8.1 Integrability properties of Killing fields

Our aim here is to discuss the circularity problem in some more detail. The task is to use the symmetry properties of the matter model in order to establish the orthogonal-integrability conditions for the Killing fields. 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 )], (8.1 )
and Einstein’s equations, implying ξ ∧ R (ξ) = 8π[ξ ∧ T(ξ)]. This follows from the definition of the twist and the Ricci identity for Killing fields, Δ ξ = − 2R (ξ), where R(ξ) is the one-form with components ν [R (ξ )]μ := Rμνξ; see, e.g., [151], Chapter 2. For a stationary and axisymmetric spacetime with Killing fields (one-forms) k and m, Eq. (8.1View Equation) implies
dg(m, ωk) = − 8π ∗ [m ∧ k ∧ T(k)], (8.2 )
and similarly for k ↔ m. Eq. (8.2View Equation) is an identity up to a term involving the Lie derivative of the twist of the first Killing field with respect to the second one (since dg(m, ωk) = Lm ωk − imd ωk). In order to establish Lm ωk = 0, it is sufficient to show that k and m commute in an asymptotically-flat spacetime. This was first achieved by Carter [44] and later, under more general conditions, by Szabados [304].

The following is understood to also apply for k ↔ m: By virtue of Eq. (8.2View Equation) – and the fact that the condition m ∧ k ∧ dk = 0 can be written as g (m, ωk ) = 0 – the circularity problem is reduced to the following two tasks:

Show that dg (m, ωk) = 0 implies g (m, ωk ) = 0.
Establish m ∧ k ∧ T(k) = 0 from the stationary and axisymmetric matter equations.

(i) Since g (m,ωk ) is a function, it is locally constant if its derivative vanishes. As m vanishes on the rotation axis, this implies g(m, ωk) = 0 in every connected 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 ω = 0 k that ω k vanishes is more involved, since ω k is a one-form. However, using the Stokes’ theorem to integrate an identity for the twist [152Jump 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. While this proves the staticity theorem for vacuum and self-gravitating scalar fields [152Jump To The Next Citation Point], it does not solve the electrovacuum case. It should be noted that in the context of the proof of uniqueness the strictly stationary property follows from staticity [72] and not the other way around (compare Figure 3View Image).

(ii) While m ∧ k ∧ T (k) = 0 follows from the symmetry conditions for electro-magnetic fields [43] and for scalar fields [150], it cannot be established for non-Abelian gauge fields [152]. This implies that the usual foliation of spacetime used to integrate the stationary and axisymmetric Maxwell equations is too restrictive to treat the EYM system. This is seen as follows: In Section 6.3 we have derived the formula (6.17View Equation). By virtue of Eq. (6.3View 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 = V ¯∗( ¯F + ϕkf ):

[ ˆ ] d ωk + 4tr (ϕkik ∗ F) = 0, (8.3 )
where ˆtr() is a suitably normalized trace (see Eq. (6.15View Equation)). Contracting this relation with the axial Killing field m, and using again the fact that the Lie derivative of ω k with respect to m vanishes, yields immediately
dg (m, ωk) = 0 ⇐⇒ tˆr (ϕk (∗F )(k,m )) = 0. (8.4 )
The difference between the Abelian and the non-Abelian case is due to the fact that the Maxwell equations automatically imply that the (km )-component of ∗F vanishes, whereas this does not follow from the Yang–Mills equations. In fact, the Maxwell equation d ∗ F = 0 and the symmetry property Lk ∗ F = ∗LkF = 0 imply the existence of a magnetic potential, d ψ = (∗F )(k, ⋅ ), thus, (∗F )(k,m ) = imdψ = Lm ψ = 0. Moreover, the latter do not imply that the Lie algebra valued scalars ϕk and (∗F )(k,m ) are orthogonal. Hence, circularity is an intrinsic property of the EM system, whereas it imposes additional requirements on non-Abelian gauge fields.

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

T = PABd ϕA ⊗ d ϕB + P g, (8.5 )
where the functions PAB 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 L ξϕA = 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 (8.1View Equation), this implies that the twist of ξ is closed. Hence, the staticity and the circularity issue for self-gravitating scalar mappings can be established, under appropriate conditions, as in the vacuum case. From this one concludes that (strictly) stationary non-rotating black-hole configuration of self-gravitating scalar fields are static if LkϕA = 0, while stationary and axisymmetric ones are circular if Lk ϕA = Lm ϕA = 0.
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