The theory of black holes was initiated by the pioneering work of Chandrasekhar [33], [34] 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.
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
], [8
], and black holes with Skyrme [50
], [97
], Higgs [11
] or dilaton fields [124], [77].
In fact, black hole solutions with hair were already known
before 1989: The first example was the Bekenstein solution [6], [7], describing a conformally coupled scalar field in an extreme
Reissner-Nordström spacetime. Since the horizon has vanishing
surface gravity,
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 [73
]. 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 [115]; in fact, they need not even be axisymmetric [150
]. Moreover, some new studies also indicate that non-rotating
black holes need not be static [21
]. 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 [18
], 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. [14
], [45
], [67
], and references therein).
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Stationary Black Holes: Uniqueness and Beyond
Markus Heusler http://www.livingreviews.org/lrr-1998-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |