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Consequences of a Killing symmetry in spacetime's local structure

In this paper we discuss the consequences of a Killing symmetry on the local geometrical structure of four-dimensional spacetimes. We have adopted the point of view introduced in recent works where the exterior derivative of the Killing plays a fundamental role. Then, we study some issues related to this approach and clarify why in many circumstances its use has advantages with respect to other approaches. We also extend the formalism developed in the case of vacuum spacetimes to the general case of an arbitrary energy–momentum content. Finally, we illustrate our framework with the case of spacetimes with a gravitating electromagnetic field.