New Exact and Asymptotic Solutions of the Stationary Vacuum Einstein Equations.

This thesis consists of two distinct but mutually complementary parts. In the first part a projection tensor calculus is described for treating spacetimes with two commuting Killing vectors and its use is illustrated by two examples--an alternative proof of a sufficient condition for such a spacetime to be orthogonally transitive (Papapetrou's Theorem) and an application to the Rainich-Misner-Wheeler (RMW) geometrization of the electromagnetic field. A special null tetrad is introduced which is naturally adapted to the group orbits and the components of the Weyl tensor was examined in this basis. This framework is applied to investigate the admissible Petrov types in the special case when the group orbits admit orthogonal two-surfaces. It is proved in particular that all vacuum fields with two commuting orthogonally transitive Killing vectors (xi)(,0)('(mu)) and (xi)(,1)('(mu)) satisfying det ((xi)(,a)('(mu))(xi)(,b(mu))) (NOT=) const. are type I or type D in the Petrov-Pirani classification with the only exception of the van Stockum exterior metric (which is in general type II). Finally, the new formalism is employed to obtain a class of "noncanonical" stationary axisymmetric vacuum metrics whose Killing orbits do not admit orthogonal two-surfaces. A new metric of this category is found which is Petrov type II with expansion -, twist-, and the shear-free geodesic rays and which includes the van Stockum solution as a limiting case. In the second part of this thesis a multipole expansion scheme is introduced for a wide class of stationary, asymptotically flat, vacuum solutions of Einstein's equations using the conformal techniques of Geroch and Hansen. An intrinsic choice of the conformal factor and suitable asymptotic flatness conditions enable one to express the rescaled gravitational mass and angular momentum potentials and the rescaled spatial metric as power series in normal coordinates around a point (LAMDA) representing the spatial infinity on the conformal manifold. From a close examination of the situation in the Newtonian case, the square of the geodesic arc length from (LAMDA) emerges as the natural candidate for the conformal factor. The coefficients of the expansion representing the asymptotic solution of stationary vacuum field equations turn out to be certain nonlinear combinations of the Geroch-Hansen multipole moments. As an example the Schwarzschild metric is discussed in the present framework.