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December 3, 2003

Marolf on Entropy Bounds

Don Marolf was in town this week, and talked about his work with Sorkin on Entropy Bounds. The Bekenstein bound,

(1)S<αRE/ℏc S \lt \alpha R E /\hbar c

on the entropy of an object of energy EE and size RR, (α\alpha is some dimensionless number) and the Holographic bound

(2)S<Ac 3/4ℏG S \lt Ac^3/4\hbar G

on the entropy in a region of area AA (a more precise formulation of this latter bound is due to Bousso) have been much bruited about since they were proposed.

The validity of these bounds ultimately rests on an argument involving the consistency of the generalized Second Law. Were it possible to have an object which violated one of these bounds, one could either drop it into a blackhole, or accrete matter onto it to form a blackhole and the resulting process would produce a net decrease of the total entropy in the universe.

What Marolf and Sorkin note is that such highly entropic object are necessarily an important component of the Hawking radiation. In a thermal ensemble at temperature T BH=(4πR BH) −1T_{\mathrm{BH}}= (4\pi R_{\mathrm{BH}})^{-1}, the probablility of finding such an object is proportional to e −βFe^{-\beta F}, where FF is the free energy, F=E−TSF=E-TS. Even when the temperature of the blackhole is low, whenever SS is large enough to potentially violate one of these bounds, these objects become an important component of the thermal ensemble.

According to their analysis, any process in which — if one neglected the Hawking radiation — the Generalized Second Law would be violated, will either be forbidden or will be accompanied by the emission of enough of these guys in the Hawking radiation that the total entropy increases, and the Second Law is satisfied.

Admittedly, I’ve never been a big fan of Holography as a putative fundamental principle of quantum gravity, but Marolf and Sorkin’s argument seems to have unhitched it from its grounding in the Generalized Second Law. That connection was, heretofore, the main reason for believing in Holography.

Posted by distler at December 3, 2003 11:33 PM

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Re: Marolf on Entropy Bounds

There is an interesting proposal by Andreas Karch. With certain assumtions, he showed that the entropy bound implies the viscosity bound. Moreover, this relation seems true even beyond the assumptions that he made. An interesting point is that the tabletop experiments could test this. The viscosity of superfluid He4 misses the bound only by a factor of 10.

Posted by: Sungho Hong on December 6, 2003 7:38 PM | Permalink | Reply to this

If it ain’t one bound, it’s another

Indeed.

One thing is clear. You can use one bound to prove another. For instance, the Generalized Covariant Entropy Bound can be used to prove the Bekenstein Bound (and vice-versa).

What’s not so clear is which, if any, actually follow from some fundamental microscopic description. The Generalized Second Law, it is widely agreed, is likely to be true, regardless of the microscopic description. If you could derive an entropy bound from it, then you’d be in business.

Failing that, you’ve just got a bunch of statements which can be shown to be equivalent …

Posted by: Jacques Distler on December 7, 2003 5:09 PM | Permalink | Reply to this

Re: Marolf on Entropy Bounds

One thing I never understood about all those entropy bounds is how one could ever give an upper bound on an entropy as one cannot measure absolute entropy but only changes in entropy.

You might now say that the thrid law helps you here as it sets S=0 at T=0 but that is not entirely true if one deals with effective theories: For example the CRC tables list entropies for various substances but only under the assumption you are comparing with the same substance at T=0. But liquid He is not really matter in its ground state at T=0: It’s energetically favourable to fuse the He nuclei to iron and this will spontaneously happen if you wait long enough.

So S=0 for He at T=0 only if you ignore nuclear physics. If you include nuclear physics He is only a local minimum and there is still a lot of entropy to be accounted for.

Of course, there is no reason to stop at nuclear physics, one could successively include higher energy degrees of freedom and find more and more entrooy.

Note that for thermodynamics the height of energy barriers is not important as long as the final state has lower energy than the initial one.

Thus, an entropy bound could only possibly be true in the final theory and not in any effective theory (as chemistry or SM or perfect fluid (as Karch considers)).

You could turn this argument around and use a black hole to test if you are already dealing with the final theory: Just measure the increase in area when you drop something in and check whether you accounted for all degrees of freedom.

Very strange indeed (why shouldn’t string theory be excluded from being an effective theory, the QCD string at least is?) especially given that it was initially derived from semi-classical gravity a theory that is definitely an effective theory and not fundamental.

Posted by: Robert Helling on December 12, 2003 12:17 PM | Permalink | Reply to this

Re: Marolf on Entropy Bounds

Given enough time, liquid helium will become a plasma of iron ions and electrons, which has a much larger entropy. But the viscosity of this plasma will also be much larger compared to helium. The ratio of the viscosity and the entropy density of the new state should be much larger than hbar/(4*pi).

One can include, e.g., neutrinos into the entropy of liquid helium. The same inclusion will make the viscosity very large due to the diverging mean free path of neutrinos and will only make the viscosity-entropy ratio bigger.

In my understanding, Karch’s construction was aimed to show that, by assuming the validity of the generalized covariant entropy bound in a theory which embeds a certain metastable state or a certain effective theory, one can make a statement about the viscosity-entropy ratio in this metastable state or this effective theory. This would turn a claim about the abstract final theory into an experimentally falsifiable statement.

Posted by: Dam T Son on December 15, 2003 5:09 PM | Permalink | Reply to this

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