Gravity and Strings

236 The Reissner–Nordström black hole
9. If we shift the radial coordinate by r = ρ + r± in the RN solution Eq. (8.75), it takes
the following form:
ds 2
= 1 + r±
−2 1 +
ρ
±2r0
dt
ρ
2
− 1 + r±
2 1 +
ρ
±2r0
−1 dρ
ρ
2 + ρ 2 d 2
(2) ,
A ′ µ =−δµt
4G (4)
N q
1 + r±
−1
− 1 . (8.91)
ρ
r±
The RN metric looks in this form (taking the minus sign) like a Schwarzschild
metric with mass r0/G (4)
N “dressed” with some factors related to the gauge potentials
or, alternatively, as the ERN solution dressed with some Schwarzschild-like factors.
The Schwarzschild component of this metric completely disappears in the extreme
limit, leaving an ERN isotropic metric. This form of charged BH metric is quite
common and occurs, as we will see, in various contexts, rewritten in this way:
ds2 = H −2Wdt2 − H 2W −1dρ2 + ρ2d 2
(2) ,
Aµ = δµtα H −1 − 1 ,
H = 1 + h ρ , W = 1 + ω ρ , ω= h 1 − (α/2) 2 .
(8.92)
We will obtain many solutions in this form. Afterwards, we will identify the
integration constants that appear in them in terms of the physical constants:
α =−4G (4)
N q/r±, h = r±, ω=±2r0. (8.93)
10. Another useful coordinate system for charged BHs [446], which covers the BH exterior
and in which the radial coordinate τ takes values between −∞ on the horizon and
0atspatial infinity, can be obtained by the transformation of the coordinate ρ above,
−r0τ r0e
ρ =− , (8.94)
sinh(r0τ)
so the metric takes the form
ds2 = e2U dt2 − e−2U
r 4 0
sinh 4 (r0τ) dτ 2 r
+
2 0
sinh 2 (r0τ) d2
(2) ,
e 2U =
1 + r−
−
2r0
r−
e
2r0
2r0τ
−2
e 2r0τ .
(8.95)