Algebraic Foundations of Colombeau Generalized Lorentz Geometry

We introduce a concept of causality in the framework of generalized pseudo-Riemannian Geometry in the sense of Colombeau. This is motivated by a generalized point value characterization of generalized pseudoRiemannian metrics due to M. Kunzinger et al. We prove an appropriate version of the inverse Cauchy-Schwarz inequality. As an application, we establish a dominant energy condition for some energy tensors as put forward in Hawking and Ellis’s book ”The large scale structure of space-time”. Most of the statements are shown by means of a new characterization of free elements in R̃n, the n-dimensional module over the ring of generalized numbers R̃. We also show that any free submodule of R̃n admits a direct summand; however R̃n fails to be semisimple. A valuable by-product of the present work is a new characterization of invertibility and strict positivity of generalized functions.