with time changing exponents , but otherwise constrained by . Because this discrete mapping of Kasner epochs is chaotic, the Mixmaster dynamics is presumed to be chaotic as well.

Mixmaster behavior can be studied in the context of Hamiltonian dynamics, with the Hamiltonian [44]

and a semi-bounded potential arising from the spatial scalar curvature (whose level curves are plotted in Figure 2)

where and are the scale factor and anisotropies, and and are the corresponding conjugate variables.

A solution of this Hamiltonian system is an infinite sequence of Kasner epochs with parameters that change when the phase space trajectories bounce off the potential walls, which become exponentially steep as the system evolves towards the singularity. Several numerical calculations of the dynamical equations arising from (2) and (3) have indicated that the Liapunov exponents of the system vanish, in apparent contradiction with the discrete maps [17, 31]. However, it has since been shown that the usual definition of the Liapunov exponents is ambiguous in this case as it is not invariant under time reparametrizations, and that with a different time variable one obtains positive exponents [14, 27].

Although BLK conjectured that local Mixmaster oscillations might be the generic behavior for singularities in more general classes of spacetimes [12], this conjecture has yet to be established. However, Weaver et al. [40] have, through numerical investigations, established evidence that Mixmaster dynamics can occur in (at least a restricted class of) inhomogeneous spacetimes which generalize the Bianchi type VI with a magnetic field and two-torus symmetry.

Considering inhomogeneous spacetimes, Isenberg and Moncrief [33] proved that the singularity in the polarized Gowdy model is
AVTD type. This has also been conjectured to be the case for more
general Gowdy models, and numerical simulations of
one-dimensional plane symmetric Gowdy spacetimes support the
notion. Furthermore, a symplectic numerical method has been
applied to investigate the AVTD conjecture in even more general
spacetimes, namely
spacetimes with
*U*
(1) symmetry [13]. Although there are concerns about the solutions due to
difficulties in resolving the steep spatial gradients near the
singularity, the numerical calculations find no evidence of BLK
oscillations. Berger [13] attributes this to several possibilities: 1) the BLK conjecture
is false; 2) the simulations have not been run long enough; 3)
Mixmaster behavior is present but hidden in the variables; or 4)
the initial data is insufficiently generic. In any case, further
investigations are needed to confirm either the BLK or AVTD
conjectures.

Computational Cosmology: from the Early Universe to the
Large Scale Structure
Peter Anninos
http://www.livingreviews.org/lrr-1998-9
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |