] (BKL) described the singularity approach of vacuum Bianchi IX
cosmologies as an infinite sequence of Kasner [171] epochs whose indices change when the scalar curvature terms in
Einstein's equations become important. They were able to describe
the dynamics approximately by a map evolving a discrete set of
parameters from one Kasner epoch to the next [22,
74]. For example, the Kasner indices for the power law dependence
of the anisotropic scale factors can be parametrized by a single
variable
. BKL determined that
The subtraction in the denominator for
yields the sensitivity to initial conditions associated with
Mixmaster dynamics (MD). Misner [187] described the same behavior in terms of the model's volume and
anisotropic shears. A multiple of the scalar curvature acts as an
outward moving potential in the anisotropy plane. Kasner epochs
become straight line trajectories moving outward along a
potential corner while bouncing from one side to the other. A
change of corner ends a BKL era when
. Numerical evolution of Einstein's equations was used to explore
the accuracy of the BKL map as a descriptor of the dynamics as
well as the implications of the map [193,
223,
225,
28]. Rendall has studied analytically the validity of the BKL map
as an approximation to the true trajectories [218].
Later, the BKL sensitivity to initial conditions was discussed
in the language of chaos [11,
172]. An extended application of Bernoulli shifts and Farey trees
was given by Rugh [224] and repeated by Cornish and Levin [85]. However, the chaotic nature of Mixmaster dynamics was
questioned when numerical evolution of the Mixmaster equations
yielded zero Lyapunov exponents (LE's) [102,
62,
147]. (The LE measures the divergence of initially nearby
trajectories. Only an exponential divergence, characteristic of a
chaotic system, will yield a positive exponent.) Other numerical
studies yielded positive LE's [216]. This issue was resolved when the LE was shown numerically and
analytically to depend on the choice of time variable [223,
27,
99]. Although MD itself is well-understood, its characterization as
chaotic or not had been quite controversial [148].
LeBlanc et al. [178] have shown (analytically and numerically) that MD can arise in
Bianchi VI
models with magnetic fields (see also [181]). In essence, the magnetic field provides the wall needed to
close the potential in a way that yields the BKL map for
u
[29]. A similar study of magnetic Bianchi I has been given by
LeBlanc [177]. Jantzen has discussed which vacuum and electromagnetic
cosmologies could display MD [168].
Cornish and Levin (CL) [86] identified a coordinate invariant way to characterize MD.
Sensitivity to initial conditions can lead to qualitatively
distinct outcomes from initially nearby trajectories. While the
LE measures the exponential divergence of the trajectories, one
can also ``color code'' the regions of initial data space
corresponding to particular outcomes. A chaotic system will
exhibit a fractal pattern in the colors. CL defined the following
set of discrete outcomes: During a numerical evolution, the BKL
parameter
u
is evaluated from the trajectories. The first time
u
> 7 appears in an approximately Kasner epoch, the trajectory
is examined to see which metric scale factor has the largest time
derivative. This defines three outcomes and thus three colors for
initial data space. However, one can easily invent prescriptions
other than that given by Cornish and Levin [86] which would yield discrete outcomes. The fractal nature of
initial data space should be common to all of them. It is not
clear how the value of the fractal dimension as measured by
Cornish and Levin would be affected. The CL prescription has been
criticized because it requires only the early part of a
trajectory for implementation [194]. Actually, this is the greatest strength of the prescription
for numerical work. It replaces a single representative,
infinitely long trajectory by (easier to compute) arbitrarily
many trajectory fragments.
] is used to generate a Mixmaster trajectory with more than 250
bounces. The trajectory is shown in the rescaled anisotropy plane
with axes
. The rescaling fixes (asymptotically) the location of the
bounces.
To study the CL fractal and ergodic properties of Mixmaster
evolution [86], one could take advantage of a new numerical algorithm due to
Berger, Garfinkle, and Strasser [38]. Symplectic methods are used to allow the known exact solution
for a single Mixmaster bounce [236] to be used in the ODE solver. Standard ODE solvers [214] can take large time steps in the Kasner segments but must slow
down at the bounce. The new algorithm patches together exactly
solved bounces. Tens of orders of magnitude improvement in speed
are obtained while the accuracy of machine precision is
maintained. In [38], the new algorithm was used to distinguish Bianchi IX and
magnetic Bianchi VI
bounces. This required an improvement of the BKL map (for
parameters other than
u) to take into account details of the exponential potential.
So far, most recent effort in MD has focused on diagonal (in
the frame of the
SU
(2) 1-forms) Bianchi IX models. Long ago, Ryan [226] showed that off-diagonal metric components can contribute
additional MSS potentials (e.g. a centrifugal wall). This has
been further elaborated by Jantzen [169].
]), there was very little rigorous information on the nature of
these solutions. Recently, the existence of a strong singularity
(curvature blowup) was proved for Bianchi VIII and IX collapse by
Ringström [221,
222] and for magnetic Bianchi VI
by Weaver [248]. A remaining open question is how closely an actual Mixmaster
evolution is approximated by a single BKL sequence [218,
222]. Since the Berger et al. algorithm [38] achieves machine level accuracy, it can be used to collect
numerical evidence on this topic. For example, it has been shown
that a given Mixmaster trajectory ceases to track the
corresponding sequence of integers obtained from the BKL
map (11
) at the point where there have been enough era-ending (mixing)
bounces to lose all the information encoded in finite precision
initial data [38].
]). Recently, Coley has considered Bianchi IX brane-world models
and found them not to be chaotic [84].
Finally, we remark on a successful application of numerical Regge calculus in 3+1 dimensions. Gentle and Miller have been able to evolve the Kasner solution [117].
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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 |
