The Wave Equation on Singular Space-times

The first part of my thesis lays the foundations to generalized Lorentz geometry. The basic algebraic structure of finite-dimensional modules over the ring of generalized numbers is investigated. This includes a new characterization of invertibility in the ring of generalized numbers as well as a characterization of free elements inside the n-dimensional module e R. The index of symmetric bilinear forms is introduced; this new concept enables a (generalized) pointwise characterization of generalized pseudo Riemannian metrics on smooth manifolds as introduced by M. Kunzinger and R. Steinbauer. It is shown that free submodules have direct summands, however e R turns out not to be semisimple. Applications of these new concepts are a generalized notion of causality, the generalized inverse Cauchy Schwarz inequality for time-like or null vectors, constructions of pseudo Riemannian metrics as well as generalized energy tensors. The motivation for this part of my thesis evolved from the main topic, the wave equation on singular space-times. The second and main part of my thesis is devoted to establishing a local existence and uniqueness theorem for the wave equation on singular space-times. The singular Lorentz metric subject to our discussion is modeled within the special algebra on manifolds in the sense of J. F. Colombeau. Inspired by an approach to generalized hyperbolicity of conical-space times due to J. Vickers and J. Wilson, we succeed in establishing certain energy estimates, which by a further elaborated equivalence of energy integrals and Sobolev norms allow us to prove existence and uniqueness of local generalized solutions of the wave equation with respect to a wide class of generalized metrics. The third part of my thesis treats three different point value resp. uniqueness questions in algebras of generalized functions. The first one, posed by Michael Kunzinger, reads as follows: Is the theorem by Albeverio et al., that elements of the so-called p-adic Colombeau Egorov algebra are determined uniquely on standard points, a p-adic scenario? We answer this problem by means of a counterexample which shows that the statement in fact does not hold. We further show that elements of an Egorov algebra of generalized functions on a locally compact ultrametric space allow a point-value characterization if and only if the metric induces the discrete topology. Secondly, we prove that the ring of generalized (real or complex) numbers endowed with the sharp norm does not admit nested sequences of closed balls to have an empty intersection. As an application we outline a possible version of the Hahn-Banach Theorem as well as the ultrametric Banach fixed point theorem. Finally, we establish that scaling invariant generalized functions on the real line are constant and we prove several new characterizations of locally constant generalized functions.