Gravity and Strings

228 The Reissner–Nordström black hole
The ± corresponds to the two possible signs of the electric charge. The metric cannot
depend on this sign because the action is invariant under the (admittedly rather trivial)
duality symmetry F →−F, g → g. The solution one obtains in this way is the Reissner–
Nordström (RN) solution 12 [735, 802] and can be conveniently written as follows:
ds 2 = f (r)dt 2 − f −1 (r)dr 2 − r 2 d 2 (2) ,
Ftr = 4G(4)
N q
r 2
f (r) = (r − r+)(r − r−)
r 2
,
,
r± = G (4)
N M ± r0, r0 = G (4)
N
M 2 − 4q 2 1 2 ,
where q is the electric charge, normalized as in Eq. (8.59), and M is the ADM mass.
Some remarks are necessary.
(8.75)
1. This metric describes the gravitational and electromagnetic field created by a spherical
(or point-like), electrically charged object of total mass M and electric charge q as
seen from far away by a static observer to which the coordinates {t, r,θ,ϕ} (that we
can keep calling “Schwarzschild coordinates”) are adapted. Schwarzschild’s solution
is contained as the special case q = 0.
Included in the (total) mass is the energy associated with the presence of an electromagnetic
field. We cannot covariantly separate the energy associated with “matter”
from the energy associated with the electromagnetic field and the gravitational
field, but we must keep in mind that the mass of the spacetime contains all these
contributions.
2. The vector field that gives the above field strength and whose local existence is
guaranteed by the fact that F satisfies the Bianchi identity is
4G
Aµ = δµt
(4)
N q
. (8.76)
r
3. There is a generalization of Birkhoff’s theorem for RN BHs (see exercise 32.1 of
[707]): RN is the only spherically symmetric family of solutions (that includes
Schwarzschild’s) of the Einstein–Maxwell system.
4. The metric above is a solution for any values of the parameters M and q and,
therefore, of r±, including complex ones.
5. The metric is singular at r = 0and also at r− and r+,ifr+ and r− are real. At r = r±
the signature changes and, in the region between r+ and r−, r is timelike and t is
spacelike and in that region the metric is not static as in Schwarzschild’s horizon
12 The Reissner–Nordström solution is also a particular case (the spherically symmetric case) of the general
static axisymmetric electrovacuum solutions discovered independently by Weyl in [949, 950] and should
also bear his name.