Sectional curvature and plane waves in general relativity
This thesis considers several problems in general relativity which involve the sectional curvature function σ. The causal structure defined by Lorentzian metrics on space-times makes the behaviour of σ more interesting in general relativity than in classical (positive-definite) geometry and comparisons between results in classical and Lorentzian geometry are made which illustrate this point. The thesis as a whole emphasises the geometrical rather than the physical aspects of the theory. There are three principal areas of study contained in this thesis: A theorem by J.L. Synge which relates sectional and Gaussian curvature along geodesics in classical geometry is introduced and generalised to Lorentzian geometry in a straight-forward manner. A result due to J. Beem and P. Parker concerning the existence of non-destructive null directions (along which gravitational tidal accelerations are bounded), in vacuum space-times is extended to arbitrary space-times in an elegant way. It is known that if a sectional curvature function σ is specified on a manifold then it is possible for two distinct, conformally-related generalised plane wave metrics to both give rise to σ. To investigate those symmetries of these space-times that might preserve σ, the concept of a sectional curvature preserving vector field on these plane waves is introduced and it is shown that these vector fields form a subalgebra of the conformal Lie algebra. A basis of this conformal algebra is then computed and used to establish conditions under which non-trivial sectional curvature preserving vector fields may exist. Other subsidiary results are also obtained as a consequence of these investigations.