Gravity and Strings

232 The Reissner–Nordström black hole
(c) The relative values of their charge and mass are such that, if we have two of
them, M1 = 2|q1| and M2 = 2|q2|,itwill always happen that
G (4)
N M1M2 = 4G (4)
N |q1q2|, (8.80)
and, if both charges have the same sign and we divide by the relative distance
between them, we obtain
F12 =−G (4) M1M2
N
r 2 12
+ 4G (4) q1q2
N
r 2 = 0. (8.81)
12
This is nothing but the force between two point-like, massive, charged,
non-relativistic objects on account of Eqs. (3.140) and (8.61) and it vanishes,
so they will be in equilibrium. Then, this suggests that it should be possible to
find static solutions describing two (or many) ERN BHs in equilibrium.
(d) On shifting the radial coordinate r = ρ + G (4)
N M of the ERN metric, it becomes
ds 2
= 1 + G(4)
N M
ρ
−2
dt 2 −
1 + G(4)
N M
ρ
2 dρ 2 2 2
+ ρ d(2) . (8.82)
On defining new Cartesian coordinates x3 = (x 1 , x 2 , x 3 ) such that |x3|=ρ and
d x 2
3 = dρ2 + ρ2d 2 (2) ,weobtain a new form of the ERN solution:
ds 2 = H −2 dt 2 − H 2 d x 2
3 ,
Aµ =−2δµt sign(q) H −1 − 1 ,
H = 1 + 2G(4)
N |q|
|x3|
N M
G(4)
= 1 +
|x3|
.
(8.83)
Observe that, in this case, due to the shift in the radial coordinate, the event
horizon is placed at x3 = 0, which in flat Minkowski spacetime is just a point.
It is, though, easy to see that the surface labeled by x3 = 0 isnot just a point
but isasphere of finite area because in the limit ρ → 0 one has to take into
account the ρ 2 factor of d 2 (2) that cancels out the poles in H 2 ,sothe induced
metric in the ρ = 0, t = constant hypersurface is, indeed,
ds 2 =−(G (4)
N M)2d 2 (2) . (8.84)
H is a harmonic function in the three-dimensional Euclidean space spanned
by the coordinates x3, i.e. it satisfies
∂i∂i H = 0. (8.85)
This fact could just be a coincidence but, if we use Eq. (8.83) as an Ansatz
in the equations of motion without imposing any particular form for H, we