A proof of the Bekenstein bound for any strength of gravity through holography
Résumé
The universal entropy bound of Bekenstein is considered, at any strength of the gravitational interaction. A proof of it is given, provided the considered general-relativistic spacetimes allow for a meaningful and inequivocal definition of the quantities which partecipate to the bound (such as system's energy and radius). This is done assuming as starting point that, for assigned statisticalmechanical local conditions, a lower-limiting scale l * to system's size definitely exists, being it required by holography through its semiclassical formulation as given by the generalized covariant entropy bound. An attempt is made also to draw some possible general consequences of the l * assumption with regards to the proliferation of species problem and to the viscosity to entropy density ratio. Concerning the latter, various fluids are considered including systems potentially relevant, to some extent, to the quark-gluon plasma case. In the description of thermodynamic systems, recent works [ 1, 2 ] have shown that the generalized covariant entropy bound (GCEB) [ 3 ], which can be considered as the most general formulation of the holographic principle for semiclassical circumstances, is universally satisfied if and only if the statistical-mechanical description is characterized by a lower-limiting spatial scale l *, determined by the assigned local thermodynamic conditions. 1 In [ 5 ] the consequence has been drawn that l * entails also a lower-limit to the temporal scale, and this supports, both for its existence and value, the recently proposed universal bound [ 6 ] to relaxation times of perturbed thermodynamic systems, bound attained by black holes. 2 All this amounts to say that the GCEB is satisfied if and only if a fundamental discreteness is present in the spatio-temporal description of statistical-mechanical systems, with a value changing from point to point being determined by local thermodynamic conditions. The exact value of l * is set from the GCEB [ 1, 2 ]. This does not necessarily mean however that l * has to depend on gravity. In effect, the derivation of l * from the GCEB is through Raychaudhuri equation so that gravity cancels out [ 2 ]. Indeed, this scale appears intrinsically unrelated to gravity (in particular l * must not be thought as something around the Planck scale ; it is in general much larger than this) since, as we will discuss later, it can be fully described simply as a consequence or expression of the flat-space quantum description of matter [ 2 ]. Among the various entropy bounds, the GCEB appears the most general one, subsuming in some way all other previous bounds at the conditions in which they are supposed to hold. The Bekenstein universal entropy bound (UEB) [ 10 ], the first proposed among them, has been shown to be implied by GCEB for weak field conditions (a strengthened version of it, actually, in case of non-spherical systems) [ 11 ]. This bound sets an upper limit to the S/E ratio (S is entropy and E is total mass-energy) for a given arbitrary system, in terms of its size. The original motivation was a gedanken experiment, in the context of black hole physics, in which the variations of total entropy are considered when objects with negligible self-gravity are deposited at the horizon. In spite of how it was derived, it has been clear since the beginning that gravity should not play any role in establishing the UEB [ 10 ]. Since, besides the systems with negligible self-gravity, also the objects with the strongest gravitational effects, namely the black holes, appear to satisfy the bound (actually they attain it), the original proposal stressed moreover the UEB should hold true for whatever strength of the gravitational
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