2.2 The uniqueness theorems

The main task of the uniqueness program is to show that the static electrovac black hole space-times are described by the Reissner–Nordström metric, while the circular ones are represented by the Kerr–Newman metric. In combination with the SRT and the staticity and circularity theorems, this implies that all stationary black hole solutions to the EM equations (with non-degenerate horizon) are parametrized by their mass, angular momentum and electric charge.

In the non-rotating case it was Israel who, in his pioneering work, showed that both static vacuum [99Jump To The Next Citation Point] and electrovac [100Jump To The Next Citation Point] 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. [137138151Jump To The Next Citation Point153]). 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 [29Jump To The Next Citation Point] (See also [31Jump To The Next Citation Point] and [33Jump To The Next Citation Point].). In the vacuum case, Robinson was able to construct an amazing identity, by virtue of which the uniqueness of the Kerr metric followed [152Jump To The Next Citation Point]. The uniqueness problem with electro-magnetic fields remained open until Mazur [131Jump To The Next Citation Point] and, independently, Bunting [25Jump To The Next Citation Point] were able to obtain a generalization of the Robinson identity in a systematic way: The Mazur identity (see also [132Jump To The Next Citation Point133Jump To The Next Citation Point]) is based on the observation that the EM equations in the presence of a Killing field describe a non-linear σ-model with coset space G ∕H = SU (1,2 )∕S (U (1) × U (2)) (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, G ∕H = SU (1,1)∕U (1).

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