### 3.3 The no-hair theorem

#### 3.3.1 The rigidity theorem

Throughout this section we will assume that the spacetime is -regular, as made precise by Definition 2.1.

To prove uniqueness of connected, analytic, non-degenerate configurations, it remains to show that every such black hole is either static or axially symmetric. The first step for this is provided by Hawking’s strong rigidity theorem (SRT) [143, 238, 60, 107], which relates the global concept of the event horizon to the independently-defined, and logically-distinct, local notion of the Killing horizon. Assuming analyticity, SRT asserts that the event horizon of a stationary black-hole spacetime is a Killing horizon. (In this terminology [151], the weak rigidity theorem is the existence, already discussed above, of prehorizons for static or stationary and axisymmetric configurations.)

A Killing horizon is called non-rotating if it is generated by the stationary Killing field, and rotating otherwise. At this stage the argument branches-off, according to whether at least one of the horizons is rotating, or not.

In the rotating case, Hawking’s theorem actually provides only a second Killing vector field defined near the Killing horizon, and to continue one needs to globalize the Killing vector field, to prove that its orbits are complete, and to show that there exists a linear combination of Killing vector fields with periodic orbits and an axis of rotation. This is done in [60], assuming analyticity, drawing heavily on the results in [253, 57, 18].

The existing attempts in the literature to construct a second Killing vector field without assuming analyticity have only had limited success. One knows now how to construct a second Killing vector in a neighborhood of non-degenerate horizons for electrovacuum black holes [2, 174, 327], but the construction of a second Killing vector throughout the d.o.c. has only been carried out for vacuum near-Kerr non-degenerate configurations so far [3] (compare [326]).

In any case, sufficiently regular analytic stationary electro-vacuum spacetimes containing a rotating component of the event horizon are axially symmetric as well, regardless of degeneracy and connectedness assumptions (for more on this subject see Section 3.4.2). One can then finish the uniqueness proof using Theorem 3.2. Note that the last theorem requires neither analyticity nor connectedness, but leaves open the question of the existence of naked singularities in non-connected candidate solutions.

In the non-rotating case, one continues by showing [84] that the domain of outer communications contains a maximal Cauchy surface. This has been proven so far only for non-degenerate horizons, and this is the only missing step to include situations with degenerate components of the horizon. This allows one to prove the staticity theorem [302, 303], that the stationary Killing field of a non-rotating, electrovacuum black-hole spacetime is hypersurface orthogonal. (Compare [134, 136, 143, 141] for previous partial results.) One can then finish the argument using Theorem 3.1.

#### 3.3.2 The uniqueness theorem

All this leads to the following precise statement, as proven in [76, 79] in vacuum and in [217, 79] in electrovacuum:

Theorem 3.3 Let be a stationary, asymptotically-flat, -regular, electrovacuum, four-dimensional analytic spacetime. If the event horizon is connected and either mean non-degenerate or rotating, then is isometric to the domain of outer communications of a Kerr–Newman spacetime.

The structure of the proof can be summarized in the flow chart of Figure 3. This is to be compared with Figure 4, which presents in more detail the weaker hypotheses needed for various parts of the argument.

The hypotheses of analyticity and non-degeneracy are highly unsatisfactory, and one believes that they are not needed for the conclusion. One also believes that, in vacuum, the hypothesis of connectedness is spurious, and that all black holes with more than one component of the event horizon are singular, but no proof is available except for some special cases [216, 319, 249]. Indeed, Theorem 3.3 should be compared with the following conjecture, it being understood that both the Minkowski and the Reissner–Nordström spacetimes are members of the Kerr–Newman family:
Conjecture 3.4 Let be an electrovacuum, four-dimensional spacetime containing a spacelike, connected, acausal hypersurface , such that is a topological manifold with boundary, consisting of the union of a compact set and of a finite number of asymptotically-flat ends. Suppose that there exists on a complete stationary Killing vector , that is globally hyperbolic, and that . Then is isometric to the domain of outer communications of a Kerr–Newman or MP spacetime.

#### 3.3.3 A uniqueness theorem for near-Kerrian smooth vacuum stationary spacetimes

The existing results on rigidity without analyticity require one to assume either staticity, or a near-Kerr condition on the spacetime geometry (see Section 3.3.1), which is quantified in terms of a smallness condition of the Mars–Simon tensor [223, 293]. The results in [3] together with Theorems 3.13.2, and a version of the Rácz–Wald Theorem [107, Proposition 4.1], lead to:

Theorem 3.5 Let be a stationary asymptotically-flat, -regular, smooth, vacuum four-dimensional spacetime. Assume that the event horizon is connected and mean non-degenerate. If the Mars–Simon tensor and the Ernst potential of the spacetime satisfy

for a small enough , then is isometric to the domain of outer communications of a Kerr spacetime.