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 ). The Bekenstein bound has been extended to systems with electric charge by Zaslavskii  and to rotating systems by Hod  (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[190, 271]. (It is interesting to note that an analogous lower bound for the relaxation time of any perturbed system, derived for nongravitational systems in , is saturated by extremal Reissner–Nordström black holes.)
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 . 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 . 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 .
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.
Living Rev. Relativity 12, (2009), 4
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