Gravity and Strings

240 The Reissner–Nordström black hole
8.5 Thermodynamics of RN black holes
As we said in the discussion of Schwarzschild BH thermodynamics, most of the results
can be generalized to BHs containing charges or angular momentum. In particular, the
zeroth and second laws of BH thermodynamics take exactly the same form and so do the
identifications between the surface gravity and temperature and horizon, Eq. (7.45), and
between area and entropy, Eq. (7.46). The first law requires the addition of a new term
that takes into account the possible changes in the BH mass due to changes in the charge
( = c = 1),
dM =
1
8πG (4)
N
κdA+ φ h dq, (8.110)
where φh is the electrostatic potential on the horizon. In this case
T = r0
2πr 2 =
+
1
2πG (4)
M2 − 4q2
N M + M2 − 4q2 , 2
S = πr 2 +
G (4)
N
φ h = φ(r+) = 4G(4)
and the Smarr formula takes the form
= πG (4)
N M + M2 − 4q2 2 ,
N q
r+
.
(8.111)
M = 2TS+ qφ h . (8.112)
It is worth stressing that the above formulae have been obtained using a generic RN
metric (i.e. non-extremal). However, we know that the limit in which we approach the ERN
solution with M = 2|q| is not continuous: the topology of the ERN, its causal structure, is
different from that of any non-extreme RN BH, no matter how close to the extreme limit
it is. Furthermore, it seems that the extreme limit cannot be approached by a finite series
of physical processes (the third law of BH thermodynamics) and it has also been argued
that the thermodynamical description of the RN BH breaks down when we approach the
extreme limit [790] (see also [653]): close enough to the extreme limit, the emission of
asingle quantum with energy equal to the Hawking temperature would take the mass of
the RN BH beyond the extreme limit. Then, the change in the spacetime metric caused by
Hawking radiation would be very big and Hawking’s calculation in which backreaction of
the metric to the radiation is ignored becomes inconsistent.
For all these reasons we may expect surprises if we naively take the limit M →|q| in
the above formulae, but this seems not to happen: in that limit the temperature vanishes
and the entropy remains finite and, if we calculate both directly on the ERN solution, we
find the same result. In any case, this is a very important issue because essentially these are
the only BHs for which a statistical computation of the entropy based on string theory has
been performed, and we should try to compute both by other methods, for instance using