Critical behavior was originally found by Choptuik [78] in a numerical study of the collapse of a spherically symmetric
massless scalar field. For recent reviews see [125,
130]. 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
such that
yields a BH. He found
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, where
r
is the radial coordinate, 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 [145
].) The numerical demonstration of this feature of the critical
solution was provided by Hamadé and Stewart [135
]. This result caused Hawking to pay off a bet to Preskill and
Thorne [61,
170].
Soon after this discovery, scaling and critical phenomena were
found in a variety of contexts. Abrahams and Evans [1] 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 [109] confirmed Choptuik's results with a completely different
algorithm that does not require AMR. He used Goldwirth and
Piran's [119
] method of simulating Christodoulou's [80] 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 [135] 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:
where
Thus
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. (While this procedure allowed
Hirschmann and Eardley to obtain
for the complex scalar field solution, they later found [146
] that, in this regime, the complex scalar field has 3 unstable
modes. This means [129
,
126] that the Eardley-Hirschmann solution is not a critical
solution. A perturbation analysis indicates that the critical
solution for the complex scalar field is the discretely
self-similar one found for the real scalar field [129
].) Koike et al. [176
] obtain
for the Evans-Coleman solution. Although the similarities among
the critical exponents
in the collapse computations suggested a universal value,
Maison [183] used these same scaling-perturbation methods to show that
depends on the equation of state
of the fluid in the Evans-Coleman solution. Gundlach [127] used a similar approach to locate Choptuik's critical solution
accurately. This is much harder due to its discrete
self-similarity. He reformulates the model as 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 [176] and applied it to the continuously-self-similar case [136
]. (For an application of renormalization group methods to
cosmology see [162].)
Choptuik et al. [79] 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 [14]. 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 [128]. The Type I behavior arises when the collapsing system has a
metastable static solution in addition to the Choptuik critical
one [132
].
Brady, Chambers and Gonçalves [71,
52] 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 [229]. The massive scalar field can be treated as collapsing dust to
yield a criterion for BH formation [120].
The Choptuik solution has also been found to be the critical
solution for charged scalar fields [132,
151]. 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 [217,
201]. Bizon and Chmaj [47] have considered the critical collapse of skyrmions.
An astrophysical application of BH critical phenomena has been considered by Niemeyer and Jedamzik [199] and Yokoyama [252]. They consider its implications for primordial BH formation and suggest that it could be important.
An interesting ``toy model'' for general relativity in many
contexts has been wave maps, also known as nonlinear
models. One of these contexts is critical collapse [146]. Recently and independently, Bizon et al. [48] and Liebling et al. [180] evolved wave maps from the base space of 3+1 Minkowski space to
the target space
. They found critical behavior separating singular and
nonsingular solutions. For some families of initial data, the
critical solution is self-similar and is an intermediate
attractor. For others, a static solution separates the singular
and nonsingular solutions. However, the static solution has
several unstable modes and is therefore not a critical solution
in the usual sense. Bizon and Tabor [49] have studied Yang-Mills fields in
D
+ 1 dimensions and found that generic solutions with
sufficiently large initial data blow up in a finite time and that
the mechanism for blowup depends on
D
. Husa et al. [158] then considered the collapse of
SU
(2) nonlinear sigma models coupled to gravity and found a
discretely self-similar critical solution for sufficiently large
dimensionless coupling constant. They also observe that for
sufficiently small coupling constant values, there is a
continuously self-similar solution. Interestingly, there is an
intermediate range of coupling constant where the discrete
self-similarity is intermittent [238].
Until recently, only Abrahams and Evans [1] had ventured beyond spherical symmetry. The first additional
departure has been made by Gundlach [131]. He considered spherical and non-spherical perturbations of
perfect fluid collapse. Only the original (spherical) growing
mode survived.
Recently, critical phenomena have been explored in 2+1 gravity. Pretorius and Choptuik [215] numerically evolved circularly symmetric scalar field collapse in 2+1 anti-de Sitter space. They found a continuously self-similar critical solution at the threshold for black hole formation. The BH's which form have BTZ [8] exteriors with strong curvature, spacelike singularities in the interior. Remarkably, Garfinkle obtained an analytic critical solution by assuming continuous self-similarity which agrees quite well with the one obtained numerically [112].
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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 Problems/Comments to livrev@aei-potsdam.mpg.de |