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13.4 Entropy bounds

13.4.1 On Bekenstein’s bounds for the entropy

Having associated the entropy Sbh := [kc3/(4Gh)] Area(S) to the (spacelike cross section S of the) event horizon, it is natural to expect the generalized second law (GSL) of thermodynamics to hold, i.e. the sum S + S m bh 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 [5253]. (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 [385390].) In traditional units this upper bound is given by Sm/E < [2pk/(hc)]R, where E and Sm are, respectively, the total energy and entropy of the system, and R 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 [29429529354] (see also [81Jump To The Next Citation Point]). The Bekenstein bound has been extended for systems with electric charge by Zaslavskii [409], and for rotating systems by Hod [201] (see also [55166]). 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 (95View Equation) holds: Identifying E with 2 mADMc and letting R be a radius for which 2 4pR is not less than the area of the event horizon of the black hole, Equation (95View Equation) can be rewritten in the traditional units as

V ~ ------------ 2 2 hc- 2 2p (RE) - J > k Sbh + pq . (96)
Obviously, the Kerr-Newman solution saturates this inequality, and in the q = 0 = J, J = 0, and q = 0 special cases (96View Equation) reduces to the upper bound given, respectively, by Bekenstein, Zaslavskii, and Hod. One should stress, however, that in general curved spacetimes the notion of energy, angular momentum, and radial distance appearing in Equation (96View Equation) 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.

13.4.2 On the holographic hypothesis

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 [36635081Jump To The Next Citation Point] 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 2 3 L P := Gh/c. This expectation is formulated in the (spacelike) holographic entropy bound [81Jump To The Next Citation Point]: Let S be a compact spacelike hypersurface with boundary S. Then the entropy S(S) of the system in S should satisfy S(S) < k Area(S)/(4L2 ) P. Formally, this bound can be obtained from the Bekenstein bound with the assumption that 4 2E < Rc /G, i.e. that R is not less than the Schwarzschild radius of E. Also, as with the Bekenstein bounds, this inequality can be violated in specific situations (see also [392Jump To The Next Citation Point81Jump To The Next Citation Point]).

On the other hand, there is another formulation of the holographic entropy bound, due to Bousso [8081Jump To The Next Citation Point]. 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 N is an everywhere non-contracting (or non-expanding) null hypersurface with spacelike cuts S1 and S2, then, assuming that the local entropy density of the matter is bounded by its energy density, the entropy flux SN through N between the cuts S1 and S2 is bounded: 2 SN < k|Area(S2) - Area(S1)|/(4L P). For a detailed discussion see [39281]. For still another, quasi-local formulation of the holographic principle see Section 2.2.5 and [365].

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