2.2 The uniqueness theorems

In the non-rotating case it was Israel who, in his pioneering work, showed that both static vacuum [99] and electrovac [100] 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 [160], Bunting and Masood-ul-Alam [26], and Ruback [154] 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 [129].)

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 [29] (See also [31] and [33].). In the vacuum case, Robinson was able to construct an amazing identity, by virtue of which the uniqueness of the Kerr metric followed [152]. The uniqueness problem with electro-magnetic fields remained open until Mazur [131] and, independently, Bunting [25] 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, .