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For stationary, asymptotically flat solutions of Einstein's equations, covariant functionals of the metric variables are defined which characterize the Kerr metric uniquely. For instance, we obtain a generalization of the “Bach tensor” to stationary metrics, which vanishes if and only if the solution is Kerr. We also give a new interpretation of the “Schwarzschild-to-Kerr-transformation.” Our results might be applicable to simplify the proof of the uniqueness theorem for stationary black holes.

We study conformal transformations in four-dimensional manifolds. In particular, we present a new set of two necessary and sufficient conditions for a space to be conformal to an Einstein space. The first condition defines the class of spaces conformal to C spaces, whereas the last one (the vanishing of the Bach tensor) gives the particular subclass of C spaces which are conformally related to Einstein spaces. © 1985 Plenum Publishing Corporation.