This thesis is about the structure of the boundary of the universe, i.e., points where the geometric structures of spacetime cannot be continued. In partic­ular, we study the structure of the b-boundary by B. Schmidt. It has been known for some time that the b-boundary construction has several drawbacks, perhaps the most severe being that it is often not Hausdorff separated from interior points in spacetime. In other words, the topology makes it impossible to distinguish which points in spacetime are near the singularity and which points are ‘far’ from it. The non-Hausdorffness of the b-completion is closely related to the concept of fibre degeneracy of the fibre in the frame bundle over a b-boundary point, the fibre being smaller than the whole structure group in a specific sense. Fibre degeneracy is to be expected for many realistic spacetimes, as was proved by C. J. S. Clarke. His proofs contain some errors however, and the purpose of paper I is to reestablish the results of Clarke, under somewhat different conditions. It is found that under some conditions on the Riemann curvature tensor, the boundary fibre must be totally degenerate (i.e., a single point). The conditions are essentially that the components of the Riemann tensor and its first derivative, expressed in a parallel frame along a curve ending at the singularity, diverge sufficiently fast. We also demonstrate the applicability of the conditions by verifying them for a number of well known spacetimes. In paper II we take a different view of the b-boundary and the b-length func­tional, and study the Riemannian geometry of the frame bundle. We calculate the curvature Rof the frame bundle, which allows us to draw two conclusions. Firstly, if some component of the curvature of spacetime diverges along a horizontal curve ending at a singularity, R must tend to — oo. Secondly, if the frame bundle is extendible through a totally degenerate boundary fibre, the spacetime must be a conformally flat Einstein space asymptotically at the corresponding b-boundary point. We also obtain some basic results on the isometries and the geodesics of the frame bundle, in relation to the corresponding structures on spacetime. The first part of paper III is concerned with imprisoned curves. In Lorentzian geometry, the situation is qualitatively different from Riemannian geometry in that there may be incomplete endless curves totally or partially imprisoned in a compact subset of spacetime. It was shown by B. Schmidt that a totally imprisoned curve must have a null geodesic cluster curve. We generalise this result to partially im­prisoned incomplete endless curves. We also show that the conditions for the fibre degeneracy theorem in paper I does not apply to imprisoned curves. The second part of paper III is concerned with the properties of the b-length functional. The b-length concept is important in general relativity because the presence of endless curves with finite b-length is usually taken as the definition of a singular spacetime. It is also closely related to the b-boundary definition. We study the structure of b-neighbourhoods, i.e., the set of points reachable from a fixed point in spacetime on (horizontal) curves with b-length less than some fixed number e > 0. This can then be used to understand how the geometry of spacetime is encoded in the frame bundle geometry, and as a tool when studying the structure of the b-boundary. We also give a result linking the b-length of a general curve in the frame bundle with the b-length of the corresponding horizontal curve. / digitalisering@umu

In this work we investigate an exact solution of Einstein's equations which is described by the Pleba'nski-Demia'nski metric. This metric represents type D space-times and contains seven free parameters, including electric and magnetic charges and a cosmological constant. We study geometrical and phy- sical properties of these space-times in the case when repeated principal null congruences have zero expansion. Therefore, first we study de Sitter universe and anti-de Sitter universe in the Pleba'nski-Demia'nski coordinates, and we care- fully analyze the corresponding parametrizations of (anti-)de Sitter hyperboloid in five-dimensional flat space-time, unknown so far, we draw the respective con- formal diagrams, and we find transformations to various known forms. After that, we investigate the more general case of the B metrics with a cosmological con- stant, and we do a basic analysis of its geometrical properties. We summarize the article by Gott from 1974, where he interprets the BI metric as a part of space-time with a tachyon singularity, and we generalize his results for the case of non-zero cosmological constant. Finally, we analyze even more general cases of the Pleba'nski-Demia'nski metric with more non-zero parameters. In particular, we study the electromagnetic field in the case of non-zero...