All vacuum Bianchi I metrics with a homothety

Abstract

All vacuum metrics with a four-parameter homothety group containing a three-dimensional Abelian subgroup are found. These include the Kasner family of metrics, metrics related to the Kasner family by complex transformations (which include a family of vacuum metrics found by Lewis in 1932 and a metric found by Petrov in 1962 to be the only vacuum metric which admits a simply-transitive four-parameter isometry group), together with algebraically degenerate van Stockum, pp-wave and plane-wave metrics. Apart from the plane-wave metric, all the other metrics are written here as one general family.