6.2 Boundary Value Formulation6 Stationary and Axisymmetric Space-Times6 Stationary and Axisymmetric Space-Times

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 Sect. 2.1 and Sect. 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 tex2html_wrap_inline3669, say,

  equation1221

and Einstein's equations, implying tex2html_wrap_inline4495 . Popup Footnote For a stationary and axisymmetric spacetime with Killing fields (one-forms) k and m, Eq. (49Popup Equation) implies Popup Footnote

  equation1239

and similarly for tex2html_wrap_inline4515 . Popup Footnote By virtue of Eq. (50Popup Equation) - and the fact that the Frobenius condition tex2html_wrap_inline4519 can be written as tex2html_wrap_inline4521 - the circularity problem is reduced to the following two tasks:

(i) Since tex2html_wrap_inline4529 is a function, it must be constant if its derivative vanishes. As m vanishes on the rotation axis, this implies tex2html_wrap_inline4521 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 tex2html_wrap_inline4535 that tex2html_wrap_inline4537 vanishes is more involved, since tex2html_wrap_inline4537 is a one-form . However, using Stoke's theorem to integrate an identity for the twist [88Jump To The Next Citation Point In The Article] shows that a strictly stationary - not necessarily simply connected - domain of outer communication must be static if tex2html_wrap_inline4537 is closed. Popup Footnote)

(ii) While tex2html_wrap_inline4527 follows from the symmetry conditions for electro-magnetic fields [26] 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 Sect.(4.4) we have derived the formula (22Popup Equation). By virtue of Eq. (10Popup Equation) this becomes an expression for the derivative of the twist in terms of the electric Yang-Mills potential tex2html_wrap_inline4545 (defined with respect to the stationary Killing field k) and the magnetic one-form tex2html_wrap_inline4549 :

  equation1270

Contracting this relation with the axial Killing field m, and using again the fact that the Lie derivative of tex2html_wrap_inline4537 with respect to m vanishes, yields immediately

  equation1274

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 tex2html_wrap_inline4395 vanishes, Popup Footnote whereas this does not follow from the Yang-Mills equations. Moreover, the latter do not imply that the Lie algebra valued scalars tex2html_wrap_inline4545 and tex2html_wrap_inline4571 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 tex2html_wrap_inline4573 with Lagrangian tex2html_wrap_inline4575 . The stress energy tensor is of the form

  equation1281

where the functions tex2html_wrap_inline4579 and P may depend on tex2html_wrap_inline4045, tex2html_wrap_inline4585, the spacetime metric tex2html_wrap_inline3873 and the target metric tex2html_wrap_inline4589 . If tex2html_wrap_inline4045 is invariant under the action of a Killing field tex2html_wrap_inline3669 - in the sense that tex2html_wrap_inline4595 for each component tex2html_wrap_inline4597 of tex2html_wrap_inline4045 - then the one-form tex2html_wrap_inline3741 becomes proportional to tex2html_wrap_inline3669 : tex2html_wrap_inline4605 . By virtue of the Killing field identity (49Popup Equation), this implies that the twist of tex2html_wrap_inline3669 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 tex2html_wrap_inline4609, while stationary and axisymmetric ones are circular if tex2html_wrap_inline4611 .



6.2 Boundary Value Formulation6 Stationary and Axisymmetric Space-Times6 Stationary and Axisymmetric Space-Times

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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