(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 [88] 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 [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 (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 :

Contracting this relation with the axial Killing field , and using again the fact that the Lie derivative of with respect to vanishes, yields immediately The difference between the Abelian and the non-Abelian case lies in the circumstance that the Maxwell equations automatically imply that the -component of vanishes,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

where the functions and may depend on , , the spacetime metric and the target metric . If is invariant under the action of a Killing field – in the sense that for each component of – then the one-form becomes proportional to : . By virtue of the Killing field identity (49), this implies that the twist of 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 , while stationary and axisymmetric ones are circular if .http://www.livingreviews.org/lrr-1998-6 |
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