Mathematics – Logic
Scientific paper
Sep 1987
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1987phdt........86b&link_type=abstract
Thesis (PH.D.)--UNIVERSITY OF SUSSEX (UNITED KINGDOM), 1987.Source: Dissertation Abstracts International, Volume: 49-08, Secti
Mathematics
Logic
2
Scientific paper
Available from UMI in association with The British Library. This thesis is divided into three, essentially self-contained, parts. In the first part we examine the structure of classical three-dimensional space-times. Here, we review and extend what is known about the gravitational theories in these models. We investigate the non-existence of a Newtonian limit to the relativistic theories showing that in the presence of certain matter terms, Newtonian gravity can be obtained as a suitable weak-field limit. We present a number of new, exact static and non-static solutions to the equations of three-dimensional general relativity with scalar field and perfect fluid sources. We comment on the relationship between the stiff perfect fluid and the scalar field. Motivated by the Kaluza-Klein procedure of dimensional reduction we find some exact scalar field solutions which have analogues in four-dimensions. We also present classification schemes based on the group of motions of homogeneous space-times and on the Cotton -York tensor. The description of the general cosmological solution in the vicinity of the singularity is given in terms of the number of arbitrary spatial functions independently specified on a space-like hypersurface. We also study a series approximation to the space-time in the vicinity of the cosmological singularity. Some conjectures are made concerning the space-time singularities. We present two exact cosmological solutions containing self-interacting scalar fields. The models exhibit an inflationary behaviour. We also present an anisotropic cosmological model. The second part of the thesis contains a study of certain cosmological models which have self-interacting scalar fields obeying an exponential potential. We use the techniques of phase portrait analysis to study the N-dimensional cosmological models as well as certain anisotropic models. The latter involves the analysis of a three-dimensional system of equations and we review the relevant theory in the Appendix. Most of the models exhibit power-law inflation and we examine the limits on the potential imposed by the constraints on the inflationary period. The last section of the thesis involves a review of the cosmic string picture of galaxy formation. We examine the structure of the string from the viewpoint of the fields. We conjecture that the solutions are stable against small time-dependent perturbations and minimise the energy. We extend this analysis to straight strings in expanding space -times and find that at late times the fields settle down to a steady solution. At early times the fields oscillate and the stress-tensor of the fields is non-diagonal. The oscillations are probably not important since they die away rapidly but the form of the stress-tensor may have to be taken into account when considering the back reaction of the string on the space-time shortly after the string is formed.
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