arXiv subjects

Roland Steinbauer

Publications and source records attributed to Roland Steinbauer.

63 records · Page 4Linked to original sources

arXiv2001

Foundations of a nonlinear distributional geometry

Co lombeau's 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.

math.FA↗

arXiv2001

Diffeomorphism invariant Colombeau Algebras. Part I: Local theory

This contribution is the first in a series of three: it reports on the construction of (a fine sheaf of) diffeomorphism invariant Colombeau algebras on open sets of Eucildean space, which completes earlier approaches. Part II and III will show, among others, the way to an intrinsic definition of Colombeau algebras on manifolds which, locally, reproduces the local diffeomorphism invariant theory.

math.FA↗

arXiv2001

Nonlinear distributional geometry and general relativity

This work reports on the construction of a nonlinear distributional geometry (in the sense of Colombeau's special setting) and its applications to general relativity with a special focus on the distributional description of impulsive gravitational waves.

math-ph↗

arXiv1999

On the foundations of nonlinear generalized functions I

We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given.

math.FA↗

arXiv1999

A global theory of algebras of generalized functions

We present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of $\hat{\mathcal G}(M)$. $\hat{\mathcal G}(M)$ is a{\em differential} algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of ${\mathcal D}'(M)$ into $\hat{\mathcal G}(M)$ that renders ${\mathcal C}^\infty (M)$ a faithful subalgebra of $\hat{\mathcal G}(M)$. Finally, it is shown that this embedding commutes with Lie derivatives. Thus $\hat{\mathcal G}(M)$ retains all the distinguishing properties of the local theory in a global context.

math.FA↗

arXiv1999

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 $(θ\de$, $θ^2\de$, $\de^2$). 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.

gr-qc↗

arXiv1999

A note on the Penrose junction conditions

Impulsive pp-waves are commonly described either by a distributional spacetime metric or, alternatively, by a continuous one. The transformation $T$ relating these forms clearly has to be discontinuous, which causes two basic problems: First, it changes the manifold structure and second, the pullback of the distributional form of the metric under $T$ is not well defined within classical distribution theory. Nevertheless, from a physical point of view both pictures are equivalent. In this work, after calculating $T$ als well as the ''Rosen''-form of the metric in the general case of a pp-wave with arbitrary wave profile we give a precise meaning to the term ``physically equivalent'' by interpreting $T$ as the distributional limit of a suitably regularized sequence of diffeomorphisms. Moreover, it is shown that $T$ provides an example of a generalized coordinate transformation in the sense of Colombeau's generalized functions.

gr-qc↗

arXiv1998

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 $\de$-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 $\de$-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 mathematical rigorous fashion. We also treat the Jacobi equation which, despite being linear in the deviation vector field, involves even more delicate singular expressions, like the ``square'' of the Dirac $\de$-distribution. Again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization-independent distributional limit. Hence it is concluded that the geometry of impulsive pp-waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill-defined products.

gr-qc↗

arXiv1997

The ultrarelativistic Reissner-Nordstrom field in the Colombeau algebra

The electromagnetic field of the ultrarelativistic Reissner-Nordstrom solution shows the physically highly unsatisfactory property of a vanishing field tensor but a nonzero, i.e., delta-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 aviod such situations when dealing with distributional valued field tensors.

gr-qc↗