2.3 Black Holes with Degenerate 2 Classification of Stationary Electrovac 2.1 RigidityStaticity and Circularity

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 In The Article] and electrovac [100Jump To The Next Citation Point In The Article] 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], [151Jump To The Next Citation Point In The Article], [153]). Later on, Simon [160], Bunting and Masood-ul-Alam [25], and Ruback [154] were able to improve on the original method, taking advantage of the positive energy theorem. Popup Footnote (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 [28Jump To The Next Citation Point In The Article] (See also [30] and [32].). 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 In The Article]. The uniqueness problem with electro-magnetic fields remained open until Mazur [131Jump To The Next Citation Point In The Article] and, independently, Bunting [24Jump To The Next Citation Point In The Article] were able to obtain a generalization of the Robinson identity in a systematic way: The Mazur identity (see also [132Jump To The Next Citation Point In The Article], [133Jump To The Next Citation Point In The Article]) is based on the observation that the EM equations in the presence of a Killing field describe a non-linear tex2html_wrap_inline3625 -model with coset space tex2html_wrap_inline3637 (provided that the dimensional reduction of the EM action is performed with respect to the axial Killing field Popup Footnote). 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).



2.3 Black Holes with Degenerate 2 Classification of Stationary Electrovac 2.1 RigidityStaticity and Circularity

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
http://www.livingreviews.org/lrr-1998-6
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de