Gravity and Strings

7.3 Thermodynamics 205
and so the first law of BH thermodynamics and Smarr’s formula take the forms
dMc 2 = TdS, Mc 2 = 2TS. (7.49)
How can a BH from which nothing can ever escape (classically) radiate? The physical
mechanism behind the Hawking radiation seems to be the process of Schwinger-pair creation
in strong background fields [195, 737], which was originally discovered for electric
fields [824], rather than quantum tunneling across the horizon, which would violate causality.
In the electric-field case, the background field gives energy to the particles of a virtual
pair, separating them. In the BH case, one of the particles in the pair is produced inside the
event horizon and the other outside the event horizon. The net effect is a loss of BH mass
and the “emission” of radiation by the BH.
The same effect causes the spontaneous discharge of charged bodies (such as a positively
charged sphere, say) left in vacuum: if the electric field is strong enough, the electron
and positron of a virtual pair can be separated. The electron will move toward the sphere
being captured by it, while the positron will be accelerated to infinity. From far away, one
would observe a radiation of positrons coming from the charged sphere, whose charge
would diminish little by little. In fact, this process is believed to cause the discharge of
Reissner–Nordström BHs [218, 289, 429, 686, 716, 737, 750] and was discovered before
the publication of Hawking’s results. 24 The energy spectrum of the charged pairs produced
in an electric field is also thermal [866], but only charged particles are produced and the
temperature is different depending on the kind of charged particles considered (electron–
positron, proton–antiproton, etc.), whereas in the gravitational case, due to the universal
coupling of gravity to all forms of energy, all kinds of particles are produced with thermal
spectra with a common Hawking temperature.
The thermodynamics of BHs has several problems or peculiarities.
1. The temperature of a Schwarzschild BH (and of all known BHs far from the extreme
limit which we will define and discuss later) decreases as the mass (the energy) increases
(see Figure 7.5) and therefore a Schwarzschild BH has a negative specific
heat (Figure 7.7)
C −1 = ∂T
∂ M =−
c3
8πG (4) < 0, (7.50)
N
M2
and becomes colder when it absorbs matter instead of when it radiates (as ordinary
thermodynamical systems do). Thus, a BH cannot be put into equilibrium with an
infinite heat reservoir because it would absorb the energy and grow without bounds.
2. The temperature grows when the mass decreases (in the evaporation, for instance)
and diverges near zero mass. 25 At the same time the specific heat becomes bigger
24 It should also be pointed out that the production of particles in the gravitational field of a rotating BH was
also discovered before [698, 861, 917, 968], but this is not a purely quantum-mechanical effect, but the
quantum translation of the well-known classical super-radiance effect.
25 Precisely when the metric becomes (apparently, smoothly) Minkowski’s. The temperature of the Minkowski
spacetime is zero, rather than being infinite like the M → 0limit of the BH temperature. This result is, at first
sight, paradoxical, but similar results are, though, very frequent and we will soon meet another one (see the