black hole, one could simply add more mass until a black hole formed, at which point the entropy
would go down to
, violating the GSL. Thus the entropy must have been less than
to begin
with.
’t Hooft argued that the inescapable implication of this is that the true space of quantum states in
a finite region must be finite dimensional and associated with the two-dimensional boundary of
the region rather than the volume.
Thus it is not enough even if the system is like a fermion
field on a lattice of finite spacing. Rather, the states in the region must be somehow determined
by a finite-state system on a boundary lattice! ’t Hooft made the analogy to a hologram, and the
idea was dubbed by Susskind the holographic hypothesis.
From a classical viewpoint, the holographic hypothesis may correspond to a statement about the
phase space of a gravitating system surrounded by a surface of area A that is not inside a black
hole. It is not inconceivable that this phase space is compact with a volume that scales as the
area. If something like this is true, then the holographic hypothesis could just be a straightforward
consequence of quantising a gravitating system.
On the other hand, it has been suggested by ‘t Hooft and Susskind that the holographic
hypothesis can only be incorporated into physics with a radical change in the foundations of the
subject. If so, it provides a tantalising hint as to the nature of that change. There are some
suggestions that string theory might be headed in the required direction, or perhaps something
very di
ff
erent like a cellular automaton model is correct. For the remainder of this section I will
ignore the holographic hint however, and continue to discuss the problem from the point of view
of local field theory.
4.2 Formational degeneracy
Bekenstein’s original idea was that the entropy of a black hole is the logarithm of the number of
ways it could have formed. This is closely related to the Boltzmann definition of entropy as the
number of micro-states compatible with the macro-state.
A
4
ℏ
G
= 1
A
A
4
A
4
A
4
[5]
[6]

Page
of
13
14
Report
u6609679
Hawking noted that a potential problem arises if one contemplates increasing the number of
species of fundamental fields. There would seem to be more ways of forming the black hole,
however the entropy is fixed at
. Hawking’s resolution of this was that the black hole will also
radiate faster because of the extra species, so that there would be less phase space per species
available for forming the hole. Presuming these two e
ff
ects balance each other, the puzzle would
be resolved.
4.3 Entanglement entropy
Another proposal is that the black hole entropy is a measure of the information hidden in
correlations between degrees of freedom on either side of the horizon. This entropy is sometimes
called
entanglement entropy.
(It has also been called
geometric entropy
.)
Summary
The report can be concluded by saying that black-hole thermodynamics is more general than
black holes—that cosmological event horizons also have an entropy and temperature.