Integration in the GHP Formalism IV: A New Lie Derivative Operator Leading to an Efficient Treatment of Killing Vectors

Abstract

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordinates. The key result is that a vector field ξ is a Killing vector field if and only if there exists a tetrad which is Lie derived with respect to ξ; this result is translated into the GHP formalism using a new generalised Lie derivative operator Ł _ξ with respect to a vector field ξ. We identify a class of it intrinsic GHP tetrads, which belongs to the class of GHP tetrads which is generalised Lie derived by this new generalised Lie derivative operator Ł _ξ in the presence of a Killing vector field ξ. This new operator Ł _ξ also has the important property that, with respect to an intrinsic GHP tetrad, it commutes with the usual GHP operators if and only if ξ is a Killing vector field. Practically, this means, for any spacetime obtained by integration in the GHP formalism using an intrinsic GHP tetrad, that the Killing vector properties can be deduced from the tetrad or metric using the Lie-GHP commutator equations, without a detailed additional analysis. Killing vectors are found in this manner for a number of special spaces.