with a
U
(1) symmetry [142
]. The non-commuting Killing vectors of local MD can be
constructed since only one Killing vector is already present. The
two commuting Killing vectors of the even simpler plane symmetric
Gowdy cosmologies [88,
18] preclude their use to test the conjecture. However, these
models are interesting in their own right since they have been
conjectured to possess an AVTD singularity [90].
Polarized plane symmetric cosmologies have been evolved
numerically using standard techniques by Anninos, Centrella, and
Matzner [2,
3]. A long-term project involving Berger, Garfinkle, and Moncrief
and our students to study the generic cosmological singularity
numerically has been reviewed in detail elsewhere [26] but will be discussed briefly here.
is described by gravitational wave amplitudes
and
which propagate in a spatially inhomogeneous background universe
described by
. (We note that the physical behavior of a Gowdy spacetime can
be computed from the effect of the metric evolution on a test
cylinder [29].) We impose
and periodic boundary conditions. The time variable
measures the area in the symmetry plane with
a curvature singularity.
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 [140]
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 [18]. For this case, it has been proven that the singularity is AVTD
[119]. This has also been conjectured to be true for generic Gowdy
models [90].
AVTD behavior is defined in [119] as follows: Solve the Gowdy wave equations neglecting all terms
containing spatial derivatives. This yields the AVTD solution [32]. If the approach to the singularity is AVTD, the full solution
comes arbitrarily close to an AVTD solution at each spatial point
as
. As
, the AVTD 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
[90]. We have shown that, except at isolated spatial points, the
nonlinear terms in the wave equation for
P
drive
v
into this range [25,
26]. 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 [32,
25]. We conclude that generic Gowdy cosmologies have an AVTD
singularity except at isolated spatial points [25,
26]. A claim to the contrary [107] is incorrect [27]. Recently, it has been proven analytically that Gowdy solutions
with 0 <
v
< 1 and AVTD behavior almost everywhere are generic [126].
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 [181].
with a spatial
U
(1) symmetry
can be described by five degrees of freedom
and their respective conjugate momenta
. All variables are functions of spatial variables
u,
v
and time,
. Einstein's equations can be obtained by variation of
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. 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 [150] is used. A spectral evaluation of derivatives [74] which has been shown to work in Gowdy simulations [17] 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 [26]. Currently, except as discussed below, the constraints are
monitored but not explicitly enforced.
Despite the current limitations of the
U
(1) code, conclusions can be confidently drawn for polarized
U
(1) models in our restricted class of initial data. No numerical
difficulties arise in this case. Polarized models have
. Grubisic and Moncrief [91] have conjectured that these polarized models are AVTD. The
numerical simulations provide strong support for this conjecture
[26,
31].
in (14
) acts as a Mixmaster-like potential to drive the system away
from AVTD behavior in generic
U
(1) models [16]. Numerical simulations provide support for this suggestion [26,
30]. Whether this potential term grows or decays depends on a
function of the field momenta. This in turn is restricted by the
Hamiltonian constraint. However, failure to enforce the
constraints can cause an erroneous relationship among the momenta
to yield qualitatively wrong behavior. There is numerical
evidence that this error tends to suppress Mixmaster-like
behavior leading to apparent AVTD behavior in extended spatial
regions [22,
23]. In fact, it has been found very recently [30] that when the Hamiltonian constraint is enforced at every time
step, the predicted local oscillatory behavior of the approach to
the singularity is observed.
Mixmaster simulations with the new algorithm [28] 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. A similar
conclusion for gravitational wave equations was obtained by
Gundlach and Pullin [99]. Although this particular analysis may be incorrect [47], it is still likely that constraint enforcement will be
essential for sufficiently long simulations. An explicitly
constraint enforcing
U
(1) code was developed some years ago by Ove (see [153] and references therein).
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Numerical Approaches to Spacetime Singularities
Beverly K. Berger http://www.livingreviews.org/lrr-1998-7 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
