4.3 Collisionless matter

A cloud of collisionless particles can be described by the Vlasov equation, i.e., the Boltzmann equation without collision term. This matter model differs from field theories by having a much larger number of matter degrees of freedom: The matter content is described by a statistical distribution f (xμ,pν) on the point particle phase space, instead of a finite number of fields ϕ (x μ). When restricted to spherical symmetry, individual particles move tangentially as well as radially, and so individually have angular momentum, but the stress-energy tensor averages out to a spherically symmetric one, with zero total angular momentum. The distribution f is then a function r 2 f (r,t,p ,L ) of radius, time, radial momentum and (conserved) angular momentum.

Several numerical simulations of critical collapse of collisionless matter in spherical symmetry have been published to date, and remarkably no type II scaling phenomena has been discovered. Indications of type I scaling have been found, but these do not quite fit the standard picture of critical collapse. Rein et al. [183] find that black hole formation turns on with a mass gap that is a large part of the ADM mass of the initial data, and this gap depends on the initial matter condition. No critical behavior of either type I or type II was observed. Olabarrieta and Choptuik [168] find evidence of a metastable static solution at the black hole threshold, with type I scaling of its life time as in Equation (13View Equation). However, the critical exponent depends weakly on the family of initial data, ranging from 5.0 to 5.9, with a quoted uncertainty of 0.2. Furthermore, the matter distribution does not appear to be universal, while the metric seems to be universal up to an overall rescaling, so that there appears to be no universal critical solution. More precise computations by Stevenson and Choptuik [192], using finite volume HRSC methods, have confirmed the existence of static intermediate solutions and non-universal scaling with exponents ranging now from 5.27 to 11.65.

Martín-García and Gundlach [155] have constructed a family of CSS spherically symmetric solutions for massless particles that is generic by function counting. There are infinitely many solutions with different matter configurations but the same stress-energy tensor and spacetime metric, due to the existence of an exact symmetry: Two massless particles with energy-momentum pμ in the solution can be replaced by one particle with 2p μ. A similar result holds for the perturbations. As the growth exponent λ of a perturbation mode can be determined from the metric alone, this means that there are infinitely many perturbation modes with the same λ. If there is one growing perturbative mode, there are infinitely many. Therefore a candidate critical solution (either static or CSS) cannot be isolated or have only one growing mode. This argument rules out the existence of both type I and type II critical phenomena (in their standard form, i.e., including universality) for massless particles in the complete system, but some partial form of criticality could still be found by restricting to sections of phase space in which that symmetry is broken, for example by prescribing a fixed form for the dependence of the distribution function f on angular momentum L, as those numerical simulations have done.

A recent investigation of Andréasson and Rein [8] with massive particles has confirmed again the existence of a mass gap and the existence of metastable static solutions at the black hole threshold, though there is no estimation of the scaling of their life-times. More interestingly, they show that the subcritical regime can lead to either dispersion or an oscillating steady state depending on the binding energy of the system. They also conclude, based on perturbative arguments, that there cannot be an isolated universal critical solution.

More numerical work is still required, but current evidence suggests that there are no type II critical phenomena, and that there is a continuum of critical solutions in type I critical phenomena and hence only limited universality.

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