(i) Show that implies .
(ii) Establish from the stationary and axisymmetric matter equations.
(i) Since is a function, it must be constant if its derivative vanishes. As vanishes on the rotation axis, this implies 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 that vanishes is more involved, since is a one-form. However, using Stoke’s theorem to integrate an identity for the twist  shows that a strictly stationary – not necessarily simply connected – domain of outer communication must be static if is closed.60)
(ii) While follows from the symmetry conditions for electro-magnetic fields  and for scalar fields , it cannot be established for non-Abelian gauge fields . 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 (22). By virtue of Eq. (10) this becomes an expression for the derivative of the twist in terms of the electric Yang–Mills potential (defined with respect to the stationary Killing field ) and the magnetic one-form :61 whereas this does not follow from the Yang–Mills equations. Moreover, the latter do not imply that the Lie algebra valued scalars and 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 with Lagrangian . The stress energy tensor is of the form
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