Centrella and Matzner [22, 23] studied a class of plane symmetric cosmologies representing gravitational inhomogeneities in the form of shocks or discontinuities separating two vacuum expanding Kasner cosmologies. By a suitable choice of parameters, the constraint equations can be satisfied at the initial time with an Euclidean 3-surface and an algebraic matching of parameters across the different Kasner regions that gives rise to a discontinuous extrinsic curvature tensor. They performed both numerical calculations and analytical estimates using a Green's function analysis to establish and verify (despite the numerical difficulties in evolving discontinuous data) certain aspects of the solutions, including gravitational wave interactions, the formation of tails, and the singularity behavior of colliding waves in expanding vacuum cosmologies.
Shortly thereafter, Centrella and Wilson [24,
25] developed a more general plane symmetric code for cosmology,
adding also hydrodynamic sources. In order to simplify the
resulting differential equations, they adopted a diagonal
3-metric of the form
, which is maintained in time with a proper choice of shift
vector. The hydrodynamic equations are solved using artificial
viscosity methods for shock capturing and Barton's method for
monotonic transport [62]. 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. [2,
4], implementing more robust numerical methods and an improved
parametric treatment of the initial value problem.
In applications of these codes, Centrella [21] investigated nonlinear gravity 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 also used to model the transition into the nonlinear
regime. Anninos et al. [3] considered small and large perturbations in the axisymmetric
Kasner models. 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 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.
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Physical and Relativistic Numerical Cosmology
Peter Anninos http://www.livingreviews.org/lrr-1998-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |