Further applications of harmonic mappings of Riemannian manifolds to gravitational fields

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Abstract
We consider static, spherically‐symmetric and stationary, axially‐symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton–Jacobi techniques, we obtain the geodesics and construct the resulting space–time geometries. For static space–times with spherical symmetry we obtain the well‐known solutions of Schwarzschild, Reissner–Nordström, and Janis–Newman–Winicour, where in the latter two cases the source for the geometry is the static, source‐free Maxwell field and the massless scalar field respectively. When we consider the source to be a triad of massless scalar fields with the internal symmetry SU(2) × SU(2), we find that the space–time geometry is once again that of Janis–Newman–Winicour. Finally we formulate the stationary, axisymmetric gravitational field problem in terms of composite mappings and obtain the Weyl, Papapetrou, and the generalized Lewis and van Stockum classes of axisymmetric solutions.

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