Complex recurrent space-times

Abstract

It is shown that the complex recurrent space-times are of Petrov type D or N. The spaces of type D are non-empty and decomposable into the product of two 2-dimensional spaces of arbitrary curvature, the recurrence vector being necessarily real. It is established that the problem of determining the type N complex recurrent spaces is equivalent to that of determining the spaces of type N admitting a recurrent null vector field. A coordinate system is given in which the metric of such spaces is determined up to three arbitrary functions and a constant. The spaces fall into four invariant classes according to the permitted algebraic form of the Ricci tensor. The Einstein-Maxwell equations are solved, the solutions obtained providing examples for three of the classes. It seems that the remaining class cannot contain any physically reasonable solutions of Einstein’s equations. The general metric of the complex recurrent space of Petrov type N with recurrence vector proportional to the principal null vector of the Weyl tensor is also given and some of its properties are discussed. The space of plane gravitational waves admitting a singular electromagnetic field is the only physically significant space-time of this class. The conformally recurrent spaces and the conformal symmetric spaces are also determined.