3.2 Stationary-axisymmetric solutions

3.2.1 Topology

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 I+-regularity, one can invoke the positive energy theorem to show [18Jump To The Next Citation Point, 19] that some linear combination of the Killing vectors, say m, must have periodic orbits, and an axis of rotation, i.e., a two-dimensional totally-geodesic submanifold of M on which the Killing vector m 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 [76Jump To The Next Citation Point]. The bottom line is that the spacetime is the product of ℝ with 3 ℝ from which a finite number of aligned balls, each corresponding to a black hole, has been removed. Moreover, the U (1 ) component of the group of isometries acts by rotations of ℝ3. 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.

3.2.2 Candidate metrics

The only known I+-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 [23Jump To The Next Citation Point, 22Jump To The Next Citation Point] (compare [268Jump To The Next Citation Point]). 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 [198Jump To The Next Citation Point]. 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 [322Jump To The Next Citation Point], 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 [319Jump To The Next Citation Point, 216Jump To The Next Citation Point, 249Jump To The Next Citation Point, 70Jump To The Next Citation Point] on the axis, but the general case remains open.

Finally, the Israel–Wilson–Perjés (IWP) metrics [267Jump To The Next Citation Point, 179Jump To The Next Citation Point], 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 [80Jump To The Next Citation Point] 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.

3.2.3 The reduction

Returning to the classification question, the analysis continues with the circularity theorem of Papapetrou [264] and Kundt and Trümper [201] (compare [43Jump To The Next Citation Point]), 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 [45Jump To The Next Citation Point] (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 [47Jump To The Next Citation Point] and [50Jump To The Next Citation Point].) 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 [76Jump To The Next Citation Point]; a simpler argument has been provided in [72Jump To The Next Citation Point].

In essence, Carter’s reduction proceeds through a global manifestly–conformally-flat (“isothermal”) coordinate system (ρ,z) 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 (ρ,z) coordinates, with sufficient control of the boundary conditions so that the uniqueness proof goes through, has been given in [76Jump To The Next Citation Point], drawing heavily on [64Jump To The Next Citation Point], assuming that all horizons are non-degenerate. A streamlined argument has been presented in [79Jump To The Next Citation Point], 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 3 ℝ ∖ 𝒜, where 𝒜 is the rotation axis {x = y = 0}, 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.

3.2.4 The Robinson–Mazur proof

In the vacuum case, Robinson was able to construct an amazing identity, by virtue of which the uniqueness of the Kerr metric followed [280Jump To The Next Citation Point]. The uniqueness problem with electro-magnetic fields remained open until Mazur [228Jump To The Next Citation Point] obtained a generalization of the Robinson identity in a systematic way: The Mazur identity (see also [229Jump To The Next Citation Point, 230Jump To The Next Citation Point, 48, 31Jump To The Next Citation Point, 168Jump To The Next Citation Point, 167Jump To The Next Citation Point]) is based on the observation that the EM equations in the presence of a Killing field describe a non-linear σ-model with coset space G ∕H = SU (1,2 )∕S (U (1) × U (2)). 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, G ∕H = SU (1,1 )∕U (1).

Reduction of the EM action with respect to the time-like Killing field yields, instead, H = S (U(1,1 ) × U (1)), 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.

3.2.5 The Bunting–Weinstein harmonic-map argument

At about the same time, and independently of Mazur, Bunting [41Jump To The Next Citation Point] 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 [322Jump To The Next Citation Point].

All the uniqueness results presented above require precise asymptotic control of the harmonic map and its derivatives at the singular set 𝒜. This is an annoying technicality, as no detailed study of the behavior of the derivatives has been presented in the literature. The approach in [75Jump To The Next Citation Point, 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.

3.2.6 The Varzugin–Neugebauer–Meinel argument

The third strategy to conclude the uniqueness proof has been advocated by Varzugin [306Jump To The Next Citation Point, 307Jump To The Next Citation Point] and, independently, by Neugebauer and Meinel [251Jump To The Next Citation Point]. The idea is to exploit the properties of the linear problem associated with the harmonic map equations, discovered by Belinski and Zakharov [23Jump To The Next Citation Point, 22Jump To The Next Citation Point] (see also [268Jump To The Next Citation Point]). 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 [249Jump To The Next Citation Point] (compare [70Jump To The Next Citation Point]).

3.2.7 The axisymmetric uniqueness theorem

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

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

  Go to previous page Go up Go to next page