Singular general relativity. Colombeau approach

This book is an exposition of "Singular Semi-Riemannian Geometry" - the study of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary signature. This book also dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry. The regularizations of singularities presented in some Colombeau solutions of the Einstein equations are an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions and not classically. In this book essentially new class Colombeau solutions to Einstein field equations is obtained. We leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. Using nonlinear distributional geometry and Colombeau generalized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates.