1.1 General remarks

Our conception of black holes has experienced several dramatic changes during the last two hundred years: While the “dark stars” of Michell [235] and Laplace [210] were merely regarded as peculiarities of Newton’s law of gravity and his corpuscular theory of light, black holes are nowadays widely believed to exist in our universe (for a review on the evolution of the subject the reader is referred to Israel’s comprehensive account [178]; see also [52, 51]). Although the observations are necessarily indirect, the evidence for both stellar and galactic black holes has become compelling [275, 232, 233, 247, 242, 231]. There seems to be consensus [276, 197, 234, 248] that the two most convincing supermassive black-hole candidates are the galactic nuclei of NGC 4258 and of our own Milky Way [123].

The theory of black holes was initiated by the pioneering work of Chandrasekhar [53, 54] in the early 1930s. (However, the geometry of the Schwarzschild solution [290, 291] was misunderstood for almost half a century; the misconception of the “Schwarzschild singularity” was retained until the late 1950s.) Computing the Chandrasekhar limit for neutron stars [8], Oppenheimer and Snyder [257], and Oppenheimer and Volkoff [258] were able to demonstrate that black holes present the ultimate fate of sufficiently-massive stars. Modern black-hole physics started with the advent of relativistic astrophysics, in particular with the discovery of pulsars in 1967.

One of the most intriguing outcomes of the mathematical theory of black holes is the uniqueness theorem, applying to a class of stationary solutions of the Einstein–Maxwell equations. Strikingly enough, its consequences can be made into a test of general relativity [285]. The assertion, that all (four-dimensional) electrovacuum black-hole spacetimes are characterized by their mass, angular momentum and electric charge, is strangely reminiscent of the fact that a statistical system in thermal equilibrium is described by a small set of state variables as well, whereas considerably more information is required to understand its dynamical behavior. The similarity is reinforced by the black-hole–mass-variation formula [9Jump To The Next Citation Point] and the area-increase theorem [143Jump To The Next Citation Point, 69Jump To The Next Citation Point], which are analogous to the corresponding laws of ordinary thermodynamics. These mathematical relationships are given physical significance by the observation that the temperature of the black body spectrum of the Hawking radiation [142] is equal to the surface gravity of the black hole. There has been steady interest in the relationship between the laws of black hole mechanics and the laws of thermodynamics. In particular, computations within string theory seem to offer a promising interpretation of black-hole entropy [171]. The reader interested in the thermodynamic properties of black holes is referred to the review by Wald [316].

There has been substantial progress towards a proof of the celebrated uniqueness theorem, conjectured by Israel, Penrose and Wheeler in the late sixties [76Jump To The Next Citation Point, 79Jump To The Next Citation Point, 217Jump To The Next Citation Point] during the last four decades (see, e.g., [58Jump To The Next Citation Point] and [59Jump To The Next Citation Point] for previous reviews). Some open gaps, notably the electrovacuum staticity theorem [302Jump To The Next Citation Point, 303Jump To The Next Citation Point] and the topology theorems [109Jump To The Next Citation Point, 110Jump To The Next Citation Point, 85Jump To The Next Citation Point], have been closed (see [59Jump To The Next Citation Point, 73Jump To The Next Citation Point, 65Jump To The Next Citation Point] for related new results). Early on, the theorem led to the expectation that the stationary–black-hole solutions of other self-gravitating matter fields might also be parameterized 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 [14Jump To The Next Citation Point], a variety of new black hole configurations have been found, which violate the generalized no-hair conjecture, that suitably regular black-hole spacetimes are classified by a finite set of asymptotically-defined global charges. These solutions include non-Abelian black holes [310Jump To The Next Citation Point, 208Jump To The Next Citation Point, 24Jump To The Next Citation Point], as well as black holes with Skyrme [94Jump To The Next Citation Point, 161Jump To The Next Citation Point], Higgs [28Jump To The Next Citation Point, 254Jump To The Next Citation Point, 140] or dilaton fields [212, 132].

In fact, black-hole solutions with hair were already known before 1989: in 1982, Gibbons found a black-hole solution with a non-trivial dilaton field, within a model occurring in the low energy limit of N = 4 supergravity [126Jump To The Next Citation Point].

While the above counterexamples to the no-hair conjecture consist of static, spherically-symmetric configurations, there exists numerical evidence that static black holes are not necessarily spherically symmetric [192Jump To The Next Citation Point, 93]; in fact, they might not even need to be axisymmetric [278Jump To The Next Citation Point]. Moreover, some studies also indicate that non-rotating black holes need not be static [38Jump To The Next Citation Point]. 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 σ-model, effectively coupled to three-dimensional gravity [250Jump To The Next Citation Point]. (σ-models are a special case of harmonic maps, and we will use both terminologies interchangeably in our context.) Since this property is not shared by models with non-Abelian gauge fields [35Jump To The Next Citation Point], it is, with hindsight, not too surprising that the Einstein–Yang–Mills system admits black holes with hair.

However, there exist other black hole solutions, which are likely to be subject to a generalized version of the uniqueness theorem. These solutions appear in theories with self-gravitating massless scalar fields (moduli) coupled to Abelian gauge fields. The expectation that uniqueness results apply to a variety of these models arises from the observation that their dimensional reduction (with respect to a Killing symmetry) yields a σ-model with symmetric target space (see, e.g., [31Jump To The Next Citation Point, 86Jump To The Next Citation Point, 120Jump To The Next Citation Point], and references therein).


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