As already mentioned in section
2, after the area theorem was proven, Bekenstein [14,
15
] proposed a way out of this difficulty: Assign an entropy,
, to a black hole given by a numerical factor of order unity
times the area,
A, of the black hole in Planck units. Define the
generalized entropy,
S
', to be the sum of the ordinary entropy,
S, of matter outside of a black hole plus the black hole
entropy
Finally, replace the ordinary second law of thermodynamics by the generalized second law (GSL): The total generalized entropy of the universe never decreases with time,
Although the ordinary second law will fail when matter is dropped into a black hole, such a process will tend to increase the area of the black hole, so there is a possibility that the GSL will hold.
Bekenstein's proposal of the GSL was made prior to the discovery of Hawking radiation. When Hawking radiation is taken into account, a serious problem also arises with the second law of black hole mechanics (i.e., the area theorem): Conservation of energy requires that an isolated black hole must lose mass in order to compensate for the energy radiated to infinity by the Hawking process. Indeed, if one equates the rate of mass loss of the black hole to the energy flux at infinity due to particle creation, one arrives at the startling conclusion that an isolated black hole will radiate away all of its mass within a finite time. During this process of black hole ``evaporation'', A will decrease. Such an area decrease can occur because the expected stress-energy tensor of quantum matter does not satisfy the null energy condition - even for matter for which this condition holds classically - in violation of a key hypothesis of the area theorem.
However, although the second law of black hole mechanics fails
during the black hole evaporation process, if we adjust the
numerical factor in the definition of
to correspond to the identification of
as temperature in the first law of black hole mechanics - so
that, as in Eq. (9
) above, we have
in Planck units - then the GSL continues to hold: Although
A
decreases, there is at least as much ordinary entropy generated
outside the black hole by the Hawking process. Thus, although the
ordinary second law fails in the presence of black holes and the
second law of black hole mechanics fails when quantum effects are
taken into account, there is a possibility that the GSL may
always hold. If the GSL does hold, it seems clear that we must
interpret
as representing the
physical
entropy of a black hole, and that the laws of black hole
mechanics must truly represent the ordinary laws of
thermodynamics as applied to black holes. Thus, a central issue
in black hole thermodynamics is whether the GSL holds in all
processes.
It was immediately recognized by Bekenstein [14] (see also [12]) that there is a serious difficulty with the GSL if one
considers a process wherein one carefully lowers a box containing
matter with entropy
S
and energy
E
very close to the horizon of a black hole before dropping it in.
Classically, if one could lower the box arbitrarily close to the
horizon before dropping it in, one would recover all of the
energy originally in the box as ``work'' at infinity. No energy
would be delivered to the black hole, so by the first law of
black hole mechanics, Eq. (7
), the black hole area,
A, would not increase. However, one would still get rid of all of
the entropy,
S, originally in the box, in violation of the GSL.
Indeed, this process makes manifest the fact that in classical
general relativity, the physical temperature of a black hole is
absolute zero: The above process is, in effect, a Carnot cycle
which converts ``heat'' into ``work'' with
efficiency [49]. The difficulty with the GSL in the above process can be viewed
as stemming from an inconsistency of this fact with the
mathematical assignment of a finite (non-zero) temperature to the
black hole required by the first law of black hole mechanics if
one assigns a finite (non-infinite) entropy to the black
hole.
Bekenstein proposed a resolution of the above difficulty with
the GSL in a quasi-static lowering process by arguing [14,
15] that it would not be possible to lower a box containing
physically reasonable matter close enough to the horizon of the
black hole to violate the GSL. As will be discussed further in
the next sub-section, this proposed resolution was later refined
by postulating a universal bound on the entropy of systems with a
given energy and size [16]. However, an alternate resolution was proposed in [94
], based upon the idea that, when quantum effects are taken into
account, the physical temperature of a black hole is no longer
absolute zero, but rather is the Hawking temperature,
. Since the Hawking temperature goes to zero in the limit of a
large black hole, it might appear that quantum effects could not
be of much relevance in this case. However, despite the fact that
Hawking radiation at infinity is indeed negligible for large
black holes, the effects of the quantum ``thermal atmosphere''
surrounding the black hole are not negligible on bodies that are
quasi-statically lowered toward the black hole. The temperature
gradient in the thermal atmosphere (see Eq. (12
)) implies that there is a pressure gradient and, consequently, a
buoyancy force on the box. This buoyancy force becomes infinitely
large in the limit as the box is lowered to the horizon. As a
result of this buoyancy force, the optimal place to drop the box
into the black hole is no longer the horizon but rather the
``floating point'' of the box, where its weight is equal to the
weight of the displaced thermal atmosphere. The minimum area
increase given to the black hole in the process is no longer
zero, but rather turns out to be an amount just sufficient to
prevent any violation of the GSL from occurring in this
process [94
].
The analysis of [94] considered only a particular class of gedankenexperiments for
violating the GSL involving the quasi-static lowering of a box
near a black hole. Of course, since one does not have a general
proof of the ordinary second law of thermodynamics - and, indeed,
for finite systems, there should always be a nonvanishing
probability of violating the ordinary second law - it would not
be reasonable to expect to obtain a completely general proof of
the GSL. However, general arguments within the semiclassical
approximation for the validity of the GSL for arbitrary
infinitesimal quasi-static processes have been given in [105
,
90
,
101
]. These arguments crucially rely on the presence of the thermal
atmosphere surrounding the black hole. Related arguments for the
validity of the GSL have been given in [48
,
82
]. In [48], it is assumed that the incoming state is a product state of
radiation originating from infinity (i.e., IN modes) and
radiation that would appear to emanate from the white hole region
of the analytically continued spacetime (i.e., UP modes), and it
is argued that the generalized entropy must increase under
unitary evolution. In [82], it is argued on quite general grounds that the (generalized)
entropy of the state of the region exterior to the black hole
must increase under the assumption that it undergoes autonomous
evolution.
Indeed, it should be noted that if one could violate the GSL
for an infinitesimal quasi-static process in a regime where the
black hole can be treated semi-classically, then it also should
be possible to violate the ordinary second law for a
corresponding process involving a self-gravitating body. Namely,
suppose that the GSL could be violated for an infinitesimal
quasi-static process involving, say, a Schwarzschild black hole
of mass
M
(with
M
much larger than the Planck mass). This process might involve
lowering matter towards the black hole and possibly dropping the
matter into it. However, an observer doing this lowering or
dropping can ``probe'' only the region outside of the black hole,
so there will be some
such that the detailed structure of the black hole will directly
enter the analysis of the process only for
. Now replace the black hole by a shell of matter of mass
M
and radius
, and surround this shell with a ``
real
'' atmosphere of radiation in thermal equilibrium at the Hawking
temperature (10
) as measured by an observer at infinity. Then the ordinary
second law should be violated when one performs the same process
to the shell surrounded by the (``real'') thermal atmosphere as
one performs to violate the GSL when the black hole is present.
Indeed, the arguments of [105,
90,
101
] do not distinguish between infinitesimal quasi-static processes
involving a black hole as compared with a shell surrounded by a
(``real'') thermal atmosphere at the Hawking temperature.
In summary, there appear to be strong grounds for believing in the validity of the GSL.
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The Thermodynamics of Black Holes
Robert M. Wald http://www.livingreviews.org/lrr-2001-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |