Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups

When the identity component of the full isometry group of a four-dimensional spacetime acts simply transitively, the unique Ricci-flat metric is 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. The construction of the closed timelike curves is new even in the four-dimensional Petrov solution, as it requires no identification of coordinates.