Gravity and Strings

328 The Kaluza–Klein black hole
Lower-dimensional S dualities and generation of KK branes. In this example, we are
also going to study a three-step mechanism for generating new solutions that exploits the
existence of an S-duality symmetry in the four-dimensional KK theory, as discussed in
Section 11.2.4.
1. Reduction of a purely gravitational five-dimensional solution to a four-dimensional
KK-theory solution.
2. S dualization of the four-dimensional KK-theory solution.
3. Oxidation of the S-dualized KK-theory solution to a new purely gravitational fivedimensional
solution.
In particular, we are going to apply this recipe to the “black pp-wave” solutions given
in Eqs. (11.153). Using the transformation Eq. (11.93) on the four-dimensional solution
Eq. (11.154), we immediately obtain
ds2 KK = Wdt2
− H W −1dr2 + r 2d2
(2) ,
d ˜s 2 E = H − 1 2 Wdt2 − H 1
2 W −1dr2 + r 2d2
(2) ,
˜F = αhd 2 ,
˜k = H − 1 2 ,
H = 1 + h r , W = 1 + ω r , ω= h(1 − α2 ).
(11.155)
The solution is naturally given in terms of the field strength ˜F. Finding the potential
is equivalent to solving the Dirac-monopole problem, which we already solved in Section
8.7.2. Here we simply quote the result: in spherical coordinates the non-vanishing
components of ˜F are
˜Fθϕ = αh sin θ = ∂θ Ãϕ, ⇒ Ãϕ =−αh cos θ, (11.156)
up to gauge transformations. This potential is singular at θ = 0 and θ = π and the solution
to this problem is to define the potential in two different patches à (±)
ϕ related by a gauge
transformation:
à (±)
ϕ =±αh(1 ∓ cos θ). (11.157)
It is useful to rewrite the equation that the untilded A has to satisfy (the Dirac-monopole
equation) in a coordinate-independent way that will allow us to generalize the solutions,
∂[i A j] = αk −1 1
0 2ɛijk∂k H. (11.158)
All the properties that depend only on the modified Einstein-frame metric (singularities,
horizons, causality, extremality, thermodynamics etc.) are the same as in the electric case.
The characteristic features of the magnetic BH appear in the KK frame and in ˆd dimensions.