Having associated the entropy to the (spacelike cross section
of the)
event horizon, it is natural to expect the generalized second law (GSL) of thermodynamics to hold, i.e. the
sum
of the entropy of the matter and the black holes cannot decrease in any process.
However, as Bekenstein pointed out, it is possible to construct thought experiments (e.g. the
so-called Geroch process) in which the GSL is violated, unless a universal upper bound for the
entropy-to-energy ratio for bounded systems exists [52, 53]. (For another resolution of the
apparent contradiction to the GSL, based on the calculation of the buoyancy force in the thermal
atmosphere of the black hole, see [385, 390].) In traditional units this upper bound is given by
, where
and
are, respectively, the total energy and entropy of the system,
and
is the radius of the sphere that encloses the system. It is remarkable that this inequality
does not contain Newton’s constant, and hence it can be expected to be applicable even for
non-gravitating systems. Although this bound is violated for several model systems, for a wide class of
systems in Minkowski spacetime the bound does hold [294, 295, 293, 54] (see also [81
]). The
Bekenstein bound has been extended for systems with electric charge by Zaslavskii [409], and for
rotating systems by Hod [201] (see also [55, 166]). Although these bounds were derived for test
bodies falling into black holes, interestingly enough these Bekenstein bounds hold for the black
holes themselves provided the generalized Gibbons-Penrose inequality (95
) holds: Identifying
with
and letting
be a radius for which
is not less than the area of
the event horizon of the black hole, Equation (95
) can be rewritten in the traditional units as
In the literature there is another kind of upper bound for the entropy for a localized system, the so-called
holographic bound. The holographic principle [366, 350, 81] says that, at the fundamental (quantum) level,
one should be able to characterize the state of any physical system located in a compact spatial domain by
degrees of freedom on the surface of the domain too, analogously to the holography by means of which a
three dimensional image is encoded into a 2-dimensional surface. Consequently, the number of physical
degrees of freedom in the domain is bounded from above by the area of the boundary of the domain instead
of its volume, and the number of physical degrees of freedom on the 2-surface is not greater one-fourth
of the area of the surface measured in Planck-area units
. This expectation is
formulated in the (spacelike) holographic entropy bound [81
]: Let
be a compact spacelike
hypersurface with boundary
. Then the entropy
of the system in
should satisfy
. Formally, this bound can be obtained from the Bekenstein bound with the
assumption that
, i.e. that
is not less than the Schwarzschild radius of
.
Also, as with the Bekenstein bounds, this inequality can be violated in specific situations (see
also [392
, 81
]).
On the other hand, there is another formulation of the holographic entropy bound, due to
Bousso [80, 81]. Bousso’s so-called covariant entropy bound is much more quasi-local than the previous
formulations, and is based on spacelike 2-surfaces and the null hypersurfaces determined by the 2-surfaces in
the spacetime. Its classical version has been proved by Flanagan, Marolf, and Wald [140]: If
is an
everywhere non-contracting (or non-expanding) null hypersurface with spacelike cuts
and
, then,
assuming that the local entropy density of the matter is bounded by its energy density, the entropy flux
through
between the cuts
and
is bounded:
.
For a detailed discussion see [392, 81]. For still another, quasi-local formulation of the holographic principle
see Section 2.2.5 and [365].
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