where is the mass of the eventual BH. The constant depends on the parameter of the initial data that is selected, but is the same for all choices. Furthermore, in terms of logarithmic variables , ( is the proper time of an observer at r = 0 with the finite proper time at which the critical evolution concludes and is a constant which scales r), the waveform X repeats (echoes) at intervals of in if is rescaled to , i.e. . The scaling behavior (1) demonstrates that the minimum BH mass (for bosons) is zero. The critical solution itself is a counter-example to cosmic censorship, (since the formation of the zero mass BH causes high curvature regions to become visible at ). (See, e.g., the discussion in Hirschmann and Eardley .) The numerical demonstration of this feature of the critical solution was provided by Hamadé and Stewart . This result caused Hawking to pay off a bet to Preskill and Thorne [44, 123].
Soon after this discovery, scaling and critical phenomena were found in a variety of contexts. Abrahams and Evans  discovered the same phenomenon in axisymmetric gravitational wave collapse with a different value of and, to within numerical error, the same value of . (Note that the rescaling of r with required Choptuik to use adaptive mesh refinement (AMR) to distinguish subsequent echoes. Abrahams and Evans' smaller () allowed them to see echoing with their 2+1 code without AMR.) Garfinkle  confirmed Choptuik's results with a completely different algorithm that does not require AMR. He used Goldwirth and Piran's  method of simulating Christodoulou's  formulation of the spherically symmetric scalar field in null coordinates. This formulation allowed the grid to be automatically rescaled by choosing the edge of the grid to be the null ray that just hits the central observer at the end of the critical evolution. (Missing points of null rays that cross the central observer's world line are replaced by interpolation between those that remain.) Hamadé and Stewart  have also repeated Choptuik's calculation using null coordinates and AMR. They are able to achieve greater accuracy and find .
where . At any fixed t, larger a implies larger . Equivalently, any fixed amplitude will be reached faster for larger eventual . Scaling arguments give the dependence of on the time at which any fixed amplitude is reached:
Therefore, one need only identify the growth rate of the unstable mode to obtain an accurate value of . It is not necessary to undertake the entire dynamical evolution or probe the space of initial data. Hirschmann and Eardley obtain for the complex scalar field solution, while Koike et al  obtain for the Evans-Coleman solution. Although the similarities among the critical exponents in the collapse computations suggested a universal value, Maison  used these same scaling-perturbation methods to show that depends on the equation of state of the fluid in the Evans-Coleman solution. Gundlach  used a similar approach to locate Choptuik's critical solution accurately. This is much harder, due to its discrete self-similarity. Gundlach reformulates the model as a nonlinear hyperbolic boundary value problem with eigenvalue and finds . As with the self-similar solutions described above, the critical solution is found directly without the need to perform a dynamical evolution or explore the space of initial data. Hara et al extended the renormalization group approach of  to the discretely-self-similar case . (For a recent application of renormalization group methods to cosmology see .)
Choptuik et al  have generalized the original Einstein-scalar field calculations to the Einstein-Yang-Mills (EYM) (for SU (2)) case. Here something new was found. Two types of behavior appeared depending on the initial data. In Type I, BH formation had a non-zero mass threshold. The critical solution is a regular, unstable solution to the EYM equations found previously by Bartnik and McKinnon . In Type II collapse, the minimum BH mass is zero with the critical solution similar to that of Choptuik (with a different , ). Gundlach has also looked at this case with the same results . The Type I behavior arises when the collapsing system has a metastable static solution in addition to the Choptuik critical one .
Brady, Chambers and Gonçalves [54, 37] conjectured that addition of a mass to the scalar field of the original model would break scale invariance and might yield a distinct critical behavior. They found numerically the same Type I and II ``phases'' seen in the EYM case. The Type II solution can be understood as perturbations of Choptuik's original model with a small scalar field mass . Here small means that where is the spatial extent of the original nonzero field region. (The scalar field is well within the Compton wavelength corresponding to .) On the other hand, yields Type I behavior. The minimum mass critical solution is an unstable soliton of the type found by Seidel and Suen . The massive scalar field can be treated as collapsing dust to yield a criterion for BH formation .
The Choptuik solution has also been found to be the critical solution for charged scalar fields [98, 113]. As , for the black hole. Q obeys a power law scaling. Numerical study of the critical collapse of collisionless matter (Einstein-Vlasov equations) has yielded a non-zero minimum BH mass . Bizon and Chmaj  have considered the critical collapse of skyrmions.
An astrophysical application of BH critical phenomena has been considered by Nimeyer and Jedamzik  and Yokoyama . They consider its implications for primordial BH formation and suggest that it could be important.
Until recently, only Abrahams and Evans  had ventured beyond spherical symmetry. The first additional departure has been made by Gundlach . He considered spherical and non-spherical perturbations of perfect fluid collapse. Only the original (spherical) growing mode survived.
|Numerical Approaches to Spacetime Singularities
Beverly K. Berger
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