Stationary Black Holes: Uniqueness and Beyond
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Abstract
The spectrum of known blackhole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all blackholeequilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electrovacuum blackhole spacetimes ceases to exist in selfgravitating nonlinear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the EinsteinMaxwell system.
Keywords
Black Hole Event Horizon Killing Vector Black Ring Killing Vector Field1 Introduction
1.1 General remarks
Our conception of black holes has experienced several dramatic changes during the last two hundred years: While the “dark stars” of Michell [235] and Laplace [210] were merely regarded as peculiarities of Newton’s law of gravity and his corpuscular theory of light, black holes are nowadays widely believed to exist in our universe (for a review on the evolution of the subject the reader is referred to Israel’s comprehensive account [178]; see also [52, 51]). Although the observations are necessarily indirect, the evidence for both stellar and galactic black holes has become compelling [275, 232, 233, 247, 242, 231]. There seems to be consensus [276, 197, 234, 248] that the two most convincing supermassive blackhole candidates are the galactic nuclei of NGC 4258 and of our own Milky Way [123].
The theory of black holes was initiated by the pioneering work of Chandrasekhar [53, 54] in the early 1930s. (However, the geometry of the Schwarzschild solution [290, 291] was misunderstood for almost half a century; the misconception of the “Schwarzschild singularity” was retained until the late 1950s.) Computing the Chandrasekhar limit for neutron stars [8], Oppenheimer and Snyder [257], and Oppenheimer and Volkoff [258] were able to demonstrate that black holes present the ultimate fate of sufficientlymassive stars. Modern blackhole physics started with the advent of relativistic astrophysics, in particular with the discovery of pulsars in 1967.
One of the most intriguing outcomes of the mathematical theory of black holes is the uniqueness theorem, applying to a class of stationary solutions of the EinsteinMaxwell equations. Strikingly enough, its consequences can be made into a test of general relativity [285]. The assertion, that all (fourdimensional) electrovacuum blackhole spacetimes are characterized by their mass, angular momentum and electric charge, is strangely reminiscent of the fact that a statistical system in thermal equilibrium is described by a small set of state variables as well, whereas considerably more information is required to understand its dynamical behavior. The similarity is reinforced by the blackholemassvariation formula [9] and the areaincrease theorem [143, 69], which are analogous to the corresponding laws of ordinary thermodynamics. These mathematical relationships are given physical significance by the observation that the temperature of the black body spectrum of the Hawking radiation [142] is equal to the surface gravity of the black hole. There has been steady interest in the relationship between the laws of black hole mechanics and the laws of thermodynamics. In particular, computations within string theory seem to offer a promising interpretation of blackhole entropy [171]. The reader interested in the thermodynamic properties of black holes is referred to the review by Wald [316].
There has been substantial progress towards a proof of the celebrated uniqueness theorem, conjectured by Israel, Penrose and Wheeler in the late sixties [76, 79, 217] during the last four decades (see, e.g., [58] and [59] for previous reviews). Some open gaps, notably the electrovacuum staticity theorem [302, 303] and the topology theorems [109, 110, 85], have been closed (see [59, 73, 65] for related new results). Early on, the theorem led to the expectation that the stationaryblackhole solutions of other selfgravitating matter fields might also be parameterized by their mass, angular momentum and a set of charges (generalized nohair conjecture). However, ever since Bartnik and McKinnon discovered the first selfgravitating YangMills soliton in 1988 [14], a variety of new black hole configurations have been found, which violate the generalized nohair conjecture, that suitably regular blackhole spacetimes are classified by a finite set of asymptoticallydefined global charges. These solutions include nonAbelian black holes [310, 208, 24], as well as black holes with Skyrme [94, 161], Higgs [28, 254, 140] or dilaton fields [212, 132].
In fact, blackhole solutions with hair were already known before 1989: in 1982, Gibbons found a blackhole solution with a nontrivial dilaton field, within a model occurring in the low energy limit of N = 4 supergravity [126].
While the above counterexamples to the nohair conjecture consist of static, sphericallysymmetric configurations, there exists numerical evidence that static black holes are not necessarily spherically symmetric [192, 93]; in fact, they might not even need to be axisymmetric [278]. Moreover, some studies also indicate that nonrotating black holes need not be static [38]. The rich spectrum of stationaryblackhole configurations demonstrates that the matter fields are by far more critical to the properties of blackhole solutions than expected for a long time. In fact, the proof of the uniqueness theorem is, at least in the axisymmetric case, heavily based on the fact that the EinsteinMaxwell equations in the presence of a Killing symmetry form a σmodel, effectively coupled to threedimensional gravity [250]. (σmodels are a special case of harmonic maps, and we will use both terminologies interchangeably in our context.) Since this property is not shared by models with nonAbelian gauge fields [35], it is, with hindsight, not too surprising that the EinsteinYangMills system admits black holes with hair.
However, there exist other black hole solutions, which are likely to be subject to a generalized version of the uniqueness theorem. These solutions appear in theories with selfgravitating massless scalar fields (moduli) coupled to Abelian gauge fields. The expectation that uniqueness results apply to a variety of these models arises from the observation that their dimensional reduction (with respect to a Killing symmetry) yields a σmodel with symmetric target space (see, e.g., [31, 86, 120], and references therein).
1.2 Organization
The purpose of this text is to review some features of fourdimensional stationary asymptoticallyflat blackhole spacetimes. Some blackhole solutions with nonzero cosmological constant can be found in [313, 36, 323, 286, 271, 15]. It should be noted that the discovery of fivedimensional black rings by Emparan and Reall [99] has given new life to the overall subject (see [100, 101] and references therein) but here we concentrate on fourdimensional spacetimes with mostly classical matter fields.
For detailed introductions into the subject we refer to Chandrasekhar’s book on the mathematical theory of black holes [56], the classic textbook by Hawking and Ellis [143], Carter’s review [50], Chapter 12 of Wald’s book [314], the overview [63] and the monograph [151].
The first part of this report is intended to provide a guide to the literature, and to present some of the main issues, without going into technical details. We start by collecting the significant definitions in Section 2. We continue, in Section 3, by recalling the main steps leading to the uniqueness theorem for electrovacuum blackhole spacetimes. The classification scheme obtained in this way is then reexamined in the light of solutions, which are not covered by nohair theorems, such as stationary KaluzaKlein black holes (Section 4) and the EinsteinYangMills black holes (Section 5).
The second part reviews the main structural properties of stationary blackhole spacetimes. In particular, we discuss the dimensional reduction of the field equations in the presence of a Killing symmetry in more detail (Section 6). For a variety of matter models, such as selfgravitating Abelian gauge fields, the reduction yields a σmodel, with symmetric target manifold, coupled to threedimensional gravity. In Section 7 we discuss some aspects of this structure, namely the Mazur identity and the quadratic mass formulae, and we present the IsraelWilson class of metrics.
The third part is devoted to stationary and axisymmetric blackhole spacetimes (Section 8). We start by recalling the circularity problem for nonAbelian gauge fields and for scalar mappings. The dimensional reduction with respect to the second Killing field leads to a boundary value problem on a fixed, twodimensional background. As an application, we outline the uniqueness proof for the KerrNewman metric.
2 Definitions
It is convenient to start with definitions, which will be grouped together in separate sections.
2.1 Asymptotic flatness
2.2 KaluzaKlein asymptotic flatness
2.3 Stationary metrics
An asymptoticallyflat, or KKasymptoticallyflat, spacetime (M, g) will be called stationary if there exists on M a complete Killing vector field k, which is timelike in the asymptotic region ℐ_{ext}; such a Killing vector will be sometimes called stationary as well. In fact, in most of the literature it is implicitly assumed that stationary Killing vectors satisfy g(k, k) < −ϵ < 0 for some ϵ and for all r large enough. This uniformity condition excludes the possibility of a timelike vector, which asymptotes to a null one. This involves no loss of generality in wellbehaved asymptoticallyflat spacetimes: indeed, this uniform timelikeness condition always holds for Killing vectors, which are timelike for all large distances if the conditions of the positive energy theorem are met [17, 77].
Note that this definition assumes that the Killing vector k is complete, which means that for every p ∈ M the orbit ϕ_{t}[k](p) of k is defined for all t ∈ ℝ. The question of completeness of Killing vectors is an important issue, which needs justifying in some steps of the uniqueness arguments [57, 59].
2.4 Domains of outer communications, event horizons
For t ∈ ℝ let ϕ_{t}[k] : M → M denote the oneparameter group of diffeomorphisms generated by k; we will write ϕ_{t} for ϕ_{t}[k] whenever ambiguities are unlikely to occur.
Recall that I^{−}(Ω), respectively J^{−}(Ω), is the set covered by pastdirected timelike, respectively causal, curves originating from Ω, while İ^{−} denotes the boundary of I^{−}, etc. The sets I^{+}, etc., are defined as I^{−}, etc., after changing timeorientation. See [143, 16, 256, 236, 266, 66] and references therein for details of causality theory.
By standard causality theory, an event horizon is the union of Lipschitz null hypersurfaces. It turns out that event horizons in stationary spacetimes satisfying energy conditions are as smooth as the metric allows [76, 69]; thus, smooth if the metric is smooth, analytic if the metric is.
2.5 Killing horizons
2.5.1 Bifurcate Killing horizons
The Schwarzschild black hole has an event horizon with a specific structure, which is captured by the following definition: A set is called a bifurcate Killing horizon if it is the union of a a smooth spacelike submanifold S of codimension two, called the bifurcation surface, on which a Killing vector field k vanishes, and of four smooth null embedded hypersurfaces obtained by following null geodesics in the four distinct null directions normal to S.
For example, the Killing vector xd_{t} + td_{x} in Minkowski spacetime has a bifurcate Killing horizon, with the bifurcation surface {t = x = 0}. As already mentioned, another example is given by the set {r = 2m} in SchwarzschildKruskalSzekeres spacetime with positive mass parameter m.
In the spirit of the previous definition, we will refer to the union of two null hypersurfaces, which intersect transversally on a 2dimensional spacelike surface as a bifurcate null surface.
The reader is warned that a bifurcate Killing horizon is not a Killing horizon, as defined in Section 2.5, since the Killing vector vanishes on S. If one thinks of S as not being part of the bifurcate Killing horizon, then the resulting set is again not a Killing horizon, since it has more than one component.
2.5.2 Killing prehorizons
One of the key steps of the uniqueness theory, as described in Section 3, forces one to consider “horizon candidates” with local properties similar to those of a proper event horizon, but with global behavior possibly worse: A connected, not necessarily embedded, null hypersurface H_{0} ⊂ \({\mathscr N}\left[ k \right]\) to which k is tangent is called a Killing prehorizon. In this terminology, a Killing horizon is a Killing prehorizon, which forms a embedded hypersurface, which coincides with a connected component of \({\mathscr N}\left[ k \right]\). The Minkowskian Killing vector ∂_{t} − ∂_{x} provides an example where \({\mathscr N}\) is not a hypersurface, with every hyperplane t + x = const being a prehorizon.
The Killing vector k = ∂_{t} + Y on \({\mathbb R} \times {{\mathbb T}^n}\), equipped with the flat metric, where \({{\mathbb T}^n}\) is an ndimensional torus, and where Y is a unit Killing vector on \({{\mathbb T}^n}\) with dense orbits, admits prehorizons, which are not embedded. This last example is globally hyperbolic, which shows that causality conditions are not sufficient to eliminate this kind of behavior.
 A theorem due to Vishveshwara [308] gives a characterization of the Killing horizon H[k] in terms of the twist ω of k:^{1} A connected component of the set \({\mathscr N}\,: = \left\{{g(k,k) = 0,\,k \ne 0} \right\}\) is a (nondegenerate) Killing horizon whenever$$\omega = 0\quad {\rm{and}}\quad {i_k}{\rm{d}}k \ne 0\quad {\rm{on}}\quad {\mathscr N}.$$(2.14)

The following characterization of Killing prehorizons is often referred to as the VishveshwaraCarter Lemma [46, 43] (compare [61, Addendum]): Let (M, g) be a smooth spacetime with complete, static Killing vector k. Then the set {p ∈ M ∣ g(k, k)∣_{p} = 0, k(p) = 0} is the union of integral leaves of the distribution k^{⊥}, which are totally geodesic within M {k = 0}.
2.5.3 Surface gravity: degenerate, nondegenerate and meannondegenerate horizons
The Killing equation implies dN = −2∇_{k}k; we see that the surface gravity measures the extent to which the parametrization of the geodesic congruence generated by k is not affine.
The proof of (2.16) given in [314] generalizes to all spacetime dimensions n +1 ≥ 4; the result also follows in all dimensions from the analysis in [165] when the horizon has compact spacelike sections.
Yet another situation of interest is a spacetime with two commuting Killing vector fields k and m, with a Killing horizon H[ξ] associated to a Killing vector ξ = k + Ωm. Such a spacetime is said to be circular if the distribution of planes spanned by k and m is hypersurfaceorthogonal. Equivalently, the metric can be locally written in a 2+2 blockdiagonal form, with one of the blocks defined by the orbits of k and m. In the circular case one shows that g(m, ω_{ξ}) = g(ξ, ω_{m}) = 0 implies dω_{ξ} = 0 on the horizon generated by ξ; see [151], Chapter 7 for details.
A significant observation is that of Kay and Wald [184], that κ must be constant on bifurcate Killing horizons, regardless of the matter content. This is proven by showing that the derivative of the surface gravity in directions tangent to the bifurcation surface vanishes. Hence, κ cannot vary between the nullgenerators. But it is clear that κ is constant along the generators.
 (i)
The dominant energy condition holds;
 (ii)
the domain of outer communications is static;
 (iii)
the domain of outer communications is circular;
 (iv)
H[k] is a bifurcate Killing horizon.
See [270] for some further observations concerning (2.16).
A Killing horizon is called degenerate if κ vanishes, and nondegenerate otherwise.
A key theorem of Rácz and Wald [270] asserts that nondegenerate horizons (with a compact cross section and constant surface gravity) are “essentially bifurcate”, in the following sense: Given a spacetime with such a nondegenerate Killing horizon, one can find another spacetime, which is locally isometric to the original one in a onesided neighborhood of a subset of the horizon, and which contains a bifurcate Killing horizon. The result can be made global under suitable conditions.
A smooth null hypersurface, not necessarily a Killing horizon, with a smooth compact crosssection S such that κ = 0 is said to be mean nondegenerate.
2.6 I^{+}regularity
The classification theory of stationary black holes requires that the spacetime under consideration satisfies various global regularity conditions. These are captured by the following definition:
The “I^{+}” of the name is due to the I^{+} appearing in (2.23).
Some comments about the definition are in order. First, one requires completeness of the orbits of the stationary Killing vector to have an action of ℝ on M by isometries. Next, global hyperbolicity of the domain of outer communications is used to guarantee its simple connectedness, to make sure that the area theorem holds, and to avoid causality violations as well as certain kinds of naked singularities in 〈〈M_{ext}〉〉. Further, the existence of a wellbehaved spacelike hypersurface is a prerequisite to any elliptic PDEs analysis, as is extensively needed for the problem at hand. The existence of compact crosssections of the future event horizon prevents singularities on the future part of the boundary of the domain of outer communications, and eventually guarantees the smoothness of that boundary. The requirement Eq. (2.23) might appear somewhat unnatural, as there are perfectly wellbehaved hypersurfaces in, e.g., the Schwarzschild spacetime, which do not satisfy this condition, but there arise various technical difficulties without this condition. Needless to say, all those conditions are satisfied by the KerrNewman and the MajumdarPapapetrou (MP) solutions.
3 Towards a classification of stationary electrovacuum black hole spacetimes
For definiteness, from now on we assume that all spacetimes are I^{+}regular. We note that the slightly weaker global conditions spelledout in Theorem 3.1 suffice for the analysis of static spacetimes, or for various intermediate steps of the uniqueness theory, but those weaker conditions are not known to suffice for the Uniqueness Theorem 3.3.
The main task of the uniqueness program is to show that the domains of outer communications of sufficiently regular stationary electrovacuum blackhole spacetimes are exhausted by the KerrNewman or the MP spacetimes.
The starting point is the smoothness of the event horizon; this is proven in [76, Theorem 4.11], drawing heavily on the results in [69].
One proves, next, that connected components of the event horizon are diffeomorphic to ℝ × S^{2}. This was established in [85], taking advantage of the topological censorship theorem of Friedman, Schleich and Witt [106]; compare [141] for a previous partial result. (Related versions of the topology theorem, applying to globallyhyperbolic, notnecessarilystationary, spacetimes, have been established by Jacobson and Venkataramani [180], and by Galloway [108, 109, 110, 112]; the strongesttodate version, with very general asymptotic hypotheses, can be found in [73].)
3.1 Static solutions
The results concerning static black holes are stronger than the general stationary case, and so this case deserves separate discussion. In any case, the proof of uniqueness for stationary black holes branches out at some point and one needs to consider separately uniqueness for static configurations.
In pioneering work, Israel showed that both static vacuum [176] and electrovacuum [177] blackhole spacetimes satisfying a set of restrictive conditions are spherically symmetric. Israel’s ingenious method, based on differential identities and Stokes’ theorem, triggered a series of investigations devoted to the static uniqueness problem (see, e.g., [244, 245, 279, 281, 294]). A breakthrough was made by Bunting and MasoodulAlam [42], who showed how to use the positive energy theorem^{2} to exclude nonconnected configurations (compare [61]).^{3}
The annoying hypothesis of analyticity, which was implicitly assumed in the above treatments, has been removed in [72]. The issue here is to show that the Killing vector field cannot become null on the domain of outer communications. The first step to prove this is the VishveshwaraCarter lemma (see Section 2.5.2 and [308, 43]), which shows that null orbits of static Killing vectors form a prehorizon, as defined in Section 2.5.2. To finish the proof one needs to show that prehorizons cannot occur within the d.o.c. This presents no difficulty when analyticity is assumed. Now, analyticity of stationary electrovacuum metrics is a standard property [245, 243] when the Killing vector is timelike, but timelikeness throughout the d.o.c. is not known yet at this stage of the argument. The nonexistence of prehorizons within the d.o.c. for smooth metrics requires more work, and is the main result in [72].
In the static vacuum case the remainder of the argument can be simplified by noting that there are no static solutions with degenerate horizons, which have spherical crosssections [81]. This is not true anymore in the electrovacuum case, where an intricate argument to handle nondegenerate horizons is needed [83] (compare [284, 295, 225, 62] for previous partial results).
All this can be summarized in the following classification theorem:
Theorem 3.1 Let (M, g) be an electrovacuum, fourdimensional spacetime containing a spacelike, connected, acausal hypersurface \({\mathscr I}\), such that \(\bar {\mathscr I}\) is a topological manifold with boundary consisting of the union of a compact set and of a finite number of asymptoticallyflat ends. Suppose that there exists on M a complete hypersurfaceorthogonal Killing vector, that the domain of outer communication 〈〈M_{ext}〉〉 is globally hyperbolic, and that \(\partial \bar {\mathscr I} \subset M\backslash \langle \langle {M_{{\rm{ext}}}}\rangle \rangle\). Then 〈〈M_{ext}〉〉 is isometric to the domain of outer communications of a ReissnerNordström or a MP spacetime.
3.2 Stationaryaxisymmetric solutions
3.2.1 Topology
A second class of spacetimes where reasonably satisfactory statements can be made is provided by stationaryaxisymmetric 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 [18, 19] that some linear combination of the Killing vectors, say m, must have periodic orbits, and an axis of rotation, i.e., a twodimensional totallygeodesic 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 simpleconnectedness and structure theorems [76]. The bottom line is that the spacetime is the product of ℝ with ℝ^{3} 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 ℝ^{3}. Equivalently, the quotient space is a halfplane from which one has removed a finite number of disjoint halfdiscs centered on points lying on the boundary of the halfplane.
3.2.2 Candidate metrics
The only known I^{+}regular stationary and axisymmetric solutions of the EinsteinMaxwell equations are the KerrNewman metrics and the MP metrics. However, it should be kept in mind that candidate solutions for nonconnected blackhole configurations exist:
First, there are the multisoliton metrics constructed using inverse scattering methods [23, 22] (compare [268]). Closely related (and possibly identical, see [148]), are the multiKerr solutions constructed by successive Backlund transformations starting from Minkowski spacetime; a special case is provided by the NeugebauerKramer doubleKerr 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 IsraelWilsonPerjés (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 nonrotating 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 nonconnected 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 [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 blockdiagonal form.
The next key observation of Carter is that the stationary and axisymmetric EM equations can be reduced to a twodimensional 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 heavyduty 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 manifestlyconformallyflat (“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 [76], drawing heavily on [64], assuming that all horizons are nondegenerate. 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 harmonictype equations on \({{\mathbb R}^3}\backslash {\mathscr A}\), where \({\mathscr A}\) 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 harmonictype equations on a halfplane 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 RobinsonMazur proof
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 electromagnetic 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 nonlinear σmodel 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 timelike 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 BuntingWeinstein harmonicmap argument
At about the same time, and independently of Mazur, Bunting [41] gave a proof of uniqueness of the relevant harmonicmap 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 multihorizon 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 \({\mathscr A}\). This is an annoying technicality, as no detailed study of the behavior of the derivatives has been presented in the literature. The approach in [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.
3.2.6 The VarzuginNeugebauerMeinel argument
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 blackhole solution must necessarily be one of the multisoliton solutions constructed by the inversescattering 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 nonexistence of twocomponent configurations [249] (compare [70]).
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 asymptoticallyflat, I^{+}regular, electrovacuum fourdimensional spacetime. Then the domain of outer communications 〈〈M_{ext}〉〉 is isometric to one of the Weinstein solutions. In particular, if the event horizon is connected, then 〈〈M_{ext}〉〉 is isometric to the domain of outer communications of a KerrNewman spacetime.
3.3 The nohair theorem
3.3.1 The rigidity theorem
Throughout this section we will assume that the spacetime is I^{+}regular, as made precise by Definition 2.1.
To prove uniqueness of connected, analytic, nondegenerate 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 independentlydefined, and logicallydistinct, local notion of the Killing horizon. Assuming analyticity, SRT asserts that the event horizon of a stationary blackhole 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 nonrotating if it is generated by the stationary Killing field, and rotating otherwise. At this stage the argument branchesoff, 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 nondegenerate 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 nearKerr nondegenerate configurations so far [3] (compare [326]).
In any case, sufficiently regular analytic stationary electrovacuum 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 nonconnected candidate solutions.
In the nonrotating 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 nondegenerate 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 nonrotating, electrovacuum blackhole 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 (M, g) be a stationary, asymptoticallyflat, I^{+}regular, electrovacuum, fourdimensional analytic spacetime. If the event horizon is connected and either mean nondegenerate or rotating, then 〈〈M_{ext}〉〉 is isometric to the domain of outer communications of a KerrNewman spacetime.
The hypotheses of analyticity and nondegeneracy 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 ReissnerNordström spacetimes are members of the KerrNewman family:
Conjecture 3.4 Let (M, g) be an electrovacuum, fourdimensional spacetime containing a spacelike, connected, acausal hypersurface \({\mathscr I}\), such that \(\bar {\mathscr I}\) is a topological manifold with boundary, consisting of the union of a compact set and of a finite number of asymptoticallyflat ends. Suppose that there exists on M a complete stationary Killing vector k, that 〈〈M_{ext}〉〉 is globally hyperbolic, and that \(\partial \bar {\mathscr I} \subset M\backslash \langle \langle {M_{{\rm{ext}}}}\rangle \rangle\). Then 〈〈M_{ext}〉〉 is isometric to the domain of outer communications of a KerrNewman or MP spacetime.
3.3.3 A uniqueness theorem for nearKerrian smooth vacuum stationary spacetimes
The existing results on rigidity without analyticity require one to assume either staticity, or a nearKerr condition on the spacetime geometry (see Section 3.3.1), which is quantified in terms of a smallness condition of the MarsSimon tensor [223, 293]. The results in [3] together with Theorems 3.1–3.2, and a version of the RáczWald Theorem [107, Proposition 4.1], lead to:
3.4 Summary of open problems
For the convenience of the reader, we summarize here the main open problems left in the nohair theorem.
3.4.1 Degenerate horizons
We recall that there exist no vacuum static spacetimes containing degenerate horizons with compact spherical sections [81]. On the other hand, MP [220, 262] black holes provide the only electrovacuum static examples with nonconnected degenerate horizons. See [78, 154] and references therein for a discussion of the geometry of MP black holes.
Under the remaining hypotheses of Theorem 3.3, the only step where the hypothesis of nondegeneracy enters is the proof of existence of a maximal hypersurface \({\mathscr I}\) in the blackhole spacetime under consideration, such that \({\mathscr I}\) is Cauchy for the domain of outer communications. The geometry of Cauchy surfaces in the case of degenerate horizons is well understood by now [62, 79], and has dramatically different properties when compared to the nondegenerate case. A proof of existence of maximal hypersurfaces in this case would solve the problem, but requires new insights. A key missing element is an equivalent of Bartnik’s a priori height estimate [10], established for asymptoticallyflat ends, that would apply to asymptoticallycylindrical ends.
3.4.2 Rigidity without analyticity
Analyticity enters the current argument at two places: First, one needs to construct the second Killing vector near the horizon. This can be done by first constructing a candidate at the horizon, and then using analyticity to extend the candidate to a neighborhood of the horizon. Next, the Killing vector has to be extended to the whole domain of outer communications. This can be done using analyticity and a theorem by Nomizu [253], together with the fact that I^{+}regular domains of outer communications are simply connected. Finally, analyticity can be used to provide a simple argument that prehorizons do not intersect 〈〈M_{ext}〉〉 (but this is not critical, as a proof is available now within the smooth category of metrics [72]).
A partiallysuccessful strategy to remove the analyticity condition has been invented by Alexakis, Ionescu and Klainerman in [2]. Their argument applies to nondegenerate nearKerrian configurations, but the general case remains open.
The key to the approach in [2] is a unique continuation theorem near bifurcate Killing horizons proven in [174], which implies the existence of a second Killing vector field, say m, in a neighborhood of the horizon. One then needs to prove that m extends to the whole domain of outer communications. This is established via another unique continuation theorem [175] with specific convexity conditions. These lead to nontrivial restrictions, and so far the argument has only been shown to apply to nearKerrian configurations.
A unique continuation theorem across more general timelike surfaces would be needed to obtain the result without smallness restrictions.
It follows from what has been said in [72] that the boundary of the set where two Killing vector fields are defined cannot become null within a domain of outer communications; this fact might be helpful in solving the full problem.
3.4.3 Many components?
The only known examples of singularityfree stationary electrovacuum black holes with more than one component are provided by the MP family. (Axisymmetric MP solutions are possible, but MP metrics only have one Killing vector in general.) It has been suspected for a very long time that these are the only such solutions, and that there are thus no such vacuum configurations. This should be contrasted with the fivedimensional case, where the Black Saturn solutions of Elvang and Figueras [97] (compare [71, 305]) provide nontrivial twocomponent examples.
It might be convenient to summarize the general facts known about fourdimensional multicomponent solutions.^{4} In case of doubts, I^{+}regularity should be assumed.
We start by noting that the static solutions, whether connected or not, have already been covered in Section 3.1.
A multicomponent electrovacuum configuration with all components nondegenerate and nonrotating would be, by what has been said, static, but then no such solutions exist (all components of an MP black hole are degenerate). On the other hand, the question of existence of a multiblackhole configuration with components of mixed type, none of which rotates, is open; what’s missing is the proof of existence of maximal hypersurfaces in such a case. Neither axisymmetry nor staticity is known for such configurations.
Analytic multiblackhole solutions with at least one rotating component are necessarily axisymmetric; this leads one to study the corresponding harmonicmap equations, with candidate solutions provided by Weinstein or by inverse scattering techniques [198, 322, 23, 22]. The Weinstein solutions have no singularities away from the axes, but they are not known in
explicit form, which makes difficult the analysis of their behavior on the axis of rotation. The multiblackhole metrics constructed by multisoliton superpositions or by Bäcklund transformation techniques are obtained as rational functions with several parameters, with explicit constraints on the parameters that lead to a regular axis [222], but the analysis of the zeros of their denominators has proved intractable so far. It is perplexing that the five dimensional solutions, which are constructed by similar methods [268], can be completely analyzed with some effort and lead to regular solutions for some choices of parameters, but the fourdimensional case remains to be understood.
The problem with the argument so far is the lack of justification of the subextremality condition. Fortunately, this condition can be avoided altogether using ideas of [88] concerning the inequality (3.1) and appealing to the results in [96, 6, 73] concerning marginallyoutertrapped surfaces (MOTS): Using existence results of weakly stable MOTS together with various aspects of the candidate Weyl metrics, one can adapt the argument of [145] to show [70] that the area inequality (3.1), with “less than” there replaced by “less than or equal to”, would hold for those components of the horizon, which have nonzero surface gravity, assuming an I^{+}regular metric of the Weyl form, if any existed. The calculations of Hennig and Neugebauer [249] together with a contradiction argument lead then to
Theorem 3.6 I^{+}regular twoKerr black holes do not exist.
The case of more than two horizons is widely open.
4 Classification of stationary toroidal KaluzaKlein black holes
In this work we are mostly interested in uniqueness results for fourdimensional black holes. This leads us naturally to consider those vacuum KaluzaKlein spacetimes with enough symmetries to lead to fourdimensional spacetimes after dimensional reduction, providing henceforth fourdimensional black holes. It is convenient to start with a very short overview of the subject; the reader is referred to [101, 172] and references therein for more information. Standard examples of KaluzaKlein black holes are provided by the Schwarzschild metric multiplied by any spatially flat homogeneous space (e.g., a torus). Nontrivial examples can be found in [272, 211]; see also [200, 172] and reference therein.
4.1 Black holes in higher dimensions
The study of spacetimes with dimension greater then four is almost as old as general relativity itself [183, 195]. Concerning black holes, while in dimension four all explicitlyknown asymptomaticallyflat and regular solutions of the vacuum Einstein equations are exhausted by the Kerr family, in spacetime dimension five the landscape of known solutions is richer. The following I^{+}regular, stationary, asymptoticallyflat, vacuum solutions are known in closed form: the MyersPerry black holes, which are higherdimensional generalizations of the Kerr metric with sphericalhorizon topology [246]; the celebrated EmparanReall black rings with S^{2} × S^{1} horizon topology [99]; the PomeranskySenkov black rings generalizing the previous by allowing for a second angularmomentum parameter [269]; and the “Black Saturn” solutions discovered by Elvang and Figueras, which provide examples of regular multicomponent black holes where a spherical horizon is surrounded by a black ring [97].^{5}
Inspection of the basic features of these solutions challenges any naive attempt to generalize the classification scheme developed for spacetime dimension four: One can find black rings and MyersPerry black holes with the same mass and angular momentum, which must necessarily fail to be isometric since the horizon topologies do not coincide. In fact there are nonisometric black rings with the same Poincaré charges; consequently a classification in terms of mass, angular momenta and horizon topology also fails. Moreover, the Black Saturns provide examples of regular vacuum multiblackhole solutions, which are widely believed not to exist in dimension four; interestingly, there exist Black Saturns with vanishing total angular momentum, a feature that presumably distinguishes the Schwarzschild metric in four dimensions.
Nonetheless, results concerning 4dimensional black holes either generalize or serve as inspiration in higher dimensions. This is true for landmark results concerning blackhole uniqueness and, in fact, classification schemes exist for classes of higher dimensional blackhole spacetimes, which mimic the symmetry properties of the “static or axisymmetric” alternative, upon which the uniqueness theory in fourdimensions is built.
For instance, staticity of I^{+}regular, vacuum, asymptoticallyflat, nonrotating, nondegenerate black holes remains true in higher dimensions^{6}. Also, Theorem 3.1 remains valid for vacuum spacetimes of dimension n + 1, n ≥ 3, whenever the positive energy theorem applies to an appropriate doubling of \({\mathscr I}\) (see [72], Section 3.1 and references therein). Moreover, the discussion in Section 3.1 together with the results in [282, 283] suggest that an analogous generalization to electrovacuum spacetimes exists, which would lead to uniqueness of the higherdimensional ReissnerNordström metrics within the class of static solutions of the EinsteinMaxwell equations, for all n ≥ 3 (see also [101, Section 8.2], [173] and references therein).
Rigidity theorems are also available for (n + 1)dimensional, asymptoticallyflat and analytic blackhole spacetimes: the nondegenerate horizon case was established in [165] (compare [239]), and partial results concerning the degenerate case were obtained in [163]. These show that stationary rotating (analytic) black holes are “axisymmetric”, in the sense that their isometry group contains ℝ × U(1); the ℝ factor corresponds to the action generated by the stationary vector, while the circle action provides an “extra” axial Killing vector. A conjecture of Reall [274], supported by the results in [98], predicts the existence of 5dimensional black holes with exactly ℝ × U(1) isometry group; in particular, it is conceivable that the rigidity results are sharp when providing only one “axial” Killing vector. The results in [133, 92, 166] are likely to be relevant in this context.
So we see that, assuming analyticity and asymptotic flatness, the dichotomy provided by the rigidity theorem remains valid but its consequences appear to be weaker in higher dimensions. A gap appears between the two favorable situations encountered in dimension four: one being the already discussed staticity and the other corresponding to black holes with cohomogeneitytwo Abelian groups of isometries. We will now consider this last scenario, which turns out to have connections to the four dimensional case.
4.2 Stationary toroidal KaluzaKlein black holes
The fourdimensional vacuum Einstein equations simplify considerably in the stationary and axisymmetric setting by reducing to a harmonic map into the hyperbolic plane (see Sections 8 and 3.2.3). A similar such reduction in (n + 1)dimensions works when the isometry group includes \({\mathbb R} \times {{\mathbb T}^{n  2}}\), i.e., besides the stationary vector there exist n − 2 commuting axial Killing vectors.
4.3 Topology of the event horizon
A theorem of Galloway and Schoen [111] shows that compact crosssections of the horizon must be of positive Yamabe type, i.e., admit metrics of positive scalar curvature. In spacetime dimension five, the positive Yamabe property restricts the horizon to be a finite connected sum of spaces with S^{3}, S^{2} × S^{1} and lensspace L(p, q) topologies. Such results require no symmetry assumptions but by further assuming stationarity and the existence of one axial Killing vector more topological restrictions, concerning the allowed factors in the connected sum, appear in five dimensions [162].
In the toroidal KaluzaKlein case, the existence of a toroidal action leads to further restrictions [169]; for instance, for N = 4 and n ≥ 4, each connected component of the horizon has necessarily one of the following topologies: \({S^3} \times {{\mathbb T}^{n  4}}\), S^{2} × T^{n−3} and \(L(p,q) \times {{\mathbb T}^{n  4}}\).
It should be noted that no asymptoticallyflat or KaluzaKlein black holes with lensspace topology of the horizon are known. Constructing a black lens, or establishing nonexistence, appears to be a challenging problem.
4.4 Orbit space structure
As already discussed, basic properties of black rings show that a classification of KKblack holes in terms of mass, angular momenta and horizon topology is not possible. But, as argued by Hollands and Yazadjiev [169], the angular momenta and the structure of the orbit space characterize such black holes if one further assumes nondegeneracy of the event horizon. This orbit space structure is in turn determined by the interval structure of the boundary of the quotient manifold, a concept related to Harmark’s rod structure developed in [137] (see also [101, Section 5.2.2.1]). Note that the interval structure codifies the horizon topology.
4.5 KK topological censorship
Blackhole uniqueness in fourdimensions uses simple connectedness of the event horizon extensively. But the Schwarzschild metric multiplied by a flat torus shows that simple connectedness does not hold for general domains of outer communications of KaluzaKlein black holes. Fortunately, simple connectedness of the orbit space \(\langle \langle {M_{{\rm{ext}}}}\rangle \rangle/\left({{\mathbb R} \times {{\mathbb T}^{n  2}}} \right)\) suffices: for instance, to prove that the (n − 1)dimensional orbit generated by the stationary and axial vectors is timelike in 〈〈M_{ext}〉〉 away from the axes (which in turn is essential to the construction of Weyl coordinates), to guarantee the existence of global twist potentials [65] and to exclude the existence of exceptional orbits of the toroidal action (see Section 4.4). The generalized topological censorship theorem of [73] shows that this property follows from the simple connectedness of the orbit space in the asymptotic end \({{\mathscr I}_{{\rm{ext}}}}/{{\mathbb T}^{n  2}} \approx {{\mathbb R}^N}\backslash B\).
4.6 Classification theorems for KKblack holes
The remaining classification results assume cohomogeneitytwo isometry actions [169]:
Theorem 4.1 Let (M_{i}, g_{i}), i = 1, 2, be two I^{+}regular, (n + 1)dimensional, n ≥ 3, stationary toroidal KaluzaKlein spacetimes, with five asymptoticallylarge dimensions (N = 4). Assume, moreover, that the event horizon is connected and mean nondegenerate. If the interval structure and the set of angular momenta coincide, then the domains of outer communications are isometric.
This theorem generalizes previous results by the same authors [167, 168] as well as a uniqueness result for a connected spherical black hole of [240].
It should be noted that this provides a variation on Mazur’s and the harmonic map methods (see Sections 3.2.4 and 3.2.5), which avoids some of their intrinsic difficulties. Indeed, the integration by parts argument based on the Mazur identity requires detailed knowledge of the maps under consideration at the singular set {ρ = 0}, while the harmonic map approach requires finding, and controlling, the distance function for the target manifold. (In some simple cases ψ is the desired distance function, but whether this is so in general is unclear.) The result then follows by a careful analysis of the asymptotic behavior of the relevant fields; such analysis was also carried out in [169].
In this context, the degenerate horizons suffer from the supplementary difficulty of controlling the behavior of the fields near the horizon. One expects that an exhaustive analysis of nearhorizon geometries would allow one to settle the question; some partial results towards this can be found in [204, 203, 202, 164].
5 Beyond EinsteinMaxwell
The purpose of this section is to reexamine the various steps leading to the classification of electrovacuum blackhole spacetimes for other matter models. In particular, it will be seen that several steps in Figure 3 cease to hold in the presence of nonAbelian gauge fields. Unfortunately, this implies that we are far from having a classification of all stationary blackhole spacetimes with physicallyinteresting sources.
5.1 Spherically symmetric black holes with hair
One can find in the literature the naive expectation that — within a given matter model — the stationary blackhole solutions are uniquely characterized by a set of global charges; this will be referred to as the generalized nohair conjecture. A model in which this might possibly be correct is provided by the static sector of the EMdilaton theory, discussed at the beginning of Section 4.6.
The failure of this generalized nohair conjecture is demonstrated by the EinsteinYangMills (EYM) theory: According to the conjecture, any static solution of the EYM equations should either coincide with the Schwarzschild metric or have some nonvanishing YangMills charges. This turned out not to be the case when, in 1989, various authors [310, 208, 24] found a family of static blackhole solutions with vanishing global YangMills charges (as defined, e.g., in [74]); these were originally constructed by numerical means and rigorous existence proofs were given later in [299, 297, 298, 29, 227]; for a review see [311]. These solutions violate the generalized nohair conjecture.
As the nonAbelian black holes are unstable [301, 329, 315], one might adopt the view that they do not present actual threats to the generalized nohair conjecture. (The reader is referred to [37] for the general structure of the pulsation equations, [309, 40], to [27] for the sphaleron instabilities of the particlelike solutions, and to [292] for a review on sphalerons.) However, various authors have found stable black holes, which are not characterized by a set of asymptotic flux integrals. For instance, there exist stable blackhole solutions with hair of the static, sphericallysymmetric EinsteinSkyrme equations [94, 156, 157, 161, 241] and to the EYM equations coupled to a Higgs triplet [28, 30, 214, 1]; it should be noted that the solutions of the EYMHiggs equations with a Higgs doublet are unstable [27, 324]. Hence, the restriction of the generalized nohair conjecture to stable configurations is not correct either.

First, there exist no purely gravitational solitons, that is, the only globallyregular, asymptoticallyflat, static vacuum solution to the Einstein equations with finite energy is Minkowski spacetime. This is the Lichnerowicz theorem, which nowadays can be obtained from the positive mass theorem and the Komar expression for the total mass of an asymptoticallyflat, stationary spacetime [170]; see, e.g., [127] or [152]. A rather strong version thereof, which does not require asymptotic conditions other than completeness of the space metric, has been established by Anderson [5], see also [4].

Next, there are no nontrivial static solutions of the EYM equations near Minkowski spacetime [221].

Further, both Deser’s energy argument [90] and Coleman’s scaling method [87] show that there do not exist pure YM solitons in flat spacetime.

Moreover, the EM system admits no soliton solutions. (This follows by applying Stokes’ theorem to the static Maxwell equations; see, e.g., [151].)

Finally, Deser [91] proved that the threedimensional EYM equations admit no soliton solutions. The argument takes advantage of the fact that the magnetic part of the YangMills field has only one nonvanishing component in 2+1 dimensions.
All this shows that it was conceivable to conjecture a nonexistence theorem for soliton solutions of the EYM equations in 3+1 dimensions, and a nohair theorem for the corresponding black hole configurations. On the other hand, none of the above examples takes care of the full nonlinear EYM system, which bears the possibility to balance the gravitational and the gauge field interactions. In fact, a closer look at the structure of the EYM action in the presence of a Killing symmetry dashes the hope to generalize the uniqueness proof along the lines used in the Abelian case: The Mazur identity owes its existence to the σmodel formulation of the EM equations. The latter is, in turn, based on scalar magnetic potentials, the existence of which is a peculiarity of Abelian gauge fields (see Section 6).
5.2 Static black holes without spherical symmetry
The above counterexamples to the generalized nohair conjecture are static and spherically symmetric. The famous Israel theorem guarantees that spherical symmetry is, in fact, a consequence of staticity, provided that one is dealing with vacuum [176] or electrovacuum [177] blackhole spacetimes. The task to extend the Israel theorem to more general selfgravitating matter models is, of course, a difficult one. In fact, the following example proves that spherical symmetry is not a generic property of static black holes.
In [213], Lee et al. reanalyzed the stability of the ReissnerNordström (RN) solution in the context of SU(2) EYMHiggs theory. It turned out that — for sufficiently small horizons — the RN black holes develop an instability against radial perturbations of the YangMills field. This suggested the existence of magneticallycharged, sphericallysymmetric black holes with hair, which were also found by numerical means [28, 30, 214, 1].
Motivated by these solutions, Ridgway and Weinberg [277] considered the stability of the magnetically charged RN black holes within a related model; the EM system coupled to a charged, massive vector field. Again, the RN solution turned out to be unstable with respect to fluctuations of the massive vector field. However, a perturbation analysis in terms of spherical harmonics revealed that the fluctuations cannot be radial (unless the magnetic charge assumes an integer value), as discussed in Weinberg’s comprehensive review on magneticallycharged black holes [317]. In fact, the work of Ridgway and Weinberg shows that static black holes with magnetic charge need not even be axially symmetric [278]. Axisymmetric, static black holes without spherical symmetry appear to exist within the pure EYM system and the EYMdilaton model [194].
This shows that static black holes may have considerably more structure than one might expect from the experience with the EM system: Depending on the matter model, they may allow for nontrivial fields outside the horizon and, moreover, they need not be spherically symmetric. Even more surprisingly, there exist static black holes without any rotational symmetry at all.
5.3 The Birkhoff theorem
The Birkhoff theorem shows that the domain of outer communication of a sphericallysymmetric blackhole solution to the vacuum or the EM equations is static. The result does not apply to many other matter models: dust, fluids, scalar fields, EinsteinVlasov, etc., and it is natural to raise the question for nonAbelian gauge fields. Now, the Einstein YangMills equations have a wellposed Cauchy problem, so one needs to make sure that the constraint equations admit nonstationary solutions: Bartnik [11] has indeed proved existence of such initial data. The problem has also been addressed numerically in [328, 329], where sphericallysymmetric solutions of the EYM equations describing the explosion of a gauge boson star or its collapse to a Schwarzschild black hole have been found. A systematic study of the problem for the EYM system with arbitrary gauge groups was performed by Brodbeck and Straumann [39]. Extending previous results of Künzle [205] (see also [206, 207]), the authors of [39] were able to classify the principal bundles over spacetime, which — for a given gauge group — admit SO(3) as symmetry group, acting by bundle automorphisms. It turns out that the Birkhoff theorem can be generalized to bundles, which admit only SO(3)invariant connections of Abelian type. We refer to [39] for the precise formulation of the statement in terms of Stiefel diagrams, and to [33, 34, 138, 255] for a classification of EYM solitons. The results in [104, 13] concerning particlelike EYM solutions are likely to be relevant for the corresponding blackhole problem, but no detailed studies of this exist so far.
5.4 The staticity problem
Going back one step further in the left half of the classification scheme displayed in Figure 3, one is led to the question of whether all black holes with nonrotating horizon are static. For nondegenerate EM black holes this issue was settled by Sudarsky and Wald [302, 303, 84],^{7} while the corresponding vacuum problem was solved quite some time ago [143]; the degenerate case remains open. Using a slightly improved version of the argument given in [143], the staticity theorem can be generalized to selfgravitating stationary scalar fields and scalar mappings [152] as, for instance, the EinsteinSkyrme system. (See also [158, 149, 160], for more information on the staticity problem). It should also be noted that the proof given in [152] works under less restrictive topological assumptions, since it does not require the global existence of a twist potential.
While the vacuum and the scalar staticity theorems are based on differential identities and integration by parts, the approach due to Sudarsky and Wald takes advantage of the ADM formalism and the existence of a maximal slicing [84]. Along these lines, the authors of [302, 303] were able to extend the staticity theorem to topologicallytrivial nonAbelian blackhole solutions. However, in contrast to the Abelian case, the nonAbelian version applies only to configurations for which either all components of the electric YangMills charge or the electric potential vanish asymptotically. This leaves some room for stationary black holes, which are nonrotating and not static. Moreover, the theorem implies that such configurations must be charged. On a perturbative level, the existence of these charged, nonstatic black holes with vanishing total angular momentum was established in [38].
5.5 Rotating black holes with hair
So far we have addressed the ramifications occurring on the “nonrotating half” of the classification diagram of Figure 3: We have argued that nonrotating black holes need not be static; static ones need not be spherically symmetric; and sphericallysymmetric ones need not be characterized by a set of global charges. The righthandside of the classification scheme has been studied less intensively so far. Here, the obvious questions are the following: Are all stationary black holes with rotating Killing horizons axisymmetric (rigidity)? Are the stationary and axisymmetric Killing fields orthogonallytransitive (circularity)? Are the circular black holes characterized by their mass, angular momentum and global charges (nohair)?
Let us start with the first issue, concerning the generality of the strong rigidity theorem (SRT). The existence of a second Killing vector field to the future of a bifurcation surface can be established by solving a characteristic Cauchy problem [107], which makes it clear that axial symmetry will hold for a large class of matter models satisfying the, say, dominant energy condition.
The counterpart to the staticity problem is the circularity problem: As general nonrotating black holes are not static, one expects that the axisymmetric ones need not be circular. This is, indeed, the case: While circularity is a consequence of the EM equations and the symmetry properties of the electromagnetic field, the same is not true for the EYM system. In the Abelian case, the proof rests on the fact that the field tensor satisfies F(k, m) = (_{*}F)(k, m) = 0, k and m being the stationary and the axial Killing field, respectively; for YangMills fields these conditions do no longer follow from the field equations and their invariance properties (see Section 8.1 for details). Hence, the familiar Papapetrou ansatz for a stationary and axisymmetric metric is too restrictive to take care of all stationary and axisymmetric degrees of freedom of the EYM system. However, there are other matter models for which the Papapetrou metric is sufficiently general: the proof of the circularity theorem for selfgravitating scalar fields is, for instance, straightforward [150]. Recalling the key simplifications of the EM equations arising from the (2+2)splitting of the metric in the Abelian case, an investigation of noncircular EYM equations is expected to be rather awkward. As rotating black holes with hair are most likely to occur already in the circular sector (see the next paragraph), a systematic investigation of the EYM equations with circular constraints is needed as well.
The static subclass of the circular sector was investigated in studies by Kleihaus and Kunz (see [194] for a compilation of the results). Since, in general, staticity does not imply spherical symmetry, there is a possibility for a static branch of axisymmetric black holes without spherical symmetry. Using numerical methods, Kleihaus and Kunz have constructed blackhole solutions of this kind for both the EYM and the EYMdilaton system [192]. The related axisymmetric soliton solutions without spherical symmetry were previously obtained by the same authors [190, 191]; see also [193] for more details. The new configurations are purely magnetic and parameterized by their winding number and the node number of the relevant gauge field amplitude. In the formal limit of infinite node number, the EYM black holes approach the ReissnerNordström solution, while the EYMdilaton black holes tend to the GibbonsMaeda black hole [126, 131]. The solutions themselves are neutral and not spherically symmetric; however, their limiting configurations are charged and spherically symmetric. Both the soliton and the blackhole solutions of Kleihaus and Kunz are unstable and may, therefore, be regarded as gravitating sphalerons and black holes inside sphalerons, respectively.
Existence of slowly rotating regular blackhole solutions to the EYM equations was established in [38]. Using the reduction of the EYM action in the presence of a stationary symmetry reveals that the perturbations giving rise to nonvanishing angular momentum are governed by a selfadjoint system of equations for a set of gauge invariant fluctuations [35]. With a soliton background, the solutions to the perturbation equations describe charged, rotating excitations of the BartnikMcKinnon solitons [14]. In the blackhole case the excitations are combinations of two branches of stationary perturbations: The first branch comprises charged black holes with vanishing angular momentum,^{8} whereas the second one consists of neutral black holes with nonvanishing angular momentum. (A particular combination of the charged and the rotating branch was found in [312].) In the presence of bosonic matter, such as Higgs fields, the slowly rotating solitons cease to exist, and the two branches of blackhole excitations merge to a single one with a prescribed relation between charge and angular momentum [35]. More information about the EYMHiggs system can be found in [209, 254].
6 Stationary Spacetimes
For physical reasons, the blackhole equilibrium states are expected to be stationary. Spacetimes admitting a Killing symmetry exhibit a variety of interesting features, some of which will be discussed in this section. In particular, the existence of a Killing field implies a canonical local 3+1 decomposition of the metric. The projection formalism arising from this structure was developed by Geroch in the early seventies [125, 124], and can be found in Chapter 16 of the book on exact solutions by Kramer et al. [199].
A slightly different, rather powerful approach to stationary spacetimes is obtained by taking advantage of their KaluzaKlein (KK) structure. As this approach is less commonly used in the present context, we will discuss the KK reduction of the EinsteinHilbert(Maxwell) action in some detail, the more so as this yields an efficient derivation of the Ernst equations and the Mazur identity. Moreover, the inclusion of nonAbelian gauge fields within this framework [35] reveals a decisive structural difference between the EinsteinMaxwell (EM) and the EinsteinYangMills (EYM) system.
6.1 Reduction of the EinsteinHilbert action
By definition, a stationary spacetime (M, g) admits an asymptoticallytimelike Killing field, that is, a vector field k with L_{k}g = 0, L_{k} denoting the Lie derivative with respect to k. At least locally, M has the structure Σ × G, where G ≈ ∝ denotes the onedimensional group generated by the Killing symmetry, and Σ is the threedimensional quotient space M/G. A stationary spacetime is called static, if the integral trajectories of k are orthogonal to Σ.
The notation t suggests that t is a time coordinate, g(∂t, ∂t) < 0, but this restriction does not play any role in the local form of the equations that we are about to derive. Similarly the local calculations that follow remain valid regardless of the causal character of k, provided that k is not null everywhere, and then one only considers the region where g(k, k) ≡ −V does not change sign. On any connected component of this region k is either spacelike or timelike, as determined by the sign of V, and then the metric ḡ is Lorentzian, respectively Riemannian, there. In any case, both the parameterization of the metric and the equations become singular at places where V has zeros, so special care is required wherever this occurs.
6.2 The coset structure of vacuum gravity
6.3 Stationary gauge fields
The reduction of the EinsteinHilbert action in the presence of a Killing field yields a σmodel, which is effectively coupled to threedimensional gravity. While this structure is retained for the EM system, it ceases to exist for selfgravitating nonAbelian gauge fields. In order to perform the dimensional reduction for the EM and the EYM equations, we need to recall the notion of a symmetric gauge field.
In mathematical terms, a gauge field (with gauge group G, say) is a connection in a principal bundle P(M, G) over spacetime M. A gauge field is symmetric with respect to the action of a symmetry group S of M, if it is described by an Sinvariant connection on P(M, G). Hence, finding the symmetric gauge fields involves the task of classifying the principal bundles P(M, G), which admit the symmetry group S, acting by bundle automorphisms. This program was carried out by Brodbeck and Straumann for arbitrary gauge and symmetry groups [33], (see also [34, 39]), generalizing earlier work of Harnad et al. [138], Jadczyk [181] and Künzle [207].
 (i)
The only perturbations of the static, purely magnetic EYM solutions, which can contribute the ADM angular momentum are the purely nonstatic, purely electric ones, δa and δϕ.
 (ii)
In firstorder perturbation theory, the relevant fluctuations, δa and δϕ, decouple from the remaining metric and matter perturbations.
The second observation follows from the fact that the magnetic YangMills equation (6.18) and the Einstein equations for V and ḡ become background equations, since they contain no linear terms in δa and δϕ. Therefore, the purely electric, nonstatic perturbations are governed by the twist equation (6.17) and the electric YangMills equation (obtained from variations of S_{eff} with respect to ϕ).
Using Eq. (6.17) to introduce the twist potential Y, the fluctuation equations for the firstorder quantities δY and δϕ assume the form of a selfadjoint system [35]. Considering perturbations of sphericallysymmetric configurations, one can expand δY and δϕ in terms of isospin harmonics. In this way one obtains a SturmLiouville problem, the solutions of which reveal the features mentioned in the last paragraph of Section 5.5 [38].
6.4 The stationary EinsteinMaxwell system
7 Some Applications
The σmodel structure is responsible for various distinguished features of the stationary EM system and related selfgravitating matter models. This section is devoted to a brief discussion of some applications. We show how the Mazur identity [230], the quadratic mass formulae [153] and the IsraelWilsonPerjés class of stationary black holes [179, 267] arise from the σmodel structure of the stationary field equations.
7.1 The Mazur identity
In the presence of a second Killing field, the EM equations (6.26) experience further, considerable simplifications, which will be discussed later. In this section we will not yet require the existence of an additional Killing symmetry. The Mazur identity [230], which is the key to the uniqueness theorem for the KerrNewman metric [228, 229], is a consequence of the coset structure of the field equations. Note, however, that while the derivation of the general form of this identity only requires one Killing vector, its application to the uniqueness argument uses two; we will return to this issue shortly.
Of decisive importance to the uniqueness proof for the KerrNewman metric is the fact that the righthand side of the above relation is nonnegative. In order to achieve this one needs two Killing fields: The requirement that Φ be represented in the form gg^{†} forces the reduction of the EM system with respect to a spacelike Killing field; otherwise the coset is SU(2, 1)/S(U(1, 1) × U(1)), which is not of the desired form. As a consequence of the spacelike reduction, the threemetric ḡ is not Riemannian, and the righthand side of Eq. (7.3) is indefinite, unless the matrix valued oneform \({\mathcal M}\) is spacelike. This is the case if there exists a timelike Killing field with L_{k}Φ = 0, implying that the currents are orthogonal to \(k:{\mathcal J}\left(k \right) = {i_k}{\Phi ^{ 1}}d\Phi = {\Phi ^{ 1}}{L_k}\Phi = 0\). The reduction of Eq. (7.3) with respect to the second Killing field and the integration of the resulting expression will be discussed in Section 8.
7.2 Mass formulae
The stationary vacuum Einstein equations describe a twodimensional σmodel coupled to threedimensional gravity. The target manifold is the pseudosphere SO(2, 1)/SO(2) ≈ SU(1, 1)/U(1), which is parameterized in terms of the norm and the twist potential of the Killing field (see Section 6.2). The symmetric structure of the target space persists for the stationary EM system, where the fourdimensional coset, SU(2, 1)/S(U(1, 1) × U(1)), is represented by a hermitian matrix Φ, comprising the two electromagnetic scalars, the norm of the Killing field and the generalized twist potential (see Section 6.4).
The coset structure of the stationary field equations is shared by various selfgravitating matter models with massless scalars (moduli) and Abelian vector fields. For scalar mappings into a symmetric target space \(\bar {G}/\bar {H}\), say, Breitenlohner et al. [31] have classified the models admitting a symmetry group, which is sufficiently large to comprise all scalar fields arising on the effective level^{10} within one coset space, G/H. A prominent example of this kind is the EMdilatonaxion system, which is relevant to N = 4 supergravity and to the bosonic sector of fourdimensional heterotic string theory: The pure dilatonaxion system has an SL(2, ℝ) symmetry, which persists in dilatonaxion gravity with an Abelian gauge field [114]. Like the EM system, the model also possesses an SO(1, 2) symmetry, arising from the dimensional reduction with respect to the Abelian isometry group generated by the Killing field. However, Gal’tsov and Kechkin [116, 117] have shown that the full symmetry group is larger than SL(2, ℝ) × SO(1, 2): The target space for dilatonaxion gravity with a U(1) vector field is the coset SO(2, 3)/(SO(2) × SO(1, 2)) [113]. Using the fact that SO(2, 3) is isomorphic to Sp(4,ℝ), Gal’tsov and Kechkin [118] were also able to give a parametrization of the target space in terms of 4 × 4 (rather than 5 × 5) matrices. The relevant coset space was shown to be Sp(4, ℝ)/U(1, 1); for the generalization to the dilatonaxion system with multiple vector fields we refer to [119, 121].
In the “extreme” case, the Bogomol’nyiPrasadSommerfield (BPS) bound [128] for the static EMdilatonaxion system, 0 = M^{2} + D^{2} + A^{2}−Q^{2}−P^{2}, was previously obtained by constructing null geodesics of the target space [86]. For sphericallysymmetric configurations with nondegenerate horizons (κ ≡ 0), Eq. (7.5) was derived by Breitenlohner et al. [31]. In fact, many of the sphericallysymmetric blackhole solutions with scalar and vector fields [126, 131, 122] are known to fulfill Eq. (7.5), where the lefthand side is expressed in terms of the horizon radius (see [120] and references therein). Using the generalized first law of blackhole thermodynamics, Gibbons et al. [130] obtained Eq. (7.5) for sphericallysymmetric solutions with an arbitrary number of vector and moduli fields.
7.3 The IsraelWilsonPerjés class
Nonstatic members of the IsraelWilsonPerjés class were constructed as well [179, 267]. However, these generalizations of the MP multiblackhole solutions share certain unpleasant properties with NUT spacetime [252] (see also [32, 237]). In fact, the results of [81] (see [139, 78, 154] for previous results) suggest that — except the MP solutions — all configurations obtained by the IsraelWilsonPerjés technique either fail to be asymptotically flat or have naked singularities.
8 Stationary and Axisymmetric Spacetimes
The presence of two Killing symmetries yields a considerable simplification of the field equations. In fact, for certain matter models the latter become completely integrable [219], provided that the Killing fields satisfy the orthogonalintegrability conditions. Spacetimes admitting two Killing fields provide the framework for both the theory of colliding gravitational waves and the theory of rotating black holes [56]. Although dealing with different physical subjects, the theories are mathematically closely related. We refer the reader to Chandrasekhar’s comparison between corresponding solutions of the Ernst equations [55].
This section reviews the structure of the stationary and axisymmetric field equations. We start by recalling the circularity problem. It is argued that circularity is not a generic property of asymptoticallyflat, stationary and axisymmetric spacetimes. However, if the symmetry conditions for the matter fields do imply circularity, then the reduction with respect to the second Killing field simplifies the field equations drastically. The systematic derivation of the KerrNewman metric and the proof of its uniqueness provide impressive illustrations of this fact.
8.1 Integrability properties of Killing fields
 (i)
Show that dg(m, ω_{k}) = 0 implies g(m, ω_{k}) =0.
 (ii)Establish m Λ k Λ T(k) = 0 from the stationary and axisymmetric matter equations.
 (i)
Since g(m, ω_{k}) is a function, it is locally constant if its derivative vanishes. As m vanishes on the rotation axis, this implies g(m, ω_{k}) = 0 in every connected domain of spacetime intersecting the axis. (At this point it is worthwhile to recall that the corresponding step in the staticity theorem requires more effort: Concluding from dω_{k} = 0 that ω_{k} vanishes is more involved, since ω_{k} is a oneform. However, using the Stokes’ theorem to integrate an identity for the twist [152] shows that a strictly stationary — not necessarily simply connected — domain of outer communication must be static if ω_{k} is closed. While this proves the staticity theorem for vacuum and selfgravitating scalar fields [152], it does not solve the electrovacuum case. It should be noted that in the context of the proof of uniqueness the strictly stationary property follows from staticity [72] and not the other way around (compare Figure 3).
 (ii)While m ∧ k ∧T(k) = 0 follows from the symmetry conditions for electromagnetic fields [43] and for scalar fields [150], it cannot be established for nonAbelian gauge fields [152]. This implies that the usual foliation of spacetime used to integrate the stationary and axisymmetric Maxwell equations is too restrictive to treat the EYM system. This is seen as follows: In Section 6.3 we have derived the formula (6.17). By virtue of Eq. (6.3) this becomes an expression for the derivative of the twist in terms of the electric YangMills potential ϕ_{k} (defined with respect to the stationary Killing field k) and the magnetic oneform \({i_k}{\ast}F = V\bar {\ast}(\bar F + {\phi _k}f)\):where \(\hat{{\rm{tr}}}\) is a suitably normalized trace (see Eq. (6.15)). Contracting this relation with the axial Killing field m, and using again the fact that the Lie derivative of ω_{k} with respect to m vanishes, yields immediately$${\rm{d}}[{\omega _k} + 4\widehat{{\rm{tr}}}\,({\phi _k}\,{i_k}*F)] = 0,$$(8.3)The difference between the Abelian and the nonAbelian case is due to the fact that the Maxwell equations automatically imply that the (km)component of *F vanishes, whereas this does not follow from the YangMills equations. In fact, the Maxwell equation d * F = 0 and the symmetry property L_{k} * F = *L_{k}F = 0 imply the existence of a magnetic potential, dψ = (*F)(k, ·), thus, (*F)(k, m) = i_{m}dψ = L_{m}ψ = 0. Moreover, the latter do not imply that the Lie algebra valued scalars ϕ_{k} and (*F) (k, m) are orthogonal. Hence, circularity is an intrinsic property of the EM system, whereas it imposes additional requirements on nonAbelian gauge fields.$${\rm{d}}\,g(m,\,{\omega _k}) = 0 \Leftrightarrow \widehat{{\rm{tr}}}\,({\phi _k}\,(*F)\,(k,\,m))] = 0.$$(8.4)
 (i)
8.2 Twodimensional elliptic equations

(Σ, ḡ) is a static pseudoRiemannian threedimensional manifold with metric \(\bar g =  {\rho ^2}{\rm{d}}{t^2} + \tilde g\);

the connection a is orthogonal to the twodimensional Riemannian manifold \(\tilde {\Sigma}, \tilde {g}\), that is, a = a_{t} dt;

the functions a_{t} and \({\tilde {g}_{ab}}\) do not depend on the coordinates t and ψ.
8.3 The Ernst equations
8.3.1 A derivation of the KerrNewman metric
The solution derived so far is the “conjugate” of the KerrNewman solution [56]. In order to obtain the KerrNewman metric itself, one has to perform a rotation in the tφplane: The spacetime metric is invariant under t → φ, φ → −t, if e^{−2λ}, a_{t} and e^{2h} are replaced by αe^{−2λ}, α^{−1}a_{t} and αe^{2h}, where \(\alpha : = a_t^2  {e^{4\lambda}}{\rho ^2}\). This additional step in the derivation of the KerrNewman metric is necessary because the Ernst potentials were defined with respect to the axial Killing field ∂_{φ}. If, on the other hand, one uses the stationary Killing field ∂_{t}, then the Ernst equations are singular at the boundary of the ergoregion.
8.4 The uniqueness theorem for the KerrNewman solution
In order to establish uniqueness of the KerrNewman metric among the stationary and axisymmetric blackhole configurations, one has to show that two solutions of the Ernst equations (8.19) are equal if they are subject to blackhole boundary conditions on \(\delta {\mathcal S}\), where \({\mathcal S}\) is the halfplane \({\mathcal S} = \{(\rho, z)\vert \rho \geq 0\}\). Carter proved nonexistence of linearized vacuum perturbations near Kerr by means of a divergence identity [45], which Robinson generalized to electrovacuum spacetimes [279].
8.4.1 Divergence identities
Considering two arbitrary solutions of the Ernst equations, Robinson was able to construct an identity [280], the integration of which proved the uniqueness of the Kerr metric. The complicated nature of the Robinson identity dashed the hope of finding the corresponding electrovacuum identity by trial and error methods (see, e.g., [47]). The problem was eventually solved when Mazur [228, 230] and Bunting [41] independently derived divergence identities useful for the problem at hand. Bunting’s approach, applying to a general class of harmonic mappings between Riemannian manifolds, yields an identity, which enables one to establish the uniqueness of a harmonic map if the target manifold has negative curvature. We refer the reader to Sections 3.2.5 and 8.4.2 (see also [49]) for discussions related to Bunting’s method.

the fact that the integrand on the righthand side is nonnegative, and

the fact that the boundary at infinity on the lefthand side vanishes for two solutions with the same mass, electric charge and angular momentum, and

the expectation that the integral over the axis and horizons, where the integrand becomes singular, vanishes for blackhole configurations with the same quotientspace structure.
In order to establish that ρ Trace (dΨ) = 0 on the boundary \(\partial {\mathcal S}\) of the halfplane,^{11} one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. One expects that ρ Trace (dΨ) vanishes on the horizon, the axis and at infinity, provided that the solutions have the same mass, charge and angular momentum, but no complete analysis of this has been presented in the literature; see [318] for some partial results. Fortunately, the supplementary difficulties arising from the need to control the derivatives of the fields disappear when the distancefunction approach described in the next Section 8.4.2 is used.
8.4.2 The distance function argument
Using this observation, the key to uniqueness is provided by the following nonstandard version of the maximum principle:
Footnotes
 1.
 2.
 3.
Nonexistence of certain static nbody configurations (possibly, but not necessarily, black holes) was established in [21, 20]). These results rely on the positive energy theorem and exclude, in particular, suitably regular configurations with a reflection symmetry across a noncompact surface, which is disjoint from the matter regions.
 4.
 5.
 6.
It should be noted that, although formulated for 4dimensional spacetimes, the results in [84] remain valid without changes in higherdimensional spacetimes.
 7.
 8.
 9.
 10.
In addition to the actual scalar fields, the effective action comprises two gravitational scalars (the norm and the generalized twist potential) and two scalars for each stationary Abelian vector field (electric and magnetic potentials).
 11.
Notes
Acknowledgments
MH takes pleasure in thanking Othmar Brodbeck, Gary Gibbons, Domenico Giulini, Dieter Maison, Gernot Neugebauer, Norbert Straumann, Michael Volkov and Robert Wald for helpful discussions. PTC acknowledges bibliographical advice from Michael Volkov.
This work was supported by SNSF Grant P2141840.94 (MH), by a Polish Ministry of Science and Higher Education grant Nr N N201 372736 grant (PTC), by projects PTDC/MAT/108921/2008 and CERN/FP/116377/2010, and by CAMSDG, through FCT plurianual funding (JLC).
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