The generalized Taub-NUT congruence in Minkowski space
As it is known, the vacuum Einstein field equations are essentially simplified by the existence of a null shear-free geodesic congruence. General solutions have been obtained for the special case defined by the vanishing of the Cartan relative invariant of lowest order. The optical geometry of these spaces is isomorphic to a 3-dimensional Cauchy-Riemann geometry and the deviation form of the congruence defines a special direction by picking out an integrable distribution of complex two-spaces in the considered Cauchy-Riemann geometry. In principle, the major goal of the present paper is the formulation of the rigorous method of lifting the Cauchy-Riemann spaces, characterized by the vanishing of the lowest order relative invariant, to Ricci flat space-times of particular algebraic types. The flat metrics built around the one-forms associated with the Robinson congruence and Taub-NUT solutions are derived by using the theorem of the `Minkowski metric construction' from a given null shear-free geodesic one-form. The resulting one-parameter family of solutions combined with the deviation one-form, with prescribed choice of the direction, provide the lifting to Minkowski space of all the Cauchy-Riemann spaces which are locally equivalent to the hypersphere. Thus, considering the special complex two- spaces as being associated with either the Taub-NUT geometry or with the Hauser type \(N\) geometry, the form of the most general solution is obtained.
scientific article; zbMATH DE number 205367