

Requiring spherical symmetry, the task to prove the no-hair
theorem for the Einstein-Maxwell (EM) system becomes almost
trivial. However, not even this part of the uniqueness proof can
be generalized: The first black hole solution demonstrating the
failure of the no-hair conjecture was obtained by Gibbons in 1982
[73
] within EM-dilaton theory.
The fact that the Gibbons solution carries no dilatonic charge
makes it asymptotically indistinguishable from a
Reissner-Nordström black hole with the same mass and electric
charge. However, since the latter is not a consistent solution of
the EM-dilaton equations, one might expect that - within a
given
matter model - the stationary black hole solutions are still
characterized by a set of global charges (generalized no-hair
conjecture). In fact, the Gibbons black hole supports the
generalized no-hair conjecture; its uniqueness within EM-dilaton
theory was established by Masood-ul-Alam in 1992 [130].
However, neither the original nor the generalized no-hair
conjecture are correct. For instance, the latter fails to be
valid within Einstein-Yang-Mills (EYM) theory: According to the
generalized version, any static solution of the EYM equations
should either coincide with the Schwarzschild metric or have some
non-vanishing Yang-Mills charges. This turned out not to be the
case, when, in 1989, various authors [174
], [122
], [8
] found a family of static black hole solutions with vanishing
Yang-Mills charges.
Since these solutions are asymptotically indistinguishable from
the Schwarzschild solution, and since the latter
is
a particular solution of the EYM equations, the non-Abelian
black holes violate the generalized no-hair conjecture.
As the non-Abelian black holes are not stable [166], [186
] [179],
one might adopt the view that they do not present actual threats
to the generalized no-hair 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 Einstein-Skyrme equations [50], [92], [93], [97] and to the EYM equations coupled to a Higgs triplet [11
], [13
], [180
], [1
].
Hence, the restriction of the generalized no-hair conjecture to
stable configurations is not correct either.
One of the reasons why it was not until 1989 that black hole
solutions with self-gravitating 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. [74] 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
three-dimensional
EYM equations admit
no soliton solutions
. The argument takes advantage of the fact that the magnetic
part of the Yang-Mills field has only one non-vanishing
component in
dimensions.
All this shows that it was conceivable to conjecture a
nonexistence theorem for soliton solutions of the EYM equations
(in
dimensions), and a no-hair 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
-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).


|
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
|