Gravity and Strings

7.3 Thermodynamics 203
This relation turns out to be true. The coefficient of proportionality can be determined
[95, 241, 857] and the first law of BH thermodynamics takes the form
dM =
1
8πG (4)
N
κdA. (7.43)
There is an integral version of this relation that can be checked immediately (the Smarr
formula [857]) by simple substitution of the values of κ and A for the Schwarzschild BH:
M =
1
4πG (4)
N
κ A. (7.44)
The above two relations (conveniently generalized to include other conserved quantities
such as the electric charge and the angular momentum) seem to hold under very general
conditions [85] (see also [446, 534, 935]).
This surprising set of analogies suggests the identification between the area of the BH
horizon A and the BH entropy and between the surface gravity κ and the BH temperature.
Stimulated by these ideas, the authors of [85] conjectured, giving some plausibility arguments,
a “third law of BH thermodynamics,” namely that “it is impossible by any procedure,
no matter how idealized, to reduce κ to zero by a finite sequence of operations.” Several
specific examples were studied by Wald in [930]. We will comment more on this in the case
of the Reissner–Nordström BH.
The analogy is, though, not sufficient to make a full identification. Indeed, as the authors
of [85] say,
It can be seen that κ/(8π) is analogous to the temperature in the same way that A is
analogous to the entropy. It should, however, be emphasized that κ/(8π) and A are
distinct from the temperature and entropy of the BH.
In fact the effective temperature of a BH is absolute zero. One way of seeing this is to
note that a BH cannot be in equilibrium with black-body radiation at any non-zero temperature,
because no radiation could be emitted from the hole whereas some radiation
would always cross the horizon into the BH.
On the other hand, in the identification A ∼ S, κ∼ T it is not clear what the proportionality
constants should be (apart from what the dimensional analysis dictates).
Hawking’s discovery [510, 511] that, when the quantum effects produced by the existence
of an event horizon are taken into account, 22 BHs radiate as if they were black bodies
22 This was originally done in a semiclassical calculation in which the background geometry is classical and
fixed and there are quantum fields around the BH. The existence of an event horizon gives rise to the Hawking
radiation but the effect of the Hawking radiation on the BH horizon (backreaction) isnot taken into
account. A pedagogical review of this calculation can be found in [907].