Mathematics – Logic
Scientific paper
Jun 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006phdt.........7h&link_type=abstract
Proquest Dissertations And Theses 2006. Section 0328, Part 0405 293 pages; [Ph.D. dissertation].Canada: Dalhousie University (C
Mathematics
Logic
11
Dynamical Systems, Spherically Symmetric, Cosmological
Scientific paper
In this thesis we present a study of the timelike self-similar spherically symmetric cosmological models with two scalar fields with exponential potentials. We first define precisely the timelike self-similar spherically symmetric (TSS) spacetimes. We write the TSS metric in a conformally isometric form in a coordinate system adapted to the geometry of the spacetime manifold. In this coordinate system, both the metric functions of the TSS spacetimes and the potential functions of the scalar fields can be simplified to four undetermined functions of a single coordinate. As a result, the Einstein field equations reduce to an autonomous system of first-order ODEs and polynomial constraints in terms of these undetermined functions. By introducing new bounded variables as well as a new independent variable and solving the constraints, we are able to apply the theory of dynamical systems to study the properties of the TSS solutions. By finding invariant sets and associated monotonic functions, by applying the LaSalle Invariance Principle and the Monotonicity Principle, by applying the [straight phi] t -connected property of a limit set, and using other theorems, we prove that all of the TSS trajectories are heteroclinic trajectories. In addition, we conduct numerical simulations to confirm and support the qualitative analysis. We obtain all possible types of TSS solutions, by analyzing the qualitative behavior of the original system of ODES from those of the reduced one. We obtain asymptotic expressions for the TSS solutions (e.g., the asymptotic expressions for the metric functions, the source functions and the Ricci scalar). In particular, self-similar flat Friedmann-Robertson-Walker spacetimes are examined in order to obtain insights into the issues related to the null surface in general TSS spacetimes in these coordinates. A discussion of the divergence of the spacetime Ricci scalar and the possible extension of the TSS solutions across the null boundary is presented. Our results provide a mathematical setting for further physical study.
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