6.3 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 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 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 S 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 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

L ξA = D 𝒱ξ, (6.13 )
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 can be parameterized in terms of a one-form ¯A orthogonal to k, in the sense that ¯ A (k) = 0, and a Lie algebra valued potential ϕ,

¯ A = ϕ(dt + a) + A, (6.14 )
where we recall that a is the non-static part of the metric (6.1View Equation). For the sake of simplicity we adopt a gauge where 𝒱k vanishes.9 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 (6.5View Equation) for the vacuum action, one easily finds that the EYM action,
∫ S = (∗R − 2ˆtr(F ∧ ∗F )), (6.15 ) EYM
where R and ∗ refer to the 4-dimensional space-time metric g and ˆ tr () denotes a suitably normalized trace (e.g., 1 ˆtr(τaτb) = 2δab where the σa’s are the Pauli matrices), gives rise to the effective action
∫ ( 2 ) ¯ ¯ -1-- 2 V-- 2 2- ¯ 2 ¯ 2 Seff = ∗ R − 2V 2|dV | + 2 |f| + V |D ϕ| − 2V |F + ϕf| , (6.16 )
where ¯D is the gauge covariant derivative with respect to ¯A, and where the inner product also involves the trace: ¯∗|F¯|2 := ˆtr(F¯∧ ¯∗ ¯F). The above action describes two scalar fields, V and ϕ, and two vector fields, a and ¯ A, which are minimally coupled to three-dimensional gravity with metric ¯g. Similarly to the vacuum case, the connection a enters Seff only via the field strength f. Again, this gives rise to a differential conservation law,
d¯∗ [V 2f − 4Vtˆr (ϕ( ¯F + ϕf ))] = 0, (6.17 )
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: For non-Abelian gauge groups, ¯ F is no longer an exact two-form, and the gauge covariant derivative of ϕ introduces source terms in the corresponding Yang–Mills equation:

D¯[V ¯∗ (¯F + ϕf )] = V −1¯∗[ϕ, ¯Dϕ ]. (6.18 )
Hence, the scalar magnetic potential – which can be introduced in the Abelian case according to dψ := V ¯∗ (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 V, the electric Yang–Mills potential ϕ and the three-metric ¯g.

As an application, we note that the effective action (6.16View Equation) is particularly suited for analyzing stationary perturbations of static (a = 0), purely magnetic (ϕ = 0) configurations [35Jump To The Next Citation Point], such as the Bartnik–McKinnon solitons [14] and the corresponding black-hole solutions [310, 208, 24]. The two crucial observations in this context are [35Jump To The Next Citation Point, 312]:

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 δϕ.
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 (6.18View Equation) and the Einstein equations for V and g¯ become background equations, since they contain no linear terms in δa and δϕ. Therefore, the purely electric, non-static perturbations are governed by the twist equation (6.17View Equation) and the electric Yang–Mills equation (obtained from variations of Seff with respect to ϕ).

Using Eq. (6.17View 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 [35]. 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 5.5 [38].

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