Throughout this section we will assume that the spacetime is -regular, as made precise by
Definition 2.1.
To prove uniqueness of connected, analytic, non-degenerate configurations, it remains to
show that every such black hole is either static or axially symmetric. The first step for this is
provided by Hawking’s strong rigidity theorem (SRT) [143, 238, 60
, 107
], which relates the global
concept of the event horizon to the independently-defined, and logically-distinct, local notion of
the Killing horizon. Assuming analyticity, SRT asserts that the event horizon of a stationary
black-hole spacetime is a Killing horizon. (In this terminology [151
], the weak rigidity theorem is the
existence, already discussed above, of prehorizons for static or stationary and axisymmetric
configurations.)
A Killing horizon is called non-rotating if it is generated by the stationary Killing field, and rotating otherwise. At this stage the argument branches-off, according to whether at least one of the horizons is rotating, or not.
In the rotating case, Hawking’s theorem actually provides only a second Killing vector field defined near
the Killing horizon, and to continue one needs to globalize the Killing vector field, to prove that its orbits
are complete, and to show that there exists a linear combination of Killing vector fields with periodic orbits
and an axis of rotation. This is done in [60], assuming analyticity, drawing heavily on the results
in [253, 57, 18].
The existing attempts in the literature to construct a second Killing vector field without assuming
analyticity have only had limited success. One knows now how to construct a second Killing vector in a
neighborhood of non-degenerate horizons for electrovacuum black holes [2, 174
, 327], but the construction
of a second Killing vector throughout the d.o.c. has only been carried out for vacuum near-Kerr
non-degenerate configurations so far [3
] (compare [326]).
In any case, sufficiently regular analytic stationary electro-vacuum spacetimes containing a rotating component of the event horizon are axially symmetric as well, regardless of degeneracy and connectedness assumptions (for more on this subject see Section 3.4.2). One can then finish the uniqueness proof using Theorem 3.2. Note that the last theorem requires neither analyticity nor connectedness, but leaves open the question of the existence of naked singularities in non-connected candidate solutions.
In the non-rotating case, one continues by showing [84] that the domain of outer communications
contains a maximal Cauchy surface. This has been proven so far only for non-degenerate horizons, and this
is the only missing step to include situations with degenerate components of the horizon. This allows one to
prove the staticity theorem [302
, 303
], that the stationary Killing field of a non-rotating, electrovacuum
black-hole spacetime is hypersurface orthogonal. (Compare [134, 136, 143
, 141] for previous partial
results.) One can then finish the argument using Theorem 3.1.
All this leads to the following precise statement, as proven in [76, 79
] in vacuum and in [217, 79
] in
electrovacuum:
Theorem 3.3 Let be a stationary, asymptotically-flat,
-regular, electrovacuum,
four-dimensional analytic spacetime. If the event horizon is connected and either mean
non-degenerate or rotating, then
is isometric to the domain of outer communications of
a Kerr–Newman spacetime.
The structure of the proof can be summarized in the flow chart of Figure 3. This is to be compared with
Figure 4
, which presents in more detail the weaker hypotheses needed for various parts of the
argument.
Theorem 3.5 Let be a stationary asymptotically-flat,
-regular, smooth, vacuum
four-dimensional spacetime. Assume that the event horizon is connected and mean non-degenerate.
If the Mars–Simon tensor
and the Ernst potential
of the spacetime satisfy
http://www.livingreviews.org/lrr-2012-7 |
Living Rev. Relativity 15, (2012), 7
![]() This work is licensed under a Creative Commons License. E-mail us: |