Gravity and Strings

11.3 KK reduction and oxidation of solutions 317
This puts in our hands an incredibly powerful tool for generating new solutions both of
the higher- and of the lower-dimensional theories.
The simplest use consists in taking a solution of the higher-dimensional theory that does
not depend on one coordinate, which we identify as the compact one, and reducing it using
the relations between higher- and lower-dimensional fields; or taking a solution of the
lower-dimensional theory and uplifting (oxidizing)ittoasolution of the higher-dimensional
theory.
Amore sophisticated use combines reduction and oxidation with a duality transformation
of the lower-dimensional solution or a GCT of the higher-dimensional solution.
In this section we are going to see the most important examples of these techniques.
11.3.1 ERN black holes
Periodic arrays and reduction. Let us consider the Einstein–Maxwell theory in Rd × S1 .
The action is given in Eq. (11.100) and is no different from the action in Rd+1 and, thus,
the equations of motion admit the same solutions, but now we have to impose different
boundary conditions, namely periodicity in the coordinate z. Obviously, solutions that do
not depend on the coordinate z are trivially periodic, but we are interested primarily in
solutions that do depend on z.
The Einstein–Maxwell theory has MP-type solutions, Eq. (8.229), in any dimension,
which depend on a completely arbitrary harmonic function H. Harmonic functions with a
point-like singularity that tend to 1 at infinity give asymptotically flat ERN BHs. We can
also require the harmonic function to be periodic in the coordinate z in order to obtain
an ERN solution in Rd × S1 . There is a systematic way to construct a harmonic function
periodic in z with a point-like singularity [712] that makes use of the fact that we can
construct solutions with an arbitrary number of ERN BHs by taking harmonic functions
with that many point-like singularities. The idea is to place an infinite number of ERN
BHs with identical masses at regular intervals along the z axis. The corresponding solution
is physically equivalent to one with a single ERN BH and a periodic z coordinate. The
harmonic function is given by the series
n=+∞
H = 1 + h
n=−∞
1
(|x ˆd−2 |2 + (z + 2πnRz) 2 |) ˆd−3
2
, (11.117)
where we have assumed for simplicity that z ∈ [0, 2π Rz] and it is (if it converges 17 )a
periodic function of z with a pole in x ˆd−2 = z = 0inthe interval [0, 2π Rz], as we wanted.
Now that we have a solution of the Einstein–Maxwell theory in R d × S 1 ,wecan follow
the standard procedure: expand in Fourier series, take the z-independent zero mode, and
use the relation between higher- and lower-dimensional fields to obtain a d-dimensional
solution of the action Eq. (11.110). For ˆd = 5, this is done in Appendix G, but for general
17 It certainly does converge for ˆd = 5. In fact, this procedure was first developed in [498] in order to obtain
harmonic functions on R 3 × S 1 and periodic SU(2) instanton solutions using the ’t Hooft Ansatz Eq. (9.22).
Some related calculations can be found in Appendix G.