Holographic Bound From Second Law

Abstract

The holographic bound that the entropy (log of number of quantum states) of a system is bounded from above by a quarter of the area of a circumscribing surface measured in Planck areas is widely regarded a desideratum of any fundamental theory, but some exceptions occur. By suitable black hole gedanken experiments I show that the bound follows from the generalized second law for two broad classes of isolated systems: generic weakly gravitating systems composed of many elementary particles, and quiescent, nonrotating strongly gravitating configurations well above Planck mass. These justify an early claim by Susskind. What is the holographic bound ?The holographic principle, first enunciated by 't Hooft, 1 claims the physical equivalence between pairs of physical theories. One member of a pair, T1, describes a bulk system U in a spacetime; the allied theory, T2, describes a boundary of that spacetime. For instance, string theory in the D = 10 spacetime AdS 5 × S 5 is known to be equivalent to a supersymmetric gauge theory on the boundary. Some more examples are known, mostly in D > 4. Many regard the holographic principle as a criterion for good physics.An obvious consistency requirement on the holographic principle is that the boundary of any system should be able to encode as much information as required to catalogue the quantum states of the bulk system: T2 should allow at least as many quantum states to reside on the boundary as T1 allows for the bulk (otherwise, so to speak, T2 does not know enough to be equivalent to T1). Suppose we go to D = 4 and take the logarithm of this inequality. On the one hand we get the entropy S of U and its gravitational field; on the other we get the entropy of the boundary, which analogy with black hole entropy suggests to quantify by one quarter of its 2-D area A in Planck units. Thus we have the guess (the holographic bound 1,2 here denoted HB; Planck units used throughout !)How do we know that this bound is true ? As support for the HB, Susskind 2 described a gedanken experiment in which a system violating the HB is forced to collapse to a black hole by adding to it extra entropy-free matter. Susskind interprets the ensuing apparent violation of the generalized second law (GSL) as evidence that the envisaged system cannot really exist. In the clearer reformulation of Wald, 3 one imagines the system as a spherically symmetric one of radius R, energy E (with R > 2E, of course) and entropy S which violates the HB: S > πR 2 . A spherically symmetric and concentric shell of mass R/2 − E is dropped on the system; by Birkhoff's theorem the total mass is now R/2. If the outermost surface of the shell reaches Schwarzschild radial coordinate 1

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