Gravity and Strings

330 The Kaluza–Klein black hole
The RN–KK dyon. Let us consider the dyonic MP solutions Eqs. (8.204). A quick calculation
gives
F 2 = 8(cos 2 α − sin 2 α)∂i H −1 ∂i H −1 , (11.163)
which vanishes for α =±π/4. For this value of the charges (whose signs we can still
change, preserving F 2 = 0) the dyonic MP solutions are also solutions of the KK action
with constant KK scalar k = k0 (=1 for simplicity) and can be uplifted to a purely gravitational
five-dimensional solution [621]:
d ˆs 2 = H −2 dt 2 − H 2 d x 2 3 − dz + √ 2 αq H −1 dt + αp Aidx i 2 ,
∂[i A j] = αp 1
2ɛijk∂k H, ∂k∂k H = 0, α2 q = α2p = 1,
where αq and αp are the possible signs of the electric and magnetic charges.
(11.164)
Skew KK reduction and generation of fluxbranes. Our last example, “skew KK reduction”
[326, 327] shows the power of the KK techniques to generate new solutions from “almost
nothing.” The general setup is the following. 21 Let us consider a metric that admits two
isometries, one compact (a U(1)), associated with the coordinate θ, and one non-compact
(an R), associated with the coordinate z with a metric of the product form
d ˆs 2 =−dz 2 − f 2 dθ + fmdx m 2 + fmndx m dx n , (11.165)
where we have normalized the period of θ ∈ [0, 2π]. We want now to construct a new
spacetime by identifying points in the above spacetime according to
z + 2π Rz z
∼
. (11.166)
θ θ − 2π B
To apply the standard Scherk–Schwarz formalism, we need to use a coordinate independent
of z and thus we define a new coordinate θ ′ adapted to the above identifications,
θ ′ = θ − B
z, (11.167)
adapted to the Killing vector Rz∂z − B∂θ and rewrite the metric, adapting it to KK reduction
in the direction z. The lower-dimensional fields are
ds 2 KK =−Rz
B k−2 dθ ′ + fmdx m 2 + fmndx m dx n ,
Rz
B
Aθ ′ = k
Rz
−2 , Am = B
k
Rz
−2 fm,
k 2 = 1 + B2
R 2 z
f 2 .
21 Here we follow [325], where more uses of this technique to construct new solutions can be found.
(11.168)