2.1 Naked singularities and the hoop conjecture

2.1.1 Overview

Perhaps, the first numerical approach to study the cosmic censorship conjecture consisted of attempts to create naked singularities. Many of these studies were motivated by Thorne’s “hoop conjecture” [239Jump To The Next Citation Point] that collapse will yield a black hole only if a mass M is compressed to a region with circumference C ≤ 4πM in all directions. (As is discussed by Wald [245Jump To The Next Citation Point], one runs into difficulties in any attempt to formulate the conjecture precisely. For example, how does one define C and M, especially if the initial data are not at least axially symmetric? Schoen and Yau defined the size of an arbitrarily shaped mass distribution in [228]. A non-rigorous prescription was used in a numerical study by Chiba [75Jump To The Next Citation Point].) If the hoop conjecture is true, naked singularities may form if collapse can yield C ≥ 4πM in some direction. The existence of a naked singularity is inferred from the absence of an apparent horizon (AH) which can be identified locally. Although a definitive identification of a naked singularity requires the event horizon (EH) to be proven to be absent, to identify an EH requires knowledge of the entire spacetime. While one finds an AH within an EH [166167], it is possible to construct a spacetime slicing which has no AH even though an EH is present [246Jump To The Next Citation Point]. Methods to find an EH in a numerically determined spacetime have only recently become available and have not been applied to this issue [179Jump To The Next Citation Point184Jump To The Next Citation Point]. A local prescription, applicable numerically, to identify an “isolated horizon” is under development by Ashtekar et al. (see for example [7]).

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Figure 1: Heuristic illustration of the hoop conjecture.

2.1.2 Naked spindle singularities?

In the best known attempt to produce naked singularities, Shapiro and Teukolsky (ST) [230Jump To The Next Citation Point] considered collapse of prolate spheroids of collisionless gas. (Nakamura and Sato [195] had previously studied the collapse of non-rotating deformed stars with an initial large reduction of internal energy and apparently found spindle or pancake singularities in extreme cases.) ST solved the general relativistic Vlasov equation for the particles along with Einstein’s equations for the gravitational field. They then searched each spatial slice for trapped surfaces. If no trapped surfaces were found, they concluded that there was no AH in that slice. The curvature invariant I = R Rμνρσ μνρσ was also computed. They found that an AH (and presumably a BH) formed if C ≤ 4πM < 1 everywhere, but no AH (and presumably a naked singularity) in the opposite case. In the latter case, the evolution (not surprisingly) could not proceed past the moment of formation of the singularity. In a subsequent study, ST [231] also showed that a small amount of rotation (counter rotating particles with no net angular momentum) does not prevent the formation of a naked spindle singularity. However, Wald and Iyer [246Jump To The Next Citation Point] have shown that the Schwarzschild solution has a time slicing whose evolution approaches arbitrarily close to the singularity with no AH in any slice (but, of course, with an EH in the spacetime). This may mean that there is a chance that the increasing prolateness found by ST in effect changes the slicing to one with no apparent horizon just at the point required by the hoop conjecture. While, on the face of it, this seems unlikely, Tod gives an example where a trapped surface does not form on a chosen constant time slice – but rather different portions form at different times. He argues that a numerical simulation might be forced by the singularity to end before the formation of the trapped surface is complete. Such a trapped surface would not be found by the simulations [241Jump To The Next Citation Point]. In response to such a possibility, Shapiro and Teukolsky considered equilibrium sequences of prolate relativistic star clusters [232]. The idea is to counter the possibility that an EH might form after the time when the simulation must stop. If an equilibrium configuration is non-singular, it cannot contain an EH since singularity theorems say that an EH implies a singularity. However, a sequence of non-singular equilibria with rising I ever closer to the spindle singularity would lend support to the existence of a naked spindle singularity since one can approach the singular state without formation of an EH. They constructed this sequence and found that the singular end points were very similar to their dynamical spindle singularity. Wald believes, however, that it is likely that the ST slicing is such that their singularities are not naked – a trapped surface is present but has not yet appeared in their time slices [245].

Another numerical study of the hoop conjecture was made by Chiba et al. [76Jump To The Next Citation Point]. Rather than a dynamical collapse model, they searched for AH’s in analytic initial data for discs, annuli, and rings. Previous studies of this type were done by Nakamura et al. [196] with oblate and prolate spheroids and by Wojtkiewicz [251] with axisymmetric singular lines and rings. The summary of their results is that an AH forms if C ≤ 4πM ≤ 1.26. (Analytic results due to Barrabès et al. [10Jump To The Next Citation Point9] and Tod [241Jump To The Next Citation Point] give similar quantitative results with different initial data classes and (possibly) an alternative definition of C.)

There is strong analytical evidence against the development of spindle singularities. It has been shown by ChruÅ›ciel and Moncrief that strong cosmic censorship holds in AF electrovac solutions which admit a G2 symmetric Cauchy surface [35Jump To The Next Citation Point]. The evolutions of these highly nonlinear equations are in fact non-singular.

2.1.3 Recent results

Garfinkle and Duncan [115] report preliminary results on the collapse of prolate configurations of Brill waves [60]. They use their axisymmetric code to explore the conjecture of Abrahams et al [2] that prolate configurations with no AH but large I in the initial slice will evolve to form naked singularities. Garfinkle and Duncan find that the configurations become less prolate as they evolve suggesting that black holes (rather than naked singularities) will form eventually from this type of initial data. Similar results have also been found by Hobill and Webster [149].

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Figure 2: This figure is based on Figure 1 of [208Jump To The Next Citation Point]. The vertical axis is time. The blue curve shows the singularity and the red curve the outermost marginally trapped surface. Note that the singularity forms at the poles (indicated by the blue arrow) before the outermost marginally trapped surface forms at the equator (indicated by the red arrow).

Pelath et al. [208] set out to generalize previous results [246241] that formation of a singularity in a slice with no AH did not indicate the absence of an EH. They looked specifically at trapped surfaces in two models of collapsing null dust, including the model considered by Barrabès et al. [10]. They indeed find a natural spacetime slicing in which the singularity forms at the poles before the outermost marginally trapped surface (OMTS) (which defines the AH) forms at the equator. Nonetheless, they also find that whether or not an OMTS forms in a slice closely (or at least more closely than one would expect if there were no relevance to the hoop conjecture) follows the predictions of the hoop conjecture.

2.1.4 Going further

Motivated by ST’s results [230], Echeverria [96] numerically studied the properties of the naked singularity that is known to form in the collapse of an infinite, cylindrical dust shell [239]. While the asymptotic state can be found analytically, the approach to it must be followed numerically. The analytic asymptotic solution can be matched to the numerical one (which cannot be followed all the way to the collapse) to show that the singularity is strong (an observer experiences infinite stretching parallel to the symmetry axis and squeezing perpendicular to the symmetry axis). A burst of gravitational radiation emitted just prior to the formation of the singularity stretches and squeezes in opposite directions to the singularity. This result for dust conflicts with rigorously nonsingular solutions for the electrovac case [35]. One wonders then if dust collapse gives any information about singularities of the gravitational field.

One useful result from dust collapse has been the study of gravitational waves which might be associated with the formation of a naked singularity. Such a program has been carried out by Harada, Iguchi, and Nakao [160161198137138159].

Nakamura et al. (NSN) [197] conjectured that even if naked spindle singularities could exist, they would either disappear or become black holes. This demise of the naked singularity would be caused by the back reaction of the gravitational waves emitted by it. While NSN proposed a numerical test of their conjecture, they believed it to be beyond the current generation of computer technology.

Chiba [75] extended previous results [76] to search for AH’s in spacetimes without axisymmetry but with a discrete symmetry. The discrete symmetry is used to identify an analog of a symmetry axis to allow a prescription for an analog of the circumference. Given this construction, it is possible to formulate the hoop conjecture in this case and to explore its validity in numerically constructed momentarily static spacetimes. Explicit application was made to multiple black holes distributed along a ring. It was found that, if the quantity š¯’˛ defined as the circumference is less than approximately 1.168, a common apparent horizon surrounds the multi-black-hole configuration.

The results of all these searches for naked spindle singularities are controversial but could be resolved if the presence or absence of the EH could be determined. One could demonstrate numerically whether or not Wald’s view of ST’s results is correct by using existing EH finders [179184] in a relevant simulation. Of course, this could only be effective if the simulation covered enough of the spacetime to include (part of) the EH.

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