Cartan ideal, prolongation, and Bäcklund transformations for Einstein’s equations

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Abstract
Einstein’s equations in the Newman–Penrose formalism for vacuum, vacuum with cosmological constant, and electrovacuum fields are expressed as Cartan ideals. Two different prolongations of these ideals are obtained. These two types of prolonged ideals generalize previous prolongations for vacuum fields to vacuum with cosmological constant and electrovacuum fields. Some Bäcklund transformations are obtained for vacuum, vacuum with cosmological constant, and electrovacuum fields. These Bäcklund transformations include the generalized Kerr–Schild (GKS) transformation, and a two-parameter generalization of the GKS transformation. GKS transformations are studied in detail. Expressions for the transformation of Newman–Penrose quantities are given and algebraic properties are discussed. It is shown that the GKS transformation cannot give algebraically general and asymptotically flat vacuum and electrovacuum space-time metrics.

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