Regularity of axisymmetric space-times in general relativity

Wilson, Jonathan Peter (1997) Regularity of axisymmetric space-times in general relativity. University of Southampton, Doctoral Thesis.

Abstract

Concepts of regularity are proposed, in the senses of differentiability of the metric, limiting behaviour of parallel propagation and to what extent the curvature may be interpreted as a distribution. These concepts are investigated in the context of axisymmetric space-times and their relationships with elementary flatness are established.

Certain classes of stationary cylindrically symmetric dust solution are analysed for these types of regularity and for the behaviour of the curvature near the axis, enabling the type of singularity as defined by parallel propagation to be determined. It is shown that regularity and singularity type are concepts independent of each other.

It is shown how one can assign a distributional curvature to a space-time, whose curvature has a support of codimension two, which is the case for the conical singularity representing a thin cosmic string in Minkowski space-time, by using Colombeau's theory of generalised functions to overcome the ambiguity resulting from the multiplication of distributions. It is shown that the energy-momentum tensor density takes a delta function form, and hence a concept of mass per unit length may be defined. These calculations are shown to be invariant under C^∞ coordinate transformations in the (x, y) 2-surface.

Generalised function techniques are subsequently applied to determine the distributional curvature of space-times representing thin cosmic strings on curved backgrounds, for which similar results are obtained. In particular the mass per unit length of a radiating cosmic string is calculated and it is shown to be equivalent to the mass at null infinity, thus giving an insight into global energy conservation.

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