3.6 Regge Calculus Model3 RELATIVISTIC COSMOLOGY3.4 Nucleosynthesis

3.5 Plane Symmetric Gravitational Waves 

Gravitational waves are an inevitable product of the general Einstein equations, and one can expect a wide spectrum of wave signals propagating throughout our Universe due to shear anisotropies, primordial metric and matter fluctuations, collapsing matter structures, ringing black holes, and colliding neutron stars, for example. The discussion here is restricted to the pure vacuum field dynamics and the fundamental nonlinear behavior of gravitational waves in numerically generated cosmological spacetimes.

Centrella and Matzner [20, 21] studied a class of plane symmetric cosmologies representing gravitational inhomogeneities in the form of shocks or discontinuities separating two vacuum expanding Kasner cosmologies (1Popup Equation). 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 [22, 23] 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

  equation171

which is maintained in time with a proper choice of shift vector. The metric (6Popup Equation) allows an overall conformal factor A to simplify the initial value problem, and a dynamical transverse wave component in the variable h . The hydrodynamic equations are solved using artificial viscosity methods for shock capturing and Barton's method for monotonic transport [57]. 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. [1, 3], implementing more robust numerical methods and an improved parametric treatment of the initial value problem.

In applications of these codes, Centrella [19] 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. [2] considered small and large perturbations in the two degenerate Kasner models: image or 2/3, and image or -1/3 respectively, where image are parameters in the Kasner metric (1Popup Equation). 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 image, 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.



3.6 Regge Calculus Model3 RELATIVISTIC COSMOLOGY3.4 Nucleosynthesis

image 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