Causal propagation of geometrical fields in relativistic cosmology

TL;DR

This paper develops a hyperbolic evolution system for geometrical fields in relativistic cosmology, analyzing wave propagation velocities of matter and gravitational disturbances, with implications for understanding irregularities in cosmological models.

Contribution

It introduces a 44-dimensional symmetric hyperbolic system for geometrical variables in relativistic cosmology, detailing characteristic wave velocities of matter and gravitational fields.

Findings

Weyl curvature accounts for Coulomb-like and transverse wave eigenfields.

Propagation velocities include 0, 1, and 1/2, depending on the eigenfield.

Application to locally rotationally symmetric cosmological models.

Abstract

We employ the extended 1+3 orthonormal frame formalism for fluid spacetime geometries $({\cal M}, {\bf g}, {\bf u})$, which contains the Bianchi field equations for the Weyl curvature, to derive a 44-D evolution system of first-order symmetric hyperbolic form for a set of geometrically defined dynamical field variables. Describing the matter source fields phenomenologically in terms of a barotropic perfect fluid, the propagation velocities $v$ (with respect to matter-comoving observers that Fermi-propagate their spatial reference frames) of disturbances in the matter and the gravitational field, represented as wavefronts by the characteristic 3-surfaces of the system, are obtained. In particular, the Weyl curvature is found to account for two (non-Lorentz-invariant) Coulomb-like characteristic eigenfields propagating with $v = 0$ and four transverse characteristic eigenfields propagating with $|v| = 1$, which are well known, and four (non-Lorentz-invariant) longitudinal characteristic eigenfields propagating with $|v| = \sfrac{1}{2}$. The implications of this result are discussed in some detail and a parallel is drawn to the propagation of irregularities in the matter distribution. In a worked example, we specialise the equations to cosmological models in locally rotationally symmetric class II and include the constraints into the set of causally propagating dynamical variables.