A colourful paper out today by Christian Maes attacks one of my favourite polemical issues, that of the relationship between gravity and thermodynamics

C. Maes, *No information or horizon paradoxes for Th. Smiths*, arXiv:1505.01456

The protagonist of this very informal writing is Th. Smiths, and improbable acronym for the average Statistical Mechanician (a previous paper by Maes featured none-the-less than Shere Khan, in the role of a motor protein! I’m longing for the next…). A few years ago I studied in some detail black hole thermodynamics, Hawking radiation, Bekenstein-Hawking entropy, Bousso bound, holography etc. Some of this material was part of a Ph.D. course held by Alessandro Pesci, a physicist mainly involved in the ALICE collaboration who developed a personal passion and had an extremely deep and personal understanding of the field (see arXiv:1002.1257 and arXiv:0708.3729). Not that I remember much, but sure I know very well the eerie feeling that this kind of literature instils in the Statistical Mechanician. In fact, while these studies are well-founded and sound in their gravitational sector, with all the machinery of General Relativity in the due place, when it comes to thermodynamics and statistical mechanics things become extremely flurry and fuzzy, to say the least. No proper definition of thermodynamic ensembles or entropy. No mention of dissipative dynamics, H theorems, etc. Often, words such as “equilibrium”, “entropy”, “second law of thermodynamics” taken out of context and manipulated at will *, jointly with fancy ideas like throwing pianos into black holes (see below). Indeed, a very fertile ground for people looking for paradoxes. Take for example one random paper

Aguirre, Carroll, Johnson, *Out of equilibrium: understanding cosmological evolution to lower-entropy states,* arXiv:1108.0417

On p.2 one finds:

Does it? “Thermodynamics”, as the name suggests, means motion of heat. What latent heat is supposed to melt the ice cube if the glass is completely isolated, implying no radiation whatsoever? The point here, as also emphasized by Maes, is that thermodynamics is not about isolated systems, but rather open systems exchanging energy and matter with the environment. Unfortunately, the common wikipedia formulation of the second law of thermodynamics “entropy of an isolated system cannot decrease” is very misleading, as the entropy of an isolated system, if it even makes sense to talk of such object, is constant and nothing very exciting happens.

Maes does a good job in dotting the i’s and crossing the t’s, showing that many alleged paradoxes might well be due to “sloppy thinking”. I am quite convinced by his explanation of the horizon problem that there is no need for disconnected patches of the Universe that once have communicated to have a uniform temperature in the cosmic microwave radiation, equilibrium statistical mechanics suffices. (Moreover, I have this very naive doubt that any cosmologist could easily object to: won’t causally disconnected events in the universe be connected to a common event to which they are both causally connected? Wouldn’t this be enough to thermalize the whole universe, or are we again falling in Democritos’s paradox, as described by Maes?).

As regards the information problem, I don’t understand all of Maes’s arguments (especially the one on the thermal state), nor I share all of his concerns. I also have the impression that there is a communication gap between the gravitist and the statistical mechanician: the latter thinks in classical terms such as time-space, where time goes along with irreversibility; the former sees time as another coordinate. When Maes writes “After all, the size of the horizon can be arbitrarily weakly curved when the mass of the black hole is large enough. I do not believe that quantum gravity effects can be relevant at such large length scales” he confuses space curvature with space-time curvature, which at the black-hole horizon is huge (so much so that it mixes space-time coordinates).

Anyway, the main point that I agree with is that thermodynamics deals with the process of equilibration, the dynamical evolution of an open system, while all predictions from GR on the black hole metric and hence any conclusion that can be drawn from it refer to stationary properties of a particular solution of the Einstein equations for a completely isolated system (one always considers one individual black hole as if it was the only object in the universe). From this the above misunderstanding that thermodynamics has anything to say about an isolated ice cube.

To Maes’s arguments I would add my favourite, which I also treated here. Thermodynamics, as all of physics after all, has a highly operational character; any quantity is to be referred to an observer and to an act of observation. There is no such thing as an absolute “entropy” until we don’t specify with respect to which observer (the observer’s apparatus establishes which degrees of freedom are relevant and which are not), like there is no absolute position of an object if we don’t specify a reference system. So, for example, in the literature one can find sentences such as

This has not much meaning. Why only considering the entropy at the molecular level? Why not including the nucleus’s structure, the entropy of radiation within the molecules, gluons, neutrinos etc.? Why not considering all the possible melodies that can be played by a piano? There is no limit to this procedure, because the entropy of a piano is only defined with respect to a process which the piano is subjected to.

My personal point is that in thermodynamics it makes only sense to talk about things that can be measured by an observer, and of processes rather than static properties. What is at stake is the consistency of his observations, not the “true reality”, so much so as in Quantum Mechanics it doesn’t make much sense to talk of the wave function as a real object, but only as a tool to understand the “Symbolism of Atomic Measurements”, to put it with Schwinger. So, my little proposal out of the information paradox is: always take one real observer’s point of view, and try to make sense of his observations. Now, the funny thing about black holes is that an observer that is not falling into the black hole, in his proper time, will never see the black hole formed, because it takes an infinite time for this to happen**. So no paradox here. What about an observer that is falling into the black hole? In his proper time he will cross the black hole in finite time, but there won’t be much difference between the inside and the outside, to himself. Frankly I don’t know what exactly he would perceive of the Hawking radiation of matter crossing like himself the event horizon. I suspect nothing strange, again.

So, my impression is that some of these paradoxes might be generated when people try asking what “really” happens. I am much more prudent, I don’t think physics is about the “real thing”; and this might well be enough to file the problem. Finally, I very much agree with Maes that “there appear to be new opportunities for joined initiatives, including work and discussions on a fluctuation theory for gravitation with special challenges regarding the statistical mechanics of the Big Bang and black holes”.

* Jaynes once remarked that most paradoxes about entropy were due to the fact that people were using the same word for different things.

** Though, for all practical results the light emitted by matter clustering towards a black hole will be so dim that it will be absorbed in its way to the observer so that, we do observe a giant black spot where in infinite time there will be a black hole.

“the entropy of an isolated system […] is constant ”

What about free expansion? I know it’s a non-equilibrium process, so that we cannot trace the entropy values throughout the expansion, but is it true or not that both the system is isolated and that in the end the entropy is higher than at the beginning?

Hi hronir, welcome back. Free expansion is indeed a problematic example because people often employ different notions of “entropy”. If one only resolves the spatial distribution, yes, one would typically see an entropy increase. But that’s because the observer is “half-blind”, as he’s not including momenta in the description. If one instead follows the phase space density evolving with the Liouville equation (which is the object that truly describes all of the underlying degrees of freedom that evolve by Hamilton’s equations) and calculates its entropy, one obtains a costant of motion. So, the first system is not truly isolated because, indeed, by neglecting some details one is sort-of identifying an open system (the spatial distribution) and the environment (momenta).

Thanks for the reply — and for the post in the first place.

Thermodynamics and statistical mechanics are full of subtleties indeed, I’m always intrigued: like in QM, there’s always a fuzzy line between the system and the observer, “hidden” in the trace over the degrees of freedom the observer is *choosing* to ignore…

Yep, the situation is similar to that in QM. In fact, I like to reverse the perspective. There is no truly isolated system, and unitary/hamiltonian evolution arises only in very specific situations where one makes a reasonable job in eliminating as much as possible the external influence, always keeping in mind that the nature of physics is to describe consistent measurements on a system, hence there is always an ultimate interaction with the environment in the form of the observer.