On Hawking's Local Rigidity Theorem for Charged Black Holes

We show the existence of the Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein-Maxwell space-time without assuming the underlying space-time is analytic. This extends a result of Friedrich et al. (Commun Math Phys 204:691-707, 1971), which holds in the interior of the black hole region. Moreover, we also show, in the presence of an additional Killing vector field T which is tangent to the horizon and not vanishing on the bifurcate sphere, then space-time must be locally axially symmetric without the analyticity assumption. This axial symmetry plays a fundamental role in the classification theory of stationary black holes.