Gravity and Strings

8.3 The electric Reissner–Nordström solution 237
11. Finally, BH solutions for an action containing several different vector fields A I µ ,
I = 1,...,N, can easily be found. Let us consider the action
S[gµν, A I µ] =
1
16πG (4)
N
d 4 x |g|
R − 1
4
I
=N
I
F
2
. (8.96)
This action is invariant under global O(N) rotations of the N vector field strengths.
This is a simple example of duality symmetry. Now, any solution of the Einstein–
Maxwell theory (one vector field) is a solution of this theory with the remaining
N − 1vector fields equal to zero, and, by performing general O(N) rotations, one
can generate new solutions in which the N vector fields are non-trivial. It is clear
that, if the original solution had the electric charge q1, the electric charges of the
new solution qi will satisfy N 2
i=1
q′
i = q2 1 .This duality symmetry does not act on
the metric and, therefore, all one has to do is to replace q2 1 by N 2
i=1
q′
i in it.
For example, had we started from the RN solution (8.92), we would have obtained
by this procedure a RN solution with many Abelian electric charges:
and
where now
I =1
ds2 = H −2Wdt2 − H 2W −1dρ2 + ρ2d 2
(2) ,
A I µ = δµtα I H −1 − 1 ,
H = 1 + h/ρ, W = 1 + ω ρ ,
I =N
I
2
α
ω = h 1 −
2
,
I =1
(8.97)
α I =−4G (4)
N q I /r±, h = r±, ω=±2r0, (8.98)
r± = G (4)
N M ± r0, r0 = G (4)
N
M 2 I =N
− 4
I =1
q 2 I
1 2
. (8.99)
This is the first and simplest example of the use of duality symmetries as solutiongenerating
symmetries. We will find more-complex examples later on, but the main
ideas are the same.
Observe that, in this procedure of generating new solutions out of known ones, the
new solutions are expressed at the beginning in terms of the old physical parameters
and the parameters of the duality transformation (in this case, O(N) and sines and