2.1 Rigidity, staticity and circularity

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Figure 1: Classification of stationary electrovac black hole space-times

It relates the global concept of the event horizon to the independently defined – and logically distinct – local notion of the Killing horizon: Requiring that the fundamental matter fields obey well behaved hyperbolic equations, and that the stress-energy tensor satisfies the weak energy condition,⁷ the first part of the SRT asserts that the event horizon of a stationary black hole spacetime is a Killing horizon.⁸ The latter is called non-rotating if it is generated by the stationary Killing field, and rotating otherwise. In the rotating case, the second part of the SRT implies that spacetime is axisymmetric.⁹

The subdivision provided by the SRT is, unfortunately, not sufficient to apply the uniqueness theorems for the Reissner–Nordström and the Kerr–Newman metric: The latter are based on the stronger requirements that the domain of outer communication (DOC) is either static (non-rotating case) or circular (axisymmetric case). Hence, in both cases one has to establish the Frobenius integrability conditions for the Killing fields beforehand (staticity and circularity theorems).

The circularity theorem, due to Carter [27], and Kundt and Trümper [118], implies that the metric of a vacuum or electrovac spacetime can, without loss of generality, be written in the well-known Papapetrou (2+2)-split. The staticity theorem, implying that the stationary Killing field of a non-rotating, electrovac black hole spacetime is hyper-surface orthogonal, is more involved than the circularity problem: First, one has to establish strict stationarity, that is, one needs to exclude ergo-regions. This problem, first discussed by Hájíček [78, 79], and Hawking and Ellis [84], was solved only recently by Sudarsky and Wald [167, 168], assuming a foliation by maximal slices.¹⁰ If ergo-regions are excluded, it still remains to prove that the stationary Killing field satisfies the Frobenius integrability condition. In the vacuum case, this was achieved by Hawking [82], who was able to extend a theorem due to Lichnerowicz [126] to black hole space-times. In the presence of Maxwell fields the problem was solved only a couple of years ago [167, 168], by means of a generalized version of the first law of black hole physics.