1.1 General

The theory of black holes was initiated by the pioneering work of Chandrasekhar [34, 35] in the early 1930s. Computing the Chandrasekhar limit for neutron stars [2], Oppenheimer and Snyder [141], and Oppenheimer and Volkoff [142] were able to demonstrate that black holes present the ultimate fate of sufficiently massive stars. Modern black hole physics started with the advent of relativistic astrophysics, in particular with the discovery of the pulsars in 1967. (The geometry of the Schwarzschild solution [157, 158] was, for instance, not understood for almost half a century; the misconception of the “Schwarzschild singularity” was retained until the late 1950s.)

One of the most intriguing outcomes of the mathematical theory of black holes is the uniqueness theorem, applying to the stationary solutions of the Einstein–Maxwell equations. Asserting that all electrovac black hole space-times are characterized by their mass, angular momentum and electric charge, the theorem bears a striking resemblance to the fact that a statistical system in thermal equilibrium is described by a small set of state variables as well, whereas considerably more information is required to understand its dynamical behavior. The similarity is reinforced by the black hole mass variation formula [3] and the area increase theorem [84], which are analogous to the corresponding laws of ordinary thermodynamics. These mathematical relationships are given physical significance by the observation that the temperature of the black body spectrum of the Hawking radiation [83] is equal to the surface gravity of the black hole.³

The proof of the celebrated uniqueness theorem, conjectured by Israel, Penrose and Wheeler in the late sixties, has been completed during the last three decades (see, e.g. [38] and [39] for reviews). Some open gaps, notably the electrovac staticity theorem [167, 168] and the topology theorem [57, 58, 44], have been closed recently (see [39] for new results). The beauty of the theorem provided support for the expectation that the stationary black hole solutions of other self-gravitating matter fields are also parametrized by their mass, angular momentum and a set of charges (generalized no-hair conjecture). However, ever since Bartnik and McKinnon discovered the first self-gravitating Yang–Mills soliton in 1988 [4], a variety of new black hole configurations which violate the generalized no-hair conjecture have been found. These include, for instance, non-Abelian black holes [174, 122, 9], and black holes with Skyrme [50, 97], Higgs [12] or dilaton fields [124, 77].

In fact, black hole solutions with hair were already known before 1989: The first example was the Bekenstein solution [7, 8], describing a conformally coupled scalar field in an extreme Reissner–Nordström spacetime. Since the horizon has vanishing surface gravity,⁴ and since the scalar field is unbounded on the horizon, the status of the Bekenstein solution gives still rise to some controversy [169]. In 1982, Gibbons found a new black hole solution within a model occurring in the low energy limit of supergravity [72]. The Gibbons solution, describing a Reissner–Nordström spacetime with a nontrivial dilaton field, must be considered the first flawless black hole solution with hair.

While the above counterexamples to the no-hair conjecture consist in static, spherically symmetric configurations, more recent investigations have revealed that static black holes are not necessarily spherically symmetric [114]; in fact, they need not even be axisymmetric [150]. Moreover, some new studies also indicate that non-rotating black holes need not be static [22]. The rich spectrum of stationary black hole configurations demonstrates that the matter fields are by far more critical to the properties of black hole solutions than expected for a long time. In fact, the proof of the uniqueness theorem is, at least in the axisymmetric case, heavily based on the fact that the Einstein–Maxwell equations in the presence of a Killing symmetry form a -model, effectively coupled to three-dimensional gravity [139]. Since this property is not shared by models with non-Abelian gauge fields [19], it is, with hindsight, not too surprising that the Einstein–Yang–Mills system admits black holes with hair.

There exist, however, other black hole solutions which are likely to be subject to a generalized version of the uniqueness theorem. These solutions appear in theories with self-gravitating massless scalar fields (moduli) coupled to Abelian vector fields. The expectation that uniqueness results apply to a variety of these models arises from the observation that their dimensional reduction (with respect to a Killing symmetry) yields a -model with symmetric target space (see, e.g. [15, 45, 67], and references therein).