List of 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. 14–15, 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 [179].
4 Like the Papapetrou–Majumdar solution [143Jump To The Next Citation Point, 128Jump To The Next Citation Point], the Bekenstein solution is not a convincing counter-example to Wheeler’s no-hair conjecture, since the classical uniqueness results do not apply to black holes with degenerate horizons.
5 Some “cosmological” black hole solutions can be found in [176, 20] and references therein.
6 We refer to [38] and [39Jump To The Next Citation Point] for detailed discussions of the open problems.
7 The original proof of the SRT [84Jump To The Next Citation Point] 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 [40Jump To The Next Citation Point]; see also [39Jump To The Next Citation Point], 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× S2. This was established only recently by Chruściel and Wald [44Jump To The Next Citation Point], taking advantage of the topological censorship theorem [55]. A related version of the topology theorem, applying to globally hyperbolic – but not necessarily stationary – space-times was obtained by Jacobson and Venkataramani [103], and Galloway [56, 57, 58, 59]. We refer to [39Jump To The Next Citation Point], Sect. 2 for a detailed discussion.
9 See [6Jump To The Next Citation Point] for a complete classification of the isometry groups.
10 The existence of a foliation by maximal slices was established by Chruściel and Wald [43Jump To The Next Citation Point].
11 The positive energy conjecture was proven by Schoen and Yau [155, 156] and, using spinor techniques, by Witten [184]; see also [171].
12 Reduction of the EM action with respect to the time-like Killing field yields, instead, H = S(U(1,1)× U(1)).
13 The model considered by Gibbons arises naturally in the low energy limit of N = 4 supergravity; see also [76Jump To The Next Citation Point] and [69Jump To The Next Citation Point] in this context.
14 Originally, these solitons were constructed by numerical means. Existence proofs using rigorous methods were given later in [164, 162, 163] and [13].
15 See [21] for the general structure of the pulsation equations, [173, 24] and [11Jump To The Next Citation Point] for the sphaleron instabilities of the particle-like solutions, and [159] for a review on sphalerons
16 The solutions of the EYM–Higgs equations with a Higgs doublet are unstable [11, 183].
17 The stability properties are discussed in Weinberg’s comprehensive review on magnetically charged black holes [181].
18 Axisymmetric, static black black holes without spherical symmetry also exist within the pure EYM system and the EYM-dilaton model [115Jump To The Next Citation Point]; see Section ??.
19 We refer to [23Jump To The Next Citation Point] for the precise formulation of the statement in terms of Stiefel diagrams, and to [17Jump To The Next Citation Point, 18Jump To The Next Citation Point, 80Jump To The Next Citation Point] for the bundle classification of EYM solitons.
20 An early apparent success rested on a sign error [30Jump To The Next Citation Point]. Carter’s amended version of the proof was subject to a certain inequality between the electric and the gravitational potential [33Jump To The Next Citation Point]. 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 time-like Killing field.
21 The new proof given in [88Jump To The Next Citation Point] works under less restrictive topological assumptions, since it does not require the global existence of a twist potential
22 See the footnote on page 35.
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 Yang–Mills fields these conditions do no longer follow from the field equations and the invariance properties; see Section 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 self-gravitating scalar fields is, for instance, straightforward [86Jump To The Next Citation Point].
25 Although non-rotating, these configurations were not discussed in Section 3.2; in the present context I prefer to view them as particular circular configurations.
26 The related axisymmetric soliton solutions without spherical symmetry were previously obtained by the same authors [111, 113]; see also [112] for more details.
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 Section 3.4 that these black holes present counter-examples to the naive generalization of the staticity theorem, they are nice illustrations of the correct non-Abelian version of the theorem [167, 168].
29 A particular combination of the charged and the rotating branch was found in [175Jump To The Next Citation Point].
30 More precisely, the definition applies to strongly asymptotically predictable space-times; see [177Jump To The Next Citation Point], Chapter 12 for the exact statements.
31 See Section 2.1 and Section 3.5 for more information on the rigidity theorem.
32 We refer to [33] and [87Jump To The Next Citation Point], Chapter 6.3 and Chapter 6.4 for derivations and more details.
33 See, e.g. [47], p. 239 or [87Jump To The Next Citation Point], p. 92 for the proof.
34 For a compilation of Killing field identities we refer to [87Jump To The Next Citation Point], Chapter 2.
35 T (ξ) is the one-form with components [T(ξ)]μ ≡ Tμνξν.
36 The derivation of Eq. (6View Equation) is the main task; see, e.g. [177].
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 , ωk) = (k , ωm) = 0 implies d ωξ = 0 on the horizon generated by ξ = k+ Ωm; see [87Jump To The Next Citation Point], Chapter 7 for details).
39 See, e.g. [91] for the details of the derivation.
40 For arbitrary p-forms α and β the inner product is defined by ¯∗⟨α, β⟩ ≡ α∧ ¯∗β, where ¯∗ is the Hodge dual with respect to ¯g.
41 Here and in the following we use the symbol k for both the Killing field ∂t and the corresponding one-form − σ(dt+ a).
42 See, e.g. [116Jump To The Next Citation Point] or [10Jump To The Next Citation Point] for the general theory of symmetric spaces.
43 The symmetry condition (18View Equation) translates into Lkϕ = [ϕ,𝒱] and Lk ¯A = ¯D𝒱, which can be used to reduce the EYM equations in the presence of a Killing symmetry in a gauge invariant manner [94, 95].
44 tˆr{} denotes the normalized trace; e.g. ˆtr{τaτb} = δab for SU (2), where τa ≡ σa∕(2i).
45 For an arbitrary two-form β, iξβ is the one-form with components ξμβ μν.
46 The second Killing field is of crucial importance to the two-dimensional boundary value formulation of the field equations and to the integration of the Mazur identity. However, the derivation of the identity in the two-dimensional context is somewhat unnatural, since the dimensional reduction with respect to the second Killing field introduces a weight factor which is slightly veiling the σ-model structure [32Jump To The Next Citation Point].
47 We refer to [116, 51], and [10] for the theory of symmetric spaces.
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 For the generalization to the dilaton-axion system with multiple vector fields we refer to [66, 68].
50 A very familiar relation of this kind is the Smarr formula [161]; see Eq. (40View Equation) below.
51 The derivation of Eq. (37View Equation) 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 time-like 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 + ΩHm, 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 Σ), while the corresponding integral of ∗dk gives κ 𝒜∕4π; see [89Jump To The Next Citation Point] for details.
53 As we are considering stationary configurations we use the dimensional reduction with respect to the asymptotically time-like 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 [81Jump To The Next Citation Point] 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, Δ ξ = − 2R(ξ), where R (ξ) is the one-form with components ν [R (ξ)]μ ≡ R μνξ; see, e.g. [87], Chapter 2.)
58 Equation (50View Equation) 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 , ωk ) = Lm ωk − imdωk). In order to establish Lmωk = 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 While this proves the staticity theorem for vacuum and self-gravitating scalar fields [88Jump To The Next Citation Point], it does not solve the electrovac case; see Section 2.1 and Section 3.5 and references therein.
61 The Maxwell equation d ∗F = 0 and the symmetry property Lk ∗F = ∗LkF = 0 imply the existence of a magnetic potential, dψ = (∗F)(k, ⋅). Thus, (∗F)(k,m) = imd ψ = Lmψ = 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 (&tidle;Σ,&tidle;g) one has 𝒥t = 0 and ¯∗𝒥 = − ρdt∧ &tidle;∗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 [182Jump To The Next Citation Point], taking advantage of the Riemann mapping theorem (or, more precisely, Caratheodory’s extension of the theorem; see, e.g. [5]).
65 It is not hard to verify that Eq. (61View Equation) is the integrability condition for Eqs. (62View Equation); see also the end of Section 4.5.
66 Again, we consider the dimensional reduction with respect to the axial Killing field.
67 The Hodge dual in Eq. (10View Equation) now refers to the decomposition (55View Equation) with respect to the axial Killing field.
68 See, e.g. [31].
69 We refer the reader to [32] for a discussion of Bunting’s method.
70 See Section 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.