Various Matter Models in Generalised Bianchi Space-times

Abstract

In the standard treatment, a group acts by isometries on a space-time, imposing its generators as the Killing vectors of this space-time, hence the corresponding solution to Einstein's field equations bears these physical symmetries. This treatment has produced a number of results with particular interest; examples of this are the spatially homogeneous cosmological models, some inhomogeneous cosmological models, but also several solutions of gravitational radiation. However, most of these solutions are carried out in vacuum or with the utilization of a perfect fluid as source. In the cases where either classical macroscopic (Euler) matter, or kinetic microscopic (Vlasov or Boltzmann) matter is used, the analysis is usually carried out with respect to an orthogonal slicing of space-time, which is further restrictive on the freedom of the action of the Bianchi group on the space-time.

In this work, we attempt to generalise these works by assuming that the Bianchi group acts by homotheties and the quotient is any one-dimensional submanifold invariant to the action of the group. We propose that such a space-time can be constructed, given the action is free and regular, i.e., that the orbits of the group are three-dimensional submanifolds of the space-time. Moreover, we propose that the transversal vector field (1) commutes with the Bianchi group generators, and (2) is tangent to a geodesic at any point of the homogeneous hypersurface. Consequently, such a space-time may indeed be a solution of the Einstein equations.

We specify this even further, by specifying the matter fields that act as the source of the Einstein equations. Initially, we prove that vacuum solutions of this set-up exist. Second, we assume free scalar fields as the source, thus pairing the Einstein equations with the Klein-Gordon one (the Einstein-scalar field system); once again, this system is also integrable under the condition that the scalar fields propagate along the orbits of the group (that is, they inherit the homotheties of the group). Following, we assume free electromagnetic fields as the source of the Einstein equations, which are now paired by the Maxwell equations (the Einstein-Maxwell system); such a system is also integrable under the condition that the electromagnetic fields inherit the homotheties of the space-time (i.e., that the electromagnetic waves propagate along the orbits of the group). Interestingly, in both cases, these conditions can be proved. Finally, we consider the case of perfect fluids; in this case, the Einstein equations are combined with the Euler equations of fluid dynamics (the Einstein-Euler system) and are similarly integrable; the last case is particularly interesting, since, apart from the usual condition that the fluid inherits the symmetries of the space-time, it poses restrictions in the equation of state of the fluid.

All three cases are followed by realistic examples, some of which can be found in the literature. Interestingly, the particular space-times (where the Bianchi groups act freely and regularly by homotheties) reveal certain peculiarities that are not present in the usually considered (spatially or space-time) homogeneous space-times. Moreover, we can prove that solutions with such peculiarities are not unique; given initial conditions sufficiently close to them, other similar solutions (with the same peculiarities) can be found.