Gravity and Strings

234 The Reissner–Nordström black hole
In GR, a non-linear (non-Abelian, self-coupling) theory, things are quite
different. There is no need to introduce sources: the theory knows that two
Schwarzschild BHs, for instance, cannot be in static equilibrium and the corresponding
solution does not exist. The coupling to gravity makes the electromagnetic
interaction effectively non-Abelian, and it does not need the introduction
of sources to know that only ERN BHs can be in static equilibrium 15 [185]. This
coupling gives rise to many other interesting phenomena in RN backgrounds,
such as the conversion of electromagnetic into gravitational waves [740].
Since the horizon of a single ERN BH looks like a point in isotropic coordinates,
we can try harmonic functions with several point-like singularities:
H(x3) = 1 +
N
2G
i=1
(4)
N |qi|
|x3 −x3,i|
. (8.89)
The overall normalization is chosen so as to obtain an asymptotically flat
solution and the coefficients of each pole are taken positive so that H(x3) is
nowhere vanishing and the metric is non-singular. Also this choice gives a
potential like the one in Eq. (8.88) for large values of |x3|.
It can be seen [500] that each pole of H indeed corresponds to a BH horizon.
In fact, to see that there is a surface of finite area at x3,i, one simply has to shift
the origin of coordinates to that point and then examine the ρ → 0 limit as
in the single-BH case. The charge of each BH can be calculated most simply
using Eq. (8.43), where the volume encloses only one singularity (the current
is nothing but a collection of Dirac-delta terms). The charges turn out to be
sign(−α)| qi|, i.e. all the charges have the same sign.
In GR it is, however, impossible to calculate the mass of each BH because
there is no local conservation law for the mass and there is no such concept
as the mass of some region of the spacetime. Only one mass can be defined,
which is the total mass of the spacetime and this is M = 2 N i=1 |qi|. However,
the equilibrium of forces existing between the black holes suggests that the
electrostatic and gravitational interaction energies (to which GR gravity is sensitive)
cancel out everywhere. If that were true, the masses and charges would
be localized at the singularities and then we could assign a mass Mi = 2|qi| to
each black hole [185]. It is, perhaps, this localization of the mass of ERN BHs
which will allow us to find sources for them, something that turned out to be
impossible for Schwarzschild BHs. This is physically a very appealing idea,
but itiscertainly not a rigorous proof.
If we do not care about singularities, we can also take some coefficients of
the poles of the harmonic function to be negative. In this way it is possible
to obtain solutions with vanishing total mass. Here, it is intuitively clear that
15 As a matter of fact, the identity M1M2 = 4|q1q2| does not imply that both objects are ERN BHs. It can
be satisfied by a non-extremal RN BH with M1 > 2|q1| and a naked singularity with M2 < 2|q2|, butthe
corresponding static solutions (if any) are not known.