Gravity and Strings

212 The Schwarzschild black hole
The solutions Eq. (7.77) are almost straightforward generalizations of the fourdimensional
Schwarzschild solution in every sense. Their most interesting property is the
existence of event horizons at r = RS in all of them, with properties that generalize those of
the d = 4 ones and lead us to the study of their thermodynamics. The uniqueness of these
(static BH) solutions was proved in [444, 589]. There is no uniqueness for stationary BHs
in higher dimensions, as the existence of the rotating black ring of [373] shows.
7.5.1 Thermodynamics
In d dimensions, the first law of BH thermodynamics and Smarr’s formula are [706]
dMc 2 =
d − 2
2(d − 3) TdS, Mc2 d − 2
= TS, (7.80)
d − 3
where the temperature T is now given in terms of the surface gravity κ by the same expression
as in four dimensions Eq. (7.45) while κ is defined by the same formula Eq. (7.21)
in any dimension. The entropy is given in terms of the volume of the (d − 2)-dimensional
constant-time slices of the event horizon V (d−2) by
S =
V (d)
4G (d)
N
The volume and surface gravity of the event horizon are
and, therefore
V (d−2) = R d−2
S ω(d−2), κ =
T =
(d − 3)c
4π RS
, S = Rd−2
S
. (7.81)
(d − 3)c2
, (7.82)
2RS
3
ω(d−2)c
4G (d)
N
. (7.83)
Smarr’s formula can be easily checked using these results.
The temperature of the higher d-dimensional BHs can also be calculated in the Euclidean
formalism with the criterion of avoiding conical singularities of the τ–r part of the metric
on the event horizon. A Euclidean calculation of the entropy may also be done.