General approach to the study of vacuum spacetimes with an isometry

In vacuum spacetimes the exterior derivative of a Killing vector field is a 2-form (called here the Papapetrou field) that satisfies Maxwell's equations without electromagnetic sources. In this paper, using the algebraic structure of the Papapetrou field, we will set up a new formalism for the study of vacuum spacetimes with an isometry, which is suitable to investigate the connections between the isometry and the Petrov type of the spacetime. This approach has some advantages, among which are that it leads to a new classification of these spacetimes and the integrability conditions provide expressions that determine the Weyl curvature completely. These facts make the formalism useful for application to any problem or situation with an isometry and requiring the knowledge of the curvature.