[No abstract supplied with this thesis - The first page (of three) of the Introduction follows]
¶
This thesis is largely concerned with the changing representations of 'boundary' or 'ideal' points of a
pseudo-Riemannian manifold -- and our primary interest is in the space-times of general relativity. In particular, we are interested in the following question: What assumptions about the 'nature' of 'portions' of a certain 'ideal boundary' construction (essentially the 'abstract boundary' of Scott and Szekeres (1994)) allow us to define precisely the topological type of these 'portions', i.e., to show that different representations of this ideal boundary, corresponding to different embeddings of the manifold into others, have corresponding 'portions' that are homeomorphic?
¶
Certain topological properties of these 'portions' are preserved, even allowing for quite unpleasant properties of the metric (Fama and Scott 1995). These results are given in Appendix D, since they are not used elsewhere and, as well as representing the main portion of work undertaken under the supervision of Scott, which deserves recognition, may serve as an interesting example of the relative ease with which certain simple results about the abstract boundary can be obtained.
¶
An answer to a more precisely formulated version of this question appears very diffcult in general. However, we can give a rather complete answer in certain cases, where we dictate certain 'generalised regularity' requirements for our embeddings, but make no demands on the precise functional form of our metrics apart from these. For example, we get a complete answer to our question for abstract boundary sets which do not 'wiggle about' too much -- i.e., they satisfy a certain Lipschitz condition -- and through which the metric can be extended in a manner which is not required to be differentiable (C[superscript1]), but is continuous and non--degenerate. We allow similar freedoms on the interior of the manifold, thereby bringing gravitational wave space-times within our sphere of discussion. In fact, in the course of developing these results in progressively greater generality, we get, almost 'free', certain abilities to begin looking at geodesic structure on quite general pseudo-Riemannian manifolds.
¶
It is possible to delineate most of this work cleanly into two major parts. Firstly, there are results which use classical geometric constructs and can be given for the original abstract boundary construction, which requires differentiability of both manifolds and metrics, and which we summarise below. The second -- and significantly longer -- part involves extensions of those constructs and results to more general metrics.
| Identifer | oai:union.ndltd.org:ADTP/216698 |
| Date | January 1998 |
| Creators | Fama, Christopher J., - |
| Publisher | The Australian National University. School of Mathematical Sciences |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | http://www.anu.edu.au/legal/copyright/copyrit.html), Copyright Christopher J. Fama |
Page generated in 0.0019 seconds