Black hole critical phenomena: a brief update

Patrick R Brady, University of Wisconsin-Milwaukee

patrick@gravity.phys.uwm.edu

The analogy between critical phenomena in statistical physics and
interesting dynamical features observed in numerical simulations of
spherical self-gravitating scalar field collapse was introduced by
Choptuik [1]. Choptuik numerically evolved one parameter patrick@gravity.phys.uwm.edu

Different matter models, couplings and symmetries were examined in an
effort to understand the extent of the universality and scaling noted
by Choptuik. In a tremendous tour-de-force, Abrahams and
Evans [2] explored the collapse of axisymmetric
gravitational wave configurations; they found tentative evidence for
discrete self-similarity and a scaling law for black hole mass with
.
Evans and Coleman [3] considered the
collapse of radiation fluid spheres; they found that near critical
evolutions exhibit continuous self-similarity (CSS), and that the scaling
exponent for the black hole mass is
.
Evans and
Coleman went further by constructing the CSS solution which is the
intermediate attractor in perfect fluid collapse. Koike *et
al.* [4] examined the spectrum of perturbations around this solution and
demonstrated that the solution is one mode unstable. The critical exponent
is related to the growth rate
of the unstable mode as
.
The completion of this program, as suggested by Evans
and Coleman, established the origin of the mass scaling law for black hole
mass. Maison [5] used the method to argue that the scaling exponent would
be a function of *k* for equations of state with
where *P* is
pressure and
energy density.

Since the last report in *Matters of Gravity*, this field has
consolidated further. It is now well understood what constitutes a
critical solution for gravitational collapse, and several important
``analytic'' results indicate our understanding of critical phenomena is
correct. Following the work of Koike *et al.*, Gundlach applied
perturbative techniques to the massless scalar field critical solution
which he computed directly [6,7]. His computation of the critical exponent
agreed with the experimentally observed value. This, and the work of Koike
*et al.*, are excellent examples of retrodiction; having the answer,
a direct computation was performed to establish the critical exponent. But
the techniques used to identify critical solutions and to compute the
associated scaling exponents have been applied by Gundlach, Martin-Garcia,
Maison and others to discover new critical solutions and predict associated
critical exponents. This work was vindicated by subsequent numerical
computation. For example, analytic calculations predicted a periodic
wiggle superimposed on the mass scaling law for massless scalar field
collapse [7,8]; the oscillations were found numerically by Hod and
Piran [8]. Charged scalar field collapse was predicted to behave as
uncharged massless scalar field near the critical point; the charge obeying
a scaling law similar to the mass but with a critical exponent
.
This was also confirmed by Hod and Piran [9].

Researchers have continued to explore the parameter space of solutions for
a variety of matter models bringing a wealth of new phenomenology to light.
Choptuik, Chmaj and Bizon [10] studied the collapse of SU(2) Yang-Mills
fields. They found two distinct types of critical behavior. In Type II
transitions, the critical solution is discretely self-similar and black
holes of arbitrarily small mass can form. The appearance of Type I
transitions was a new feature in black hole critical phenomena; the
critical solution is the static Bartnik-McKinnon solution and black hole
formation turns on at finite mass. Type I phase transitions have also been
found for massive scalar fields [11] and *SU*(2) Skyrme models [12].
Further phenomenology has also been identified in studies of the magnetic
Yang Mills fields. Choptuik, Hirschmann and Marsa [13] found transitions
between black holes formed in Type I collapse and black holes formed in
Type II collapse; the critical solution is an unstable colored black hole.

Interestingly, numerical confirmation of Maison's predictions were not
forthcoming until late in 1997 [14]. With this came new understanding of
regular self-similar perfect fluid solutions. Folklore had it that no
regular self-similar solutions existed for *k*> 0.899 in the equations of
state .
Neilsen and Choptuik found evidence for such solutions in
their collapse simulations, and re-investigated the exact self-similar
solutions. They found that the nature of the sonic horizon changes when
*k*> 0.899 *but* regular self-similar solutions do exist. The
surprise of this was emphasized when Neilsen and Choptuik evolved stiff
fluids with *k*=1 and found a CSS critical solution. Since an irrotational
perfect fluid with *k*=1 can be recast as a scalar field with timelike
gradient, Neilsen and Choptuik had found a scalar field solution with a CSS
critical solution in contrast top Choptuik's original work. This issue is
under active investigation.

Lack of space prohibits detailed discussion of the many other lines of research that have been pursued. Astrophysical implications of formation of tiny black holes in the early universe have been considered. Attempts have been made to understand the semi-classical corrections to Type II critical phenomena. The symmetries of the critical solutions led to the development symmetry seeking coordinates [15] which might be useful in other circumstances. In a mammoth effort, Gundlach, Martin-Garcia and Garfinkle [16] have examined small deviations from spherical symmetry and predicted scaling relations for angular momentum in Type II transitions. The interested reader is referred to Gundlach's review article [17] for more details.

So what does the future hold. There is little doubt that axisymmetric (and ultimately 3-dimensional) collapse simulations will bring a wealth of new phenomenology. Significant effort is under way to produce accurate and robust codes to perform the parameter space surveys that are needed. Just as the initial study of massless scalar field collapse required the introduction of new techniques into numerical relativity, ongoing research should foster further developments.

*References:*

[1]
M. W. Choptuik, * Phys. Rev. Letters* **70**, 9-12 (1993).

[2]
A. M. Abrahams and C. R. Evans, *Phys. Rev. Letters* **70**, 2980 (1993).

[3]
C. R. Evans and J. S. Coleman, *Phys. Rev. Letters* **
72**, 1782 (1994).

[4]
T. Koike, T. Hara, and S. Adachi,
*Phys. Rev. Lett.* **74**, 5170 (1995),
`gr-qc/9503007`.

[5]
D. Maison, *Phys. Lett. B* **366**, 82 (1996).

[6]
C. Gundlach, *Phys.
Rev. Lett.* **75**, 3214 (1995),
`gr-qc/9507054`.

[7]
C. Gundlach, *Phys.
Rev. D* **55**, 695 (1997),
`gr-qc/9604019`.

[8]
S. Hod and T. Piran,
*Phys. Rev. D* **55**, 440 (1997),
`gr-qc/9606087`.

[9]
S. Hod and T. Piran, *Phys. Rev. D* **55**,
3485 (1997), `
gr-qc/9606093`
http://xxx.lanl.gov/abs/gr-qc/9606093.

[10]
M. W. Choptuik, T. Chmaj, P. Bizon, *Phys. Rev. Lett.* **77**,
424 (1996), `gr-qc/9603051`;
P. Bizon and T. Chmaj, *Acta Phys. Polon.* **B29**, 1071 (1998),
`gr-qc/9802002`.

[11]
P. R. Brady, C. M. Chambers, and S. M. C. V. Goncalves, *Phys. Rev. D*
**56**, 6057 (1997),
`gr-qc/9709014`.

[12]
P. Bizon, T. Chmaj, *Phys. Rev.*
**D58**, 041501 (1998), `
gr-qc/9801012`

http://xxx.lanl.gov/abs/gr-qc/9801012.

[13]
M. W. Choptuik, E. W. Hirschmann, and R. L. Marsa, *Phys. Rev.* **
D60**, 124011 (1999), `gr-qc/9903081`.

[14]
D. W. Neilsen and M. W. Choptuik, *Class. Quant. Grav.* **17**,
761 (2000),
`gr-qc/9812053`; P. Brady
and M. J. Cai, in *Proceedings of the 8th Marcel Grossman Meeting*, T. Piran, ed.

World Scientific, Singapore, 1999.

[15]
D. Garfinkle and C. Gundlach, * Class. Quant. Grav.* **16** 4111 (1999),
`gr-qc/9908016`.

[16]
C. Gundlach, ``Critical gravitational collapse of a perfect fluid with
p=k*rho: Nonspherical perturbations'',
`gr-qc/9906124`;
C. Gundlach and J. M. Martin-Garcia,
``Gauge-invariant and coordinate-independent perturbations of
stellar collapse. I: The interior,''
`gr-qc/9906068`;
D. Garfinkle, C. Gundlach, and J. M. Martin-Garcia, *Phys. Rev.* **
D59**, 104012 (1999), `gr-qc/9811004`.

[17]
C. Gundlach, ``Critical phenomena in gravitational collapse,'' to appear
in Living Reviews in Relativity,
`gr-qc/0001046`.