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 [73] for the Einstein-Yang-Mills system, and by Gundlach and
Martín-García [78] 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 [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
,
,
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 [78], 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
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
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