6.1 Integrability properties of Killing fields

Our aim here is to discuss the circularity problem in some more detail. We refer the reader to Section 2.1 and Section 3.5 for the general context and for references concerning the staticity and the circularity issues. In both cases, the task is to use the symmetry properties of the matter model in order to establish the Frobenius integrability conditions for the Killing field(s). The link between the relevant components of the stress-energy tensor and the integrability conditions is provided by a general identity for the derivative of the twist of a Killing field ξ, say,
dωξ = ∗ [ξ ∧ R (ξ )], (49 )
and Einstein’s equations, implying ξ ∧ R (ξ ) = 8 π[ξ ∧ T (ξ)].57 For a stationary and axisymmetric spacetime with Killing fields (one-forms) k and m, Eq. (49View Equation) implies58
d(m , ωk) = − 8π ∗ [m ∧ k ∧ T(k)], (50 )
and similarly for k ↔ m.59 By virtue of Eq. (50View Equation) – and the fact that the Frobenius condition m ∧ k ∧ dk = 0 can be written as (m , ωk) = 0 – the circularity problem is reduced to the following two tasks:

(i) Show that d (m , ωk) = 0 implies (m , ωk ) = 0.

(ii) Establish m ∧ k ∧ T (k) = 0 from the stationary and axisymmetric matter equations.

(i) Since (m , ωk) is a function, it must be constant if its derivative vanishes. As m vanishes on the rotation axis, this implies (m , ωk ) = 0 in every domain of spacetime intersecting the axis. (At this point it is worthwhile to recall that the corresponding step in the staticity theorem requires more effort: Concluding from d ωk = 0 that ωk vanishes is more involved, since ωk is a one-form. However, using Stoke’s theorem to integrate an identity for the twist [88Jump To The Next Citation Point] shows that a strictly stationary – not necessarily simply connected – domain of outer communication must be static if ωk is closed.60)

(ii) While m ∧ k ∧ T (k) = 0 follows from the symmetry conditions for electro-magnetic fields [27] and for scalar fields [86], it cannot be established for non-Abelian gauge fields [88]. This implies that the usual foliation of spacetime used to integrate the stationary and axisymmetric Maxwell equations is too restrictive to treat the Einstein–Yang–Mills (EYM) system. This is seen as follows: In Section (4.4) we have derived the formula (22View Equation). By virtue of Eq. (10View Equation) this becomes an expression for the derivative of the twist in terms of the electric Yang–Mills potential ϕk (defined with respect to the stationary Killing field k) and the magnetic one-form ¯ ¯ ik ∗ F = σ∗(F + ϕf):

[ ] d ωk + 4tˆr {ϕk ik ∗ F } = 0. (51 )
Contracting this relation with the axial Killing field m, and using again the fact that the Lie derivative of ωk with respect to m vanishes, yields immediately
d (m , ωk ) = 0 ⇐ ⇒ tˆr {ϕk (∗F) (k,m )} = 0. (52 )
The difference between the Abelian and the non-Abelian case lies in the circumstance that the Maxwell equations automatically imply that the (km )-component of ∗F vanishes,61 whereas this does not follow from the Yang–Mills equations. Moreover, the latter do not imply that the Lie algebra valued scalars ϕk and (∗F )(k,m ) are orthogonal. Hence, circularity is a generic property of the Einstein–Maxwell (EM) system, whereas it imposes additional requirements on non-Abelian gauge fields.

Both the staticity and the circularity theorems can be established for scalar fields or, more generally, scalar mappings with arbitrary target manifolds: Consider a self-gravitating scalar mapping ϕ : (M, g) → (N, G ) with Lagrangian L[ϕ,d ϕ,g,G ]. The stress energy tensor is of the form

T = PAB dϕA ⊗ d ϕB + P g, (53 )
where the functions P AB and P may depend on ϕ, dϕ, the spacetime metric g and the target metric G. If ϕ is invariant under the action of a Killing field ξ – in the sense that A L ξϕ = 0 for each component ϕA of ϕ – then the one-form T (ξ) becomes proportional to ξ: T(ξ) = P ξ. By virtue of the Killing field identity (49View Equation), this implies that the twist of ξ is closed. Hence, the staticity and the circularity issue for self-gravitating scalar mappings reduce to the corresponding vacuum problems. From this one concludes that stationary non-rotating black hole configuration of self-gravitating scalar fields are static if A Lk ϕ = 0, while stationary and axisymmetric ones are circular if Lk ϕA = Lm ϕA = 0.
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