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Distributional Methods in General Relativity

Roland Steinbauer
Ph.D Thesis, University of Vienna, 2000

ABSTRACT: The aim of this work is a detailed study of applicability and applications of distributional concepts and methods---with a special focus on the theory of algebras of generalized functions---in the theory of general relativity. Idealizations play a crucial role in modelling physical phenomena: In many cases, they are indispensable for making the latter accessible to a theoretical treatment. As typical examples, think of point particles and point charges. On describing these idealizations mathematically one is naturally led to L. Schwartz' theory of distributions. Unfortunately this theory is only linear, a fact that seriously limitates its range of applicability in nonlinear physical theories. In the present work, after reviewing the theory of distribution valued sections in vector bundles (chapter 1), we investigate its usefulness in the inherently nonlinear theory of general relativity. Following Geroch and Traschen, in chapter 2 we draw the conclusion that a mathematically rigorous and physically sensible framework based upon linear distribution theory excludes the description of such interesting spacetimes as cosmic strings and impulsive gravitational waves. At this stage the theory of algebras of generalized functions as developed by J. F. Colombeau throughout the 1980s enters the field. In this approach one constructs associative and commutative differential algebras canonically containing the vector space of distributions as a subspace and the algebra of smooth functions as a faithful subalgebra. Hence, according to L. Schwartz' so-called ``impossibility result,'' it combines all favorable differential algebraic properties with a maximum of consistency properties with respect to classical operations. Apart from being a valuable tool in the analysis of nonlinear partial differential equations involving singular data or coefficients, the usefulness of algebras of generalized functions for geometric applications in the beginning was seriously restricted due to its lack of diffeomorphism invariance; a flaw that has ultimately been removed only recently. In this work we introduce algebras of generalized functions in chapter3 and devote the entire chapter 4 to the construction of generalized sections in vector bundles. In particular, we construct a generalized curvature framework well suited to the needs of general relativity. The final chapter 5 provides a detailed distributional description of the geometry of impulsive gravitational waves. We treat the geodesic as well as the geodesic deviation equation for this class of singular spacetimes in the previously developed generalized setting. Moreover, we carry out a detailed mathematical analysis of the discontinuous change of coordinates frequently applied to the impulsive wave metric in physical literature. We conclude this work with an outlook to promising lines of further research.

Mathematics Subject Classification 2000: 46F30, 83C35, 83C15, 46F10, 46F05, 83C75
1999 Physics and Astronomy Classification Scheme: 04.20.Cv, 02.30.Sa, 04.20.Jb, 04.30.-w 02.30.Hq

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