Gravity and Strings

7.3 Thermodynamics 207
If no information is carried by Hawking’s radiation and the BH evaporates indefinitely,
the information about the initial state from which the BH originated is completely
lost forever and the theory of quantum gravity governing all these processes
is non-unitary, in contrast to all the other physical theories. This is, for instance,
Hawking’s own viewpoint.
There is a third group that proposes that the information is not carried out of the BH
by Hawking radiation but the evaporation process stops at some point, leaving a BH
remnant containing that information.
There is a little-explored fourth possibility, which is consistent with the classical
results on stability of BHs and the no-hair conjecture; namely that the information
never enters BHs.
There is, however, no conclusive solution for the BH information problem. In the
models based on string theory that we will explain here, BHs are standard quantummechanical
systems and information is always recovered (even if after a long time).
5. Concerning the BH entropy problem, the statistical-mechanical entropy of systems
of fixed energy E is given by
S(E) = ln ρ(E), (7.51)
where ρ(E) is the density of states of the system whose energy is E. IfaBHisjust
another quantum-mechanical system with E = M,agood theory of quantum gravity
should allow us to calculate the Bekenstein–Hawking entropy S from knowledge of
the density of BH microstates ρ(M).Also, if that theory exists and the above relation
is justified, our knowledge of the Bekenstein–Hawking entropy can be used to find
ρ(M) for large values of M (when the quantum corrections are small),
ρ(M) ∼ exp M 2 . (7.52)
We see that the number of BH states with a given mass must grow extremely fast if
it is to explain the BH’s huge entropy (for a solar-mass BH, ρ ∼ 101076). The thermodynamical
description of systems whose densities of states grow so fast with the
energy is, however, very complicated: the canonical partition function
Z(T ) ∼ dEρ(E)e − E T (7.53)
diverges whenever ρ(E) grows like eE or faster. For instance, the density of states
of any string theory grows exponentially with the mass and the partition function
diverges above Hagedorn’s temperature (see e.g. [31]). For p-branes [38]
ρ(M) ∼ exp λM 2p
p+1 , (7.54)
and for p > 1 the partition function diverges already at zero temperature. The density
of states of BHs must grow faster than that of any of these theories.