A second class of spacetimes where reasonably satisfactory statements can be made is provided by stationary-axisymmetric solutions. Here one assumes from the outset that, in addition to the stationary Killing vector, there exists a second Killing vector field. Assuming -regularity, one can invoke the positive energy theorem to show [18, 19] that some linear combination of the Killing vectors, say , must have periodic orbits, and an axis of rotation, i.e., a two-dimensional totally-geodesic submanifold of on which the Killing vector vanishes. The description of the quotient manifold is provided by the deep mathematical results concerning actions of isometry groups of [259, 273], together with the simple-connectedness and structure theorems [76]. The bottom line is that the spacetime is the product of with from which a finite number of aligned balls, each corresponding to a black hole, has been removed. Moreover, the component of the group of isometries acts by rotations of . Equivalently, the quotient space is a half-plane from which one has removed a finite number of disjoint half-discs centered on points lying on the boundary of the half-plane.

The only known -regular stationary and axisymmetric solutions of the Einstein–Maxwell equations are the Kerr–Newman metrics and the MP metrics. However, it should be kept in mind that candidate solutions for non-connected black-hole configurations exist:

First, there are the multi-soliton metrics constructed using inverse scattering methods [23, 22] (compare [268]). Closely related (and possibly identical, see [148]), are the multi-Kerr solutions constructed by successive Bäcklund transformations starting from Minkowski spacetime; a special case is provided by the Neugebauer–Kramer double-Kerr solutions [198]. These are explicit solutions, with the metric functions being rational functions of coordinates and of many parameters. It is known that some subsets of those parameters lead to metrics, which are smooth at the axis of rotation, but one suspects that those metrics will be nakedly singular away from the axis. We will return to that question in Section 3.4.3.

Next, there are the solutions constructed by Weinstein [322], obtained from an abstract existence theorem for suitable harmonic maps. The resulting metrics are smooth everywhere except perhaps at some components of the axis of rotation. It is known that some Weinstein solutions have conical singularities [319, 216, 249, 70] on the axis, but the general case remains open.

Finally, the Israel–Wilson–Perjés (IWP) metrics [267, 179], discussed in more detail in Section 7.3, provide candidates for rotating generalizations of the MP black holes. Those metrics are remarkable because they admit nontrivial Killing spinors. The Killing vector obtained from the Killing spinor is causal everywhere, so the horizons are necessarily non-rotating and degenerate. It has been shown in [80] that the only regular IWP metrics with a Killing vector timelike throughout the d.o.c. are the MP metrics. A strategy for a proof of timelikeness has been given in [80], but the details have yet to be provided. In any case, one expects that the only regular IWP metrics are the MP ones.

Some more information concerning candidate solutions with non-connected horizons can be found in Section 3.4.3.

Returning to the classification question, the analysis continues with the circularity theorem of Papapetrou [264] and Kundt and Trümper [201] (compare [43]), which asserts that, locally and away from null orbits, the metric of a vacuum or electrovacuum spacetime can be written in a 2+2 block-diagonal form.

The next key observation of Carter is that the stationary and axisymmetric EM equations can be reduced to a two-dimensional boundary value problem [45] (see Sections 6.1 and 8.2 for more details), provided that the area density of the orbits of the isometry group can be used as a global spacelike coordinate on the quotient manifold. (See also [47] and [50].) Prehorizons intersecting the d.o.c. provide one of the obstructions to this, and a heavy-duty proof that such prehorizons do not arise was given in [76]; a simpler argument has been provided in [72].

In essence, Carter’s reduction proceeds through a global manifestly–conformally-flat (“isothermal”) coordinate system on the quotient manifold. One also needs to carefully monitor the boundary conditions satisfied by the fields of interest. The proof of existence of the coordinates, with sufficient control of the boundary conditions so that the uniqueness proof goes through, has been given in [76], drawing heavily on [64], assuming that all horizons are non-degenerate. A streamlined argument has been presented in [79], where the analysis has also been extended to cover configurations with degenerate components.

So, at this stage one has reduced the problem to the study of solutions of harmonic-type equations on , where is the rotation axis , with precise boundary conditions at the axis. Moreover, the solution is supposed to be invariant under rotations. Equivalently, one has to study a set of harmonic-type equations on a half-plane with specific singularity structure on the boundary.

There exist today at least three arguments that finish the proof, to be described in the following subsections.

In the vacuum case, Robinson was able to construct an amazing identity, by virtue of which the uniqueness of the Kerr metric followed [280]. The uniqueness problem with electro-magnetic fields remained open until Mazur [228] obtained a generalization of the Robinson identity in a systematic way: The Mazur identity (see also [229, 230, 48, 31, 168, 167]) is based on the observation that the EM equations in the presence of a Killing field describe a non-linear -model with coset space . The key to the success is Carter’s idea to carry out the dimensional reduction of the EM action with respect to the axial Killing field. Within this approach, the Robinson identity loses its enigmatic status – it turns out to be the explicit form of the Mazur identity for the vacuum case, .

Reduction of the EM action with respect to the time-like Killing field yields, instead, , but the resulting equations become singular on the ergosurface, where the Killing vector becomes null.

More information on this subject is provided in Sections 7.1 and 8.4.1.

At about the same time, and independently of Mazur, Bunting [41] gave a proof of uniqueness of the relevant harmonic-map equations exploiting the fact that the target space for the problem at hand is negatively curved. A further systematic PDE study of the associated harmonic maps has been carried out by Weinstein: as already mentioned, Weinstein provided existence results for multi-horizon configurations, as well as uniqueness results [322].

All the uniqueness results presented above require precise asymptotic control of the harmonic map and its derivatives at the singular set . This is an annoying technicality, as no detailed study of the behavior of the derivatives has been presented in the literature. The approach in [75, Appendix C] avoids this problem, by showing that a pointwise control of the harmonic map is enough to reach the desired conclusion.

For more information on this subject consult Section 8.4.2.

The third strategy to conclude the uniqueness proof has been advocated by Varzugin [306, 307] and, independently, by Neugebauer and Meinel [251]. The idea is to exploit the properties of the linear problem associated with the harmonic map equations, discovered by Belinski and Zakharov [23, 22] (see also [268]). This proceeds by showing that a regular black-hole solution must necessarily be one of the multi-soliton solutions constructed by the inverse-scattering methods, providing an argument for uniqueness of the Kerr solution within the class. Thus, one obtains an explicit form of the candidate metric for solutions with more components, as well as an argument for the non-existence of two-component configurations [249] (compare [70]).

What has been said so far can be summarized as follows:

Theorem 3.2 Let be a stationary, axisymmetric asymptotically-flat, -regular, electrovacuum four-dimensional spacetime. Then the domain of outer communications is isometric to one of the Weinstein solutions. In particular, if the event horizon is connected, then is isometric to the domain of outer communications of a Kerr–Newman spacetime.

Living Rev. Relativity 15, (2012), 7
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