Introducing suitable dimensionless first-order variables Z (such as a, , , and for the spherically symmetric scalar field), one can write the field equations as a first order system:
Every appearance of m gives rise to an appearance of . If the field equations contain only positive integer powers of m, one can make an ansatz for the critical solution of the form
This is an expansion around a scale-invariant solution - obtained by setting , in powers of (scale on which the solution varies)/(scale set by the field equations).
After inserting the ansatz into the field equations, each is calculated recursively from the preceding ones. For large enough (on spacetime scales small enough, close enough to the singularity), this expansion is expected to converge. A similar ansatz can be made for the linear perturbations of , and solved again recursively. Fortunately, one can calculate the leading order background term on its own, and obtain the exact echoing period in the process (in the case of DSS). Similarly, one can calculate the leading order perturbation term on the basis of alone, and obtain the exact value of the critical exponent in the process. This procedure was carried out by Gundlach  for the Einstein-Yang-Mills system, and by Gundlach and Martín-García  for massless scalar electrodynamics. Both systems have a single scale 1/ e (in units c = G =1), where e is the gauge coupling constant.
The leading order term in the expansion of the self-similar critical solution obeys the equation
Clearly, this leading order term is independent of the overall scale L . The critical exponent depends only on , and is therefore also independent of L . There is a region in the space of initial data where in fine-tuning to the black hole threshold the scale L becomes irrelevant, and the behaviour is dominated by the critical solution . In this region, the usual type II critical phenomena occur, independently of the value of L in the field equations. In this sense, all systems with a single length scale L in the field equations are in one universality class [82, 78]. The massive scalar field, for any value of m, or massless scalar electrodynamics, for any value of e, are in the same universality class as the massless scalar field.
It should be stressed that universality classes with respect to a dimensionful parameter arise in regions of phase space (which may be large). Another region of phase space may be dominated by an intermediate attractor that has a scale proportional to L . This is the case for the massive scalar field with mass m : In one region of phase space, the black hole threshold is dominated by the Choptuik solution and type II critical phenomena occur, in another, it is dominated by metastable oscillating boson stars, whose mass is 1/ m times a factor of order 1 .
This notion of universality classes is fundamentally the same as in statistical mechanics. Other examples include modifications to the perfect fluid equation of state that do not affect the limit of high density. The SU (2) Yang-Mills and SU (2) Skyrme models, in spherical symmetry, also belong to the same universality class .
If there are several scales , , etc. present in the problem, a possible approach is to set the arbitrary scale in (12) equal to one of them, say , and define the dimensionless constants from the others. The size of the universality classes depends on where the appear in the field equations. If a particular appears in the field equations only in positive integer powers, the corresponding appears only multiplied by , and will be irrelevant in the scaling limit. All values of this therefore belong to the same universality class. As an example, adding a quartic self-interaction to the massive scalar field, gives rise to the dimensionless number , but its value is an irrelevant (in the language of renormalization group theory) parameter. All self-interacting scalar fields are in fact in the same universality class. Contrary to the statement in , I would now conjecture that massive scalar electrodynamics, for any values of e and m, forms a single universality class in a region of phase space where type II critical phenomena occur. Examples of dimensionless parameters which do change the universality class are the k of the perfect fluid, the of the 2-dimensional sigma model, or a conformal coupling of the scalar field.
|Critical Phenomena in Gravitational Collapse
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