8 - Integrability and divergence identities

Summary

The strong rigidity theorem implies that stationary black hole spacetimes are either axisymmetric or have a nonrotating horizon. The uniqueness theorems are, however, based on stronger assumptions: In the nonrotating case, staticity is required whereas the uniqueness of the Kerr–Newman family is established for circular spacetimes. The first purpose of this chapter is therefore to discuss the circumstances under which the integrability conditions can be established.

Our second aim is to present a systematic approach to divergence identities for spacetimes with one Killing field. In particular, we consider the stationary Einstein–Maxwell equations and derive a mass formula for nonrotating - not necessarily static - electrovac black hole spacetimes.

The chapter is organized as follows: In the first section we recall that the two Killing fields in a stationary and axisymmetric domain fulfil the integrability conditions if the Ricci–circularity conditions hold. As an application, we establish the circularity theorem for electrovac spacetimes.

The second section is devoted to the staticity theorem. As mentioned earlier, the staticity issue is considerably more involved than the circularity problem. The original proof of the staticity theorem for black hole spacetimes applied to the vacuum case (Hawking and Ellis 1973). Here we present a different proof which establishes the equivalence of staticity and Ricci–staticity for a strictly stationary domain. Since our reasoning involves no potentials, it is valid under less restrictive topological conditions. We conclude this section with some comments on the electrovac staticity theorem, which is still subject to investigations.