4.5 The Stationary Einstein-Maxwell System4 Stationary Space-Times4.3 The Coset Structure of

4.4 Stationary Gauge Fields 

The reduction of the Einstein-Hilbert action in the presence of a Killing field yields a tex2html_wrap_inline3625 -model which is effectively coupled to three-dimensional gravity. While this structure is retained for the EM system, it ceases to exist for self-gravitating non-Abelian gauge fields. In order to perform the dimensional reduction for the EM and the EYM equations, we need to recall the notion of a symmetric gauge field.

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 tex2html_wrap_inline3669, say, if there exists a Lie algebra valued function tex2html_wrap_inline4027, such that

  equation611

where tex2html_wrap_inline4027 is the generator of an infinitesimal gauge transformation, tex2html_wrap_inline4031 denotes the Lie derivative, and tex2html_wrap_inline4033 is the gauge covariant exterior derivative, tex2html_wrap_inline4035 .

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 tex2html_wrap_inline4039 orthogonal to k, tex2html_wrap_inline4043, and a Lie algebra valued potential tex2html_wrap_inline4045,

  equation626

where we recall that a is the non-static part of the metric (8Popup Equation). For the sake of simplicity we adopt a gauge where tex2html_wrap_inline4049 vanishes. Popup Footnote By virtue of the above decomposition, the field strength becomes tex2html_wrap_inline4055, where tex2html_wrap_inline4057 is the Yang-Mills field strength for tex2html_wrap_inline4039 and tex2html_wrap_inline4061 . Using the expression (12Popup Equation) for the vacuum action, one easily finds that the EYM action,

  equation638

gives rise to the effective action Popup Footnote

  equation646

where tex2html_wrap_inline4071 is the gauge covariant derivative with respect to tex2html_wrap_inline4039, and where the inner product also involves the trace: tex2html_wrap_inline4075 . The above action describes two scalar fields, tex2html_wrap_inline3625 and tex2html_wrap_inline4045, and two vector fields, a and tex2html_wrap_inline4039, which are minimally coupled to three-dimensional gravity with metric tex2html_wrap_inline3837 . Like in the vacuum case, the connection a enters tex2html_wrap_inline4089 only via the field strength tex2html_wrap_inline4061 . Again, this gives rise to a differential conservation law,

  equation663

by virtue of which one can (locally) introduce a generalized twist potential Y, defined by tex2html_wrap_inline4095 .

The main difference between the Abelian and the non-Abelian case concerns the variational equation for tex2html_wrap_inline4039, that is, the Yang-Mills equation for tex2html_wrap_inline4057 : The latter assumes the form of a differential conservation law only in the Abelian case. For non-Abelian gauge groups, tex2html_wrap_inline4057 is no longer an exact two-form, and the gauge covariant derivative of tex2html_wrap_inline4045 causes source terms in the corresponding Yang-Mills equation:

  equation667

Hence, the scalar magnetic potential - which can be introduced in the Abelian case according to tex2html_wrap_inline4105 - ceases to exist for non-Abelian Yang-Mills fields. The remaining stationary EYM equations are easily derived from variations of tex2html_wrap_inline4089 with respect to the gravitational potential tex2html_wrap_inline3625, the electric Yang-Mills potential tex2html_wrap_inline4045 and the three-metric tex2html_wrap_inline3837 .

As an application, we note that the effective action (21Popup Equation) is particularly suited for analyzing stationary perturbations of static (a = 0), purely magnetic (tex2html_wrap_inline4117) configurations [18Jump To The Next Citation Point In The Article], such as the Bartnik-McKinnon solitons [4] and the corresponding black hole solutions [174], [122], [8]. The two crucial observations in this context are [18Jump To The Next Citation Point In The Article], [175]:

The second observation follows from the fact that the magnetic Yang-Mills equation (23Popup Equation) and the Einstein equations for tex2html_wrap_inline3625 and tex2html_wrap_inline3837 become background equations, since they contain no linear terms in tex2html_wrap_inline4119 and tex2html_wrap_inline4121 . The purely electric, non-static perturbations are, therefore, governed by the twist equation (22Popup Equation) and the electric Yang-Mills equation (obtained from variations of tex2html_wrap_inline4089 with respect to tex2html_wrap_inline4045).

Using Eq. (22Popup Equation) to introduce the twist potential Y, the fluctuation equations for the first order quantities tex2html_wrap_inline4141 and tex2html_wrap_inline4121 assume the form of a self-adjoint system [18]. Considering perturbations of spherically symmetric configurations, one can expand tex2html_wrap_inline4141 and tex2html_wrap_inline4121 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].



4.5 The Stationary Einstein-Maxwell System4 Stationary Space-Times4.3 The Coset Structure of

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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