Gravity and Strings

204 The Schwarzschild black hole
T
Fig. 7.5. The temperature T versus the mass M of a Schwarzschild black hole.
S
Fig. 7.6. The entropy S versus the mass M of a Schwarzschild black hole.
with temperature 23
T = κ
2πc
M
M
(7.45)
dramatically changed this situation. On the one hand, it removed the last obstruction to a
complete identification of BHs as thermodynamical systems. On the other, the coefficient
of proportionality between κ and T was completely determined, and determined, in turn,
that between A and S:
S = Ac3
4G (4)
N
. (7.46)
Observe that this relation can be rewritten in this way:
S = 1
32π 2
A
, (7.47)
ℓ 2 Planck
that is, essentially the area of the horizon measured in Planckian units, a huge number for
astrophysical-size BHs, in agreement with our discussions about the no-hair conjecture.
Observe also that the appearance of in T makes manifest its quantum-mechanical origin.
In particular, for Schwarzschild’s BH we have (see Figures 7.5 and 7.6)
T =
c3 8πG (4)
N
4πG(4)
, S =
M c
N M2
, (7.48)
23 In our units Boltzmann’s constant is 1 and dimensionless so T has dimensions of energy, ML2T −2 or L−1 in natural units, and the entropy is dimensionless.