] has already been presented in Section
4.5
. It has been discussed in the context of critical collapse
in [18,
106], and later [121,
26]. The original, analytic, Roberts solution is cut and pasted to
obtain a new solution which has a regular center
r
=0 and which is asymptotically flat. Solutions from this family
[see Eqns. (64)] with
p
>1 can be considered as black holes, and to leading order
around the critical value
p
=1, their mass is
. The pitfall in this approach is that only perturbations within
the self-similar family are considered, so the formal critical
exponent applies only to this one, very special, family of
initial data. But the
p
=1 solution has many growing perturbations which are spherically
symmetric (but not self-similar), and is therefore not a critical
solution in the sense of being an attractor of codimension one.
This was already clear because it did not appear in collapse
simulations at the black hole threshold, but Frolov has
calculated the perturbation spectrum analytically [56,
57]. The eigenvalues of spherically symmetric perturbations fill a
sector of the complex plane, with
. All nonspherical perturbations decay. Other supposed critical
exponents that have been derived analytically are usually valid
only for a single, very special family of initial data also.
Other authors have employed analytic approximations to the
actual Choptuik solution. Pullin [109] has suggested describing critical collapse approximately as a
perturbation of the Schwarzschild spacetime. Price and
Pullin [108] have approximated the Choptuik solution by two flat space
solutions of the scalar wave equation that are matched at a
``transition edge'' at constant self-similarity coordinate
x
. The nonlinearity of the gravitational field comes in through
the matching procedure, and its details are claimed to provide an
estimate of the echoing period
. While the insights of this paper are qualitative, some of its
ideas reappear in the construction [71] of the Choptuik solution as a 1+1 dimensional boundary value
problem. Frolov [55] has suggested approximating the Choptuik solution as the
Roberts solution plus its most rapidly growing (spherical)
perturbation mode, pointing out that it oscillates in
with a period 4.44, but ignoring the fact that it also grows
exponentially. This is probably not a correct approach.
In summary, purely analytic approaches have so far remained unsuccessful in explaining critical collapse.
|
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 |