The past few years have seen the development of strong
numerical evidence in support of the BKL claims [36]. The Method of Consistent Potentials (MCP) [123
] has been used to organize the data obtained in simulations of
spatially inhomogeneous cosmologies [42
,
35
,
250
,
43
,
36
,
33
,
41
]. The main idea is to obtain a Kasner-like velocity term
dominated (VTD) solution at every spatial point by solving
Einstein's equations truncated by removing all terms containing
spatial derivatives. If the spacetime really is AVTD, all the
neglected terms will be subdominant (exponentially small in
variables where the VTD solution is linear in the time
) when the VTD solution is substituted back into them. For the
MCP to successfully predict whether or not the spacetime is AVTD,
the dynamics of the full solution must be dominated at (almost)
every spatial point by the VTD solution behavior. Surprisingly,
MCP predictions have proved valid in numerical simulations of
cosmological spacetimes with one [43
] and two [42
,
35
,
41
] spatial symmetries. In the case of
U
(1) symmetric models, a comparison between the observed
behavior [43
] and that in a vacuum, diagonal Bianchi IX model written in
terms of
U
(1) variables provides strong support for LMD [45] since the phenomenology of the inhomogeneous cosmologies can be
reproduced by this rewriting of the standard Bianchi IX MD.
Polarized plane symmetric cosmologies have been evolved
numerically using standard techniques by Anninos, Centrella, and
Matzner [5,
6]. The long-term project involving Berger, Garfinkle, and
Moncrief and their students to study the generic cosmological
singularity numerically has been reviewed in detail
elsewhere [37,
36
,
32] but will be discussed briefly here.
Einstein's equations split into two groups. The first is
nonlinearly coupled wave equations for dynamical variables
P
and
Q
(where
) obtained from the variation of [188]
where
and
are canonically conjugate to
P
and
Q
respectively. This Hamiltonian has the form required by the
symplectic scheme. If the model is, in fact, AVTD, the
approximation in the symplectic numerical scheme should become
more accurate as the singularity is approached. The second group
of Einstein equations contains the Hamiltonian and
-momentum constraints, respectively. These can be expressed as
first order equations for
in terms of
P
and
Q
. This break into dynamical and constraint equations removes two
of the most problematical areas of numerical relativity from this
model - the initial value problem and numerical preservation of
the constraints.
For the special case of the polarized Gowdy model (Q
= 0),
P
satisfies a linear wave equation whose exact solution is
well-known [26]. For this case, it has been proven that the singularity is
AVTD [165]. This has also been conjectured to be true for generic Gowdy
models [123
].
AVTD behavior is defined in [165] as follows: Solve the Gowdy wave equations neglecting all terms
containing spatial derivatives. This yields the VTD
solution [42]. If the approach to the singularity is AVTD, the full solution
comes arbitrarily close to a VTD solution at each spatial point
as
. As
, the VTD solution becomes
where
v
> 0 . Substitution in the wave equations shows that this
behavior is consistent with asymptotic exponential decay of all
terms containing spatial derivatives only if
[123]. We have shown that, except at isolated spatial points, the
nonlinear terms in the wave equation for
P
drive
v
into this range [35
,
37
]. The exceptional points occur when coefficients of the
nonlinear terms vanish and are responsible for the growth of
spiky features seen in the wave forms [42
,
35
]. We conclude that generic Gowdy cosmologies have an AVTD
singularity except at isolated spatial points [35
,
37
]. This has been confirmed by Hern and Stuart [143] and by van Putten [243]. After the nature of the solutions became clear through
numerical experiments, it became possible to use Fuchsian
asymptotic methods to prove that Gowdy solutions with 0 <
v
< 1 and AVTD behavior almost everywhere are generic [173]. These methods have recently been applied to Gowdy spacetimes
with
and
topologies with similar conclusions [233].
One striking property of the Gowdy models are the development of ``spiky features'' at isolated spatial points where the coefficient of a local ``potential term'' vanishes [42, 35]. Recently, Rendall and Weaver have shown analytically how to generate such spikes from a Gowdy solution without spikes [220].
Addition of a magnetic field to the vacuum Gowdy models (plus
a topology change) which yields the inhomogeneous generalization
of magnetic Bianchi VI
models provides an additional potential which grows
exponentially if 0 <
v
< 1 . Local Mixmaster behavior has recently been observed in
these magnetic Gowdy models [250,
247].
Garfinkle has used a vacuum Gowdy model with
spatial topology to test an algorithm for axis regularity [111]. Along the way, he has shown that these models are also AVTD
with behavior at generic spatial points that is eventually
identical to that in the
case. Comparison of the two models illustrates that topology or
other global or boundary conditions are important early in the
simulation but become irrelevant as the singularity is
approached.
Gowdy spacetimes are not the most general
symmetric vacuum cosmologies. Certain off-diagonal metric
components (the twists which are
,
in the notation of (12
)) have been set to zero [121]. Restoring these terms (see [83,
40]) yields spacetimes that are not AVTD but rather appear to
exhibit a novel type of LMD [41
,
249]. The LMD in these models is an inhomogeneous generalization of
non-diagonal Bianchi models with ``centrifugal'' MSS potential
walls [227,
169] in addition to the usual curvature walls. In [41], remarkable quantitative agreement is found between predictions
of the MCP and numerical simulation of the full Einstein
equations. A version of the code with AMR has been
developed [15]. (Asymptotic methods have been used to prove that the polarized
version of these spacetimes have AVTD solutions [163].)
Here
and
are analogous to
P
and
Q
while
is a conformal factor for the metric
in the
u
-
v
plane perpendicular to the symmetry direction. Symplectic
methods are still easily applicable. Note particularly that
contains two copies of the Gowdy
plus a free particle term and is thus exactly solvable. The
potential term
is very complicated. However, it still contains no momenta so
its equations are trivially exactly solvable. However, issues of
spatial differencing are problematic. (Currently, a scheme due to
Norton [200] is used. A spectral evaluation of derivatives [100] which has been shown to work in Gowdy simulations [23] does not appear to be helpful in the
U
(1) case.) A particular solution to the initial value problem is
used since the general solution is not available [37
].
Current limitations of the
U
(1) code do not affect simulations for the polarized case since
problematic spiky features do not develop. Polarized models have
. Grubisic and Moncrief [124] have conjectured that these polarized models are AVTD. The
numerical simulations provide strong support for this
conjecture [37
,
44]. Asymptotic methods have been used to prove that an open set of
AVTD solutions exist for this case [164].
Mixmaster simulations with the new algorithm [38] can easily evolve more than 250 bounces reaching
. This compares to earlier simulations yielding 30 or so bounces
with
. The larger number of bounces quickly reveals that it is
necessary to enforce the Hamiltonian constraint. An explicitly
constraint enforcing
U
(1) code was developed some years ago by Ove (see [207] and references therein).
It is well known [20] that a scalar field can suppress Mixmaster oscillations in
homogeneous cosmologies. BKL argued that the suppression would
also occur in spatially inhomogeneous models. This was
demonstrated numerically for magnetic Gowdy and
U
(1) symmetric spacetimes [33]. Andersson and Rendall proved that completely general
cosmological (spatially
) spacetimes (no symmetries) with sufficiently strong scalar
fields have generic AVTD solutions [3]. Garfinkle [107] has constructed a 3D harmonic code which, so far, has found
AVTD solutions with a scalar field present. Work on generic
vacuum models is in progress.
Cosmological models inspired by string theory contain higher derivative curvature terms and exotic matter fields. Damour and Henneaux have applied the BKL approach to such models and conclude that their approach to the singularity exhibits LMD [88].
Finally, there has been a study of the relationship between the ``long wavelength approximation'' and the BKL analyses by Deruelle and Langlois [90].
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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 |