Black hole critical phenomena: a brief update

Patrick R Brady, University of Wisconsin-Milwaukee
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 p families of initial data. For all families, he found a critical value of the parameter, p* say. No black hole formed in evolutions with p<p* and black holes always formed in evolutions with p>p*. Choptuik also observed two fundamental properties of solutions with $p\simeq p^*$. First, they exhibited self-similar echoing, later called discrete self-similarity, which was universal. Second, the mass of black holes formed in marginally super-critical collapse obeyed a scaling law $M_{\mathrm{BH}} \propto \vert p - p^*\vert^\gamma$ with $\gamma \simeq 0.37$ independent of the initial data family. Choptuik speculated that there was a universal solution which acted as an intermediate attractor when p=p*. Since this ground-breaking work, a considerable literature has emerged on critical phenomena in gravitational collapse.

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 $\gamma \simeq 0.37$. 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 $\gamma \simeq 0.36$. 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 $\gamma$ is related to the growth rate $\beta$ of the unstable mode as $\gamma= 1/\beta$. 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 $P=k \rho$ where P is pressure and $\rho$ 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 $\delta=0.88$. 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 $P=k \rho$. 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.


[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),

[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

[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.

Jorge Pullin