In the non-rotating case it was Israel who, in his pioneering work, showed that both static vacuum  and electrovac  black hole space-times are spherically symmetric. Israel’s ingenious method, based on differential identities and Stokes’ theorem, triggered a series of investigations devoted to the static uniqueness problem (see, e.g. [137, 138, 151, 153]). Later on, Simon , Bunting and Masood-ul-Alam , and Ruback  were able to improve on the original method, taking advantage of the positive energy theorem.11 (The “latest version” of the static uniqueness theorem can be found in .)
The key to the uniqueness theorem for rotating black holes exists in Carter’s observation that the stationary and axisymmetric EM equations reduce to a two-dimensional boundary value problem  (See also  and .). In the vacuum case, Robinson was able to construct an amazing identity, by virtue of which the uniqueness of the Kerr metric followed . The uniqueness problem with electro-magnetic fields remained open until Mazur  and, independently, Bunting  were able to obtain a generalization of the Robinson identity in a systematic way: The Mazur identity (see also [132, 133]) is based on the observation that the EM equations in the presence of a Killing field describe a non-linear -model with coset space (provided that the dimensional reduction of the EM action is performed with respect to the axial Killing field12). Within this approach, the Robinson identity looses its enigmatic status – it turns out to be the explicit form of the Mazur identity for the vacuum case, .
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