2.2 The Uniqueness Theorems2 Classification of Stationary Electrovac 2 Classification of Stationary Electrovac

2.1 Rigidity, Staticity and Circularity 

At the basis of the classification of stationary electrovac black hole space-times lies Hawking's strong rigidity theorem (SRT) [84Jump To The Next Citation Point In The Article]. It relates the global concept of the event horizon to the independently defined - and logically distinct - local notion of the Killing horizon: Requiring that the fundamental matter fields obey well behaved hyperbolic equations, and that the stress-energy tensor satisfies the weak energy condition, Popup Footnote the first part of the SRT asserts that the event horizon of a stationary black hole spacetime is a Killing horizon. Popup Footnote The latter is called non-rotating if it is generated by the stationary Killing field, and rotating otherwise. In the rotating case, the second part of the SRT implies that spacetime is axisymmetric. Popup Footnote

The subdivision provided by the SRT is, unfortunately, not sufficient to apply the uniqueness theorems for the Reissner-Nordström and the Kerr-Newman metric: The latter are based on the stronger requirements that the domain of outer communication (DOC) is either static (non-rotating case) or circular (axisymmetric case). Hence, in both cases one has to establish the Frobenius integrability conditions for the Killing fields beforehand (staticity and circularity theorems).

The circularity theorem, due to Carter [26Jump To The Next Citation Point In The Article], and Kundt and Trümper [118], implies that the metric of a vacuum or electrovac spacetime can, without loss of generality, be written in the well-known Papapetrou tex2html_wrap_inline3633 -split. The staticity theorem, implying that the stationary Killing field of a non-rotating, electrovac black hole spacetime is hyper-surface orthogonal, is more involved than the circularity problem: First, one has to establish strict stationarity, that is, one needs to exclude ergo-regions. This problem, first discussed by Hajicek [78], [79], and Hawking and Ellis [84Jump To The Next Citation Point In The Article], was solved only recently by Sudarsky and Wald [167Jump To The Next Citation Point In The Article], [168Jump To The Next Citation Point In The Article], assuming a foliation by maximal slices. Popup Footnote If ergo-regions are excluded, it still remains to prove that the stationary Killing field satisfies the Frobenius integrability condition. In the vacuum case, this was achieved by Hawking [82], who was able to extend a theorem due to Lichnerowicz [126] to black hole space-times. In the presence of Maxwell fields the problem was solved only a couple of years ago [167Jump To The Next Citation Point In The Article], [168Jump To The Next Citation Point In The Article], by means of a generalized version of the first law of black hole physics.



2.2 The Uniqueness Theorems2 Classification of Stationary Electrovac 2 Classification of Stationary Electrovac

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