Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups

When the maximal isometry group of a four-dimensional spacetime acts simply transitively, such a Ricci-flat metric is uniquely determined to be the Petrov solution. This isometry group is almost abelian; that is, its Lie algebra contains an abelian ideal of codimension one. In this paper, we study Lorentzian left-invariant metrics on almost abelian Lie groups of dimension four or higher. In particular, we construct a Ricci-flat but non-flat metric that generalizes the Petrov solution to arbitrarily high dimensions. The generalized solution is geodesically complete and admits closed timelike curves.