### 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 [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.
(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
field).
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, .