

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'' [239
] that collapse will yield a black hole only if a mass
M
is compressed to a region with circumference
in all directions. (As is discussed by Wald [244
], 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 [75
].) If the hoop conjecture is true, naked singularities may form
if collapse can yield
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 [166,
167], it is possible to construct a spacetime slicing which has no
AH even though an EH is present [246
]. Methods to find an EH in a numerically determined spacetime
have only recently become available and have not been applied to
this issue [179
,
184
]. A local prescription, applicable numerically, to identify an
``isolated horizon'' is under development by Ashtekar et al. (see
for example [7]).
Figure 1:
Heuristic illustration of the hoop conjecture.
In the best known attempt to produce naked singularities, Shapiro
and Teukolsky (ST) [230
] 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
was also computed. They found that an AH (and presumably a BH)
formed if
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 [246
] 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 [241
]. 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 [244].
Another numerical study of the hoop conjecture was made by
Chiba et al. [76
]. 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
. (Analytic results due to Barrabès et al. [10
,
9] and Tod [241
] 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 Chrusciel and
Moncrief that strong cosmic censorship holds in AF electrovac
solutions which admit a
symmetric Cauchy surface [34
]. The evolutions of these highly nonlinear equations are in fact
non-singular.
Garfinkle and Duncan [114] 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].
Figure 2:
This figure is based on Figure 1 of [208
]. 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 [246,
241] 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.
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 [34]. 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 [160,
161,
198,
137,
138,
159].
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
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 [179,
184] 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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Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-2002-1
© Max-Planck-Gesellschaft. ISSN 1433-8351
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