This is similar to Eqn. (33), but the growth of the unstable mode is now exponential in
*t*, not in
. In a close parallel to
, we define a time
by

so that the initial data at are

and so the final black hole mass is independent of . The scaling is only apparent in the lifetime of the critical solution, which we can take to be . It is

The type I critical solution can also have a discrete
symmetry, that is, can be periodic in time instead of being
static. This behavior was found in collapse situations of the
massive scalar field by Brady, Chambers and Gonçalves [22]. Previously, Seidel and Suen [113] had constructed periodic, asymptotically flat, spherically
symmetric self-gravitating massive scalar field solutions they
called oscillating soliton stars. By dimensional analysis, the
scalar field mass
*m*
sets an overall scale of 1/
*m*
(in units
*G*
=
*c*
=1). For given
*m*, Seidel and Suen found a one-parameter family of such solutions
with two branches. The more compact solution for a given ADM mass
is unstable, while the more extended one is stable to spherical
perturbations. Brady, Chambers and Gonçalves (BCG) report that
the type I critical solutions they find are from the unstable
branch of the Seidel and Suen solutions. They see a one-parameter
family of (type I) critical solutions, rather than an isolated
critical solution. BCG in fact report that the black hole mass
gap does depend on the initial data. As expected from the
discrete symmetry, they find a small wiggle in the mass of the
critical solution which is periodic in
. Whether type I or type II behavior is seen appears to depend
mainly on the ratio of the length scale of the initial data to
the length scale 1/
*m*
.

In the critical phenomena that were first observed, with an
isolated critical solution, only one number's worth of
information, namely the separation
of the initial data from the black hole threshold, survives to
the late stages of the time evolution. Recall that our definition
of a critical solution is one that has exactly one unstable
perturbation mode, with a black hole formed for one sign of the
unstable mode, but not for the other. This definition does not
exclude an
*n*
-dimensional family of critical solutions. Each solution in the
family then has
*n*
marginal modes leading to neighboring critical solutions, as
well as the one unstable mode.
*n*
+1 numbers' worth of information survive from the initial data,
and the mass gap in type I, or the critical exponent for the
black hole mass in type II, for example, depend on the initial
data through
*n*
parameters. In other words, universality exists in diminished
form. The results of BCG are an example of a one-parameter family
of type I critical solutions. Recently, Brodbeck et al. [25] have shown, under the assumption of linearization stability,
that there is a one-parameter family of stationary, rotating
solutions beginning at the (spherically symmetric)
Bartnik-McKinnon solution. This could turn out to be a second
one-parameter family of type I critical solutions, provided that
the Bartnik-McKinnon solution does not have any unstable modes
outside spherical symmetry (which has not yet been
investigated) [111].

Bizon and Chmaj have studied type I critical collapse of an
*SU*
(2) Skyrme model coupled to gravity, which in spherical symmetry
with a hedgehog ansatz is characterized by one field
*F*
(*r*,
*t*) and one dimensionless coupling constant
. Initial data
,
surprisingly form black holes for both large and small values of
the parameter
*p*, while for an intermediate range of
*p*
the endpoint is a stable static solution called a skyrmion. (If
*F*
was a scalar field, one would expect only one critical point on
this family.) The ultimate reason for this behavior is the
presence of a conserved integer ``baryon number'' in the matter
model. Both phase transitions along this one-parameter family are
dominated by a type I critical solution, that is a different
skyrmion which has one unstable mode. In particular, an
intermediate time regime of critical collapse evolutions agrees
well with an ansatz of the form (40), where
,
and
were obtained independently. It is interesting to note that the
type I critical solution is singular in the limit
, which is equivalent to
, because the known type II critical solutions for any matter
model also do not have a weak gravity limit.

Apparently, type I critical phenomena can arise even without the presence of a scale in the field equations. A family of exact spherically symmetric, static, asymptotically flat solutions of vacuum Brans-Dicke gravity given by van Putten was found by Choptuik, Hirschmann and Liebling [41] to sit at the black hole threshold and to have exactly one growing mode. This family has two parameters, one of which is an arbitrary overall scale.

Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |