4.4 Stationary gauge fields

The reduction of the Einstein–Hilbert action in the presence of a Killing field yields a σ-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 [17], (see also [1823]), 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 ξ, say, if there exists a Lie algebra valued function 𝒱 ξ, such that

LξA = D 𝒱ξ , (18 )
where 𝒱ξ is the generator of an infinitesimal gauge transformation, L ξ denotes the Lie derivative, and D is the gauge covariant exterior derivative, D 𝒱ξ = d𝒱 ξ + [A, 𝒱ξ].

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 ¯A orthogonal to k, A¯(k) = 0, and a Lie algebra valued potential ϕ,

A = ϕ(dt + a) + A¯, (19 )
where we recall that a is the non-static part of the metric (8View Equation). For the sake of simplicity we adopt a gauge where 𝒱k vanishes.43 By virtue of the above decomposition, the field strength becomes ¯ ¯ F = D ϕ ∧ (dt + a) + (F + ϕf), where F¯ is the Yang–Mills field strength for ¯A and f ≡ da. Using the expression (12View Equation) for the vacuum action, one easily finds that the EYM action,
∫ ( ) SEYM = ∗R − 2tˆr {F ∧ ∗F } , (20 )
gives rise to the effective action44
∫ ( 2 ) Seff = ¯∗ ¯R − --1-|dσ|2 + σ-|f|2 + 2|¯D ϕ|2 − 2 σ| ¯F + ϕ f|2 , (21 ) 2 σ2 2 σ
where ¯ D is the gauge covariant derivative with respect to ¯ A, and where the inner product also involves the trace: { } ¯∗|F¯|2 ≡ tˆr ¯F ∧ ¯∗F¯. The above action describes two scalar fields, σ and ϕ, and two vector fields, a and ¯ A, which are minimally coupled to three-dimensional gravity with metric ¯g. Like in the vacuum case, the connection a enters Se ff only via the field strength f ≡ da. Again, this gives rise to a differential conservation law,
[ { }] d¯∗ σ2f − 4σ ˆtr ϕ ( ¯F + ϕf ) = 0, (22 )
by virtue of which one can (locally) introduce a generalized twist potential Y, defined by − dY = ¯∗[...].

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

[ ( )] [ ] D¯ σ ¯∗ F¯ + ϕf = σ−1¯∗ ϕ, ¯Dϕ . (23 )
Hence, the scalar magnetic potential – which can be introduced in the Abelian case according to dψ ≡ σ¯∗(F¯ + ϕf ) – ceases to exist for non-Abelian Yang–Mills fields. The remaining stationary EYM equations are easily derived from variations of S eff with respect to the gravitational potential σ, the electric Yang–Mills potential ϕ and the three-metric ¯g.

As an application, we note that the effective action (21View Equation) is particularly suited for analyzing stationary perturbations of static (a = 0), purely magnetic (ϕ = 0) configurations [19Jump To The Next Citation Point], such as the Bartnik–McKinnon solitons [4] and the corresponding black hole solutions [1741229]. The two crucial observations in this context are [19Jump To The Next Citation Point175]:

(i) The only perturbations of the static, purely magnetic EYM solutions which can contribute the ADM angular momentum are the purely non-static, purely electric ones, δa and δ ϕ.

(ii) In first order perturbation theory the relevant fluctuations, δa and δϕ, decouple from the remaining metric and matter perturbations

The second observation follows from the fact that the magnetic Yang–Mills equation (23View Equation) and the Einstein equations for σ and ¯g become background equations, since they contain no linear terms in δa and δϕ. The purely electric, non-static perturbations are, therefore, governed by the twist equation (22View Equation) and the electric Yang–Mills equation (obtained from variations of Seff with respect to ϕ).

Using Eq. (22View Equation) to introduce the twist potential Y, the fluctuation equations for the first order quantities δY and δϕ assume the form of a self-adjoint system [19]. Considering perturbations of spherically symmetric configurations, one can expand δY and δϕ 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 3.5 [22].


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