Belinsky, Lifshitz, and Khalatnikov
(BLK)
[32, 33
]
and Misner
[119]
discovered that the Einstein equations in the vacuum homogeneous
Bianchi type IX (or Mixmaster) cosmology exhibit complex behavior
and are sensitive to initial conditions as the Big Bang
singularity is approached. In particular, the solutions near the
singularity are described qualitatively by a discrete map
[30, 32]
representing different sequences of Kasner spacetimes
Although BLK conjectured that local Mixmaster
oscillations might be the generic behavior for singularities in
more general classes of spacetimes
[33], it is only recently that this conjecture has begun to be
supported by numerical evidence (see Section
3.1.2
and
[37]).
Considering inhomogeneous spacetimes, Isenberg
and Moncrief
[98]
proved that the singularity in the polarized Gowdy model is AVTD
type, as are more general polarized
symmetric cosmologies
[38]
. Early numerical studies using symplectic methods confirmed AVTD
behavior and found no evidence of BLK oscillations, even in
spacetimes with
symmetry
[36
]
(although there were concerns about the solutions due to
difficulties in resolving steep spatial gradients near the
singularity
[36
], which were addressed later by Hern and Stewart
[87]
for the Gowdy
models).
However, Weaver et al.
[160]
established the first evidence through numerical simulations
that Mixmaster dynamics can occur in a class of inhomogeneous
spacetimes which generalize the Bianchi type
model with a magnetic field and two-torus symmetry. Berger and
Moncrief
[41, 42]
also demonstrated that
symmetric vacuum cosmologies exhibit local Mixmaster dynamics
consistent with the BLK conjecture, despite numerical
difficulties in resolving steep gradients (which they managed by
enforcing the Hamiltonian constraint and spatially averaging the
solutions). Another more recent example supporting the BLK
conjecture is provided by Garfinkle
[79], who finds local oscillating behavior approaching the
singularity in closed vacuum (but otherwise generic) spacetimes
with no assumed symmetry in the initial data.
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