Stationary Black Holes: Uniqueness and Beyond
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Abstract
The spectrum of known black hole solutions to the stationary Einstein equations has increased in an unexpected way during the last decade. In particular, it has turned out that not all black hole equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electrovacuum black hole spacetimes ceases to exist in selfgravitating nonlinear field theories. This text aims to review some of the recent developments and to discuss them in the light of the uniqueness theorem for the EinsteinMaxwell system.
Keywords
Black Hole Black Hole Solution Stationary Black Hole Killing Field Ernst Potential1 Introduction
1.1 General
Our conception of black holes has experienced several dramatic changes during the last two hundred years: While the “dark stars” of Michell [134] and Laplace [123] were merely regarded as peculiarities of Newton’s law of gravity and his corpuscular theory of light, black holes have nowadays achieved the status of astrophysical objects, being as real as ordinary stars.^{1} In fact, today’s technology is sufficiently advanced to enable us, for the first time, to actually detect black holes. Although the observations are necessarily indirect, the evidence for both stellar and galactic black holes has become very compelling lately.^{2}
The theory of black holes was initiated by the pioneering work of Chandrasekhar [34], [35] in the early 1930s. Computing the Chandrasekhar limit for neutron stars [2], Oppenheimer and Snyder [141], and Oppenheimer and Volkoff [142] were able to demonstrate that black holes present the ultimate fate of sufficiently massive stars. Modern black hole physics started with the advent of relativistic astrophysics, in particular with the discovery of the pulsars in 1967. (The geometry of the Schwarzschild solution [157], [158] was, for instance, not understood for almost half a century; the misconception of the “Schwarzschild singularity” was retained until the late 1950s.)
One of the most intriguing outcomes of the mathematical theory of black holes is the uniqueness theorem, applying to the stationary solutions of the EinsteinMaxwell equations. Asserting that all electrovac black hole spacetimes are characterized by their mass, angular momentum and electric charge, the theorem b ears a striking resemblance to 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 black hole mass variation formula [3] and the area increase theorem [84], 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 [83] is equal to the surface gravity of the black hole.^{3}
The proof of the celebrated uniqueness theorem, conjectured by Israel, Penrose and Wheeler in the late sixties, has been completed during the last three decades (see, e.g. [39] and [40] for reviews). Some open gaps, notably the electrovac staticity theorem [167], [168] and the topology theorem [57], [58], [44], have been closed recently (see [40] for new results). The beauty of the theorem provided support for the expectation that the stationary black hole solutions of other selfgravitating matter fields are also parametrized 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 [4], a variety of new black hole configurations which violate the generalized nohair conjecture have been found. These include, for instance, nonAbelian black holes [174], [122], [9], and black holes with Skyrme [50], [97], Higgs [12] or dilaton fields [124], [77].
In fact, black hole solutions with hair were already known before 1989: The first example was the Bekenstein solution [7], [8], describing a conformally coupled scalar field in an extreme ReissnerNordström spacetime. Since the horizon has vanishing surface gravity,^{4} and since the scalar field is unbounded on the horizon, the status of the Bekenstein solution gives still rise to some controversy [169]. In 1982, Gibbons found a new black hole solution within a model occurring in the low energy limit of N = 4 supergravity [73]. The Gibbons solution, describing a ReissnerNordström spacetime with a nontrivial dilaton field, must be considered the first flawless black hole solution with hair.
While the above counterexamples to the nohair conjecture consist in static, spherically symmetric configurations, more recent investigations have revealed that static black holes are not necessarily spherically symmetric [115]; in fact, they need not even be axisymmetric [150]. Moreover, some new studies also indicate that nonrotating black holes need not be static [22]. The rich spectrum of stationary black hole configurations demonstrates that the matter fields are by far more critical to the properties of black hole 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 [139]. Since this property is not shared by models with nonAbelian gauge fields [19], it is, with hindsight, not too surprising that the EinsteinYangMills system admits black holes with hair.
There exist, however, 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 vector 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. [15], [45], [67], and references therein).
1.2 Organization
The purpose of this text is to review some of the most important features of black hole spacetimes. Since the investigation of dynamical problems lies beyond the scope of this report, we shall mainly be concerned with stationary situations. Moreover, the concept of the event horizon requires asymptotic flatness. (Black hole solutions with cosmological constant are, therefore, not considered in this text.^{5}) Hence, we are dealing with asymptotically flat, stationary black configurations of selfgravitating classical matter fields.
The emphasis is given to the recent developments in the field and to the fundamental concepts. For detailed introductions into the subject we refer to Chandrasekhar’s book on the mathematical theory of black holes [37], the classic by Hawking and Ellis [84], Carter’s review [33], and chapter 12 of Wald’s book [178]. Some of the issues which are not raised in this text can be found in [87], others will be included in a future version.
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 recalling the main steps involved in the uniqueness theorem for electrovacuum black hole spacetimes (Sect. 2). The classification scheme obtained in this way is then reexamined in the light of the solutions which are not covered by nohair theorems, such as the EinsteinYangMills black holes (Sect. 3).
The second part reviews the main structural properties of stationary black hole spacetimes. In particular, we recall the notion of a Killing horizon, and discuss the dimensional reduction of the field equations in the presence of a Killing symmetry in some detail (Sect. 4). For a variety of matter models, such as selfgravitating Abelian gauge fields, the reduction yields a σmodel with symmetric target manifold, effectively coupled to threedimensional gravity. Particular applications of this distinguished structure are the Mazur identity, the quadratic mass formulas and the Israel Wilson class (Sect. 5).
The third part is devoted to stationary and axisymmetric black hole spacetimes (Sect. 6). 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 yields a boundary value problem on a fixed, twodimensional background, provided that the field equations assume the coset structure on the effective level. As an application we recall the uniqueness proof for the KerrNewman metric.
2 Classification of Stationary Electrovac Black Hole SpaceTimes
2.1 Rigidity, Staticity and Circularity
At the basis of the classification of stationary electrovac black hole spacetimes lies Hawking’s strong rigidity theorem (SRT) [84].
It relates the global concept of the event horizon to the independently defined — and logically distinct — local notion of the Killing horizon: Requiring that the fundamental matter fields obey well behaved hyperbolic equations, and that the stressenergy tensor satisfies the weak energy condition,^{7} the first part of the SRT asserts that the event horizon of a stationary black hole spacetime is a Killing horizon.^{8} The latter is called nonrotating if it is generated by the stationary Killing field, and rotating otherwise. In the rotating case, the second part of the SRT implies that spacetime is axisymmetric.^{9}
The subdivision provided by the SRT is, unfortunately, not sufficient to apply the uniqueness theorems for the ReissnerNordström and the KerrNewman metric: The latter are based on the stronger requirements that the domain of outer communication (DOC) is either static (nonrotating case) or circular (axisymmetric case). Hence, in both cases one has to establish the Frobenius integrability conditions for the Killing fields beforehand (staticity and circularity theorems).
The circularity theorem, due to Carter [27], and Kundt and Trümper [118], implies that the metric of a vacuum or electrovac spacetime can, without loss of generality, be written in the wellknown Papapetrou (2+2)split. The staticity theorem, implying that the stationary Killing field of a nonrotating, electrovac black hole spacetime is hypersurface orthogonal, is more involved than the circularity problem: First, one has to establish strict stationarity, that is, one needs to exclude ergoregions. This problem, first discussed by Hajicek [78], [79], and Hawking and Ellis [84], was solved only recently by Sudarsky and Wald [167], [168], assuming a foliation by maximal slices.^{10} If ergoregions are excluded, it still remains to prove that the stationary Killing field satisfies the Frobenius integrability condition. In the vacuum case, this was achieved by Hawking [82], who was able to extend a theorem due to Lichnerowicz [126] to black hole spacetimes. In the presence of Maxwell fields the problem was solved only a couple of years ago [167], [168], by means of a generalized version of the first law of black hole physics.
2.2 The Uniqueness Theorems
The main task of the uniqueness program is to show that the static electrovac black hole spacetimes are described by the ReissnerNordström metric, while the circular ones are represented by the KerrNewman metric. In combination with the SRT and the staticity and circularity theorems, this implies that all stationary black hole solutions to the EM equations (with nondegenerate horizon) are parametrized by their mass, angular momentum and electric charge.
In the nonrotating case it was Israel who, in his pioneering work, showed that both static vacuum [99] and electrovac [100] black hole spacetimes are spherically symmetric. Israel’s ingenious method, based on diferential identities and Stokes’ theorem, triggered a series of investigations devoted to the static uniqueness problem (see, e.g. [137], [138], [151], [153]). Later on, Simon [160], Bunting and MasoodulAlam [26], and Ruback [154] were able to improve on the original method, taking advantage of the positive energy theorem.^{11} (The “latest version” of the static uniqueness theorem can be found in [129].)
The key to the uniqueness theorem for rotating black holes exists in Carter’s observation that the stationary and axisymmetric EM equations reduce to a twodimensional boundary value problem [29] (See also [31] and [33].). In the vacuum case, Robinson was able to construct an amazing identity, by virtue of which the uniqueness of the Kerr metric followed [152]. The uniqueness problem with electromagnetic fields remained open until Mazur [131] and, independently, Bunting [25] were able to obtain a generalization of the Robinson identity in a systematic way: The Mazur identity (see also [132], [133]) 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) x U(2)) (provided that the dimensional reduction of the EM action is performed with respect to the axial Killing field^{12}). Within this approach, the Robinson identity looses 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).
2.3 Black Holes with Degenerate Horizons
The uniqueness theorem outlined above applies exclusively to Killing horizons with nonvanishing surface gravity. In fact, the multi black hole solutions of Papapetrou [143] and Majumdar [128] illustrate that stationary EM black holes with degenerate Killing horizons need not belong to the KerrNewman family. In order to complete the classification of stationary electrovac black hole spacetimes one has to include the PapapetrouMajumdar solutions, and to establish their uniqueness amongst the stationary configurations with degenerate, nonconnected horizons. Some progress toward this goal was recently achieved by Chruściel and Nadirashvili [42]; a complete proof is, however, not yet available (see also [90] for more information).
3 Beyond EinsteinMaxwell
The purpose of this section is to estimate the generality of the various steps leading to the classification of electrovac black hole spacetimes. In particular, we shall argue that virtually all theorems displayed in Fig. 1 cease to exist in the presence of nonAbelian gauge fields. Unfortunately, this implies that we are far from having a classification of all stationary black hole spacetimes.
3.1 Spherically Symmetric Black Holes with Hair
Requiring spherical symmetry, the task to prove the nohair theorem for the EinsteinMaxwell (EM) system becomes almost trivial. However, not even this part of the uniqueness proof can be generalized: The first black hole solution demonstrating the failure of the nohair conjecture was obtained by Gibbons in 1982 [73] within EMdilaton theory.^{13} The fact that the Gibbons solution carries no dilatonic charge makes it asymptotically indistinguishable from a ReissnerNordstruom black hole with the same mass and electric charge. However, since the latter is not a consistent solution of the EMdilaton equations, one might expect that — within a given matter model — the stationary black hole solutions are still characterized by a set of global charges (generalized nohair conjecture). In fact, the Gibbons black hole supports the generalized nohair conjecture; its uniqueness within EMdilaton theory was established by MasoodulAlam in 1992 [130].
However, neither the original nor the generalized nohair conjecture are correct. For instance, the latter fails to be valid within EinsteinYangMills (EYM) theory: According to the generalized version, 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 [174], [122], [9] found a family of static black hole solutions with vanishing YangMills charges.^{14} Since these solutions are asymptotically indistinguishable from the Schwarzschild solution, and since the latter is a particular solution of the EYM equations, the nonAbelian black holes violate the generalized nohair conjecture.
As the nonAbelian black holes are not stable [166], [186] [179],^{15} one might adopt the view that they do not present actual threats to the generalized nohair conjecture. However, during the last years, various authors have found stable black holes which are not characterized by a set of asymptotic flux integrals: For instance, there exist stable black hole solutions with hair to the static, spherically symmetric EinsteinSkyrme equations [50], [92], [93], [97] and to the EYM equations coupled to a Higgs triplet [12], [14], [180], [1].^{16} 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 globally regular, asymptotically flat, static vacuum solution to the Einstein equations with finite energy is Minkowski spacetime. (This result is obtained from the positive mass theorem and the Komar expression for the total mass of an asymptotically flat, stationary spacetime; see, e.g. [74] or [88].)

Second, both Deser’s energy argument [48] and Coleman’s scaling method [46] show that there exist no pure YM solitons in flat spacetime.

Third, the EM system admits no soliton solutions. (This follows by applying Stokes’ theorem to the static Maxwell equations; see, e.g. [87].)

Finally, Deser [49] 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 Sect. 4).
3.2 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 [99] or electrovac [100] black hole 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.
A few years ago, Lee et al. [125] reanalyzed the stability of the ReissnerNordeström (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 magnetically charged, spherically symmetric black holes with hair, which were also found by numerical means [12], [14], [180], [1].
Motivated by these solutions, Ridgway and Weinberg [149] 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).^{17} In fact, the work of Ridgway and Weinberg shows that static black holes with magnetic charge need not even be axially symmetric [150].^{18}
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.
3.3 The Birkhoff Theorem
The Birkhof theorem implies that the domain of outer communication of a spherically symmetric black hole solution to the vacuum or the EM equations is static. Like its counterpart, the Israel theorem, the Birkhof theorem admits no straightforward extension to arbitrary matter models, such as nonAbleian gauge fields: Numerical investigations have revealed spherically symmetric solutions of the EYM equations which describe the explosion of a gauge boson star or its collapse to a Schwarzschild black hole [185], [186]. A systematic study of the problem for the EYM system with arbitrary gauge groups was performed by Brodbeck and Straumann [23]. Extending previous results due to Künzle [119] (see also [120], [121]), the authors of [23] 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 Birkhof theorem can be generalized to bundles which admit only SO(3) invariant connections of Abelian type.^{19}
3.4 The Staticity Problem
Going back one step further on the left half of the classification scheme displayed in Fig. 1, one is led to the question whether all black holes with nonrotating horizon are static. For the EM system this issue was settled only recently [167], [168],^{20}whereas the corresponding vacuum problem was solved quite some time ago [84]. Using a slightly improved version of the argument given in [84],^{21}. the staticity theorem can be generalized to selfgravitating stationary scalar fields and scalar mappings [88] as, for instance, the EinsteinSkyrme system. (See also [94], [85], [96], for more information on the staticity problem.)
While the vacuum and the scalar staticity theorems are based on differential identities and Stokes’ law, the new approach due to Sudarsky and Wald takes advantage of the ADM formalism and a maximal slicing property [43]. Along these lines, the authors of [167], [168] were also able to extend the staticity theorem to nonAbelian black hole 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. As the asymptotic value of a Lie algebra valued scalar is not a gauge freedom in the nonAbelian case, the EYM staticity theorem leaves some room for stationary black holes which are nonrotating — but not static. Moreover, the theorem implies that these configurations must be charged. On the perturbative level, the existence of these charged, nonstatic black holes with vanishing total angular momentum was recently established by rigorous means [22].
3.5 Rotating Black Holes with Hair
So far we have addressed the ramifications occurring on the “nonrotating half” of the classification diagram shown in Fig. 1: We have argued that nonrotating black holes need not be static, static ones need not be spherically symmetric, and spherically symmetric ones need not be characterized by a set of global charges. The righthandside of the classification scheme has been studied less intensively until now. Here, the obvious questions are the following ones: Are all stationary black holes with rotating Killing horizons axisymmetric (rigidity)? Are the stationary and axisymmetric Killing fields hypersurface orthogonal (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). While earlier attempts to proof the theorem were flawed^{22} and subject to restrictive assumptions concerning the matter fields [84], the recent work of Chruściel [38], [40] has shown that the SRT is basically a geometric feature of stationary spacetimes. It is, therefore, conceivable to suppose that both parts of the theorem — that is, the existence of a Killing horizon and the existence of an axial symmetry in the rotating case — are generic features of stationary black hole spacetimes. (See also [6] for the classification of asymptotically flat spacetimes.)
The counterpart to the staticity problem is the circularity problem: As the nonrotating black holes are, in general, not static, one expects that the axisymmetric ones need not necessarily 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.^{23} 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.^{24} Recalling the enormous simplifications of the EM equations arising from the (2+2)split of the metric in the Abelian case, an investigation of the noncircular EYM equations will 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 recent studies by Kleihaus and Kunz (see [111] 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.^{25} Using numerical methods, Kleihaus and Kunz have constructed black hole solutions of this kind for both the EYM and the EYMdilaton system [115].^{26} The new configurations are purely magnetic and parametrized 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 [73], [76].^{27} Both the soliton and the black hole solutions of Kleihaus and Kunz are unstable and may, therefore, be regarded as gravitating sphalerons and black holes inside sphalerons, respectively.
Slowly rotating regular and black hole solutions to the EYM equations were recently established in [22]. 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 [19]. For a soliton background the solutions to the perturbation equations describe charged, rotating excitations of the BartnikMcKinnon solitons [4]. In the black hole case the excitations are combinations of two branches of stationary perturbations: The first branch comprises charged black holes with vanishing angular momentum,^{28} whereas the second one consists of neutral black holes with nonvanishing angular momentum.^{29} In the presence of bosonic matter, such as Higgs fields, the slowly rotating solitons cease to exist, and the two branches of black hole excitations merge to a single one with a prescribed relation between charge and angular momentum [19].
4 Stationary SpaceTimes
For physical reasons, the black hole 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 3+1 decomposition of the metric. The projection formalism arising from this structure was developed by Geroch in the early seventies [71], [70], and can be found in chapter 16 of the book on exact solutions by Kramer et al. [117].
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 since this yields an efficient derivation of the Ernst equations and the Mazur identity). Moreover, the inclusion of nonAbelian gauge fields within this framework [19] reveals a decisive structural difference between the EinsteinMaxwell (EM) and the EinsteinYangMills (EYM) system. Before discussing the dimensional reduction of the field equations in the presence of a Killing field, we start this section by recalling the concept of the Killing horizon.
4.1 Killing Horizons
 An immediate consequence of the above definition is the fact that ξ and dN are proportional on H[ξ]. (Note that (ξ, dN) = 0, since L_{ ξ }N = 0, and that two orthogonal null vectors are proportional.) This suggests the following definition of the surface gravity, κ,Since the Killing equation implies dN = 2∇_{ ξ }ξ, the above definition shows that the surface gravity measures the extent to which the parametrization of the geodesic congruence generated by ξ is not affine.$${\rm{d}}N =  2\kappa \xi \quad {\rm{on}}\;H\left[ \xi \right]{\rm{.}}$$(1)
 A theorem due to Vishveshwara [172] gives a characterization of the Killing horizon H[ξ] in terms of the twist ω of ξ:^{33} The surface N = (ξ, ξ) = 0 is a Killing horizon if and only if$$\omega = 0\;\;\,and\;\;\,{i_\xi }d\xi \ne \quad 0\;\;\,{\rm{on}}\;\;\,N = 0.$$(2)
 Using general identities for Killing fields^{34} one can derive the following explicit expressions for κ:Introducing the four velocity \(u = \xi /\sqrt {  N} \) for a timelike ξ, the first expression shows that the surface gravity is the limiting value of the force applied at infinity to keep a unit mass at H[ξ] in place: \(\kappa = {\rm{lim}}\left( {\sqrt {  N} \left a \right} \right),\), where a = ∇_{ u }u (see, e.g. [178]).$${\kappa ^2} =  {\left[ {\frac{1}{N}\left( {{\nabla _\xi }\xi ,{\nabla _\xi }\xi } \right)} \right]_{H\left[ \xi \right]}} =  {\left[ {\frac{1}{4}\Delta N} \right]_{H\left[ \xi \right]}}.$$(3)
 Of crucial importance to the zeroth law of black hole physics (to be discussed below) is the fact that the (ξξ)component of the Ricci tensor vanishes on the horizon,This follows from the above expressions for κ and the general Killing field identity 2NR(ξ, ξ) = 4 (∇_{ ξ }ξ, ∇_{ ξ }ξ) — NΔN — 4(ω, ω).$$R\left( {\xi ,\,\xi } \right) = 0\quad {\rm{on}}\;H\left[ \xi \right].$$(4)

(i) Einstein’s equations are fulfilled with matter satisfying the dominant energy condition.

(ii) The domain of outer communications is either static or circular.

(iii) H[ξ] is a bifurcate Killing horizon.
 (i)The original proof of the zeroth law rests on the first assumption [3]. The reasoning is as follows: First, Einstein’s equations and the fact that R(ξ, ξ) vanishes on the horizon (see above), imply that T(ξ, ξ) = 0 on H[ξ]. Hence, the oneform T(ξ)^{35} is perpendicular to ξ and, therefore, spacelike or null on H[ξ]. On the other hand, the dominant energy condition requires that T(ξ) is timelike or null. Thus, T(ξ) is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein’s equations again, ξ∧R(ξ) = 0 on H[ξ]. The result that κ is uniform over the horizon now follows from the general property^{36}$$\xi \wedge {\rm{d}}\kappa =  \xi \wedge R\left( \xi \right)\quad {\rm{on}}\;H\left[ \xi \right].$$(6)
 (ii)By virtue of Eq. (6) and the general Killing field identity dω = * [ξ∧R(ξ)], the zeroth law follows if one can show that the twist oneform is closed on the horizon [147]:While the original proof (i) takes advantage of Einstein’s equations and the dominant energy condition to conclude that the twist is closed, one may also achieve this by requiring that ω vanishes identically,^{37} which then proves the second version of the first zeroth law.^{38}$${\left[ {{\rm{d}}\omega } \right]_{H\left[ \xi \right]}} = 0\; \Rightarrow \;\kappa = {\rm{constant}}\;{\rm{on}}\;H\left[ \xi \right].$$(7)
 (iii)
The third version of the zeroth law, due to Kay and Wald [105], is obtained for bifurcate Killing horizons. Computing the derivative of the surface gravity in a direction tangent to the bifurcation surface shows that κ cannot vary between the nullgenerators. (It is clear that κ is constant along the generators.) The bifurcate horizon version of the zeroth law is actually the most general one: First, it involves no assumptions concerning the matter fields. Second, the work of Rκcz and Wald strongly suggests that all physically relevant Killing horizons are either of bifurcate type or degenerate [146], [147].
4.2 Reduction of the EinsteinHilbert Action
By definition, a stationary spacetime (M, g) admits an asymptotically timelike 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 ∑ × G, where G ≈ IR denotes the onedimensional group generated by the Killing symmetry, and ∑ is the threedimensional quotient space M/G. A stationary spacetime is called static, if the integral trajectories of k are orthogonal to ∑.
4.3 The Coset Structure of Vacuum Gravity
4.4 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 called 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 recently carried out by Brodbeck and Straumann for arbitrary gauge and symmetry groups [17], (see also [18], [23]), generalizing earlier work of Harnad et al. [80], Jadczyk [104] and Künzle [121].

(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 first order perturbation theory the relevant fluctuations, δa and δϕ, decouple from the remaining metric and matter perturbations
Using Eq. (22) to introduce the twist potential Y, the fluctuation equations for the first order quantities δY and δϕ assume the form of a selfadjoint system [19]. Considering perturbations of spherically symmetric 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 Sect. 3.5 [22].
4.5 The Stationary EinsteinMaxwell System
5 Applications of the Coset Structure
The σmodel structure is responsible for various distinguished features of the stationary EinsteinMaxwell (EM) system and related selfgravitating matter models. This section is devoted to a brief discussion of some applications: We argue that the Mazur identity [133], the quadratic mass formulas [89] and the IsraelWilson class of stationary black holes [102], [145] owe their existence to the σmodel structure of the stationary field equations.
5.1 The Mazur Identity
In the presence of a second Killing field, the EM equations (32) 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 [133], which is the key to the uniqueness theorem for the KerrNewman metric [131], [132], is a consequence of the coset structure of the field equations, which only requires the existence of one Killing field.^{46}
Of decisive importance to the uniqueness proof for the KerrNewman metric is the fact that the RHS 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 RHS of Eq. (35) 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: \({\mathcal J}(k) = {i_k}{\Phi ^{  1}}d\Phi = {\Phi ^{  1}}{L_k}\Phi = 0\). The reduction of Eq. (35) with respect to the second Killing field and the integration of the resulting expression will be discussed in Sect. 6.
5.2 Mass Formulae
The stationary vacuum Einstein equations describe a twodimensional σmodel which is effectively coupled to threedimensional gravity. The target manifold is the pseudosphere SO(2,1)/SO(2) ≈ SU(1,1)/U(1), which is parametrized in terms of the norm and the twist potential of the Killing field (see Sect. 4.3). 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 Sect. 4.5).
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 ̄G/̄H, say, Breitenlohner et al. [15] have classified the models admitting a symmetry group which is sufficiently large to comprise all scalar fields arising on the effective level^{48} within one coset space, G/H. A prominent example of this kind is the EMdilatonaxion system, which is relevant to N = 4supergravity and to the bosonic sector of fourdimensional heterotic string theory: The pure dilatonaxion system has an SL(2, IR) symmetry which persists in dilatonaxion gravity with an Abelian gauge field [61]. 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. Gal’tsov and Kechkin [63], [64] have shown that the full symmetry group is, however, larger than SL(2, IR) × SO(1, 2): The target space for dilatonaxion gravity with an U(1) vector field is the coset SO(2, 3)/(SO(2) × SO(1, 2)) [62]. Using the fact that SO(2, 3) is isomorphic to Sp(4, IR), Gal’tsov and Kechkin [65] were also able to give a parametrization of the target space in terms of 4 × 4 (rather than 5 × 5) matrices. The relevant coset was shown to be Sp(4,IR)/U(1, 1).^{49}
In the “extreme” case, the BPS bound [75] for the static EMdilatonaxion system, 0 = M^{2}+D^{2}+A^{2}Q^{2}P^{2}, was previously obtained by constructing the null geodesics of the target space [45]. For spherically symmetric configurations with nondegenerate horizons (κ ≠ 0), Eq. (37) was derived by Breitenlohner et al. [15]. In fact, many of the spherically symmetric black hole solutions with scalar and vector fields [73], [76], [69] are known to fulfill Eq. (37), where the LHS is expressed in terms of the horizon radius (see [67] and references therein). Using the generalized first law of black hole thermodynamics, Gibbons et al. [72] recently obtained Eq. (37) for spherically symmetric solutions with an arbitrary number of vector and moduli fields.
5.3 The IsraelWilson Class
Nonstatic members of the IsraelWilson class were constructed as well [102], [145]. However, these generalizations of the PapapetrouMajumdar multi black hole solutions share certain unpleasant properties with NUT spacetime [140] (see also [16], [136]). In fact, the work of Hartle and Hawking [81], and Chruściel and Nadirashvili [42] strongly suggests that — except the PM solutions — all configurations obtained by the IsraelWilson technique are either not asymptotically Euclidean or have naked singularities. In order to complete the uniqueness theorem for the PM metric among the static black hole solutions with degenerate horizon, it basically remains to establish the equality M = Q under the assumption that the horizon has some degenerate components. Until now, this has been achieved only by requiring that all components of the horizon have vanishing surface gravity and that all “horizon charges” have the same sign [90].
6 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 [127], provided that the Killing fields satisfy the Frobenius conditions. Spacetimes admitting two Killing fields provide the framework for both the theory of colliding gravitational waves and the theory of rotating black holes [37]. Although dealing with different physical subjects, the theories are mathematically closely related. As a consequence of this, various stationary and axisymmetric solutions which have no physical relevance give rise to interesting counterparts in the theory of colliding waves.^{56}
This section reviews the structure of the stationary and axisymmetric field equations. We start by recalling the circularity problem (see also Sect. 2.1 and Sect. 3.5). It is argued that circularity is not a generic property of asymptotically flat, stationary and axisymmetric spacetimes. If, however, 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.
6.1 Integrability Properties of Killing Fields

(i) Show that d(m, ͩ_{ k }) = 0 implies (m, ͩ_{ k }) = 0.

(ii) Establish m ∧ k ∧ T(k) = 0 from the stationary and axisymmetric matter equations.
(i) Since (m, ͩ_{ k }) is a function, it must be constant if its derivative vanishes. As m vanishes on the rotation axis, this implies (m, ͩ_{ k }) = 0 in every 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 dͩ_{ k } = 0 that ͩ_{ k } vanishes is more involved, since ͩ_{ k } is a oneform. However, using Stoke’s theorem to integrate an identity for the twist [88] shows that a strictly stationary — not necessarily simply connected — domain of outer communication must be static if ͩ_{ k } is closed.^{60})
6.2 Boundary Value Formulation

(∑, ḡ) is a static pseudoRiemannian threedimensional manifold with metric ḡ = ρ^{2}dt^{2} + ̃g;

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

the functions a_{ t } and ̃g_{ ab } do not depend on the coordinates t and φ.
6.3 The Ernst Equations
The solution derived in this way is the “conjugate” of the KerrNewman solution [37]. 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 X, a_{ t } and e^{2h} are replaced by kX, k^{1}a_{ t } and ke^{2h}, where k = a _{ t } ^{2} — X^{2}ρ^{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.
6.4 The Uniqueness Theorem for the KerrNewman solution
In order to establish the uniqueness of the KerrNewman metric among the stationary and axisymmetric black hole configurations, one has to show that two solutions of the Ernst equations (67) are equal if they are subject to the same boundary and regularity conditions on \(\partial \mathcal{S}\), where \({\mathcal S}\) is the semistrip \({\mathcal S} = \{ (x,\;y)\left {x \ge 1,\left y \right \le 1\} } \right\) (see Sect. 6.3.) For infinitesimally neighboring solutions, Carter solved this problem for the vacuum case by means of a divergence identity [29], which Robinson generalized to electrovac spacetimes [151].
Considering two arbitrary solutions of the Ernst equations, Robinson was able to construct an identity [152], 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 electrovac identity by trial and error methods.^{68} In fact, the problem was only solved when Mazur [131], [133] and Bunting [25] independently succeeded in deriving the desired divergence identities by using the distinguished structure of the EM equations in the presence of a Killing symmetry. 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.^{69}

the integrand on the RHS is nonnegative.

The LHS vanishes for two solutions with the same mass, electric charge and angular momentum.
The RHS is nonnegative because of the following observations: First, the inner product is definite, and ̃η is a positive volumeform, since ḡ is a Riemannian metric. Second, the factor ρ is nonnegative in \({\mathcal S}\), since \({\mathcal S}\) is the image of the upper halfplane, ρ ≥ 0. Last, the oneforms \({{\mathcal J}_\vartriangle}\) and \({\mathcal M}\) are spacelike, since the matrices Φ depend only on the coordinates of (̃∑, ̃g).
In order to establish that ρTrace{dΨ} = 0 on the boundary \(\partial {\mathcal S}\) of the semistrip, one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. A careful investigation^{71} then shows that ρTrace {dΨ} vanishes on the horizon, the axis and at infinity, provided that the solutions have the same mass, charge and angular momentum.
7 Conclusion
The fact that the stationary electrovac black holes are parametrized by their mass, angular momentum and electric charge is due to the distinguished structure of the EinsteinMaxwell equations in the presence of a Killing symmetry. In general, the classification of the stationary black hole spacetimes within a given matter model is a difficult task, involving the investigation of Einstein’s equations with a low degree of symmetries. The variety of black hole configurations in the SU(2) EinsteinYangMills system indicates that — in spite of its beautiful and intuitive content — the uniqueness theorem is a distinguished feature of electrovac spacetimes. In general, the stationary black hole solutions of selfgravitating matter fields are considerably less simple than one might have expected from the experience with the EinsteinMaxwell system.
Footnotes
 1.
For a review on the evolution of the subject the reader is referred to Israel’s comprehensive account [101].
 2.
A list of the most promising candidates was recently presented by Rees [148] at a symposium dedicated to the memory of S. Chandrasekhar (Chicago, Dec. 1415, 1996).
 3.
Lately, there has been growing interest in the fascinating relationship between the laws of black hole mechanics and the laws of thermodynamics. In particular, recent computations within string theory seem to offer a promising interpretation of black hole entropy [98]. The reader interested in the thermodynamic properties of black holes is referred to the review by Wald [177].
 4.
 5.
 6.
 7.
The original proof of the SRT [84] was based on an analyticity requirement which had no justification [41]. A precise formulation and a correct proof of the theorem were given only recently by Chruściel [38]; see also [40], Sect. 5. In particular, no energy conditions enter the new version of the SRT.
 8.
In order to prove the SRT one also needs to show that the connected components of the event horizon have the topology IR X S2. This was established only recently by Chruściel and Wald [44], taking advantage of the topological censorship theorem [55]. A related version of the topology theorem, applying to globally hyperbolic — but not necessarily stationary — spacetimes was obtained by Jacobson and Venkataramani [103], and Galloway [56], [57], [58], [59]. We refer to [40], Sect. 2 for a detailed discussion.
 9.
See [6] for a complete classification of the isometry groups.
 10.
The existence of a foliation by maximal slices was established by Chruściel and Wald [43].
 11.
 12.
Reduction of the EM action with respect to the timelike Killing field yields, instead, H = S(U(1,1) × U(1)).
 13.
 14.
 15.
 16.
 17.
The stability properties are discussed in Weinberg’s comprehensive review on magnetically charged black holes [181].
 18.
 19.
 20.
An early apparent success rested on a sign error [30]. Carter’s amended version of the proof was subject to a certain inequality between the electric and the gravitational potential [33]. The origin of this inequality has become clear only recently; the particular combination of the potentials arises naturally in the dimensional reduction of the EM system with respect to a timelike Killing field.
 21.
The new proof given in [88] works under less restrictive topological assumptions, since it does not require the global existence of a twist potential
 22.
See the footnote on page 7.
 23.
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 the invariance properties; see Sect. 6.1 for details.
 24.
There are other matter models for which the Papetrou metric is sufficiently general: The proof of the circularity theorem for selfgravitating scalar fields is, for instance, straightforward [86].
 25.
Although nonrotating, these configurations were not discussed in Sect. 3.2; in the present context I prefer to view them as particular circular configurations.
 26.
 27.
The solutions themselves are neutral and not spherically symmetric; however, their limiting configurations are charged and spherically symmetric.
 28.
We have already mentioned in Sect. 3.4 that these black holes present counterexamples to the naive generalization of the staticity theorem, they are nice illustrations of the correct nonAbelian version of the theorem [167], [168].
 29.
A particular combination of the charged and the rotating branch was found in [175].
 30.
More precisely, the definition applies to strongly asymptotically predictable spacetimes; see [178], Chap. 12 for the exact statements.
 31.
 32.
 33.
 34.
For a compilation of Killing field identities we refer to [87], Chap. 2.
 35.
T(ξ) is the oneform with components [T(ξ)]_{ μ } ≡ T_{ μν }ξ^{ ν }.
 36.
 37.
The fact that ω vanishes always on the horizon is, of course, not sufficient to conclude that dω vanishes as well on Hξ].
 38.
This is obvious for static configurations, since ξ coincides with the static Killing field. In the circular case one also needs to show that (m, ω_{ κ }) = (κ, ω_{ m }) = 0 implies dω_{ ξ } = 0 on the horizon generated by ξ = κ + Ωm; see [87], Chap. 7 for details).
 39.
See, e.g. [91] for the details of the derivation.
 40.
For arbitrary pforms α and β the inner product is defined by ̄*〈(α, β)〉 = α ∧ ̄*β, where ̄* is the Hodge dual with respect to ḡ.
 41.
Here and in the following we use the symbol k for both the Killing field ∂_{ t } and the corresponding oneform σ(d_{ t } + a).
 42.
 43.
 44.
\({\widehat {{\rm{tr}}}\left\{ {} \right\}}\) denotes the normalized trace; e.g. \(\widehat {{\rm{tr}}}\left\{ {{\tau _a}{\tau _b}} \right\} = {\delta _{ab}}\) for SU(2), where τ_{ a } ≡ σ_{ a }/(2i).
 45.
For an arbitrary twoform β, i_{ ξ }β is the oneform with components ξ^{ μ }β_{ μν }.
 46.
The second Killing field is of crucial importance to the twodimensional boundary value formulation of the field equations and to the integration of the Mazur identity. However, the derivation of the identity in the twodimensional context is somewhat unnatural, since the dimensional reduction with respect to the second Killing field introduces a weight factor which is slightly veiling the σmodel structure [32].
 47.
 48.
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).
 49.
 50.
 51.
The derivation of Eq. (37) is not restricted to static configurations. However, when evaluating the surface terms, one assumes that the horizon is generated by the same Killing field which is also used in the dimensional reduction; the asymptotically timelike Killing field k. A generalization of the method to rotating black holes requires the evaluation of the potentials (defined with respect to k) on a Killing horizon which is generated by ℓ = k+Ω_{ H }^{ m }, rather than k.
 52.
Here one uses the fact that the electric potential assumes a constant value on the horizon. The quantity Q_{ H } is defined by the flux integral of *F over the horizon (at time ∑), while the corresponding integral of *dk gives κA/4π; see [89] for details.
 53.
As we are considering stationary configurations we use the dimensional reduction with respect to the asymptotically timelike Killing field k with norm σ = — (k, k) = N.
 54.
For purely electric configurations one has F = k ∧ dϕ/σ. Staticity implies k = σdt and thus dk = k ∧ dσ/σ.
 55.
Hartle and Hawking [81] have shown that all real singularities are “hidden” behind these null surfaces.
 56.
We refer the reader to Chandrasekhar’s comparison between corresponding solutions of the Ernst equations [36].
 57.
This follows from the definition of the twist and the Ricci identity for Killing fields, Δξ = 2R(ξ), where R(ξ) is the oneform with components [R(ξ)]_{ μ } ≡ R_{ μν }ξ^{ ν }; see, e.g. [87], Chap. 2.)
 58.
Equation (50) is an identity up to a term involving the Lie derivative of the twist of the first Killing field with respect to the second one (since d(m, ω_{ k } = L_{ m }ͩ_{ k }i_{ m }dͩ_{ k }). In order to establish L_{ m }ͩ_{ k } = 0 it is sufficient to show that k and m commute in an asymptotically flat spacetime. This was first achieved by Carter [28] and later, under more general conditions, by Szabados [170].
 59.
The following is understood to apply also for k↔m.
 60.
 61.
The Maxwell equation d*F = 0 and the symmetry property L_{ k }*F = *L_{ k }F = 0 imply the existence of a magnetic potential, dͨ = (*F)(k, ·). Thus, (*F)(k, m) = i_{ m }dͨ = L_{ r }ͨ = 0.
 62.
The extension of the circularity theorem from the EM system to the coset models under consideration is straightforward.
 63.
Since Φ is a matrix valued function on (̃∑, ̃g) one has \({{\mathcal J}_t} \equiv 0\) and \(\bar * {\mathcal J} =  \rho {\rm{d}}t \wedge \tilde * {\mathcal J}\).
 64.
In order to introduce Weyl coordinates one has to exclude critical points of ρ. This was first achieved by Carter [30] using Morse theory; see, e.g. [135]. A more recent, very direct proof was given by Weinstein [182], taking advantage of the Riemann mapping theorem (or, more precisely, Caratheodory’s extension of the theorem; see, e.g. [5]).
 65.
 66.
Again, we consider the dimensional reduction with respect to the axial Killing field.
 67.
 68.
See, e.g. [31].
 69.
We refer the reader to [32] for a discussion of Bunting’s method.
 70.
See Sect. 5.1 for details and references.
 71.
We refer to [182] for a detailed discussion of the boundary and regularity conditions for axisymmetric black holes.
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