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Distributional Methods in General Relativity
Roland SteinbauerPh.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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