A spacetime characterization of the Kerr metric

We obtain a characterization of the Kerr metric among stationary, asymptoti cally flat, vacuum spacetimes, which extends the characterization in terms of the Simon tensor (defined only in the manifold of trajectories) to the w hole spacetime. More precisely, we define a three index tensor on any space time with a Killing field, which vanishes identically for Kerr and which co incides in the strictly stationary region with the Simon tensor when projec ted down into the manifold of trajectories. We prove that a stationary asym ptotically flat vacuum spacetime with vanishing spacetime Simon tensor is l ocally isometric to Kerr. A geometrical interpretation of this characteriza tion in terms of the Weyl tensor is also given. Namely, a stationary, asymp totically flat vacuum spacetime such that each principal null direction of the Killing form is a repeated principal null direction of the Weyl tensor is locally isometric to Kerr.