Given the scaling power law for the black hole mass in critical
collapse, one would like to know what happens if one takes a
generic oneparameter family of initial data with both electric
charge and angular momentum (for suitable matter), and finetunes
the parameter
p
to the black hole threshold. Does the mass still show powerlaw
scaling? What happens to the dimensionless ratios
and
Q
/
M, with
L
being the black hole angular momentum and
Q
its electric charge? Tentative answers to both questions have
been given using perturbations around spherically symmetric
uncharged collapse.
Gundlach and MartínGarcía [78] have studied scalar massless electrodynamics in spherical
symmetry. Clearly, the real scalar field critical solution of
Choptuik is a solution of this system too. Less obviously, it
remains a critical solution within massless (and in fact,
massive) scalar electrodynamics in the sense that it still has
only one growing perturbation mode within the enlarged solution
space. Some of its perturbations carry electric charge, but as
they are all decaying, electric charge is a subdominant effect.
The charge of the black hole in the critical limit is dominated
by the most slowly decaying of the charged modes. From this
analysis, a universal powerlaw scaling of the black hole charge
was predicted. The predicted value
of the critical exponent (in scalar electrodynamics) was
subsequently verified in collapse simulations by Hod and
Piran [86]. (The mass scales with
as for the uncharged scalar field.) General considerations using
dimensional analysis led Gundlach and MartínGarcía to the
general prediction that the two critical exponents are always
related, for any matter model, by the inequality
This has not yet been verified in any other matter model.
Gundlach's results on nonspherically symmetric perturbations
around spherical critical collapse of a perfect fluid [77] allow for initial data, and therefore black holes, with
infinitesimal angular momentum. All nonspherical perturbations
decrease towards the singularity. The situation is therefore
similar to scalar electrodynamics versus the real scalar field.
The critical solution of the more special model (here, the
strictly spherically symmetric fluid) is still a critical
solution within the more general model (a slightly nonspherical
and slowly rotating fluid). In particular, axial perturbations
(also called oddparity perturbations) with angular dependence
l
=1 (i.e.\ dipole) will determine the angular momentum of the
black hole produced in slightly supercritical collapse. Using a
perturbation analysis similar to that of Gundlach and
MartínGarcía [78], Gundlach [75] (see correction in [70]) has derived the angular momentum scaling law
For the range 0.123<
k
< 0.446 of equations of state, the angular momentum exponent
is related to the mass exponent
by
In particular for
k
=1/3, one gets
. An angular momentum exponent
was derived for the massless scalar field in [63] using secondorder perturbation theory. Both results have not
yet been tested against numerical collapse simulations.

Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr19994
© MaxPlanckGesellschaft. ISSN 14338351
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livrev@aeipotsdam.mpg.de
