Quantum Field Theory and Gravity: Conceptual and Mathematical ...

340 S. Hollands
In the case where Σ is the spatial section of a black hole spacetime, there
are further constraints coming from Einstein’s equations and the orientability
of the spacetime. The topological censorship theorem [12, 7] is seen to imply
[23] that the 2-dimensional orbit space ˆΣ =Σ/U(1) D−3 can neither have
any conifold points, nor holes, nor handles, and therefore has to be diffeomorphic
to an upper half-plane ˆΣ ∼ = {(r, z) | r>0}. The boundary segments
correspond to intervals on the boundary (r = 0) of this upper half-plane
and are places (“axes”) in the manifold M where a linear combination of the
rotational Killing fields vanishes, or to a horizon. Here the Killing fields do
not vanish except where an “axis” meets a horizon. Furthermore, with each
boundary segment, one can associate its length l i ≥ 0 in an invariant way. It
can be shown (see e.g. [6]) that the metric of a black hole spacetime with the
indicated symmetries globally takes the Weyl–Papapetrou form
g = − r2 dt 2
det f +e−ν (dr 2 +dz 2 )+f ij (dφ i + w i dt)(dφ j + w j dt) , (2)
where the metric coefficients only depend on r, z,andwhereφ i are 2π-periodic
coordinates. The vacuum Einstein equations lead to two sets of differential
equations. One set can be written [27] as a sigma-model field equation for a
matrix field Φ defined by
( )
(det f)
−1
−(det f)
Φ=
−1 χ i
−(det f) −1 χ i f ij +(detf) −1 , (3)
χ i χ j
where χ i are certain potentials that are defined in terms of the metric coefficients.
The other set can be viewed as determining the conformal factor
ν from Φ. The matrix field Φ obeys certain boundary conditions at r =0
related to the winding numbers and moduli l i ∈ R + labeling the i-th boundary
interval. Hence, it is evident 4 that any uniqueness theorem for the black
holes under consideration would have to involve these data in addition to the
usual invariants, such as mass and angular momentum.
In fact, what one can prove is the following theorem in D =5(modulo
technical assumptions about analyticity and certain global causal constraints)
[22]:
Theorem 1. Given are two asymptotically flat, vacuum black hole solutions,
each with a single non-extremal horizon. If the angular momenta J 1 ,J 2 coincide,
and if all the integers (“winding numbers”) 〈(p 1 ,q 1 ),...,(p N ,q N )〉 and
real numbers 〈l 1 ,...,l N 〉 (“moduli”) coincide, then the solutions are isometric.
The theorem has a generalization to higher dimensions D (with our
symmetry assumption) [23]; here the asymptotically flat condition must be
4 This observation seems to have been made first in [15]. In this paper, the importance of
the quantities l i , (p i ,q i ), and similarly in higher dimensions D>5, was emphasized, but
their properties and relation to the topology and global properties of M were not yet fully
understood.