4.4 Gravity regularizes self-similar matter4 Extensions of the basic 4.2 CSS and DSS critical

4.3 Approximate self-similarity and universality classes

As we have seen, the presence of a length scale in the field equations can give rise to static (or oscillating) asymptotically flat critical solutions and a mass gap at the black hole threshold. Depending on the initial data and on how the scale appears in the field equations, this scale can also become asymptotically irrelevant as a self-similar solution reaches ever smaller spacetime scales. This behavior was already noticed by Choptuik in the collapse of a massive scalar field, or of a more general scalar field with an arbitrary potential term generally [38], and confirmed by Brady, Chambers and Gonçalves [22Jump To The Next Citation Point In The Article]. It was also seen in the spherically symmetric EYM system [40]. In order to capture the notion of an asymptotically self-similar solution, one may set the arbitrary scale L in the definition (12Popup Equation) of tex2html_wrap_inline2337 to the scale set by the field equations, here 1/ m .

Introducing suitable dimensionless first-order variables Z (such as a, tex2html_wrap_inline2323, tex2html_wrap_inline2325, tex2html_wrap_inline2823 and tex2html_wrap_inline2825 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 tex2html_wrap_inline2829 . 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 tex2html_wrap_inline2729 - obtained by setting tex2html_wrap_inline2835, in powers of (scale on which the solution varies)/(scale set by the field equations).

After inserting the ansatz into the field equations, each tex2html_wrap_inline2837 is calculated recursively from the preceding ones. For large enough tex2html_wrap_inline2337 (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 tex2html_wrap_inline2565, and solved again recursively. Fortunately, one can calculate the leading order background term tex2html_wrap_inline2729 on its own, and obtain the exact echoing period tex2html_wrap_inline2377 in the process (in the case of DSS). Similarly, one can calculate the leading order perturbation term on the basis of tex2html_wrap_inline2729 alone, and obtain the exact value of the critical exponent tex2html_wrap_inline2289 in the process. This procedure was carried out by Gundlach [73] for the Einstein-Yang-Mills system, and by Gundlach and Martín-García [78Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline2729 in the expansion of the self-similar critical solution tex2html_wrap_inline2565 obeys the equation


Clearly, this leading order term is independent of the overall scale L . The critical exponent tex2html_wrap_inline2289 depends only on tex2html_wrap_inline2729, 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 tex2html_wrap_inline2729 . 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, 78Jump To The Next Citation Point In The Article]. 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 [22].

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 [15].

If there are several scales tex2html_wrap_inline2891, tex2html_wrap_inline2893, tex2html_wrap_inline2895 etc. present in the problem, a possible approach is to set the arbitrary scale in (12Popup Equation) equal to one of them, say tex2html_wrap_inline2891, and define the dimensionless constants tex2html_wrap_inline2899 from the others. The size of the universality classes depends on where the tex2html_wrap_inline2901 appear in the field equations. If a particular tex2html_wrap_inline2903 appears in the field equations only in positive integer powers, the corresponding tex2html_wrap_inline2901 appears only multiplied by tex2html_wrap_inline2829, and will be irrelevant in the scaling limit. All values of this tex2html_wrap_inline2901 therefore belong to the same universality class. As an example, adding a quartic self-interaction tex2html_wrap_inline2911 to the massive scalar field, gives rise to the dimensionless number tex2html_wrap_inline2913, 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 [78Jump To The Next Citation Point In The Article], 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 tex2html_wrap_inline2449 of the 2-dimensional sigma model, or a conformal coupling of the scalar field.

4.4 Gravity regularizes self-similar matter4 Extensions of the basic 4.2 CSS and DSS critical

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
© Max-Planck-Gesellschaft. ISSN 1433-8351
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