Gravity and Strings

8.7 Electric–magnetic duality 245
It has been suggested that the same is true for any extreme charged BH, not just ERN
BHs, and also that the Bekenstein–Hawking entropy formula Eq. (7.46) should be [652]
S =
χ Ac3
8G (4)
N .
(8.126)
Since one of the main successes of string theory has been the calculation of the (finite!)
entropy of the ERN BH, this result is a bit disturbing. Actually it implies that string theory
and the Euclidean path-integral approach to quantum gravity give different predictions for
the entropy of the ERN BH. It has been argued in [547] that the near-horizon ERN geometry
suffers important corrections in string theory. The reason would be that, although
the topology is that of a cylinder, the geometry is rather that of a pipette, with a radius
that tends to zero at infinity when we asymptotically approach the horizon. String theory
compactified on a circle undergoes a phase transition when the radius reaches the self-dual
value. Thus, beyond the point of the pipette at which the radius has that value, the geometry
may indeed change, 22 although no precise calculations have been done so far.
8.7 Electric-magnetic duality
As we explained in Section 8.2, the full set of sourceless Maxwell equations (the Maxwell
equation plus the Bianchi identity) is invariant (up to signs) under the replacement of the
field strength F by its dual ˜F = ⋆F F → ˜F = ⋆ F. (8.127)
This is true in flat as well as in curved spacetime. In a given frame, this transformation
corresponds to the interchange of electric and magnetic fields according to Eq. (8.38), hence
the name electric–magnetic duality. This transformation squares to (minus) the identity
and, therefore, it generates a Z2 electric–magnetic-duality group.
The Z2 can easily be extended to a continuous symmetry group: 23
˜F = aF + b ⋆ F, ⇒ ⋆ ˜F =−bF + a ⋆ F, a 2 + b 2 = 0, (8.128)
is an invertible transformation that leaves the set of the two equations invariant (up to factors).
It is convenient to define the duality vector
F
F
≡ ⋆F . (8.129)
It is subject to the constraint
⋆ F
0
=
−1
1 F,
0
(8.130)
with which the Maxwell equations can be written as
∇µ F µν = 0, (8.131)
22 See analogous discussions on page 577 about the correspondence principle.
23 For the moment, all these are classical considerations. We will see that quantum effects (in particular, charge
quantization) break the continuous symmetry to a discrete subgroup.