The presence of two Killing symmetries yields a considerable simplification of the field equations. In fact, for certain matter models the latter become completely integrable [219], provided that the Killing fields satisfy the orthogonal-integrability conditions. Spacetimes admitting two Killing fields provide the framework for both the theory of colliding gravitational waves and the theory of rotating black holes [56]. Although dealing with different physical subjects, the theories are mathematically closely related. We refer the reader to Chandrasekhar’s comparison between corresponding solutions of the Ernst equations [55].

This section reviews the structure of the stationary and axisymmetric field equations. We start by recalling the circularity problem. It is argued that circularity is not a generic property of asymptotically-flat, stationary and axisymmetric spacetimes. However, if the symmetry conditions for the matter fields do imply circularity, then the reduction with respect to the second Killing field simplifies the field equations drastically. The systematic derivation of the Kerr–Newman metric and the proof of its uniqueness provide impressive illustrations of this fact.

8.1 Integrability properties of Killing fields

8.2 Two-dimensional elliptic equations

8.3 The Ernst equations

8.3.1 A derivation of the Kerr–Newman metric

8.4 The uniqueness theorem for the Kerr–Newman solution

8.4.1 Divergence identities

8.4.2 The distance function argument

8.2 Two-dimensional elliptic equations

8.3 The Ernst equations

8.3.1 A derivation of the Kerr–Newman metric

8.4 The uniqueness theorem for the Kerr–Newman solution

8.4.1 Divergence identities

8.4.2 The distance function argument

Living Rev. Relativity 15, (2012), 7
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