Centrella and Matzner [58, 59] studied a class of plane symmetric cosmologies representing gravitational inhomogeneities in the form of shocks or discontinuities separating two vacuum expanding Kasner cosmologies (2). By a suitable choice of parameters, the constraint equations can be satisfied at the initial time with a 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 [60, 61] 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 [57] 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 behavor 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.

Two additional examples of general relativistic codes developed for the purpose of investigating dynamical behaviors in non-flat, vacuum, cosmological topologies are attributed to Holcomb [91] and Ove [129]. Holcomb considered vacuum axisymmetric models to study the structure of General Relativity and the properties of gravitational waves in non-asymptotically flat spacetimes. The code was based on the ADM 3 + 1 formalism and used Kasner matching conditions at the outer edges of the mesh, mean curvature slicing, and a shift vector to enforce the isothermal gauge in order to simplify the metric and to put it in a form that resembles quasi-isotropic coordinates. However, a numerical instability was observed in cases where the mesh domain exceeded the horizon size. This was attributed to the particular gauge chosen, which does not appear well-suited to the Kasner metric as it results in super-luminal coordinate velocities beyond the horizon scale.

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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