Roland Steinbauer

The use of generalized functions and distributions in general relativity

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Generalized pseudo-Riemannian geometry

Generalized tensor analysis in the sense of Colombeaus construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions, we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a Fundamental Lemma of (pseudo-) Riemannian geometry in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.

A Rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves

The geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions (θδ,θ2δ,δ2). A solution concept for these equations based on embedding the distributional metric into the Colombeau algebra of generalized functions is presented. Using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly. The obtained limits are regularization independent and display the physically expected behavior.

Geodesics and geodesic deviation for impulsive gravitational waves

The geometry of impulsive pp-waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric. The geodesic equation involves formally ill-defined products of distributions due to the nonlinearity of the equations and the presence of the Dirac δ-distribution in the space–time metric. Thus, strictly speaking, it cannot be treated within Schwartz’s linear theory of distributions. To cope with this difficulty we proceed by first regularizing the δ-singularity, then solving the regularized equation within classical smooth functions and, finally, obtaining a distributional limit as solution to the original problem. Furthermore, it is shown that this limit is independent of the regularization without requiring any additional condition, thereby confirming earlier results in a mathematically rigorous fashion. We also treat the Jacobi equation which, despite being linear in the deviation vector field, involves even more delicate singular expressions, li...

Foundations of a Nonlinear Distributional Geometry

Colombeaus construction of generalized functions (in its special variant) is extended to a theory of generalized sections of vector bundles. As particular cases, generalized tensor analysis and exterior algebra are studied. A point value characterization for generalized functions on manifolds is derived, several algebraic characterizations of spaces of generalized sections are established and consistency properties with respect to linear distributional geometry are derived. An application to nonsmooth mechanics indicates the additional flexibility offered by this approach compared to the purely distributional picture.

The ultrarelativistic Reissner–Nordstro/m field in the Colombeau algebra

The electromagnetic field of the ultrarelativistic Reissner–Nordstro/m solution shows the physically highly unsatisfactory property of a vanishing field tensor but a nonzero, i.e., δ-like, energy density. The aim of this work is to analyze this situation from a mathematical point of view, using the framework of Colombeau’s theory of nonlinear generalized functions. It is shown that the physically unsatisfactory situation is mathematically perfectly defined and that one cannot avoid such situations when dealing with distributional valued field tensors.

Remarks on the distributional Schwarzschild geometry

This work is devoted to a mathematical analysis of the distributional Schwarzschild geometry. The Schwarzschild solution is extended to include the singularity; the energy momentum tensor becomes a δ-distribution supported at r=0. Using generalized distributional geometry in the sense of Colombeau’s (special) construction the nonlinearities are treated in a mathematically rigorous way. Moreover, generalized function techniques are used as a tool to give a unified discussion of various approaches taken in the literature so far; in particular we comment on geometrical issues.

Intrinsic Characterization of Manifold-Valued Generalized Functions

The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (analogous to scalar-valued cases of Colombeaus construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly. Finally, a new concept of association adapted to the present setting is introduced.

The wave equation on singular space-times

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

Generalised connections and curvature

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang–Mills theory are given.