In mathematical terms, a gauge field (with gauge group , say) is a connection in a principal bundle
over spacetime
. A gauge field is symmetric with respect to the action of a symmetry group
of
, if it is described by an
-invariant connection on
. Hence, finding the symmetric
gauge fields involves the task of classifying the principal bundles
, which admit the symmetry
group
, acting by bundle automorphisms. This program was carried out by Brodbeck and Straumann for
arbitrary gauge and symmetry groups [33], (see also [34, 39]), generalizing earlier work of Harnad et al.
[138], Jadczyk [181] and Künzle [207].
The gauge fields constructed in the above way are invariant under the action of up to gauge
transformations. This is also the starting point of the alternative approach to the problem, due to
Forgács and Manton [105]. It implies that a gauge potential
is symmetric with respect to the
action of a Killing field
, say, if there exists a Lie algebra valued function
, such that
Let us now consider a stationary spacetime with (asymptotically) time-like Killing field . A
stationary gauge potential can be parameterized in terms of a one-form
orthogonal to
, in the sense
that
, and a Lie algebra valued potential
,
The main difference between the Abelian and the non-Abelian case concerns the variational equation for
, that is, the Yang–Mills equation for
: For non-Abelian gauge groups,
is no longer an exact
two-form, and the gauge covariant derivative of
introduces source terms in the corresponding
Yang–Mills equation:
As an application, we note that the effective action (6.16) is particularly suited for analyzing stationary
perturbations of static (
), purely magnetic (
) configurations [35
], such as the
Bartnik–McKinnon solitons [14] and the corresponding black-hole solutions [310, 208, 24]. The two crucial
observations in this context are [35
, 312]:
The second observation follows from the fact that the magnetic Yang–Mills equation (6.18) and the
Einstein equations for
and
become background equations, since they contain no linear terms in
and
. Therefore, the purely electric, non-static perturbations are governed by the twist
equation (6.17
) and the electric Yang–Mills equation (obtained from variations of
with respect to
).
Using Eq. (6.17) to introduce the twist potential
, the fluctuation equations for the first-order
quantities
and
assume the form of a self-adjoint system [35]. 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 Section 5.5 [38].
http://www.livingreviews.org/lrr-2012-7 |
Living Rev. Relativity 15, (2012), 7
![]() This work is licensed under a Creative Commons License. E-mail us: |