Proof of a Geroch conjecture

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Abstract
We prove that any given stationary axisymmetric vacuum space-time (SAV) can be generated from Minkowski space by at least one Kinnersley–Chitre transformation, i.e., by at least one member of the Geroch group K, provided that the metric tensor and the Killing vectors are C3 in a domain which covers at least one point of the axis at which one of the Killing vectors characterizing the space-time is timelike. We find that the set of all Kinnersley–Chitre transformations which map any given SAV into another given SAV is uniquely determined by the initial and final values of the Ernst potential on the axis. An explicit formula for these K-C transformations in terms of the initial and final axis values is given; this formula generalizes an analogous one which Xanthopoulos found for the asymptotically flat SAV’s.