Gravity and Strings

262 The Reissner–Nordström black hole
which has interesting properties: it belongs to SL(2, R) but itissymmetric. It can be seen
that it parametrizes the SL(2, R)/SO(2) cosets. Furthermore, under an SL(2, R) transformation
, ittransforms (due to the transformation of g and θ) according to
M = M T , (8.211)
so q TM−1 q is form-invariant under SL(2, R) transformations. Thus, using SL(2, R) duality,
we cannot generate any new solutions not yet contained in the above family.
The result is that we have generated a family of solutions that contains three parameters
more than the initial one by using a three-dimensional duality group. The solutions
are expressed in the simplest form when one uses objects that have good transformation
properties under the duality group: duality vectors and matrices. On the other hand, the
family covers the most general BH-type solution of the Einstein–Maxwell theory that one
can have according to the no-hair conjecture: the BH solution depends on only two conserved
charges (electric and magnetic) and two moduli parameters, which are not really
characteristic of the BH but rather of the vacuum of the theory.
This example may look quite simple, but it has the same features as some more complicated
and juicy cases.
To end this section, let us comment on a couple of subtle points.
• Electric–magnetic-duality rotations and the Wick rotation do not commute. Although
we did not stress it, the Euclidean electric RN solution has a purely imaginary
electromagnetic field. Electric–magnetic-duality rotations of the Euclidean purely
electric RN solution generate a Euclidean solution with imaginary magnetic charge
that remains imaginary when we Wick-rotate back to the Lorentzian signature. If
we Wick-rotate the dyonic RN solution, we obtain a Euclidean solution with real
magnetic charge. This gives rise to problems in the calculation of the entropy in
the Euclidean-path-integral formalism, 29 but they can be dealt with, as shown in
[196, 302, 307, 520].
• In the extreme magnetic RN BH case, we could also try to look for a source. However,
the only thing that works is to view the magnetic charge as the electric charge of the
dual vector field.
8.9 Higher-dimensional RN solutions
Just as there are higher-dimensional analogs of the Schwarzschild BH, there are also higherdimensional
analogs of the electric RN BH, which are solutions of the equations of motion
that one obtains from considering the Einstein–Maxwell action in d dimensions.
Let us first consider the higher-dimensional generalization of the Einstein-scalar system
that we considered at the beginning of this chapter:
S[gµν,ϕ] =
c3 16πG (d)
N
d d x |g| R + 2∂µϕ∂ µ ϕ . (8.212)
29 The thermodynamical quantities that one derives from the Lorentzian metric of the dyonic RN solution are
clearly S-duality-invariant.