3.2 Scale-invariance and self-similarity3 The basic scenario3 The basic scenario

3.1 The dynamical systems picture

We shall pretend that general relativity can be treated as an infinite-dimensional dynamical system. The phase space is the space of pairs of three-metrics and extrinsic curvatures (plus any matter variables) that obey the Hamiltonian and momentum constraints. In the following we restrict ourselves to asymptotically flat data. In other words, it is the space of initial data for an isolated self-gravitating system. The evolution equations are the ADM equations. They contain the lapse and shift as free fields that can be given arbitrary values. In order to obtain an autonomous dynamical system, one needs a general prescription that provides a lapse and shift for given initial data. What such a prescription could be is very much an open problem and is discussed below in Section  5.2 . That is the first gap in the dynamical systems picture. The second gap is that even with a prescription for the lapse and shift in place, a given spacetime does not correspond to a unique trajectory in phase space, but to many, depending on how the spacetime is sliced. A possibility would be to restrict the phase space further, for example to maximal slices only. The third problem is that in order to talk about attractors and repellers we need a notion of convergence on the phase space, that is a distance measure. In the following, we brazenly ignore all three gaps in order to apply some fundamental concepts of dynamical systems theory to gravitational collapse.

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 Popup Footnote .

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.


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Figure 1: The phase space picture for the black hole-dispersion threshold in the presence of a continuously self-similar (CSS) solution. The arrow lines are time evolutions, corresponding to spacetimes. The line without an arrow is not a time evolution, but a 1-parameter family of initial data that crosses the black hole threshold at tex2html_wrap_inline2215 .

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 .


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Figure 2: The phase space picture for discrete self-similarity. The plane represents the critical surface. (In reality this is a hypersurface of codimension one in an infinite-dimensional space.) The circle (fat unbroken line) is the limit cycle representing the critical solution. The thin unbroken curves are spacetimes attracted to it. The dashed curves are spacetimes repelled from it. There are two families of such curves, labeled by one periodic parameter, one forming a black hole, the other dispersing to infinity. Only one member of each family is shown.


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Figure 3: The global structure of spherically symmetric critical spacetimes. One dimension in spherical symmetry has been suppressed.

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, 6Jump To The Next Citation Point In The Article, 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.

3.2 Scale-invariance and self-similarity3 The basic scenario3 The basic scenario

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
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