4.6 Beyond spherical symmetry4 Extensions of the basic 4.4 Gravity regularizes self-similar matter

4.5 Critical phenomena and naked singularities 

Choptuik's results have an obvious bearing on the issue of cosmic censorship. (For a general review of cosmic censorship, see [119].) As we shall see in this section, the critical spacetime has a naked singularity. This spacetime can be approximated arbitrarily well up to fine-tuning of a generic parameter. A region of arbitrarily high curvature is seen from infinity as fine-tuning is improved. Critical collapse therefore provides a set of smooth initial data for naked singularity formation that has codimension one in phase space. It does not violate cosmic censorship if one states it as ``generic(!) smooth initial data for reasonable matter do not form naked singularities''.

Nevertheless, critical collapse is an interesting test of cosmic censorship. First of all, the set of data is of codimension one, certainly in the space of spherical asymptotically flat data, and apparently [77Jump To The Next Citation Point In The Article] also in the space of all asymptotically flat data. This means that one can fine-tune any generic parameter, whichever comes to hand, as long as it parameterizes a smooth curve in the space of initial data. Secondly, critical phenomena seem to be generic with respect to matter models, including realistic matter models with intrinsic scales. These two features together mean that, in a hypothetical experiment to create a Planck-sized black hole in the laboratory through a strong explosion, one could fine-tune any one design parameter of the bomb, without requiring control over its detailed effects on the explosion.

The metric of the critical spacetime is of the form tex2html_wrap_inline3111 times a regular metric. From this general form alone, one can conclude that tex2html_wrap_inline2603 is a curvature singularity, where Riemann and Ricci invariants blow up like tex2html_wrap_inline3115, and which is at finite proper time from regular points. The Weyl tensor with index position tex2html_wrap_inline3117 is conformally invariant, so that components with this index position remain finite as tex2html_wrap_inline2589 . In this property it resembles the initial singularity in Penrose's Weyl tensor conjecture rather than the final singularity in generic gravitational collapse. This type of singularity is called ``conformally compactifiable'' [116] or ``isotropic'' [69]. Is the singularity naked, and is it timelike, null or a ``point''? The answer to these questions remains confused, partly because of coordinate complications, partly because of the difficulty of investigating the singular behavior of solutions numerically.

Choptuik's, and Evans and Coleman's numerical codes were limited to the region t <0 in the Schwarzschild-like coordinates (4Popup Equation), with the origin of t adjusted so that the singularity is at t =0. Evans and Coleman conjectured that the singularity is shrouded in an infinite redshift based on the fact that tex2html_wrap_inline2323 grows as a small power of r at constant t . This is directly related to the fact that a goes to a constant tex2html_wrap_inline3135 as tex2html_wrap_inline3137 at constant t, as one can see from the Einstein equation (8Popup Equation). This in turn means simply that the critical spacetime is not asymptotically flat, but asymptotically conical at spacelike infinity, with the Hawking mass proportional to r .

Hamadé and Stewart [81] evolved near-critical scalar field spacetimes on a double null grid, which allowed them to follow the time evolution up to close to the future light cone of the singularity. They found evidence that this light cone is not preceded by an apparent horizon, that it is not itself a (null) curvature singularity, and that there is only a finite redshift along outgoing null geodesics slightly preceding it. (All spherically symmetric critical spacetimes appear to be qualitatively alike as far as the singularity structure is concerned, so that what we say about one is likely to hold for the others.)

Hirschmann and Eardley [84Jump To The Next Citation Point In The Article] were the first to continue a critical solution itself right up to the future light cone. They examined a CSS complex scalar field solution that they had constructed as a nonlinear ODE boundary value problem, as discussed in Section  4.4 . (This particular one is not a proper critical solution, but that should not matter for the global structure.) They continued the ODE evolution in the self-similar coordinate x through the coordinate singularity at t =0 up to the future light cone by introducing a new self-similarity coordinate x . The self-similar ansatz reduces the field equations to an ODE system. The past and future light cones are regular singular points of the system, at tex2html_wrap_inline3149 and tex2html_wrap_inline3151 . At these ``points'' one of the two independent solutions is regular and one singular. The boundary value problem that originally defines the critical solution corresponds to completely suppressing the singular solution at tex2html_wrap_inline3149 (the past light cone). The solution can be continued through this point up to tex2html_wrap_inline3151 . There it is a mixture of the regular and the singular solution.

We now state this more mathematically. The ansatz of Hirschmann and Eardley for the self-similar complex scalar field is (we slightly adapt their notation)


with tex2html_wrap_inline2739 a real constant. Near the future light cone they find that f is approximately of the form


with tex2html_wrap_inline3161 and tex2html_wrap_inline3163 regular at tex2html_wrap_inline3151, and tex2html_wrap_inline2613 a small positive constant. The singular part of the scalar field oscillates an infinite number of times as tex2html_wrap_inline3169, but with decaying amplitude. This means that the scalar field tex2html_wrap_inline2325 is just differentiable, and that therefore the stress tensor is just continuous. It is crucial that spacetime is not flat, or else tex2html_wrap_inline2613 would vanish. For this in turn it is crucial that the regular part tex2html_wrap_inline3175 of the solution does not vanish, as one sees from the field equations.

The only other case in which the critical solution has been continued up to the future light cone is Choptuik's real scalar field solution [74]. Let tex2html_wrap_inline3101 and tex2html_wrap_inline3179 be the ingoing and outgoing wave degrees of freedom respectively defined in (54Popup Equation). At the future light cone tex2html_wrap_inline3151 the solution has the form


where C is a positive real constant, tex2html_wrap_inline3185, tex2html_wrap_inline3187 and tex2html_wrap_inline3189 are regular real functions with period tex2html_wrap_inline2377 in their second argument, and tex2html_wrap_inline2613 is a small positive real constant. (We have again simplified the original notation.) Again, the singular part of the solution oscillates an infinite number of times but with decaying amplitude. Gundlach concludes that the scalar field, the metric coefficients, all their first derivatives, and the Riemann tensor exist, but that is as far as differentiability goes. (Not all second derivatives of the metric exist, but enough to construct the Riemann tensor.) If either of the regular parts tex2html_wrap_inline3185 or tex2html_wrap_inline3187 vanished, spacetime would be flat, tex2html_wrap_inline2613 would vanish, and the scalar field itself would be singular. In this sense, gravity regularizes the self-similar matter field ansatz. In the critical solution, it does this perfectly at the past lightcone, but only partly at the future lightcone. Perhaps significantly, spacetime is almost flat at the future horizon in both the examples, in the sense that the Hawking mass divided by r is a very small number. In the spacetime of Hirschmann and Eardley it appears to be as small as tex2html_wrap_inline2285, but not zero according to numerical work by Horne [88].

In summary, the future light cone (or Cauchy horizon) of these two critical spacetimes is not a curvature singularity, but it is singular in the sense that differentiability is lower than elsewhere in the solution. Locally, one can continue the solution through the future light cone to an almost flat spacetime (the solution is of course not unique). It is not clear, however, if such a continuation can have a regular center r =0 (for t >0), although this seems to have been assumed in [84Jump To The Next Citation Point In The Article]. A priori, one should expect a conical singularity, with a (small) defect angle at r =0.

The results just discussed were hampered by the fact that they are investigations of singular spacetimes that are only known in numerical form, with a limited precision. As an exact toy model we consider an exact spherically symmetric, CSS solution for a massless real scalar field that was apparently first discovered by Roberts [112Jump To The Next Citation Point In The ArticlePopup Equation] and then re-discovered in the context of critical collapse by Brady [18Jump To The Next Citation Point In The Article] and Oshiro et al. [106Jump To The Next Citation Point In The Article]. We use the notation of Oshiro et al. The solution can be given in double null coordinates as


with p a constant parameter. (Units G = c =1.) Two important curvature indicators, the Ricci scalar and the Hawking mass, are


The center r =0 has two branches, u =(1+ p) v in the past of u = v =0, and u =(1- p) v in the future. For 0< p <1 these are timelike curvature singularities. The singularities have negative mass, and the Hawking mass is negative in the past and future light cones. One can cut these regions out and replace them by Minkowski space, not smoothly of course, but without creating a tex2html_wrap_inline3225 -function in the stress-energy tensor. The resulting spacetime resembles the critical spacetimes arising in gravitational collapse in some respects: It is self-similar, has a regular center r =0 at the past of the curvature singularity u = v =0 and is continuous at the past light cone. It is also continuous at the future light cone, and the future branch of r =0 is again regular.

It is interesting to compare this with the genuine critical solutions that arise as attractors in critical collapse. They are as regular as the Roberts solution (analytic) at the past r =0, more regular (analytic versus continuous) at the past light cone, as regular (continuous) at the future light cone and, it is to be feared, less regular at the future branch of r =0: In contrary to previous claims [84, 72] there may be no continuation through the future sound or light cone that does not have a conical singularity at the future branch of r =0. The global structure still needs to be clarified for all known critical solutions.

In summary, the critical spacetimes that arise asymptotically in the fine-tuning of gravitational collapse to the black hole threshold have a curvature singularity that is visible at infinity with a finite redshift. The Cauchy horizon of the singularity is mildly singular (low differentiability), but the curvature is finite there. It is unclear at present if the singularity is timelike or if there exists a continuation beyond the Cauchy horizon with a regular center, so that the singularity is limited, loosely speaking, to a point. Further work should be able to clarify this. In any case, the singularity is naked and the critical solutions therefore provide counter-examples to any formulation of cosmic censorship which states only that naked singularities cannot arise from smooth initial data in reasonable matter models. The statement must be that there is no open ball of smooth initial data for naked singularities.

Recent analytic work by Christodoulou on the spherical scalar field [49] is not directly relevant to the smooth (analytic or tex2html_wrap_inline3239) initial data discussed here. Christodoulou considers a larger space of initial data that are not tex2html_wrap_inline3241 . He shows that for any data set tex2html_wrap_inline3243 in this class that forms a naked singularity there are data tex2html_wrap_inline3245 and tex2html_wrap_inline3247 such that the data sets tex2html_wrap_inline3249 do not contain a naked singularity, for any tex2html_wrap_inline3251 and tex2html_wrap_inline3253 except zero. Here tex2html_wrap_inline3245 is data of bounded variation, and tex2html_wrap_inline3247 is absolutely continuous data. Therefore, the set of naked singularity data is at least codimension two in the space of data of bounded variation, and of codimension at least one in the space of absolutely continuous data. The semi-numerical result of Gundlach claims that it is codimension exactly one in the set of smooth data. The result of Christodoulou holds for any tex2html_wrap_inline3243, including initial data for the Choptuik solution. The apparent contradiction is resolved if one notes that the tex2html_wrap_inline3245 and tex2html_wrap_inline3247 of Christodoulou are not smooth in (at least) one point, namely where the initial data surface is intersected by the past light cone of the singularity in tex2html_wrap_inline3243 Popup Footnote . The data tex2html_wrap_inline3249 are therefore not smooth.

4.6 Beyond spherical symmetry4 Extensions of the basic 4.4 Gravity regularizes self-similar matter

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
© Max-Planck-Gesellschaft. ISSN 1433-8351
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