Stationary Axially Symmetric Generalizations of the Weyl Solutions in General Relativity

Abstract

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