The memory effect in impulsive plane waves: comments, corrections, clarifications

Recently the 'memory effect' has been studied in plane gravitational waves and, in particular, in impulsive plane waves. Based on an analysis of the particle motion (mainly in Baldwin-Jeffery-Rosen coordinates) a 'velocity memory effect' is claimed to be found in [P.-M. Zhang, C. Duval, and P. A. Horvathy. Memory effect for impulsive gravitational waves. Classical Quantum Gravity, 35(6):065011, 20, 2018]. Here we point out a conceptual mistake in this account and employ earlier works to explain how to correctly derive the particle motion and how to correctly deal with the notorious distributional Brinkmann form of the metric and its relation to the continuous Rosen form.Comment: Identical to the version published as "Comment on 'Memory effect for impulsive gravitational waves'" in Class. Quantum Grav., 12 pages, LaTe

On the geometry of impulsive gravitational waves

We describe impulsive gravitational pp-waves entirely in the distributional picture. Applying Colombeau's nonlinear framework of generalized functions we handle the formally ill-defined products of distributions which enter the geodesic as well as the geodesic deviation equation. Using a universal regularization procedure we explicitly derive regularization independent distributional limits. In the special case of impulsive plane waves we compare our results with the particle motion derived from the continuous form of the metric.Comment: Unchanged from v2; Comment on publication status changed to: Prepared for "Proceedings of the 8-th National Romaninan Conference on GRG, Bistritza, June 1998", which to my best konwledge never appeare

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.Comment: Latex, 11 pages, Contribution to Proceedings of the International Conference on Generalized Functions (ICGF 2000, Guadeloupe); typos correcte

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.Comment: Latex, 12 pages, Contribution to Proceedings of the International Conference on Generalized Functions (ICGF 2000, Guadeloupe

Every Lipschitz metric has C1C^1-geodesics

We prove that the geodesic equation for any semi-Riemannian metric of regularity $C^{0,1}$ possesses $C^1$-solutions in the sense of Filippov.Comment: final version, minor correction

Geodesics in nonexpanding impulsive gravitational waves with Λ\Lambda, II

We investigate all geodesics in the entire class of nonexpanding impulsive gravitational waves propagating in an (anti-)de Sitter universe using the distributional metric. We extend the regularization approach of part I, [SSLP16] to a full nonlinear distributional analysis within the geometric theory of generalized functions. We prove global existence and uniqueness of geodesics that cross the impulsive wave and hence geodesic completeness in full generality for this class of low regularity spacetimes. This, in particular, prepares the ground for a mathematically rigorous account on the 'physical equivalence' of the continuous with the distributional `from' of the metric.Comment: 21 pages, 1 figure; v2: close to final versio

Geodesic completeness of generalized space-times

We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.Comment: 8 pages, v3: minor corrections, final versio

Generalized pseudo-Riemannian geometry

Generalized tensor analysis in the sense of Colombeau's 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.Comment: 23 pages, Latex, final for

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.Comment: 29 pages, Latex, title changed, final version, to appear in Acta Appl. Mat

On geodesics in low regularity

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value problem for geodesics for locally Lipschitz continuous metrics and generalize recent results on existence, regularity and uniqueness of solutions in the sense of Filippov.Comment: 14 pages, 5 figures; v2: minor change