Spacetimes with a preferred null direction and a two-dimensional group of isometries: the null dust case

Abstract

The authors consider spacetimes of general relativity admitting a preferred null direction lmu and a two-dimensional Abelian group of isometries G2. A null tetrad formulation of the Killing equations is given, as well as a classification of G2 according to the orientation of lmu with respect to the group transitivity surfaces. Two theorems concerning the action of the isometry group on lmu are presented: the first one deals with spacetimes admitting at least one Killing vector, while the second one deals with spacetimes admitting a G2. The Einstein field equations with a shear-free and diverging null dust source are integrated under the assumptions: (i) the spacetime admits an Abelian G2 whose transitivity surfaces are non-orthogonal to and do not contain the null dust propagation vector; (ii) there exists a Killing vector kmu whose magnitude is almost everywhere bounded at the endpoints of the null dust rays. These spacetimes are of Petrov type II, non-asymptotically flat, and the G2 is non-orthogonally transitive. By switching off the null dust the authors have explicitly obtained the underlying vacuum metrics which generalise some of the diverging Petrov type D vacuum metrics found by Kinnersley (1969).