### 2.1 Rigidity, staticity and circularity

At the basis of the classification of stationary electrovac black hole space-times lies Hawking’s strong
rigidity theorem (SRT) [84].
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,
the first part of the SRT asserts that the event horizon of a stationary black hole spacetime is a Killing
horizon.
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.

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.
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.