics before Choptuik's discoveries.
It is clear from the dynamical systems picture that the closer the initial phase point (data set) is to the critical surface, the closer the phase point will get to the critical point, and the longer it will remain close to it. Making this observation quantitative will give rise to Choptuik's mass scaling law in Section 3.3 below. But we first need to define self-similarity in GR.
|Critical Phenomena in Gravitational Collapse
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