Uniqueness properties of the Kerr metric

Abstract

We obtain a geometrical condition on vacuum, stationary, asymptotically flat spacetimes which is necessary and sufficient for the spacetime to be locally isometric to Kerr. Namely, we prove a theorem stating that an asymptotically flat, stationary, vacuum spacetime such that the so-called Killing form is an eigenvector of the self-dual Weyl tensor must be locally isometric to Kerr. Asymptotic flatness is a fundamental hypothesis of the theorem, as we demonstrate by writing down the family of metrics obtained when this requirement is dropped. This result indicates why the Kerr metric plays such an important role in general relativity. It may also be of interest in order to extend the uniqueness theorems of black holes to the non-connected and to the non-analytic case.

30 pages, LaTeX, submitted to Classical and Quantum Gravity

References in corpus (3)

• A spacetime characterization of the Kerr metric
• The gravitating $\sigma$ model in 2+1 dimensions: black hole solutions
• Segre decomposition of spacetimes

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