Stationary Axially Symmetric Generalizations of the Weyl Solutions in General Relativity

It is shown that a necessary condition that normal-hyperbolic solutions of the Einstein vacuum field equations for the metric tensor defined by the quadratic differential form ds²=fdu²-2mdudv-ldv²-e^2γ(dx²+dz²) (where f, l, m, and γ are functions of x and z, and fl+m²=x²) be of type III or N is that x^-1f, x^-1l, and x^-1m be functions of a single function μ; it is further shown that no such nonflat solutions exist. Solutions having this functional dependence are found to belong to one of three classes: the Weyl class and two classes which may be obtained from it. One of these classes is characterized by Sachs-Penrose type-I stationary solutions having one real and two distinct complex-conjugate eigenvalues. The other class is characterized by Sachs-Penrose type-II stationary solutions admitting a single shear-, twist-, and expansion-free doubly degenerate geodesic ray which is also a null, hypersurface-orthogonal Killing vector. Further invariant properties of these classes are discussed, as well as the special case where μ depends only upon x.