In mathematical terms, a gauge field (with gauge group G, say) is a connection in a principal bundle P (M, G) over spacetime M . A gauge field is called symmetric with respect to the action of a symmetry group S of M, if it is described by an S -invariant connection on P (M, G). Hence, finding the symmetric gauge fields involves the task of classifying the principal bundles P (M, G) which admit the symmetry group S, acting by bundle automorphisms. This program was recently carried out by Brodbeck and Straumann for arbitrary gauge and symmetry groups [16], (see also [17], [22]), generalizing earlier work of Harnad et al. [80], Jadczyk [104] and Künzle [121].
The gauge fields constructed in the above way are invariant
under the action of
S
up to gauge transformations. This is also the starting point of
the alternative approach to the problem, due to Forgács and
Manton [54]. It implies that a gauge potential
A
is symmetric with respect to the action of a Killing field
, say, if there exists a Lie algebra valued function
, such that
where
is the generator of an infinitesimal gauge transformation,
denotes the Lie derivative, and
is the gauge covariant exterior derivative,
.
Let us now consider a stationary spacetime with
(asymptotically) time-like Killing field
k
. A stationary gauge potential is parametrized in terms of a
one-form
orthogonal to
k,
, and a Lie algebra valued potential
,
where we recall that
a
is the non-static part of the metric (8). For the sake of simplicity we adopt a gauge where
vanishes.
By virtue of the above decomposition, the field strength becomes
, where
is the Yang-Mills field strength for
and
. Using the expression (12
) for the vacuum action, one easily finds that the EYM
action,
gives rise to the effective action
where
is the gauge covariant derivative with respect to
, and where the inner product also involves the trace:
. The above action describes two scalar fields,
and
, and two vector fields,
a
and
, which are minimally coupled to three-dimensional gravity with
metric
. Like in the vacuum case, the connection
a
enters
only via the field strength
. Again, this gives rise to a differential conservation law,
by virtue of which one can (locally) introduce a generalized
twist potential
Y, defined by
.
The main difference between the Abelian and the non-Abelian
case concerns the variational equation for
, that is, the Yang-Mills equation for
: The latter assumes the form of a differential conservation law
only in the Abelian case. For non-Abelian gauge groups,
is no longer an exact two-form, and the gauge covariant
derivative of
causes source terms in the corresponding Yang-Mills
equation:
Hence, the scalar magnetic potential - which can be introduced
in the Abelian case according to
- ceases to exist for non-Abelian Yang-Mills fields. The
remaining stationary EYM equations are easily derived from
variations of
with respect to the gravitational potential
, the electric Yang-Mills potential
and the three-metric
.
As an application, we note that the effective action (21) is particularly suited for analyzing stationary perturbations
of static (a
= 0), purely magnetic (
) configurations [18
], such as the Bartnik-McKinnon solitons [4] and the corresponding black hole solutions [174], [122], [8]. The two crucial observations in this context are [18
], [175]:
Using Eq. (22) to introduce the twist potential
Y, the fluctuation equations for the first order quantities
and
assume the form of a self-adjoint system [18]. Considering perturbations of spherically symmetric
configurations, one can expand
and
in terms of isospin harmonics. In this way one obtains a
Sturm-Liouville problem, the solutions of which reveal the
features mentioned in the last paragraph of Sect.
3.5
[21].
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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 |