Journal
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 354, Issue 10, Pages 4179-4199Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/S0002-9947-02-03058-1
Keywords
algebras of generalized functions; Colombeau algebras; generalized tensor fields; generalized metric; (generalized) pseudo-Riemannian geometry; general relativity
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Generalized tenser 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 (pspudo-) 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.
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