Studies in Lorentzian geometry and mathematical relativity

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)

First Committee Member

Gregory J. Galloway, Committee Chair


Two separate groups of results are considered. First, the concept of causal completeness first defined by Galloway is introduced. This concept, which generalizes compactness for spacelike hypersurfaces, is used to prove two results involving global hyperbolicity. First, that a causally complete spacelike hypersurface in a globally hyperbolic spacetime must be Cauchy. Second, that any chronological spacetime foliated by causally complete spacelike hypersurfaces must be globally hyperbolic. Note that both of these results are generalizations of previously known results that involved compact spacelike hypersurfaces.In a second set of results, the Lorentzian Busemann function is examined for timelike geodesically complete spacetimes. A condition is given and under this condition, continuity of the Busemann function is proven. Furthermore, the techniques used allow to prove results on the existence of maximizers, the behavior of the time-like cut locus, and the regularity and mean convexity of the Busemann level sets. Finally, these results are used to give a new (and simpler) proof of the Lorentzian Splitting Theorem, and to prove a result on the rigidity of the Hawking-Penrose Singularity theorems.



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