Abstract:
We conduct three investigations in Relativistic Cosmology that is the Einstein Field Equations applied to the largest scales with source field typically taken to be a perfect fluid and fundamental observers comoving with the preferred fluid four-velocity. We show using a tetrad analysis of the evolution equations for the dynamical variables and all the constraints these satisfy in classical General Relativity, that there are no new consistent perfect fluid cosmologies with the kinematic variables and the electric and/or magnetic parts of the Weyl curvature all rotationally symmetric about a common axis in an open neighbourhood Ա of an event. The consistent solutions of this kind are either locally rotationally symmetric, or most generally are subcases of the Szekeres model-an inhomogeneous dust model with no Killing symmetries. This result and its obvious future generalisations provides an input into the equivalence problem in cosmology necessary for a mathematically consistent understanding of probability and a measure set for universes required in quantum cosmology, for instance. We investigate such generalisations and find that similar results hold under some further assumptions dependent on the level of generalisation. In particular, we examine situations where either the electric part or the magnetic part of the free gravitational field are not rotationally symmetric, and also make a brief comment on the most general case where only the shear is rotationally symmetric. We use a tetrad analysis to show that the well-known result that holds for relativistic shear-free dust cosmologies in Einstein's classical theory either the expansion vanishes or the flow is irrotational - has an analogue in the Kaluza-Klein universe model, which has its roots presumably in string theory (or M-theory), recently proposed by Randall and Sundrum. The Big Bang singularity of General Relativity can not be avoided in these so-called brane universes in the situation where we neglect non-local tidal effects on the dynamics by allowing the vorticity to spin up as the singularity is approached in shear-free cases. Moreover, we show that in the general case of a shearing perfect fluid, the singularity at the start of the universe is approached even more strongly than in classical General Relativity in the case of no tidal interaction. Finally, we reconsider the issue of proving large scale spatial homogeneity of the universe in classical General Relativity, given isotropic observations about us and the possibility of source evolution both in numbers and luminosities. We use a spherically symmetric dust universe model (compatible with observations) for our investigation and we solve the field equations on the null cone analytically for the first time. Two theorems make precise the freedom available in constructing cosmological models that will fit the observations. They make quite clear that homogeneity cannot be proven without either a fully determinate theory of source evolution, or availability of distance measures that are independent of source evolution. We contrast this goal with the standard approach that assumes spatial homogeneity a priori, and determines source evolution functions on the basis of this assumption.
Reference:
Mustapha, N. 2000. Dynamical studies in relativistic cosmology. University of Cape Town.
Bibliography: leaves 158-167.