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

The authors consider spacetimes of general relativity admitting a preferred null direction lsup(μ) 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 lsup(μ) with respect to the group transitivity surfaces. Two theorems concerning the action of the isometry group on lsup(μ) 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 certain assumptions. By switching off the null dust, the underlying vacuum metrics are obtained which generalise some of the diverging Petrov type D vacuum metrics found by previous workers. (author)