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 $d{s}^2=fd{u}^2-2mdudv-ld{v}^2-e^{2\gamma }(d{x}^2+d{z}^2)$ (where $f$, $l$, $m$, and $\gamma$ are functions of $x$ and $z$, and $fl+m^2=x^2$) be of type III or $N$ is that $x^{-1\mathrm{}}f$, $x^{-1\mathrm{}}l$, and $x^{-1\mathrm{}}m$ be functions of a single function $\mu$; 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 $\mu$ depends only upon $x$.