On the local extension of Killing vector-fields in Ricci flat manifolds

Abstract

We revisit the extension problem for Killing vector-fields in smooth Ricci flat manifolds, and its relevance to the black hole rigidity problem. We prove both a stronger version of the main local extension result established earlier, as well as two types of results concerning non-extendibility. In particular, we show that one can find local, stationary, vacuum extensions of a Kerr solution K ( m , a ) \mathcal {K}(m,a) , 0 > a > m 0>a>m , in a future neighborhood of any point p p of the past horizon lying outside both the bifurcation sphere and the axis of symmetry, which admit no extension of the Hawking vector-field of K ( m , a ) \mathcal {K}(m,a) . This result illustrates one of the major difficulties one faces in trying to extend Hawking’s rigidity result to the more realistic setting of smooth stationary solutions of the Einstein vacuum equations; unlike in the analytic situation, one cannot hope to construct an additional symmetry of stationary solutions (as in Hawking’s Rigidity Theorem) by relying only on local information.