2.3 Nature of the Singularity 2 Singularities in AF Spacetimes2.1 Naked Singularities and the

2.2 Critical Behavior in Collapse

2.2.1 Gravitational collapse simulations

We now consider an effect that was originally found by Choptuik [59Jump To The Next Citation Point In The Article] in a numerical study of the collapse of a spherically symmetric massless scalar field. For recent reviews see [92, 93]. We note that this is the first completely new phenomenon in general relativity to be discovered by numerical simulation. In collapse of a scalar field, essentially two things can happen: Either a black hole (BH) forms, or the scalar waves pass through each other and disperse. Choptuik discovered that for any 1-parameter set of initial data labeled by p, there is a critical value tex2html_wrap_inline1280 such that tex2html_wrap_inline1282 yields a BH. He found

  equation75

where tex2html_wrap_inline1284 is the mass of the eventual BH. The constant tex2html_wrap_inline1286 depends on the parameter of the initial data that is selected, but tex2html_wrap_inline1288 is the same for all choices. Furthermore, in terms of logarithmic variables tex2html_wrap_inline1290, tex2html_wrap_inline1292 (tex2html_wrap_inline1294 is the proper time of an observer at r = 0 with tex2html_wrap_inline1298 the finite proper time at which the critical evolution concludes and tex2html_wrap_inline1300 is a constant which scales r), the waveform X repeats (echoes) at intervals of tex2html_wrap_inline1306 in tex2html_wrap_inline1308 if tex2html_wrap_inline1310 is rescaled to tex2html_wrap_inline1312, i.e. tex2html_wrap_inline1314 . The scaling behavior (1Popup Equation) 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 tex2html_wrap_inline1316). (See, e.g., the discussion in Hirschmann and Eardley [109Jump To The Next Citation Point In The Article].) The numerical demonstration of this feature of the critical solution was provided by Hamadé and Stewart [102Jump To The Next Citation Point In The Article]. 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 [1Jump To The Next Citation Point In The Article] discovered the same phenomenon in axisymmetric gravitational wave collapse with a different value of tex2html_wrap_inline1306 and, to within numerical error, the same value of tex2html_wrap_inline1320 . (Note that the rescaling of r with tex2html_wrap_inline1324 required Choptuik to use adaptive mesh refinement (AMR) to distinguish subsequent echoes. Abrahams and Evans' smaller tex2html_wrap_inline1306 (tex2html_wrap_inline1328) allowed them to see echoing with their 2+1 code without AMR.) Garfinkle [81] confirmed Choptuik's results with a completely different algorithm that does not require AMR. He used Goldwirth and Piran's [86Jump To The Next Citation Point In The Article] method of simulating Christodoulou's [61] 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 [102] have also repeated Choptuik's calculation using null coordinates and AMR. They are able to achieve greater accuracy and find tex2html_wrap_inline1332 .

2.2.2 Critical Solutions as an Eigenvalue Problem

Evans and Coleman [72] realized that self-similar rather than self-periodic collapse might be more tractable both numerically (since ODE's rather than PDE's are involved) and analytically. They discovered that a collapsing radiation fluid had that desirable property. (Note that self-similarity (homothetic motion) is incompatible with AF. However, most of the action occurs in the center so that a match of the self-similar inner region to an outer AF one should always be possible.) In a series of papers, Hirschmann and Eardley [108, 109] developed a (numerical) self-similar solution to the spherically symmetric complex scalar field equations. These are ODE's with too many boundary conditions causing a solution to exist only for certain fixed values of tex2html_wrap_inline1306 . Numerical solution of this eigenvalue problem allows very accurate determination of tex2html_wrap_inline1306 . The self-similarity also allows accurate calculation of tex2html_wrap_inline1320 as follows: The critical tex2html_wrap_inline1340 solution is unstable to a small change in p . At any time t (where t < 0 is increasing toward zero), the amplitude a of the perturbation exhibits power law growth:

  equation102

where tex2html_wrap_inline1350 . At any fixed t, larger a implies larger tex2html_wrap_inline1284 . Equivalently, any fixed amplitude tex2html_wrap_inline1358 will be reached faster for larger eventual tex2html_wrap_inline1284 . Scaling arguments give the dependence of tex2html_wrap_inline1284 on the time at which any fixed amplitude is reached:

  equation110

where

  equation114

Thus

  equation119

Therefore, one need only identify the growth rate of the unstable mode to obtain an accurate value of tex2html_wrap_inline1364 . It is not necessary to undertake the entire dynamical evolution or probe the space of initial data. Hirschmann and Eardley obtain tex2html_wrap_inline1366 for the complex scalar field solution, while Koike et al [129Jump To The Next Citation Point In The Article] obtain tex2html_wrap_inline1368 for the Evans-Coleman solution. Although the similarities among the critical exponents tex2html_wrap_inline1320 in the collapse computations suggested a universal value, Maison [135] used these same scaling-perturbation methods to show that tex2html_wrap_inline1320 depends on the equation of state tex2html_wrap_inline1374 of the fluid in the Evans-Coleman solution. Gundlach [95] 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 tex2html_wrap_inline1306 and finds tex2html_wrap_inline1378 . 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 [129] to the discretely-self-similar case [103Jump To The Next Citation Point In The Article]. (For a recent application of renormalization group methods to cosmology see [118].)

2.2.3 Recent Results

Recently, Gundlach [97] completed his eigenvalue analysis of the Choptuik solution to find the growth rate of the unstable mode to be tex2html_wrap_inline1380 . He also predicted a periodic ``wiggle'' in the Choptuik mass scaling relation. This was later observed numerically by Hod and Piran [114]. Self-similar critical behavior has been seen in string theory related axion-dilaton models [69, 101] and in the nonlinear tex2html_wrap_inline1382 -model [110]. Garfinkle and Duncan have shown that subcritical collapse of a spherically symmetric scalar field yields a scaling relation for the maximum curvature observed by the central observer with critical parameters that would be expected on the basis of those found for supercritical collapse [83].

Choptuik et al [60] 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 [9]. In Type II collapse, the minimum BH mass is zero with the critical solution similar to that of Choptuik (with a different tex2html_wrap_inline1386, tex2html_wrap_inline1388). Gundlach has also looked at this case with the same results [96]. The Type I behavior arises when the collapsing system has a metastable static solution in addition to the Choptuik critical one [98Jump To The Next Citation Point In The Article].

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 tex2html_wrap_inline1390 . Here small means that tex2html_wrap_inline1392 where tex2html_wrap_inline1394 is the spatial extent of the original nonzero field region. (The scalar field is well within the Compton wavelength corresponding to tex2html_wrap_inline1390 .) On the other hand, tex2html_wrap_inline1398 yields Type I behavior. The minimum mass critical solution is an unstable soliton of the type found by Seidel and Suen [168]. The massive scalar field can be treated as collapsing dust to yield a criterion for BH formation [87].

The Choptuik solution has also been found to be the critical solution for charged scalar fields [98, 113]. As tex2html_wrap_inline1400, tex2html_wrap_inline1402 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 [161]. Bizon and Chmaj [34] have considered the critical collapse of skyrmions.

An astrophysical application of BH critical phenomena has been considered by Nimeyer and Jedamzik [149] and Yokoyama [183]. They consider its implications for primordial BH formation and suggest that it could be important.

2.2.4 Going Further

The question is then why these critical phenomena should appear in so many collapsing gravitational systems. The discrete self-similarity of Choptuik's solution may be understood as scaling of a limit cycle [103]. (The self-similarity of other systems may be understood as scaling of a limit point.) Garfinkle [82] has conjectured that the scale invariance of Einstein's equations might provide an underlying explanation for the self-similarity and discrete-self-similarity found in collapse.

Until recently, only Abrahams and Evans [1] had ventured beyond spherical symmetry. The first additional departure has been made by Gundlach [94]. He considered spherical and non-spherical perturbations of tex2html_wrap_inline1406 perfect fluid collapse. Only the original (spherical) growing mode survived.



2.3 Nature of the Singularity 2 Singularities in AF Spacetimes2.1 Naked Singularities and the

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-1998-7
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