Stationary gauge fields

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 [18 , 23 ]), 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

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 $ϕ$ ,

where we recall that $a$ is the non-static part of the metric (8

). 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 (12

) for the vacuum action, one easily finds that the EYM action,

gives rise to the effective action⁴⁴

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,

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:

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 (21 ) is particularly suited for analyzing stationary perturbations of static ( $a = 0$ ), purely magnetic ( $ϕ = 0$ ) configurations [19 ], such as the Bartnik–McKinnon solitons [4 ] and the corresponding black hole solutions [174 , 122 , 9 ]. The two crucial observations in this context are [19 , 175 ]:

(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 (23 ) 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 (22 ) and the electric Yang–Mills equation (obtained from variations of $Seff$ with respect to $ϕ$ ).

Using Eq. (22 ) 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 ].