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.^{
$\text{14}$
}
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]
[
178]
,^{
$\text{15}$
}
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]
.^{
$\text{16}$
}
Hence, the restriction of the generalized nohair conjecture to stable configurations is not correct either.
One of the reasons why it was not until 1989 that black hole solutions with selfgravitating gauge fields were discovered was the widespread belief that the EYM equations admit
no soliton solutions. There were, at least, four reasons in support of this hypothesis.

∙
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. [
73]
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
$\sigma $
model 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
$SU\left(2\right)$
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).^{
$\text{17}$
}
In fact, the work of Ridgway and Weinberg shows that static black holes with magnetic charge need not even be axially symmetric [
150]
.^{
$\text{18}$
}
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 Birkhoff 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 Birkhoff 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
$SO\left(3\right)$
as symmetry group, acting by bundle automorphisms. It turns out that the Birkhoff theorem can be generalized to bundles which admit only
$SO\left(3\right)$
invariant connections of Abelian type.^{
$\text{19}$
}
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]
,^{
$\text{20}$
}
whereas the corresponding vacuum problem was solved quite some time ago [
84]
. Using a slightly improved version of the argument given in [
84]
,^{
$\text{21}$
}
. 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
^{
$\text{22}$
}
and subject to restrictive assumptions concerning the matter fields [
84]
, the recent work of Chruściel [
40]
, [
39]
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.^{
$\text{23}$
}
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.^{
$\text{24}$
}
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 [
114]
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.^{
$\text{25}$
}
Using numerical methods, Kleihaus and Kunz have constructed black hole solutions of this kind for both the EYM and the EYMdilaton system [
113]
.^{
$\text{26}$
}
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 [
72]
, [
76]
.^{
$\text{27}$
}
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,^{
$\text{28}$
}
whereas the second one consists of neutral black holes with nonvanishing angular momentum.^{
$\text{29}$
}
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]
.
The proof of the circularity theorem for selfgravitating scalar fields is, for instance, straightforward [
86]
.
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
non Abelian 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
The black hole region of an asymptotically flat spacetime
$(M,\text{}\mathit{g}\text{})$
is the part of
$M$
which is not contained in the causal past of future null infinity.^{
$\text{30}$
}
Hence, the event horizon, being defined as the boundary of the black hole region, is a global concept. Of crucial importance to the theory of black holes is the strong rigidity theorem, which implies that the event horizon of a stationary spacetime is a Killing horizon.^{
$\text{31}$
}
The definition of the latter is of purely local nature:
Consider a Killing field
$\xi $
, say, and the set of points where
$\xi $
is null,
$N\equiv \left(\xi ,\xi \right)=0$
. A connected component of this set which is a null hypersurface,
$\left(dN,dN\right)=0$
, is called a Killing horizon ,
$H\left[\xi \right]$
. Killing horizons possess a variety of interesting properties:^{
$\text{32}$
}

∙
An immediate consequence of the above definition is the fact that
$\xi $
and
$dN$
are proportional on
$H\left[\xi \right]$
. (Note that
$\left(\xi ,dN\right)=0$
, since
${L}_{\xi}N=0$
, and that two orthogonal null vectors are proportional.) This suggests the following definition of the surface gravity,
$\kappa $
,
$$\begin{array}{c}dN=2\kappa \xi \text{on}H\left[\xi \right]\text{}.\end{array}$$ 
(1)

Since the Killing equation implies
$dN=2{\nabla}_{\xi}\xi $
, the above definition shows that the surface gravity measures the extent to which the parametrization of the geodesic congruence generated by
$\xi $
is not affine.

∙
A theorem due to Vishveshwara [172] gives a characterization of the Killing horizon
$H\left[\xi \right]$
in terms of the twist
$\omega $
of
$\xi $
:^{
$\text{33}$
}
The surface
$N\equiv \left(\xi ,\xi \right)=0$
is a Killing horizon if and only if
$$\begin{array}{c}\omega =0\text{and}{i}_{\xi}d\xi \ne 0\text{on}N=0\text{}.\end{array}$$ 
(2)


∙
Using general identities for Killing fields^{
$\text{34}$
}
one can derive the following explicit expressions for
$\kappa $
:
$$\begin{array}{c}{\kappa}^{2}={\left[\frac{1}{N}\left({\nabla}_{\xi}\xi ,{\nabla}_{\xi}\xi \right)\right]}_{H\left[\xi \right]}={\left[\frac{1}{4}\Delta N\right]}_{H\left[\xi \right]}.\end{array}$$ 
(3)

Introducing the four velocity
$u=\xi /\sqrt{N}$
for a timelike
$\xi $
, the first expression shows that the surface gravity is the limiting value of the force applied at infinity to keep a unit mass at
$H\left[\xi \right]$
in place:
$\kappa =lim\left(\sqrt{N}\righta\left\right)$
, where
$a={\nabla}_{u}u$
(see, e.g. [177] ).

∙
Of crucial importance to the zeroth law of black hole physics (to be discussed below) is the fact that the
$\left(\xi \xi \right)$
component of the Ricci tensor vanishes on the horizon,
$$\begin{array}{c}R(\xi ,\xi )=0\text{on}H\left[\xi \right]\text{}.\end{array}$$ 
(4)

This follows from the above expressions for
$\kappa $
and the general Killing field identity
$2NR(\xi ,\xi )=4\left({\nabla}_{\xi}\xi ,{\nabla}_{\xi}\xi \right)N\Delta N4\left(\omega ,\omega \right)$
.
It is an interesting fact that the surface gravity plays a similar role in the theory of stationary black holes as the temperature does in ordinary thermodynamics. Since the latter is constant for a body in thermal equilibrium, the result
$$\begin{array}{c}\kappa =\text{constant on}H\left[\xi \right]\text{}\end{array}$$ 
(5)

is usually called the zeroth law of black hole physics [
3]
. The zeroth law can be established by different means: Each of the following alternatives is sufficient to prove that
$\kappa $
is uniform over the Killing horizon generated by
$\xi $
.

∙
(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)
$H\left[\xi \right]$
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
$R(\xi ,\xi )$
vanishes on the horizon (see above), imply that
$T(\xi ,\xi )=0$
on
$H\left[\xi \right]$
. Hence, the oneform
$T\left(\xi \right)$
^{
$\text{35}$
}
is perpendicular to
$\xi $
and, therefore, spacelike or null on
$H\left[\xi \right]$
.
On the other hand, the dominant energy condition requires that
$T\left(\xi \right)$
is timelike or null. Thus,
$T\left(\xi \right)$
is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein's equations again,
$\xi \wedge R\left(\xi \right)=0$
on
$H\left[\xi \right]$
. The result that
$\kappa $
is uniform over the horizon now follows from the general property^{
$\text{36}$
}
$$\begin{array}{c}\xi \wedge d\kappa =\xi \wedge R\left(\xi \right)\text{on}H\left[\xi \right]\text{}.\end{array}$$ 
(6)

(ii) By virtue of Eq. ( 6 ) and the general Killing field identity
$d\omega =*[\xi \wedge R(\xi \left)\right]$
, the zeroth law follows if one can show that the twist oneform is closed on the horizon [
147]
:
$$\begin{array}{c}{\left[d\omega \right]}_{H\left[\xi \right]}=0\u27f9\kappa =\text{constant on}H\left[\xi \right]\text{}.\end{array}$$ 
(7)

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
$\omega $
vanishes identically,^{
$\text{37}$
}
which then proves the second version of the first zeroth law.^{
$\text{38}$
}
(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
$\kappa $
cannot vary between the nullgenerators. (It is clear that
$\kappa $
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 Rácz and Wald strongly suggests that all physically relevant Killing horizons are either of bifurcate type or degenerate [
146]
, [
147]
.
[
177]
.
4.2 Reduction of the EinsteinHilbert Action
By definition, a stationary spacetime
$(M,\text{}\mathit{g}\text{})$
admits an asymptotically timelike Killing field, that is, a vector field
$k$
with
${L}_{k}\text{}\mathit{g}\text{}=0$
,
${L}_{k}$
denoting the Lie derivative with respect to
$k$
. At least locally,
$M$
has the structure
$\Sigma \times G$
, where
$G\approx IR$
denotes the onedimensional group generated by the Killing symmetry, and
$\Sigma $
is the threedimensional quotient space
$M/G$
. A stationary spacetime is called static, if the integral trajectories of
$k$
are orthogonal to
$\Sigma $
.
With respect to the adapted time coordinate
$t$
, defined by
$k\equiv {\partial}_{t}$
, the metric of a stationary spacetime is parametrized in terms of a threedimensional (Riemannian) metric
$\overline{\text{}\mathit{g}\text{}}\equiv {\overline{g}}_{ij}d{x}^{i}d{x}^{j}$
, a oneform
$a\equiv {a}_{i}d{x}^{i}$
, and a scalar field
$\sigma $
, where stationarity implies that
${\overline{g}}_{ij}$
,
${a}_{i}$
and
$\sigma $
are functions on
$(\Sigma ,\overline{\text{}\mathit{g}\text{}})$
:
$$\begin{array}{c}\text{}\mathit{g}\text{}=\sigma (dt+a{)}^{2}+\frac{1}{\sigma}\overline{\text{}\mathit{g}\text{}}.\end{array}$$ 
(8)

Using Cartan's structure equations (see, e.g. [
165]
), it is a straightforward task to compute the Ricci scalar for the above decomposition of the spacetime metric^{
$\text{39}$
}
. The result shows that the EinsteinHilbert action of a stationary spacetime reduces to the action for a scalar field
$\sigma $
and an Abelian vector field
$a$
, which are coupled to threedimensional gravity. The fact that this coupling is minimal is a consequence of the particular choice of the conformal factor in front of the threemetric
$\overline{\text{}\mathit{g}\text{}}$
in the decomposition ( 8 ). The vacuum field equations are, therefore, equivalent to the threedimensional Einsteinmatter equations obtained from variations of the effective action
$$\begin{array}{c}{S}_{eff}=\int \overline{*}\left(\overline{R}\frac{1}{2{\sigma}^{2}}\langle d\sigma ,d\sigma \rangle +\frac{{\sigma}^{2}}{2}\langle da,da\rangle \right),\end{array}$$ 
(9)

with respect to
${\overline{g}}_{ij}$
,
$\sigma $
and
$a$
. (Here and in the following
$\overline{R}$
and
$\langle ,\rangle $
denote the Ricci scalar and the inner product^{
$\text{40}$
}
with respect to
$\overline{\text{}\mathit{g}\text{}}$
.) It is worth noting that the quantities
$\sigma $
and
$a$
are related to the norm and the twist of the Killing field as follows:
$$\begin{array}{c}\sigma =\text{}\mathit{g}\text{}(k,k),\omega \equiv \frac{1}{2}*(k\wedge dk)=\frac{1}{2}{\sigma}^{2}\overline{*}da,\end{array}$$ 
(10)

where
$*$
and
$\overline{*}$
denote the Hodge dual with respect to
$\text{}\mathit{g}\text{}$
and
$\overline{\text{}\mathit{g}\text{}}$
, respectively^{
$\text{41}$
}
. Since
$a$
is the connection of a fiber bundle with base space
$\Sigma $
and fiber
$G$
, it behaves like an Abelian gauge potential under coordinate transformations of the form
$t\to t+\phi \left({x}^{i}\right)$
. Hence, it enters the effective action in a gaugeinvariant way, that is, only via the “Abelian field strength”,
$f\equiv da$
.
4.3 The Coset Structure of Vacuum Gravity
For many applications, in particular for the black hole uniqueness theorems, it is of crucial importance that the oneform
$a$
can be replaced by a function (twist potential). We have already pointed out that
$a$
, parametrizing the nonstatic part of the metric, enters the effective action ( 9 ) only via the field strength,
$f\equiv da$
. For this reason, the variational equation for
$a$
(that is, the offdiagonal Einstein equation) assumes the form of a sourcefree Maxwell equation,
$$\begin{array}{c}d\overline{*}\left({\sigma}^{2}da\right)=0\u27f9dY\equiv \overline{*}\left({\sigma}^{2}da\right).\end{array}$$ 
(11)

By virtue of Eq. ( 10 ), the (locally defined) function
$Y$
is a potential for the twist oneform,
$dY=2\omega $
. In order to write the effective action ( 9 ) in terms of the twist potential
$Y$
, rather than the oneform
$a$
, one considers
$f\equiv da$
as a fundamental field and imposes the constraint
$df=0$
with the Lagrange multiplier
$Y$
. The variational equation with respect to
$f$
then yields
$f=\overline{*}({\sigma}^{2}dY)$
, which is used to eliminate
$f$
in favor of
$Y$
. One finds
$\frac{1}{2}{\sigma}^{2}f\wedge \overline{*}fYdf$
$\to $
$\frac{1}{2}{\sigma}^{2}dY\wedge \overline{*}dY$
. Thus, the action ( 9 ) becomes
$$\begin{array}{c}{S}_{eff}=\int \overline{*}\left(\overline{R}\frac{\langle d\sigma ,d\sigma \rangle +\langle dY,dY\rangle}{2{\sigma}^{2}}\right),\end{array}$$ 
(12)

where we recall that
$\langle ,\rangle $
is the inner product with respect to the threemetric
$\overline{\text{}\mathit{g}\text{}}$
defined in Eq. ( 8 ).
The action (
12 ) describes a harmonic mapping into a twodimensional target space, effectively coupled to threedimensional gravity. In terms of the complex Ernst potential
$E$
[
52]
, [
53]
, one has
$$\begin{array}{c}{S}_{eff}=\int \overline{*}\left(\overline{R}2\frac{\langle dE,d\overline{E}\rangle}{(E+\overline{E}{)}^{2}}\right),E\equiv \sigma +iY.\end{array}$$ 
(13)

The stationary vacuum equations are obtained from variations with respect to the threemetric
$\overline{\text{}\mathit{g}\text{}}$
[
$\left(ij\right)$
equations] and the Ernst potential
$E$
[
$\left(0\mu \right)$
equations]. One easily finds
${\overline{R}}_{ij}=2(E+\overline{E}{)}^{2}E{,}_{i}\overline{E}{,}_{j}$
and
$\overline{\Delta}E=2(E+\overline{E}{)}^{1}\langle dE,dE\rangle $
, where
$\overline{\Delta}$
is the Laplacian with respect to
$\overline{\text{}\mathit{g}\text{}}$
.
The target space for stationary vacuum gravity, parametrized by the Ernst potential
$E$
, is a Kähler manifold with metric
${G}_{E\overline{E}}={\partial}_{E}{\partial}_{\overline{E}}ln\left(\sigma \right)$
(see [
62]
for details). By virtue of the mapping
$$\begin{array}{c}E\mapsto z=\frac{1E}{1+E},\end{array}$$ 
(14)

the semiplane where the Killing field is timelike,
$\text{Re(E)}>0$
, is mapped into the interior of the complex unit disc,
$D=\{z\in C\leftz\right<1\}$
, with standard metric
$(1z{}^{2}{)}^{2}\langle dz,d\overline{z}\rangle $
. By virtue of the stereographic projection,
$\text{Re}\left(z\right)={x}^{1}({x}^{0}+1{)}^{1}$
,
$\text{Im}\left(z\right)={x}^{2}({x}^{0}+1{)}^{1}$
, the unit disc
$D$
is isometric to the pseudosphere,
$P{S}^{2}=\left\{\right({x}^{0},{x}^{1},{x}^{2})\in I{R}^{3}({x}^{0}{)}^{2}+({x}^{1}{)}^{2}+({x}^{2}{)}^{2}=1\}$
. As the threedimensional Lorentz group,
$SO(2,1)$
, acts transitively and isometrically on the pseudosphere with isotropy group
$SO\left(2\right)$
, the target space is the coset
$P{S}^{2}\approx SO(2,1)/SO\left(2\right)$
^{
$\text{42}$
}
. Using the universal covering
$SU(1,1)$
of
$SO(2,1)$
, one can parametrize
$P{S}^{2}\approx SU(1,1)/U\left(1\right)$
in terms of a positive hermitian matrix
$\Phi \left(x\right)$
, defined by
$$\begin{array}{c}\Phi \left(x\right)=\left(\begin{array}{cc}{x}^{0}& {x}^{1}+i{x}^{2}\\ {x}^{1}i{x}^{2}& {x}^{0}\end{array}\right)=\frac{1}{1z{}^{2}}\left(\begin{array}{cc}1+z{}^{2}& 2z\\ 2\overline{z}& 1+z{}^{2}\end{array}\right).\end{array}$$ 
(15)

Hence, the effective action for stationary vacuum gravity becomes the standard action for a
$\sigma $
model coupled to threedimensional gravity [
139]
,
$$\begin{array}{c}{S}_{eff}=\int \overline{*}\left(\overline{R}\frac{1}{4}Trace\langle {\Phi}^{1}d\Phi ,{\Phi}^{1}d\Phi \rangle \right).\end{array}$$ 
(16)

The simplest nontrivial solution to the vacuum Einstein equations is obtained in the static, spherically symmetric case: For
$E=\sigma \left(r\right)$
one has
$2{\overline{R}}_{rr}=({\sigma}^{\prime}/\sigma {)}^{2}$
and
$\overline{\Delta}ln\left(\sigma \right)=0$
. With respect to the general spherically symmetric ansatz
$$\begin{array}{c}\overline{\text{}\mathit{g}\text{}}=d{r}^{2}+{\rho}^{2}\left(r\right)d{\Omega}^{2},\end{array}$$ 
(17)

one immediately obtains the equations
$4{\rho}^{\prime \prime}/\rho =({\sigma}^{\prime}/\sigma {)}^{2}$
and
$({\rho}^{2}{\sigma}^{\prime}/\sigma {)}^{\prime}=0$
, the solution of which is the Schwarzschild metric in the usual parametrization:
$\sigma =12M/r$
,
${\rho}^{2}=\sigma \left(r\right){r}^{2}$
.
4.4 Stationary Gauge Fields
The reduction of the EinsteinHilbert action in the presence of a Killing field yields a
$\sigma $
model 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
$S$
invariant 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]
.
The gauge fields constructed in the above way are invariant under the action of
$S$
up to gauge transformations. This is also the starting point of the alternative approach to the problem, due to Forgács and Manton [
54]
. It implies that a gauge potential
$A$
is symmetric with respect to the action of a Killing field
$\xi $
, say, if there exists a Lie algebra valued function
${\mathcal{V}}_{\xi}$
, such that
$$\begin{array}{c}{L}_{\xi}A=D{\mathcal{V}}_{\xi},\end{array}$$ 
(18)

where
${\mathcal{V}}_{\xi}$
is the generator of an infinitesimal gauge transformation,
${L}_{\xi}$
denotes the Lie derivative, and
$D$
is the gauge covariant exterior derivative,
$D{\mathcal{V}}_{\xi}=d{\mathcal{V}}_{\xi}+[A,{\mathcal{V}}_{\xi}]$
.
Let us now consider a stationary spacetime with (asymptotically) timelike Killing field
$k$
. A stationary gauge potential is parametrized in terms of a oneform
$\overline{A}$
orthogonal to
$k$
,
$\overline{A}\left(k\right)=0$
, and a Lie algebra valued potential
$\phi $
,
$$\begin{array}{c}A=\phi (dt+a)+\overline{A},\end{array}$$ 
(19)

where we recall that
$a$
is the nonstatic part of the metric ( 8 ). For the sake of simplicity we adopt a gauge where
${\mathcal{V}}_{k}$
vanishes.^{
$\text{43}$
}
By virtue of the above decomposition, the field strength becomes
$F=\overline{D}\phi \wedge (dt+a)+(\overline{F}+\phi f)$
, where
$\overline{F}$
is the YangMills field strength for
$\overline{A}$
and
$f\equiv da$
. Using the expression ( 12 ) for the vacuum action, one easily finds that the EYM action,
$$\begin{array}{c}{S}_{EYM}=\int \left(*R2\hat{\text{tr}}\left\{F\wedge *F\right\}\right),\end{array}$$ 
(20)

gives rise to the effective action^{
$\text{44}$
}
$$\begin{array}{c}{S}_{eff}=\int \overline{*}\left(\overline{R}\frac{1}{2{\sigma}^{2}}d\sigma {}^{2}+\frac{{\sigma}^{2}}{2}f{}^{2}+\frac{2}{\sigma}\overline{D}\phi {}^{2}2\sigma \overline{F}+\phi f{}^{2}\right),\end{array}$$ 
(21)

where
$\overline{D}$
is the gauge covariant derivative with respect to
$\overline{A}$
, and where the inner product also involves the trace:
$\overline{*}\overline{F}{}^{2}\equiv \hat{\text{tr}}\left\{\overline{F}\wedge \overline{*}\overline{F}\right\}$
. The above action describes two scalar fields,
$\sigma $
and
$\phi $
, and two vector fields,
$a$
and
$\overline{A}$
, which are minimally coupled to threedimensional gravity with metric
$\overline{\text{}\mathit{g}\text{}}$
. Like in the vacuum case, the connection
$a$
enters
${S}_{eff}$
only via the field strength
$f\equiv da$
.
Again, this gives rise to a differential conservation law,
$$\begin{array}{c}d\overline{*}\left[{\sigma}^{2}f4\sigma \hat{\text{tr}}\left\{\phi (\overline{F}+\phi f)\right\}\right]=0,\end{array}$$ 
(22)

by virtue of which one can (locally) introduce a generalized twist potential
$Y$
, defined by
$dY=\overline{*}[\dots ]$
.
The main difference between the Abelian and the nonAbelian case concerns the variational equation for
$\overline{A}$
, that is, the YangMills equation for
$\overline{F}$
: The latter assumes the form of a differential conservation law only in the Abelian case. For nonAbelian gauge groups,
$\overline{F}$
is no longer an exact twoform, and the gauge covariant derivative of
$\phi $
causes source terms in the corresponding YangMills equation:
$$\begin{array}{c}\overline{D}\left[\sigma \overline{*}\left(\overline{F}+\phi f\right)\right]={\sigma}^{1}\overline{*}\left[\phi ,\overline{D}\phi \right].\end{array}$$ 
(23)

Hence, the scalar magnetic potential – which can be introduced in the Abelian case according to
$d\psi \equiv \sigma \overline{*}(\overline{F}+\phi f)$
– ceases to exist for nonAbelian YangMills fields. The remaining stationary EYM equations are easily derived from variations of
${S}_{eff}$
with respect to the gravitational potential
$\sigma $
, the electric YangMills potential
$\phi $
and the threemetric
$\overline{\text{}\mathit{g}\text{}}$
.
As an application, we note that the effective action (
21 ) is particularly suited for analyzing stationary perturbations of static (
$a=0$
), purely magnetic (
$\phi =0$
) configurations [
19]
, such as the BartnikMcKinnon solitons [
4]
and the corresponding black hole solutions [
174]
, [
122]
, [
9]
. The two crucial observations in this context are [
19]
, [
175]
:

∙
(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,
$\delta a$
and
$\delta \phi $
.

∙
(ii) In first order perturbation theory the relevant fluctuations,
$\delta a$
and
$\delta \phi $
, decouple from the remaining metric and matter perturbations
The second observation follows from the fact that the magnetic YangMills equation ( 23 ) and the Einstein equations for
$\sigma $
and
$\overline{\text{}\mathit{g}\text{}}$
become background equations, since they contain no linear terms in
$\delta a$
and
$\delta \phi $
. The purely electric, nonstatic perturbations are, therefore, governed by the twist equation ( 22 ) and the electric YangMills equation (obtained from variations of
${S}_{eff}$
with respect to
$\phi $
).
Using Eq. (
22 ) to introduce the twist potential
$Y$
, the fluctuation equations for the first order quantities
$\delta Y$
and
$\delta \phi $
assume the form of a selfadjoint system [
19]
. Considering perturbations of spherically symmetric configurations, one can expand
$\delta Y$
and
$\delta \phi $
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
In the Abelian case, both the offdiagonal Einstein equation ( 22 ) and the Maxwell equation ( 23 ) give rise to scalar potentials, (locally) defined by
$$\begin{array}{c}d\psi \equiv \sigma \overline{*}\left(\overline{F}+\phi f\right),dY\equiv {\sigma}^{2}\overline{*}f+2\phi d\psi 2\psi d\phi .\end{array}$$ 
(24)

Like for the vacuum system, this enables one to apply the Lagrange multiplier method in order to express the effective action in terms of the scalar fields
$Y$
and
$\psi $
, rather than the oneforms
$a$
and
$\overline{A}$
. As one is often interested in the dimensional reduction of the EM system with respect to a spacelike Killing field, we give here the general result for an arbitrary Killing field
$\xi $
with norm
$N$
:
$$\begin{array}{c}{S}_{eff}=\int \overline{*}\left(\overline{R}2\frac{d\phi {}^{2}+d\psi {}^{2}}{N}\frac{dN{}^{2}+dY2\phi d\psi +2\psi d\phi {}^{2}}{2{N}^{2}}\right),\end{array}$$ 
(25)

where
$\overline{*}d\phi {}^{2}\equiv d\phi \wedge \overline{*}d\phi $
, etc. The electromagnetic potentials
$\phi $
and
$\psi $
and the gravitational scalars
$N$
and
$Y$
are obtained from the fourdimensional field strength
$F$
and the Killing field (one form) as follows:^{
$\text{45}$
}
$$\begin{array}{c}d\phi ={i}_{\xi}F,d\psi ={i}_{\xi}*F,\end{array}$$ 
(26)

$$\begin{array}{c}N=\left(\xi ,\xi \right),dY=2\left(\omega +\phi d\psi \psi d\phi \right),\end{array}$$ 
(27)

where
$2\omega \equiv *(\xi \wedge d\xi )$
. The inner product
$\langle ,\rangle $
is taken with respect to the threemetric
$\overline{\text{}\mathit{g}\text{}}$
, which becomes pseudoRiemannian if
$\xi $
is spacelike. In the stationary and axisymmetric case, to be considered in Sect. 6 , the KaluzaKlein reduction will be performed with respect to the spacelike Killing field. The additional stationary symmetry will then imply that the inner products in ( 25 ) have a fixed sign, despite the fact that
$\overline{\text{}\mathit{g}\text{}}$
is not a Riemannian metric in this case.
The action (
25 ) describes a harmonic mapping into a fourdimensional target space, effectively coupled to threedimensional gravity. In terms of the complex Ernst potentials,
$\Lambda \equiv \phi +i\psi $
and
$E\equiv N\Lambda \overline{\Lambda}+iY$
[
52]
, [
53]
, the effective EM action becomes
$$\begin{array}{c}{S}_{eff}=\int \overline{*}\left(\overline{R}2\frac{d\Lambda {}^{2}}{N}\frac{1}{2}\frac{dE+2\overline{\Lambda}d\Lambda {}^{2}}{{N}^{2}}\right),\end{array}$$ 
(28)

where
$d\Lambda {}^{2}\equiv \langle d\Lambda ,\overline{d\Lambda}\rangle $
. The field equations are obtained from variations with respect to the threemetric
$\overline{\text{}\mathit{g}\text{}}$
and the Ernst potentials. In particular, the equations for
$E$
and
$\Lambda $
become
$$\begin{array}{c}\overline{\Delta}E=\frac{\langle dE,dE+2\overline{\Lambda}d\Lambda \rangle}{N(E,\Lambda )},\overline{\Delta}\Lambda =\frac{\langle d\Lambda ,dE+2\overline{\Lambda}d\Lambda \rangle}{N(E,\Lambda )},\end{array}$$ 
(29)

where
$N=\Lambda \overline{\Lambda}+\frac{1}{2}(E+\overline{E})$
. The isometries of the target manifold are obtained by solving the respective Killing equations [
139]
(see also [
107]
, [
108]
, [
109]
, [
110]
). This reveals the coset structure of the target space and provides a parametrization of the latter in terms of the Ernst potentials. For vacuum gravity we have seen in Sect. 4.3 that the coset space,
$G/H$
, is
$SU(1,1)/U\left(1\right)$
, whereas one finds
$G/H=SU(2,1)/S\left(U\right(1,1)\times U(1\left)\right)$
for the stationary EM equations. If the dimensional reduction is performed with respect to a spacelike Killing field, then
$G/H=SU(2,1)/S\left(U\right(2)\times U(1\left)\right)$
. The explicit representation of the coset manifold in terms of the above Ernst potentials,
$E$
and
$\Lambda $
, is given by the hermitian matrix
$\Phi $
, with components
$$\begin{array}{c}{\Phi}_{AB}={\eta}_{AB}+2\text{sig(}N\text{)}{\overline{v}}_{A}{v}_{B},({v}_{0},{v}_{1},{v}_{2})\equiv \frac{1}{2\sqrt{\leftN\right}}(E1,E+1,2\Lambda ),\end{array}$$ 
(30)

where
${v}_{A}$
is the Kinnersley vector [
106]
, and
$\eta \equiv \text{diag}(1,+1,+1)$
. It is straightforward to verify that, in terms of
$\Phi $
, the effective action ( 28 ) assumes the
$SU(2,1)$
invariant form
$$\begin{array}{c}{\mathcal{S}}_{eff}=\int \overline{*}\left(\overline{R}\frac{1}{4}\text{Trace}\langle \mathcal{J},\mathcal{J}\rangle \right),\text{with}\mathcal{J}\equiv {\Phi}^{1}d\Phi ,\end{array}$$ 
(31)

where
$\text{Trace}\langle \mathcal{J},\mathcal{J}\rangle \equiv \langle {\mathcal{J}}_{B}^{A},{\mathcal{J}}_{A}^{B}\rangle \equiv {\overline{g}}^{ij}\left({\mathcal{J}}_{i}{)}_{B}^{A}\right({\mathcal{J}}_{j}{)}_{A}^{B}$
. The equations of motion following from the above action are the threedimensional Einstein equations (obtained from variations with respect to
$\overline{\text{}\mathit{g}\text{}}$
) and the
$\sigma $
model equations (obtained from variations with respect to
$\Phi $
):
$$\begin{array}{c}{\overline{R}}_{ij}=\frac{1}{4}\text{Trace}\left\{{\mathcal{J}}_{i}{\mathcal{J}}_{j}\right\},d\overline{*}\mathcal{J}=0.\end{array}$$ 
(32)

By virtue of the Bianchi identity,
${\overline{\nabla}}_{j}{\overline{G}}^{ij}=0$
, and the definition
${\mathcal{J}}_{i}\equiv {\Phi}^{1}{\overline{\nabla}}_{i}\Phi $
, the
$\sigma $
model equations are the integrability conditions for the threedimensional Einstein equations.
5 Applications of the Coset Structure
The
$\sigma $
model 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
$\sigma $
model 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.^{
$\text{46}$
}
In order to obtain the Mazur identity, one considers two arbitrary hermitian matrices,
${\Phi}_{1}$
and
${\Phi}_{2}$
. The aim is to compute the Laplacian (with respect to an arbitrary metric
$\overline{\text{}\mathit{g}\text{}}$
) of the relative difference
$\Psi $
, say, between
${\Phi}_{2}$
and
${\Phi}_{1}$
,
$$\begin{array}{c}\Psi \equiv {\Phi}_{2}{\Phi}_{1}^{1}1l.\end{array}$$ 
(33)

It turns out to be convenient to introduce the current matrices
${\mathcal{J}}_{1}={\Phi}_{1}^{1}\overline{\nabla}{\Phi}_{1}$
and
${\mathcal{J}}_{2}={\Phi}_{2}^{1}\overline{\nabla}{\Phi}_{2}$
, and their difference
${\mathcal{J}}_{\u25b3}={\mathcal{J}}_{2}{\mathcal{J}}_{1}$
, where
$\overline{\nabla}$
denotes the covariant derivative with respect to the metric under consideration. Using
$\overline{\nabla}\Psi ={\Phi}_{2}{\mathcal{J}}_{\u25b3}{\Phi}_{1}^{1}$
, the Laplacian of
$\Psi $
becomes
$$\overline{\Delta}\Psi =\langle \overline{\nabla}{\Phi}_{2},{\mathcal{J}}_{\u25b3}\rangle {\Phi}_{1}^{1}+{\Phi}_{2}\langle {\mathcal{J}}_{\u25b3},\overline{\nabla}{\Phi}_{1}^{1}\rangle +{\Phi}_{2}\left(\overline{\nabla}{\mathcal{J}}_{\u25b3}\right){\Phi}_{1}^{1}.$$
For hermitian matrices one has
$\overline{\nabla}{\Phi}_{2}={\mathcal{J}}_{2}^{\u2020}{\Phi}_{2}$
and
$\overline{\nabla}{\Phi}_{1}^{1}={\Phi}_{1}^{1}{\mathcal{J}}_{1}^{\u2020}$
, which can be used to combine the trace of the first two terms on the RHS of the above expression. One easily finds
$$\begin{array}{c}\text{Trace}\left\{\overline{\Delta}\Psi \right\}=\text{Trace}\left\{\langle {\Phi}_{1}^{1}{\mathcal{J}}_{\u25b3}^{\u2020},{\Phi}_{2}{\mathcal{J}}_{\u25b3}\rangle +{\Phi}_{2}\left(\overline{\nabla}{\mathcal{J}}_{\u25b3}\right){\Phi}_{1}^{1}\right\}.\end{array}$$ 
(34)

The above expression is an identity for the relative difference of two arbitrary hermitian matrices. If the latter are solutions of a nonlinear
$\sigma $
model with action
$\int \text{Trace}\left\{\mathcal{J}\wedge \overline{*}\mathcal{J}\right\}$
, then their currents are conserved [see Eq. ( 32 )], implying that the second term on the RHS vanishes.
Moreover, if the
$\sigma $
model describes a mapping with coset space
$SU(p,q)/S\left(U\right(p)\times U(q\left)\right)$
, then this is parametrized by positive hermitian matrices of the form
$\Phi =g{g}^{\u2020}$
.^{
$\text{47}$
}
Hence, the “onshell” restriction of the Mazur identity to
$\sigma $
models with coset
$SU(p,q)/S\left(U\right(p)\times U(q\left)\right)$
becomes
$$\begin{array}{c}\text{Trace}\left\{\overline{\Delta}\Psi \right\}=\text{Trace}\langle \mathcal{\mathcal{M}},{\mathcal{\mathcal{M}}}^{\u2020}\rangle ,\end{array}$$ 
(35)

where
$\mathcal{\mathcal{M}}\equiv {g}_{1}^{1}{\mathcal{J}}_{\u25b3}^{\u2020}{g}_{2}$
.
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
$\Phi $
be represented in the form
$g{g}^{\u2020}$
forces the reduction of the EM system with respect to a spacelike Killing field; otherwise the coset is
$SU(2,1)/S\left(U\right(1,1)\times U(1\left)\right)$
, which is not of the desired form. As a consequence of the spacelike reduction, the threemetric
$\overline{\text{}\mathit{g}\text{}}$
is not Riemannian, and the RHS of Eq. ( 35 ) is indefinite, unless the matrix valued oneform
$\mathcal{\mathcal{M}}$
is spacelike. This is the case if there exists a timelike Killing field with
${L}_{k}\Phi =0$
, 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. ( 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
$\sigma $
model which is effectively coupled to threedimensional gravity. The target manifold is the pseudosphere
$SO(2,1)/SO\left(2\right)\approx SU(1,1)/U\left(1\right)$
, 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,
$SU(2,1)/S\left(U\right(1,1)\times U(1\left)\right)$
, is represented by a hermitian matrix
$\Phi $
, 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
$\overline{G}/\overline{H}$
, 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^{
$\text{48}$
}
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
$SL(2,IR)$
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
$SL(2,IR)\times SO(1,2)$
:
The target space for dilatonaxion gravity with an
$U\left(1\right)$
vector field is the coset
$SO(2,3)/\left(SO\right(2)\times SO(1,2\left)\right)$
[
60]
. 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
$4\times 4$
(rather than
$5\times 5$
) matrices. The relevant coset was shown to be
$Sp(4,IR)/U(1,1)$
.^{
$\text{49}$
}
Common to the black hole solutions of the above models is the fact that their Komar mass can be expressed in terms of the total charges and the area and surface gravity of the horizon [
89]
. The reason for this is the following: Like the EM equations ( 32 ), the stationary field equations consist of the threedimensional Einstein equations and the
$\sigma $
model equations,
$$\begin{array}{c}{\overline{R}}_{ij}=\frac{1}{4}\text{Trace}\left\{{\mathcal{J}}_{i}{\mathcal{J}}_{j}\right\},d\overline{*}\mathcal{J}=0.\end{array}$$ 
(36)

The current oneform
$\mathcal{J}\equiv {\Phi}^{1}d\Phi $
is given in terms of the hermitian matrix
$\Phi $
, which comprises all scalar fields arising on the effective level. The
$\sigma $
model equations,
$d\overline{*}\mathcal{J}=0$
, include
$\text{dim}\left(G\right)$
differential current conservation laws, of which
$\text{dim}\left(H\right)$
are redundant. Integrating all equations over a spacelike hypersurface extending from the horizon to infinity, Stokes' theorem yields a set of relations between the charges and the horizonvalues of the scalar potentials.^{
$\text{50}$
}
The crucial observation is that Stokes' theorem provides
$\text{dim}\left(G\right)$
independent Smarr relations, rather than only
$\text{dim}(G/H)$
ones. (This is due to the fact that all
$\sigma $
model currents are algebraically independent, although there are
$\text{dim}\left(H\right)$
differential identities which can be derived from the
$\text{dim}(G/H)$
field equations.) The complete set of Smarr type formulas can be used to get rid of the horizonvalues of the scalar potentials. In this way one obtains a relation which involves only the Komar mass, the charges and the horizon quantities. For the EMdilatonaxion system one finds, for instance [
89]
,
$$\begin{array}{c}{\left(\frac{1}{4\pi}\kappa \mathcal{A}\right)}^{2}={M}^{2}+{N}^{2}+{D}^{2}+{A}^{2}{Q}^{2}{P}^{2},\end{array}$$ 
(37)

where
$\kappa $
and
$\mathcal{A}$
are the surface gravity and the area of the horizon, and the RHS comprises the asymptotic flux integrals, that is, the total mass, the NUT charge, the dilaton and axion charges, and the electric and magnetic charges, respectively.^{
$\text{51}$
}
A very simple illustration of the idea outlined above is the static, purely electric EM system.
In this case, the electrovac coset
$SU(2,1)/S\left(U\right(1,1)\times U(1\left)\right)$
reduces to
$G/H=SU(1,1)/IR$
.
The matrix
$\Phi $
is parametrized in terms of the electric potential
$\phi $
and the gravitational potential
$\sigma \equiv {k}_{\mu}{k}^{\mu}$
. The
$\sigma $
model equations comprise
$\text{dim}\left(G\right)=3$
differential conservation laws, of which
$\text{dim}\left(H\right)=1$
is redundant:
$$\begin{array}{c}d\overline{*}\left(\frac{d\phi}{\sigma}\right)=0,d\overline{*}\left(\frac{d\sigma}{\sigma}2\phi \frac{d\phi}{\sigma}\right)=0,\end{array}$$ 
(38)

$$\begin{array}{c}d\overline{*}\left(\left(\sigma +{\phi}^{2}\right)\frac{d\phi}{\sigma}\phi \frac{d\sigma}{\sigma}\right)=0.\end{array}$$ 
(39)

[It is immediately verified that Eq. ( 39 ) is indeed a consequence of the Maxwell and Einstein Eqs. ( 38 ).] Integrating Eqs. ( 38 ) over a spacelike hypersurface and using Stokes' theorem yields^{
$\text{52}$
}
$$\begin{array}{c}Q={Q}_{H},M=\frac{\kappa}{4\pi}\mathcal{A}+{\phi}_{H}{Q}_{H},\end{array}$$ 
(40)

which is the wellknown Smarr formula. In a similar way, Eq. ( 39 ) provides an additional relation of the Smarr type,
$$\begin{array}{c}Q=2{\phi}_{H}\frac{\kappa}{4\pi}\mathcal{A}+{\phi}_{H}^{2}{Q}_{H},\end{array}$$ 
(41)

which can be used to compute the horizonvalue of the electric potential,
${\phi}_{H}$
. Using this in the Smarr formula ( 40 ) gives the desired expression for the total mass,
${M}^{2}=(\kappa \mathcal{A}/4\pi {)}^{2}+{Q}^{2}$
.
In the “extreme” case, the BPS bound [
74]
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 (
$\kappa \ne 0$
), Eq. ( 37 ) was derived by Breitenlohner et al. [
15]
. In fact, many of the spherically symmetric black hole solutions with scalar and vector fields [
72]
, [
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. [
75]
recently obtained Eq. ( 37 ) for spherically symmetric solutions with an arbitrary number of vector and moduli fields.
The above derivation of the mass formula (
37 ) is neither restricted to spherically symmetric configurations, nor are the solutions required to be static. The crucial observation is that the coset structure gives rise to a set of Smarr formulas which is sufficiently large to derive the desired relation. Although the result ( 37 ) was established by using the explicit representations of the EM and EMdilatonaxion coset spaces [
89]
, similar relations are expected to exist in the general case.
More precisely, it should be possible to show that the Hawking temperature of all asymptotically flat (or asymptotically NUT) nonrotating black holes with massless scalars and Abelian vector fields is given by
$$\begin{array}{c}{T}_{H}=\frac{2}{\mathcal{A}}\sqrt{\sum ({Q}_{S}{)}^{2}\sum ({Q}_{V}{)}^{2}},\end{array}$$ 
(42)

provided that the stationary field equations assume the form ( 36 ), where
$\Phi $
is a map into a symmetric space,
$G/H$
. Here
${Q}_{S}$
and
${Q}_{V}$
denote the charges of the scalars (including the gravitational ones) and the vector fields, respectively.
5.3 The IsraelWilson Class
A particular class of solutions to the stationary EM equations is obtained by requiring that the Riemannian manifold
$(\Sigma ,\overline{\text{}\mathit{g}\text{}})$
is flat [
102]
. For
${\overline{g}}_{ij}={\delta}_{ij}$
, the threedimensional Einstein equations obtained from variations of the effective action ( 28 ) with respect to
$\overline{\text{}\mathit{g}\text{}}$
become^{
$\text{53}$
}
$$\begin{array}{c}4\sigma \Lambda {,}_{i}\overline{\Lambda}{,}_{j}=\left(E{,}_{i}+2\overline{\Lambda}\Lambda {,}_{i}\right)\left(\overline{E}{,}_{j}+2\Lambda \overline{\Lambda}{,}_{j}\right).\end{array}$$ 
(43)

Israel and Wildon [
102]
have shown that all solutions of this equation fulfill
$\Lambda ={c}_{0}+{c}_{1}E$
. In fact, it is not hard to verify that this ansatz solves Eq. ( 43 ), provided that the complex constants
${c}_{0}$
and
${c}_{1}$
are subject to
${c}_{0}{\overline{c}}_{1}+{c}_{1}{\overline{c}}_{0}=1/2$
. Using asymptotic flatness, and adopting a gauge where the electromagnetic potentials and the twist potential vanish in the asymptotic regime, one has
${E}_{\infty}=1$
and
${\Lambda}_{\infty}=0$
, and thus
$$\begin{array}{c}\Lambda =\frac{{\text{e}}^{i\alpha}}{2}\left(1E\right),\text{where}\alpha \in IR\text{.}\end{array}$$ 
(44)

It is crucial that this ansatz solves both the equation for
$E$
and the one for
$\Lambda $
: One easily verifies that Eqs. ( 29 ) reduce to the single equation
$$\begin{array}{c}\overline{\Delta}{\left(1+E\right)}^{1}=0,\end{array}$$ 
(45)

where
$\overline{\Delta}$
is the threedimensional flat Laplacian. For static, purely electric configurations the twist potential
$Y$
and the magnetic potential
$\psi $
vanish. The ansatz ( 44 ), together with the definitions of the Ernst potentials,
$E=\sigma \Lambda {}^{2}+iY$
and
$\Lambda =\phi +i\psi $
(see Sect. 4.5 ), yields
$$\begin{array}{c}1+E=2\sqrt{\sigma},\text{and}\phi =1\sqrt{\sigma}.\end{array}$$ 
(46)

Since
${\sigma}_{\infty}=1$
, the linear relation between
$\phi $
and the gravitational potential
$\sqrt{\sigma}$
implies
$(d\sigma {)}_{\infty}=(2d\phi {)}_{\infty}$
. By virtue of this, the total mass and the total charge of every asymptotically flat, static, purely electric IsraelWilson solution are equal:
$$\begin{array}{c}M=\frac{1}{8\pi}\int *dk=\frac{1}{4\pi}\int *F=Q,\end{array}$$ 
(47)

where the integral extends over an asymptotic twosphere.^{
$\text{54}$
}
The simplest nontrivial solution of the flat Poisson equation ( 45 ),
$\overline{\Delta}{\sigma}^{1/2}=0$
, corresponds to a linear combination of
$n$
monopole sources
${m}_{a}$
located at arbitrary points
${\underline{x}}_{a}$
,
$$\begin{array}{c}{\sigma}^{1/2}\left(\underline{x}\right)=1+{\sum}_{a=1}^{n}\frac{{m}_{a}}{\underline{x}{\underline{x}}_{a}}.\end{array}$$ 
(48)

This is the PapapetrouMajumdar (PM) solution [
143]
, [
128]
, with spacetime metric
$\text{}\mathit{g}\text{}=\sigma d{t}^{2}+{\sigma}^{1}d{\underline{x}}^{2}$
and electric potential
$\phi =1\sqrt{\sigma}$
. The PM metric describes a regular black hole spacetime, where the horizon comprises
$n$
disconnected components.^{
$\text{55}$
}
In Newtonian terms, the configuration corresponds to
$n$
arbitrarily located charged mass points with
$\left{q}_{a}\right=\sqrt{G}{m}_{a}$
. The PM solution escapes the uniqueness theorem for the ReissnerNordström metric, since the latter applies exclusively to spacetimes with
$M>\leftQ\right$
.
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.^{
$\text{56}$
}
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
Our aim here is to discuss the circularity problem in some more detail. We refer the reader to Sect. 2.1 and Sect. 3.5 for the general context and for references concerning the staticity and the circularity issues. In both cases, the task is to use the symmetry properties of the matter model in order to establish the Frobenius integrability conditions for the Killing field(s). The link between the relevant components of the stressenergy tensor and the integrability conditions is provided by a general identity for the derivative of the twist of a Killing field
$\xi $
, say,
$$\begin{array}{c}d{\omega}_{\xi}=*\left[\xi \wedge R\left(\xi \right)\right],\end{array}$$ 
(49)

and Einstein's equations, implying
$\xi \wedge R\left(\xi \right)=8\pi [\xi \wedge T(\xi \left)\right]$
.^{
$\text{57}$
}
For a stationary and axisymmetric spacetime with Killing fields (oneforms)
$k$
and
$m$
, Eq. ( 49 ) implies^{
$\text{58}$
}
$$\begin{array}{c}d\left(m,{\omega}_{k}\right)=8\pi *\left[m\wedge k\wedge T\left(k\right)\right],\end{array}$$ 
(50)

and similarly for
$k\leftrightarrow m$
.^{
$\text{59}$
}
By virtue of Eq. ( 50 ) – and the fact that the Frobenius condition
$m\wedge k\wedge dk=0$
can be written as
$\left(m,{\omega}_{k}\right)=0$
– the circularity problem is reduced to the following two tasks:

∙
(i) Show that
$d\left(m,{\omega}_{k}\right)=0$
implies
$\left(m,{\omega}_{k}\right)=0$
.

∙
(ii) Establish
$m\wedge k\wedge T\left(k\right)=0$
from the stationary and axisymmetric matter equations.
(i) Since
$\left(m,{\omega}_{k}\right)$
is a function , it must be constant if its derivative vanishes. As
$m$
vanishes on the rotation axis, this implies
$\left(m,{\omega}_{k}\right)=0$
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
$d{\omega}_{k}=0$
that
${\omega}_{k}$
vanishes is more involved, since
${\omega}_{k}$
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
${\omega}_{k}$
is closed.^{
$\text{60}$
}
) (ii) While
$m\wedge k\wedge T\left(k\right)=0$
follows from the symmetry conditions for electromagnetic fields [
27]
and for scalar fields [
86]
, it cannot be established for nonAbelian gauge fields [
88]
. This implies that the usual foliation of spacetime used to integrate the stationary and axisymmetric Maxwell equations is too restrictive to treat the EinsteinYangMills (EYM) system. This is seen as follows:
In Sect.(
4.4 ) we have derived the formula ( 22 ). By virtue of Eq. ( 10 ) this becomes an expression for the derivative of the twist in terms of the electric YangMills potential
${\phi}_{k}$
(defined with respect to the stationary Killing field
$k$
) and the magnetic oneform
${i}_{k}*F=\sigma \overline{*}(\overline{F}+\phi f)$
:
$$\begin{array}{c}d\left[{\omega}_{k}+4\hat{\text{tr}}\left\{{\phi}_{k}{i}_{k}*F\right\}\right]=0.\end{array}$$ 
(51)

Contracting this relation with the axial Killing field
$m$
, and using again the fact that the Lie derivative of
${\omega}_{k}$
with respect to
$m$
vanishes, yields immediately
$$\begin{array}{c}d\left(m,{\omega}_{k}\right)=0\u27fa\hat{\text{tr}}\left\{{\phi}_{k}\left(*F\right)(k,m)\right\}=0.\end{array}$$ 
(52)

The difference between the Abelian and the nonAbelian case lies in the circumstance that the Maxwell equations automatically imply that the
$\left(km\right)$
component of
$*F$
vanishes,^{
$\text{61}$
}
whereas this does not follow from the YangMills equations. Moreover, the latter do not imply that the Lie algebra valued scalars
${\phi}_{k}$
and
$\left(*F\right)(k,m)$
are orthogonal. Hence, circularity is a generic property of the EinsteinMaxwell (EM) system, whereas it imposes additional requirements on nonAbelian gauge fields.
Both the staticity and the circularity theorems can be established for scalar fields or, more generally, scalar mappings with arbitrary target manifolds: Consider a selfgravitating scalar mapping
$\phi :(M,\text{}\mathit{g}\text{})\to (N,\text{}\mathit{G}\text{})$
with Lagrangian
$L[\phi ,d\phi ,\text{}\mathit{g}\text{},\text{}\mathit{G}\text{}]$
. The stress energy tensor is of the form
$$\begin{array}{c}T={P}_{AB}d{\phi}^{A}\otimes d{\phi}^{B}+P\text{}\mathit{g}\text{},\end{array}$$ 
(53)

where the functions
${P}_{AB}$
and
$P$
may depend on
$\phi $
,
$d\phi $
, the spacetime metric
$\text{}\mathit{g}\text{}$
and the target metric
$\text{}\mathit{G}\text{}$
. If
$\phi $
is invariant under the action of a Killing field
$\xi $
– in the sense that
${L}_{\xi}{\phi}^{A}=0$
for each component
${\phi}^{A}$
of
$\phi $
– then the oneform
$T\left(\xi \right)$
becomes proportional to
$\xi $
:
$T\left(\xi \right)=P\xi $
.
By virtue of the Killing field identity (
49 ), this implies that the twist of
$\xi $
is closed. Hence, the staticity and the circularity issue for selfgravitating scalar mappings reduce to the corresponding vacuum problems. From this one concludes that stationary nonrotating black hole configuration of selfgravitating scalar fields are static if
${L}_{k}{\phi}^{A}=0$
, while stationary and axisymmetric ones are circular if
${L}_{k}{\phi}^{A}={L}_{m}{\phi}^{A}=0$
.
6.2 Boundary Value Formulation
The vacuum and the EM equations in the presence of a Killing symmetry describe harmonic mappings into coset manifolds, effectively coupled to threedimensional gravity (see Sect. 4 ). This feature is shared by a variety of other selfgravitating theories with scalar (moduli) and Abelian vector fields (see Sect. 5.2 ), for which the field equations assume the form ( 32 ):
$$\begin{array}{c}{\overline{R}}_{ij}=\frac{1}{4}\text{Trace}\left\{{\mathcal{J}}_{i}{\mathcal{J}}_{j}\right\},d\overline{*}\mathcal{J}=0,\end{array}$$ 
(54)

The current oneform
$\mathcal{J}={\Phi}^{1}d\Phi $
is given in terms of the hermitian matrix
$\Phi $
, which comprises the norm and the generalized twist potential of the Killing field, the fundamental scalar fields and the electric and magnetic potentials arising on the effective level for each Abelian vector field. If the dimensional reduction is performed with respect to the axial Killing field
$m={\partial}_{\phi}$
with norm
$X\equiv \left(m,m\right)$
, then
${\overline{R}}_{ij}$
is Ricci tensor of the pseudoRiemannian threemetric
$\overline{\text{}\mathit{g}\text{}}$
, defined by
$$\begin{array}{c}\text{}\mathit{g}\text{}=X(d\phi +a{)}^{2}+\frac{1}{X}\overline{\text{}\mathit{g}\text{}}.\end{array}$$ 
(55)

In the stationary and axisymmetric case under consideration, there exists, in addition to
$m$
, an asymptotically timelike Killing field
$k$
. Since
$k$
and
$m$
fulfill the Frobenius integrability conditions, the spacetime metric can be written in the familiar
$(2+2)$
split.^{
$\text{62}$
}
Hence, the circularity property implies that

∙
$(\Sigma ,\overline{\text{}\mathit{g}\text{}})$
is a static pseudoRiemannian threedimensional manifold with metric
$\overline{\text{}\mathit{g}\text{}}={\rho}^{2}d{t}^{2}+\stackrel{~}{\text{}\mathit{g}\text{}}$
;

∙
the connection
$a$
is orthogonal to the twodimensional Riemannian manifold
$(\stackrel{~}{\Sigma},\stackrel{~}{\text{}\mathit{g}\text{}})$
, that is,
$a={a}_{t}dt$
;

∙
the functions
${a}_{t}$
and
${\stackrel{~}{g}}_{ab}$
do not depend on the coordinates
$t$
and
$\phi $
.
With respect to the resulting Papapetrou metric [
144]
,
$$\begin{array}{c}\text{}\mathit{g}\text{}=X(d\phi +{a}_{t}dt{)}^{2}+\frac{1}{X}\left({\rho}^{2}d{t}^{2}+\stackrel{~}{\text{}\mathit{g}\text{}}\right),\end{array}$$ 
(56)

the field equations ( 54 ) become a set of partial differential equations on the twodimensional Riemannian manifold
$(\stackrel{~}{\Sigma},\stackrel{~}{\text{}\mathit{g}\text{}})$
:
$$\begin{array}{c}\stackrel{~}{\Delta}\rho =0,\end{array}$$ 
(57)

$$\begin{array}{c}{\stackrel{~}{R}}_{ab}\frac{1}{\rho}{\stackrel{~}{\nabla}}_{b}{\stackrel{~}{\nabla}}_{a}\rho =\frac{1}{4}\text{Trace}\left\{{\mathcal{J}}_{a}{\mathcal{J}}_{b}\right\},\end{array}$$ 
(58)

$$\begin{array}{c}{\stackrel{~}{\nabla}}^{a}\left(\rho {J}_{a}\right)=0,\end{array}$$ 
(59)

as is seen from the standard reduction of the Ricci tensor
${\overline{R}}_{ij}$
with respect to the static threemetric
$\overline{\text{}\mathit{g}\text{}}={\rho}^{2}d{t}^{2}+\stackrel{~}{\text{}\mathit{g}\text{}}$
.^{
$\text{63}$
}
The last simplification of the field equations is due to the circumstance that
$\rho $
can be chosen as one of the coordinates on
$(\stackrel{~}{\Sigma},\stackrel{~}{\text{}\mathit{g}\text{}})$
. This follows from the facts that
$\rho $
is harmonic (with respect to the Riemannian twometric
$\stackrel{~}{\text{}\mathit{g}\text{}}$
) and nonnegative, and that the domain of outer communications of a stationary black hole spacetime is simply connected [
44]
. The function
$\rho $
and the conjugate harmonic function
$z$
are called Weyl coordinates.^{
$\text{64}$
}
With respect to these, the metric
$\stackrel{~}{\text{}\mathit{g}\text{}}$
can be chosen to be conformally flat, such that one ends up with the spacetime metric
$$\begin{array}{c}\text{}\mathit{g}\text{}=\frac{{\rho}^{2}}{X}d{t}^{2}+X{\left(d\phi +{a}_{t}dt\right)}^{2}+\frac{1}{X}{e}^{2h}\left(d{\rho}^{2}+d{z}^{2}\right),\end{array}$$ 
(60)

the
$\sigma $
model equations
$$\begin{array}{c}{\partial}_{\rho}\left(\rho {\mathcal{J}}_{\rho}\right)+{\partial}_{z}\left(\rho {\mathcal{J}}_{z}\right)=0,\end{array}$$ 
(61)

and the remaining Einstein equations
$$\begin{array}{c}{\partial}_{\rho}h=\frac{\rho}{8}\text{Trace}\left\{{\mathcal{J}}_{\rho}{\mathcal{J}}_{\rho}{\mathcal{J}}_{z}{\mathcal{J}}_{z}\right\},{\partial}_{z}h=\frac{\rho}{4}\text{Trace}\left\{{\mathcal{J}}_{\rho}{\mathcal{J}}_{z}\right\},\end{array}$$ 
(62)

for the function
$h(\rho ,z)$
.^{
$\text{65}$
}
Since Eq. ( 58 ) is conformally invariant, the metric function
$h(\rho ,z)$
does not appear in the
$\sigma $
model equation ( 61 ). Therefore, the stationary and axisymmetric equations reduce to a boundary value problem for the matrix
$\Phi $
on a fixed, twodimensional background. Once the solution to Eq. ( 61 ) is known, the remaining metric function
$h(\rho ,z)$
is obtained from Eqs. ( 62 ) by quadrature.
[
5]
).
6.3 The Ernst Equations
The Ernst equations [
52]
, [
53]
– being the key to the KerrNewman metric – are the explicit form of the circular
$\sigma $
model equations ( 61 ) for the EM system, that is, for the coset
$SU(2,1)/S\left(U\right(2)\times U(1\left)\right)$
.^{
$\text{66}$
}
The latter is parametrized in terms of the Ernst potentials
$\Lambda =\phi +i\psi $
and
$E=X\Lambda \overline{\Lambda}+iY$
, where the four scalar potentials are obtained from Eqs. ( 26 ) and ( 27 ) with
$\xi =m$
. Instead of writing out the components of Eq. ( 61 ) in terms of
$\Lambda $
and
$E$
, it is more convenient to consider Eqs. ( 29 ), and to reduce them with respect to the static metric
$\overline{\text{}\mathit{g}\text{}}={\rho}^{2}d{t}^{2}+\stackrel{~}{\text{}\mathit{g}\text{}}$
(see Sect. 6.2 ).
Introducing the complex potentials
$\varepsilon $
and
$\lambda $
according to
$$\begin{array}{c}\varepsilon =\frac{1E}{1+E},\lambda =\frac{2\Lambda}{1+E},\end{array}$$ 
(63)

one easily finds the two equations
$$\begin{array}{c}\stackrel{~}{\Delta}\zeta +\langle d\zeta ,\frac{d\rho}{\rho}+\frac{2(\overline{\varepsilon}d\varepsilon +\overline{\lambda}d\lambda )}{1\varepsilon {}^{2}\lambda {}^{2}}\rangle =0,\end{array}$$ 
(64)

where
$\zeta $
stands for either of the complex potentials
$\varepsilon $
or
$\lambda $
, and where the Laplacian and the inner product refer to the twodimensional metric
$\stackrel{~}{\text{}\mathit{g}\text{}}$
.
In order to control the boundary conditions for black holes, it is convenient to introduce prolate spheroidal coordinates
$x$
and
$y$
, defined in terms of the Weyl coordinates
$\rho $
and
$z$
by
$$\begin{array}{c}{\rho}^{2}={\mu}^{2}({x}^{2}1)(1{y}^{2}),z=\mu xy,\end{array}$$ 
(65)

where
$\mu $
is a constant. The domain of outer communications, that is, the upper halfplane
$\rho \ge 0$
, corresponds to the semistrip
$\mathcal{S}=\left\{\right(x,y\left)\rightx\ge 1,\lefty\right\le 1\}$
. The boundary
$\rho =0$
consists of the horizon (
$x=0$
) and the northern (
$y=1$
) and southern (
$y=1$
) segments of the rotation axis.
In terms of
$x$
and
$y$
, the Riemannian metric
$\stackrel{~}{\text{}\mathit{g}\text{}}$
becomes
$({x}^{2}1{)}^{1}d{x}^{2}+(1{y}^{2}{)}^{1}d{y}^{2}$
, up to a conformal factor which does not enter Eqs. ( 64 ). The Ernst equations finally assume the form (
${\varepsilon}_{x}\equiv {\partial}_{x}\varepsilon $
, etc.)
$$\left(1\varepsilon {}^{2}\lambda {}^{2}\right)\left\{{\partial}_{x}({x}^{2}1){\partial}_{x}+{\partial}_{y}(1{y}^{2}){\partial}_{y}\right\}\zeta $$
$$\begin{array}{c}=2\left\{({x}^{2}1)\left(\overline{\varepsilon}{\varepsilon}_{x}+\overline{\lambda}{\lambda}_{x}\right){\partial}_{x}+(1{y}^{2})\left(\overline{\varepsilon}{\varepsilon}_{y}+\overline{\lambda}{\lambda}_{y}\right){\partial}_{y}\right\}\zeta ,\end{array}$$ 
(66)

where
$\zeta $
stand for
$\varepsilon $
or
$\lambda $
. A particularly simple solution to the Ernst equations is
$$\begin{array}{c}\varepsilon =px+iqy,\lambda ={\lambda}_{0},\text{where}{p}^{2}+{q}^{2}+{\lambda}_{0}^{2}=1\text{},\end{array}$$ 
(67)

with real constants
$p$
,
$q$
and
${\lambda}_{0}$
. The norm
$X$
, the twist potential
$Y$
and the electromagnetic potentials
$\phi $
and
$\psi $
(all defined with respect to the axial Killing field) are obtained from the above solution by using Eqs. ( 63 ) and the expressions
$X=\text{Re}(E)\Lambda {}^{2}$
,
$Y=\text{Im}(E)$
,
$\phi =\text{Re}(\Lambda )$
,
$\psi =\text{Im}(\Lambda )$
. The offdiagonal element of the metric,
$a={a}_{t}dt$
, is obtained by integrating the twist expression ( 10 ), where the twist oneform is given in Eq. ( 27 ).^{
$\text{67}$
}
Eventually, the metric function
$h$
is obtained from Eqs. ( 62 ) by quadrature.
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
$t\phi $
plane: The spacetime metric is invariant under
$t\to \phi $
,
$\phi \to t$
, if
$X$
,
${a}_{t}$
and
${e}^{2h}$
are replaced by
$kX$
,
${k}^{1}{a}_{t}$
and
$k{e}^{2h}$
, where
$k\equiv {a}_{t}^{2}{X}^{2}{\rho}^{2}$
. 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
${\partial}_{\phi}$
. If, on the other hand, one uses the stationary Killing field
${\partial}_{t}$
, then the Ernst equations are singular at the boundary of the ergoregion.
In terms of BoyerLindquist coordinates,
$$\begin{array}{c}r=m(1+px),cos\vartheta =y,\end{array}$$ 
(68)

one eventually finds the KerrNewman metric in the familiar form:
$$\begin{array}{c}\text{}\mathit{g}\text{}=\frac{\Delta}{\Xi}{\left[dt\alpha {sin}^{2}\vartheta d\phi \right]}^{2}+\frac{{sin}^{2}\vartheta}{\Xi}{\left[({r}^{2}+{\alpha}^{2})d\phi \alpha dt\right]}^{2}+\Xi \left[\frac{1}{\Delta}d{r}^{2}+d{\vartheta}^{2}\right],\end{array}$$ 
(69)

where the constant
$\alpha $
is defined by
${a}_{t}\equiv \alpha {sin}^{2}\vartheta $
. The expressions for
$\Delta $
,
$\Xi $
and the electromagnetic vector potential
$A$
show that the KerrNewman solution is characterized by the total mass
$M$
, the electric charge
$Q$
, and the angular momentum
$J=\alpha M$
:
$$\begin{array}{c}\Delta ={r}^{2}2Mr+{\alpha}^{2}+{Q}^{2},\Xi ={r}^{2}+{\alpha}^{2}{cos}^{2}\vartheta .\end{array}$$ 
(70)

$$\begin{array}{c}A=\frac{Q}{\Xi}r\left[dt\alpha {sin}^{2}\vartheta d\phi \right].\end{array}$$ 
(71)

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
$\mathcal{S}=\left\{\right(x,y\left)\rightx\ge 1,\lefty\right\le 1\}$
(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.^{
$\text{68}$
}
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.^{
$\text{69}$
}
The Mazur identity ( 34 ) applies to the relative difference
$\Psi ={\Phi}_{2}{\Phi}_{1}^{1}1l$
of two arbitrary hermitian matrices. If the latter are solutions of a
$\sigma $
model with symmetric target space of the form
$SU(p,q)/S\left(U\right(p)\times U(q\left)\right)$
, then the identity implies^{
$\text{70}$
}
$$\begin{array}{c}\text{Trace}\left\{\overline{\Delta}\Psi \right\}=\text{Trace}\langle \mathcal{\mathcal{M}},{\mathcal{\mathcal{M}}}^{\u2020}\rangle ,\end{array}$$ 
(72)

where
$\mathcal{\mathcal{M}}\equiv {g}_{1}^{1}{\mathcal{J}}_{\u25b3}^{\u2020}{g}_{2}$
, and
${\mathcal{J}}_{\u25b3}^{\u2020}$
is the difference between the currents.
The reduction of the EM equations with respect to the axial Killing field yields the coset
$SU(2,1)/S\left(U\right(2)\times U(1\left)\right)$
(see Sect. 4.5 ), which, reduces to the vacuum coset
$SU\left(2\right)/S\left(U\right(1)\times U(1\left)\right)$
(see Sect. 4.3 ). Hence, the above formula applies to both the axisymmetric vacuum and electrovac field equations, where the Laplacian and the inner product refer to the pseudoRiemannian threemetric
$\overline{\text{}\mathit{g}\text{}}$
defined by Eq. ( 55 ). Now using the existence of the stationary Killing symmetry and the circularity property, one has
$\overline{\text{}\mathit{g}\text{}}={\rho}^{2}d{t}^{2}+\stackrel{~}{\text{}\mathit{g}\text{}}$
, which reduces Eq. ( 72 ) to an equation on
$(\stackrel{~}{\Sigma},\stackrel{~}{\text{}\mathit{g}\text{}})$
.
Integrating over the semistrip
$\mathcal{S}$
and using Stokes' theorem immediately yields
$$\begin{array}{c}{\int}_{\partial \mathcal{S}}\rho \stackrel{~}{*}\text{Trace}\left\{d\Psi \right\}={\int}_{\mathcal{S}}\rho \text{Trace}\langle \mathcal{\mathcal{M}},{\mathcal{\mathcal{M}}}^{\u2020}\rangle \stackrel{~}{\eta},\end{array}$$ 
(73)

where
$\stackrel{~}{\eta}$
and
$\stackrel{~}{*}$
are the volume form and the Hodge dual with respect to
$\stackrel{~}{\text{}\mathit{g}\text{}}$
. The uniqueness of the KerrNewman metric follows from the facts that

∙
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
$\stackrel{~}{\eta}$
is a positive volumeform, since
$\stackrel{~}{\text{}\mathit{g}\text{}}$
is a Riemannian metric. Second, the factor
$\rho $
is nonnegative in
$\mathcal{S}$
, since
$\mathcal{S}$
is the image of the upper halfplane,
$\rho \ge 0$
. Last, the oneforms
${\mathcal{J}}_{\u25b3}$
and
$\mathcal{\mathcal{M}}$
are spacelike, since the matrices
$\Phi $
depend only on the coordinates of
$(\stackrel{~}{\Sigma},\stackrel{~}{\text{}\mathit{g}\text{}})$
.
In order to establish that
$\rho \text{Trace}\left\{d\Psi \right\}=0$
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^{
$\text{71}$
}
then shows that
$\rho \text{Trace}\left\{d\Psi \right\}$
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
$SU\left(2\right)$
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.
8 Acknowledgments
It is a pleasure to thank Othmar Brodbeck, Gary Gibbons, Domenico Giulini, Dieter Maison, Gernod Neugebauer, Norbert Straumann, Michael Volkov and Bob Wald for helpful discussions.
This work was supported by SNSF Grant P2141840.94.
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