Gravity and Strings

320 The Kaluza–Klein black hole
which never goes to a (d − 2)-sphere with finite radius. However, we know that this solution
corresponds to a solution with a regular horizon in d + 1dimensions! One possible way to
explain what is happening here is the following: the results of the dimensional–reduction
procedure are meaningful within certain approximations. In particular, we assume that the
massive modes can be ignored because their masses are very large, which means that the
compactification radius is small. In this geometry, the compactification radius, measured
by the modulus k,isnot constant over the space but depends on r, blowing up when r → 0
(the locus of the putative horizon). Thus, near this point, there are KK modes whose masses
become small enough to be taken into account, but we have not done this and the solution
cannot be considered valid near r = 0. Near r = 0 the solution is indeed ˆd-dimensional and
regular. Similar mechanisms have been proposed in other cases and in the context of string
theory to show how some singularities disappear when we take into account the higherdimensional
origin of the solution [441].
Oxidation. Dimensional oxidation is in general a much simpler operation than reduction:
we simply take a solution of the lower-dimensional theory and rewrite it in terms of the
ˆd-dimensional fields, obtaining a solution of the higher-dimensional theory that does not
depend on the compact coordinate. However, this solution may (but need not) be the zero
mode of a solution that does depend periodically on the compact coordinate and in general
we cannot know which of these possibilities is true.
In any case, the first step consists in having a solution of the lower-dimensional theory
and our problem is that the d-dimensional ERN solution (in general, the MP solutions)
is not a solution of the dimensionally reduced ˆd = (d + 1)-dimensional Einstein–Maxwell
theory. Let us examine the KK scalar equation of motion in the Einstein frame. It takes the
form
∇ 2 ln k ∼ c1k a1 F 2 + c2k a2 G 2 , (11.127)
and requires a non-trivial k if F 2 = 0orG = 0, as is the case here. Thus, the MP solutions
cannot, in general, be considered solutions of the reduced Einstein–Maxwell equations and,
thus, cannot be dimensionally oxidized.
There are, however, exceptions. For instance
1. Solutions with F 2 = 0 satisfy the KK scalar equation of motion and thus can be
oxidized to a purely gravitational solution. One example is the dyonic ERN BH with
electric and magnetic charges related by p =±16πG (4)
N q (see page 330). Another
example is provided by electromagnetic pp-waves.
2. We have seen in Section 11.2.5 that any solution of the four-dimensional Einstein–
Maxwell theory (N = 2, d = 4 SUGRA) can be oxidized to a solution of N = 2, d =
5 SUGRA using Eqs. (11.116) and we have mentioned that solutions of the latter can
be further oxidized to N = (1, 0), d = 6SUGRA.
Observe that we can oxidize the four-dimensional Einstein–Maxwell solutions with
F 2 = 0intwodifferent ways to d = 5. The second form makes use of the supersymmetric
structure of the theory and ensures that the supersymmetry properties will be preserved in
the oxidation, whereas in the first case they will not.