Having associated the entropy with 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 Geroch process) in which the GSL is violated, unless a universal upper bound for the
entropy-to-energy ratio for bounded systems exists [69, 70]. (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 [532, 537].) 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
nongravitating systems. Although this bound is violated for several model systems, for a wide class of
systems in Minkowski spacetime the bound does hold [404, 405, 406, 71] (see also [104
]). The
Bekenstein bound has been extended to systems with electric charge by Zaslavskii [579] and to
rotating systems by Hod [269] (see also [72, 226]). 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 (13.1
) holds. Identifying
with
and letting
be a radius for which
is not less than the area of
the event horizon of the black hole, Eq. (13.3
) can be rewritten in the traditional units as
One should stress, however, that in general curved spacetimes the notion of energy, angular momentum,
and radial distance appearing in Eq. (13.4) are not yet well defined. Perhaps it is just the quasi-local ideas
that should be used to make them well defined, and there is a deep connection between the
Gibbons–Penrose inequality and the Bekenstein bound. The former is the geometric manifestation of the
latter for black holes.
In the literature there is another kind of upper bound for the entropy of a localized system, the holographic
bound. The holographic principle [504, 482, 104] 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 as well, analogous to the holography by means of which a
three-dimensional image is encoded into a two-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 two-surface is not greater than
one-fourth of the area of the surface measured in Planck-area units
. This expectation is
formulated in the (spacelike) holographic entropy bound [104
]. 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 [539
, 104
]).
On the other hand, there is another formulation of the holographic entropy bound, due to
Bousso [103, 104]. Bousso’s covariant entropy bound is much more quasi-local than the previous
formulations, and is based on spacelike two-surfaces and the null hypersurfaces determined by the
two-surfaces in the spacetime. Its classical version has been proven by Flanagan, Marolf, and Wald [189]. If
is an everywhere noncontracting (or nonexpanding) 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 [539, 104]. For another, quasi-local
formulation of the holographic principle see Section 2.2.5 and [498].
Let the spacetime be static and asymptotically flat (and hence we use the notation of Section 12.3.1), and
the localized, uncollapsed body is contained in the domain with smooth, compact boundary
. Then Abreu and Visser define the surface gravity vector to be the acceleration of
the Killing observers weighted by the red-shift factor:
. However, its flux
integral on
is just
times of the Tolman energy (see Section 12.3.1). Then, by the
Gibbs–Duhem relation, the equilibrium and stability conditions of Tolman and the Unruh relation
between temperature and surface gravity, Abreu and Visser derive [2, 3] the upper bound
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