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

338 S. Hollands
asymptotically flat boundary conditions at infinity? In D = 4 dimensions
the answer to this question is provided by the famous black hole uniqueness
theorems 1 , which state that the only such solutions are provided by the
Kerr family of metrics, and these are completely specified by their angular
momentum and mass. Unfortunately, already in D = 5, the analogous statement
is demonstrably false, as there exist different asymptotically flat black
holes—having even event horizons of different topology—with the same values
of the angular momenta and charges 2 . Nevertheless, one might hope that
a classification is still possible if a number of further invariants of the solutions
are also incorporated. This turns out to be possible [22, 23], at least if
one restricts attention to solutions which are not only stationary, but moreover
have a comparable amount of symmetry as the Kerr family, namely the
symmetry group 3 R × U(1) D−3 . Such a spacetime cannot be asymptotically
flat in D>5, but it can be asymptotically Kaluza-Klein, i.e. asymptotically a
direct product e.g. of the form R 4,1 ×T D−5 . The purpose of this contribution
is to outline the nature of this classification.
Because the symmetry group has D − 2 dimensions, the metric will,
in a sense, depend non-trivially only on two remaining coordinates, and the
Einstein equations will consequently reduce to a coupled system of PDE’s
in these variables. However, before one can study these equations, one must
understand more precisely the nature of the two remaining coordinates, or,
mathematically speaking, the nature of the orbit space M/[R × U(1) D−3 ].
The quotient by R simply gets rid of a global time coordinate, so one is left
with the quotient of a spatial slice Σ by U(1) D−3 . To get an idea about the
possible topological properties of this quotient, we consider the following two
simple, but characteristic, examples in the case dim Σ = 4, i.e. D =5.
The first example is Σ = R 4 , with one factor of U(1) × U(1) acting by
rotations in the 12-plane and the other in the 34-plane. Introducing polar
coordinates (R 1 ,φ 1 )and(R 2 ,φ 2 ) in each of these planes, the group shifts
φ 1 resp. φ 2 , and the quotient is thus given simply by the first quadrant
{(R 1 ,R 2 ) ∈ R 2 | R 1 ≥ 0,R 2 ≥ 0}, which is a 2-manifold whose boundary
consists of the two semi-axes and the corner where the two axes meet. The
first axis corresponds to places in R 4 where the Killing field m 1 = ∂/∂φ 1
vanishes, the second axis to places where m 2 = ∂/∂φ 2 vanishes. On the
corner, both Killing fields vanish and the group action has a fixed point. The
second example is the cartesian product of a plane with a 2-torus, Σ = R 2 ×
T 2 . Letting (x 1 ,x 2 ) be cartesian coordinates on the plane, and (φ 1 ,φ 2 )angles
on the torus, the group action is generated by the vector fields m 1 = ∂/∂φ 1
and by m 2 = α∂/∂φ 2 +β(x 1 ∂/∂x 2 −x 2 ∂/∂x 1 ), where α, β are integers. These
1 For a recent proof dealing properly with all the mathematical technicalities, see [5]. Original
references include [3, 36, 2, 28, 16, 17, 24].
2 For a review of exact black hole solutions in higher dimensions, see e.g. [9].
3 In D = 4, this is not actually a restriction, because the rigidity theorem [16, 4, 11, 35]
shows that any stationary black hole solution has the additional U(1)-symmetry. In higher
dimensions, there is a similar theorem [18, 19, 31], but it guarantees only one additional
U(1)-factor, and not D − 3.