Shortly thereafter, Centrella and Wilson
[59, 60
]
developed a polarized plane symmetric code for cosmology, adding
also hydrodynamic sources with artificial viscosity methods for
shock capturing and Barton’s method for monotonic transport
[162
]
. The evolutions are fully constrained (solving both the momentum
and Hamiltonian constraints at each time step) and use the mean
curvature slicing condition. This work was subsequently extended
by Anninos et al.
[9, 11, 7
], implementing more robust numerical methods, an improved
parametric treatment of the initial value problem, and generic
unpolarized metrics.
In applications of these codes, Centrella
[61]
investigated nonlinear gravitational waves in Minkowski space
and compared the full numerical solutions against a first order
perturbation solution to benchmark certain numerical issues such
as numerical damping and dispersion. A second order perturbation
analysis was used to model the transition into the nonlinear
regime. Anninos et al.
[10]
considered small and large perturbations in the two degenerate
Kasner models:
or
, and
or
, respectively, where
are parameters in the Kasner metric (2
). Carrying out a second order perturbation expansion and
computing the Newman-Penrose (NP) scalars, Riemann invariants and
Bel-Robinson vector, they demonstrated, for their particular
class of spacetimes, that the nonlinear behavior is in the
Coulomb (or background) part represented by the leading order
term in the NP scalar
, and not in the gravitational wave component. For standing-wave
perturbations, the dominant second order effects in their
variables are an enhanced monotonic increase in the background
expansion rate, and the generation of oscillatory behavior in the
background spacetime with frequencies equal to the harmonics of
the first order standing-wave solution.
Expanding these investigations of the Coulomb
nonlinearity, Anninos and McKinney
[16]
used a gauge invariant perturbation formalism to construct
constrained initial data for general relativistic cosmological
sheets formed from the gravitational collapse of an ideal gas in
a critically closed FLRW “background” model. They compared
results to the Newtonian Zel’dovich
[165
]
solution over a broad range of field strengths and flows, and
showed that the enhanced growth rates of nonlinear modes (in both
the gas density and Riemann curvature invariants) accelerate the
collapse process significantly compared to Newtonian and
perturbation theory. They also computed the back-reaction of
these structures to the mean cosmological expansion rate and
found only a small effect, even for cases with long wavelengths
and large amplitudes. These structures were determined to produce
time-dependent gravitational potential signatures in the CMBR
(essentially fully relativistic Rees-Sciama effects) comparable
to, but still dominated by, the large scale Sachs-Wolfe
anisotropies. This confirmed, and is consistent with, the
assumptions built into Newtonian calculations of this effect.
Ove developed an independent code based on the ADM formalism to study cosmic censorship issues, including the nature of singular behavior allowed by the Einstein equations, the role of symmetry in the creation of singularities, the stability of Cauchy horizons, and whether black holes or a ring singularity can be formed by the collision of strong gravitational waves. Ove adopted periodic boundary conditions with 3-torus topology and a single Killing field, and therefore generalizes to two dimensions the planar codes discussed in the previous section. This code also used a variant of constant mean curvature slicing, was fully constrained at each time cycle, and the shift vector was chosen to put the metric into a (time-dependent) conformally flat form at each spatial hypersurface.
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