81 research outputs found
An algebraic approach to manifold-valued generalized functions
We discuss the nature of structure-preserving maps of varies function
algebras. In particular, we identify isomorphisms between special Colombeau
algebras on manifolds with invertible manifold-valued generalized functions in
the case of smooth parametrization. As a consequence, and to underline the
consistency and validity of this approach, we see that this generalized version
on algebra isomorphisms in turn implies the classical result on algebras of
smooth functions.Comment: 7 page
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
(\theta\de, \theta^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.Comment: RevTeX, 9 pages, final version (minor corrections, references added
Isomorphisms of algebras of Colombeau generalized functions
We show that for smooth manifolds X and Y, any isomorphism between the
special algebra of Colombeau generalized functions on X, resp. Y is given by
composition with a unique Colombeau generalized function from Y to X. We also
identify the multiplicative linear functionals from the special algebra of
Colombeau generalized functions on X to the ring of Colombeau generalized
numbers. Up to multiplication with an idempotent generalized number, they are
given by an evaluation map at a compactly supported generalized point on X.Comment: 10 page
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.Comment: Latex, 19 pages, 1 figure. Discussion of class of metrics covered by
our results and some examples added. Conclusion more detailed. Version to
appear in Communications in Mathematical Physic
Generalized solutions and distributional shadows for Dirac equations
We discuss the application of recent results on generalized solutions to the
Cauchy problem for hyperbolic systems to Dirac equations with external fields.
In further analysis we focus on the question of existence of associated
distributional limits and derive their explicit form in case of free Dirac
fields with regularizations of initial values corresponding to point-like
probability densities
A regularisation approach to causality theory for C^{1,1}Lorentzian metrics
We show that many standard results of Lorentzian causality theory remain valid if the regularity of the metric is reduced to C^{1,1}. Our approach is based on regularisations of the metric adapted to the causal structure
Singular reduction modules of differential equations
The notion of singular reduction modules, i.e., of singular modules of
nonclassical (conditional) symmetry, of differential equations is introduced.
It is shown that the derivation of nonclassical symmetries for differential
equations can be improved by an in-depth prior study of the associated singular
modules of vector fields. The form of differential functions and differential
equations possessing parameterized families of singular modules is described up
to point transformations. Singular cases of finding reduction modules are
related to lowering the order of the corresponding reduced equations. As
examples, singular reduction modules of evolution equations and second-order
quasi-linear equations are studied. Reductions of differential equations to
algebraic equations and to first-order ordinary differential equations are
considered in detail within the framework proposed and are related to previous
no-go results on nonclassical symmetries.Comment: 38 pages, advanced version. Extension of results of arXiv:0808.3577
to the case of a greater number of independent variable
Aichelburg-Sexl boost of an isolated source in general relativity
A study of the Aichelburg--Sexl boost of the Schwarzschild field is described
in which the emphasis is placed on the field (curvature tensor) with the metric
playing a secondary role. This is motivated by a description of the Coulomb
field of a charged particle viewed by an observer whose speed relative to the
charge approaches the speed of light. Our approach is exemplified by carrying
out an Aichelburg-- Sexl type boost on the Weyl vacuum gravitational field due
to an isolated axially symmetric source. Detailed calculations of the boosts
transverse and parallel to the symmetry axis are given and the results, which
differ significantly, are discussed.Comment: 25 pages, LateX2
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