Gravity and Strings

11.3 KK reduction and oxidation of solutions 323
We conclude that the extreme electric KK BH solution does indeed describe the longrange
fields of a KK mode.
The name “extreme BH” for a solution that does not have a regular event horizon needs
some justification: the reason is that this solution belongs to a larger family of BH solutions
with regular event horizons and also with Cauchy horizons, which we will construct in
Section 11.3.4. When the mass and electric charge are equal (the “extreme limit”), the event
and Cauchy horizons coincide and become singular. The general families of non-extreme
dilaton BHs will be studied in Section 12.1. Those with the right dilaton coupling can be
oxidized to one dimension more.
Finally, observe that purely gravitational pp-waves can always be oxidized to one dimension
more by taking the product with the metric of a flat line. We know that the dependence
of the harmonic functions can be extended to that coordinate. The first observation is also
true for any purely gravitational solution, which is always a solution of the KK action
Eq. (11.39). However, the dependences of the functions in the metric cannot always be extended
to the new compact coordinate. This is the case for the Schwarzschild BH solution,
as we are going to see.
11.3.3 Non-extreme Schwarzschild and RN black holes
Dimensional reduction. Paradoxically, the simplest and most fundamental BH solutions
are also the most difficult to reduce because it is also more difficult to generalize them to
the case in which one coordinate is compact. We certainly cannot construct, in a simple and
naive way, infinite periodic arrays of Schwarzschild and non-extreme RN BHs because it
is not at all clear how to construct solutions for more than one non-extreme BH, and, on
physical grounds, one does not expect them even to exist because the interaction between
non-extreme BHs is not balanced and they cannot be in static equilibrium.
Nevertheless, there are solutions describing an arbitrary number of aligned Schwarzschild
BHs: the Israel–Khan solutions [595]. They belong to Weyl’s family of axisymmetric
vacuum solutions [640, 949, 950] and, thus, they have a metric that, in Weyl’s canonical
coordinates {t,ρ,z,ϕ},takes the form
ds 2 = e 2U dt 2 − e −2U e 2k (dρ 2 + dz 2 ) + ρ 2 dϕ 2 , (11.135)
where U is a harmonic function in three-dimensional Euclidean space that is independent
of ϕ (because of axisymmetry) and k depends on U through two first-order differential
equations that can be integrated straightaway:
∂i∂iU = 0,
∂ρk = ρ[(∂ρU) 2 − (∂zU) 2 ],
∂zk = 2ρ∂ρU∂zU.
The simplest choice of U is, in spherical coordinates r 2 = ρ 2 + z 2 ,
N M
U =− G(4)
r
(11.136)
, k =− (G(4)
N M)2 sin 2 θ
, (11.137)
2r 2