Things have been hectic the last few weeks. So the detour into the Reggae calculus and associated Causal Dynamic Triangulation offshoot I am working on in preparation for the deeper dive into ‘Gravity as Entropic Force’ is still in the offing. In the meantime checkout:

Gravity Emerges from Quantum Information, Say Physicists from the fine folks at Technology Review, an MIT shop. Seems the number of pre-prints appearing has drawn their attention. Also,

Johannes Koelman has a nice suggestive write up about how statistical behavior can appear, in the large, to be a force. Would be nice if some enterprising person implemented the mikado universe with StarLogo.

First of all, the paper continues to generate interest at a healthy rate as indicated by the SPIRES citation index. The index is not complete since the indexing relies on the format of citations, but the rate continues to be about one paper a day.

Two of the pre-prints (here and here) argue that Verlinde’s entropic derivation of gravity is logically flawed. I haven’t had a chance to look closely at these papers and have no opinion yet.

Also, Nobel laureate George Smoot (et al) have put up a pre-print with a very Verlinde-ish flavor. The abstract reads:

To accommodate the observed accelerated expansion of the universe, one popular idea is to invoke a driving term in the Friedmann-Lema\^{i}tre equation of dark energy which must then comprise 70% of the present cosmological energy density. We propose an alternative interpretation which takes into account the temperature intrinsic to the information holographically stored on the screen which is the surface of the universe. Dark energy is thereby obviated and the acceleration is due to an entropic force naturally arising from the information storage on a surface screen. We consider an additional quantitative approach based upon the entropy and surface terms usually neglected in General Relativity and show that this leads to the entropic accelerating universe.

They aren’t the first to suggest that dark energy (sorry, no reference for this statement) has an “entropic” origin. And though I suspect in the end the ontology will change, the whole approach of considering dark energy as the “Hawking radiation of the cosmic horizon” has some truth to it.

In an interesting interview T. Padmanabhan has a couple of things to say about Verlinde’s paper:

Verlinde’s paper has two logical parts. One part derives Einstein’s equations from thermodynamic arguments. This is not new, and I had done this a long time ago. In the second part, and this is the part he seems most excited by, he has derived Newton’s law of motion out of entropic considerations. Unfortunately, I do not agree with this part, and I think it involves faulty, circular reasoning. Nevertheless, it has generated popular interest in this work. Hopefully, it will spur more work in the area, but in the right direction.

It will take some time but hopefully a future post will dive into:

• Looking closely at the assumptions being used in discussion (axiomization and physics ontology)
• The duality of the holographic principle (vis-a-vis operationalism and why there is confusion about ‘emergence’)
• Causal Dynamic Triangulation (CDT), the successor of the ‘regge calculus‘ – the angle of the jangle
• And the one-parameter group of automorphisms which arise with some noncommutative mathematical objects

UPDATE: Today an interesting blog post showed up where the author went through Verlinde’s paper (aided by some discussions with Verlinde) and broke down some of the underlying assumptions. Its worth a read. The author collected the notes taken while reviewing Verlinde’s pre-print in this pdf.

DARPA Mathematical Challenge Twenty-two:

Settle the Smooth Poincare Conjecture in Dimension 4

• What are the implications for space-time and cosmology? And might the answer unlock the secret of “dark energy”?

OK, where was I?

Wile E. Coyote

You know how this is going to end. The coyote is not going to catch the roadrunner. But what if the coyote’s rocket were to travel at the speed of light? And what if the roadrunner constantly accelerates but never attains the speed of light?

It turns out that if  roadrunner has a sufficient head-start (and we know coyote isn’t going to light the fuse until after roadrunner swooshes by) and roadrunner has ‘constant proper acceleration’ (though the bird will never reach the speed of light!) coyote can never catch roadrunner.  Which is a little surprising. After all, its a scenario where the coyote is always traveling at a velocity greater (namely c)  than the roadrunner.

The phenomenon is known as the “rindler horizon” and a great description of this Zeno-like ‘paradox’ is provided by the technically adept science (and science fiction) writer Greg Egan here.

In the roadrunner/coyote scenario above, the coyote cannot catch and consequently cannot influence (or cook and eat) the roadrunner. You might say the coyote is ‘causally disconnected’ from the roadrunner. In fact, the roadrunner’s constant acceleration defines a rindler horizon which is also a sort of ‘causal horizon’. Another causal horizon most people are aware of is the event horizon of a black hole. Are these horizon’s similar at a more fundamental level? Remember that Unruh wrote in 1976, when writing about his acceleration temperature/radiation:

The similarity of this case with the behavior of a detector near the black hole is brought out, and it is shown that a geodesic detector near the horizon will not see the Hawking flux of particle

Ted Jacobson and Renaud Parentani wrote a wonderful survey article which also suggests there is a common underlying framework for causal horizons. In their article they write:

Any causal horizon is endowed with a surface entropy density of 1/4.

The realization that horizon entropy is an intrinsically observer dependent notion raises the obvious question of what are the states that the horizon entropy counts? The notion that it counts the number of internal configuations, i.e. configurations behind the horizon, was argued against in [57] on various grounds. It seems only even possibly viable if the holographic conjecture[58] holds, i.e. if the entire description of the world behind any horizon can be fully described on its bounding surface. It was argued in [57] that the holographic conjecture is at best valid when there is no trapped surface behind the horizon, but it may otherwise in some sense be true. Whether or not it is true, the fact remains that, for the observers who remain confined to the “outside” of the horizon, the horizon entropy somehow captures the number of ways that the world beyond the horizon can affect the world outside.

[57] T. Jacobson, “On the nature of black hole entropy,” in General Relativity and Relativistic Astrophysics: Eighth Canadian Conference, AIP Conference Proceedings 493, C. Burgess and R.C. Myers, eds. (AIP Press, 1999), pp. 85-97.

[58] R. Bousso,“The holographic principle,” Rev. Mod. Phys. 74, 825 (2002).

They conclude their 2003 article with these sentences:

While no definitive answer as to the ultimate nature of horizon entropy seems immediately at hand, an abundance of insight has been gleaned from the three decades of work. Perhaps the time is ripe to synthesize this insight and make the leap to a new conception.

In the next post I will return to Verlinde’s paper. To be continued…

Woit has some thoughts about recent ‘trends’ in physics, including Verlinde’s contribution.

A reminder that there is a well-established standard for crackpottery.

And finally…

“They laughed at Newton. They laughed at Galileo. But they also laughed at Bozo the Clown.”
– Carl Sagan

In the previous post I compared Verlinde’s ‘revelation‘ to the hyped proposal of Garret Lisi. I also tried to lay some of the historical groundwork which is context for Verlinde’s paper. Lee Smolin, in his paper building on Verlinde’s result, provides a more concise recounting of the historical context. Check it out.

Erik Verlinde

Verlinde is not a crackpot. A postdoctoral student  and then long term research member of the Institute of Advanced Study, Verlinde has regularly published in the area of string theory. He currently teaches at the Institute for Theoretical Physics at the University of Amsterdam. But his paper has not generated a universal ‘a-ha’ among physicists. One of my favorite reactions is Robert Helling’s:

The latest paper by Eric Verlinde on gravity as an entropic force makes me wonder whether I am getting old: Let me admit it: I just don’t get it.

Helling then goes on to write that Verlinde’s paper reminds him of the following proof:

girls = time * money [obvious, a priori]

time = money [experimentally determined]

girls = money ^ 2 [substitution into line #1]

money = sqrt(evil) [biblical]

money ^ 2 = evil [square of previous line]

girls = evil [transitive property]

I have no problem with that. But I would have thought Helling more sympathetic to Verlinde’s paper. After all, Verlinde’s use of polymer elasticity as an example of ‘entropic force‘ surely would have resonated with Helling’s own investigations into the thermodynamics of protein folding. And Helling, who won a ‘Schlössmann Award’ for describing gravity as an emergent property, can’t possibly be thrown by the dethroning of gravity from the big-4 of forces. Maybe, as he says, he is just ‘getting old’.

I suspect at this stage the uncertainty about Verlinde’s paper has to do with the following:

• The derivation of Einstein’s equations has been done before using roughly the same set of assumptions (see earlier post reference to Jacobson).
• There is an incredible muddle regarding the particulars and degree of gravity emergence. For one thing, the holographic principle in its usual forms has tons of implicit assumptions about the geometry (including dimensionality) of ’space’.
• The Unruh effect/temperature buries within itself a multitude of sins, especially relativistic quantum field theory.
• Most importantly, the paper messes with people’s usual ontology. The pre-geometric physical entities which provide the semantics to an unstated, but implied (since gravity is emergent) reformulation of the holographic principle as well as the dynamics of those essentially information-theoretic entities is pretty alien.

And yet… and yet…

It seems – at this point in time – what Verlinde has done is pointed to a reaxiomization  of some basic physics. Similar to there being several different equally adequate axiomizations (given a set of basic symbols) of propositional logic, Verlinde will focus attention on perhaps redundant underlying assumptions of quantum mechanics, general relativity and thermodynamics.

To be continued…

It’s been two and half years since Garret Lisi achieved his 70 column-inches of fame with his “Exceptionally Simple Theory of Everything“[PDF]. Also a trained theoretical physicist, Lisi made headlines around the world [here, here and here for example]. The story was made-for-tv material. Bohemian surfer/snowboarder dude unwinds the Rubik’s cube of the universe.  The theory came complete with a pretty picture. But in the end his theory suffered a wipeout:

Distler had demonstrated in his blog that this is a mathematical impossibility.

The Scientific American post-mortem on Lisi’s theory concludes with this graf (my emphasis):

Today the theory is being largely but not entirely ignored. Lisi, naturally, continues to work on it, as does Smolin. Lisi says that even if what Distler claims is true, it would only be true for the variant of E8 (“real E8”) originally used in his paper and that another variant (“complex E8”) would certainly work. Smolin argues that the press coverage gave the false impression that Lisi’s proposal was a finished work. “In reality,” he says, “almost every new theoretical proposal is first presented in a way that is flawed and incomplete, with open issues that need to be filled in…. While Lisi’s proposal has exciting aspects, this is the case with it as well.”

Though covered in the Dutch press, Verlinde’s theory hasn’t achieved the notoriety of Lisi’s. The ‘Exceptionally Simple Theory of Everything’ was written about more often in magazines and newspapers than ever referenced in scientific articles. On the other hand, Verlinde’s January 2010 paper already has more citations than Lisi’s theory ever produced and is serving as the inspiration of roughly one scientific article a day.

In this and subsequent blog posts I will describe, with some context, Verlinde’s paper. And describe what I believe is going on in this small area of physics. My purpose isn’t to advocate for Verlinde’s theory. The work being done – and to come – is more about: 1) understanding the assumptions built into the theory’s underlying components and, 2) the somewhat fascinating – nearly self-referential – creation of the ontology of physics from the entropy reduction in the ’self-organizing’ system of ‘accepted physical theory’.

Some Background

Verlinde readily admits in his blog he isn’t the first person to derive Einstein’s equations from ideas involving thermodynamics, quantum mechanics and boundary assumptions. Jacobson had done this back in 1995. Even then, the interplay between quantum mechanics, thermodynamics and gravity had been “in the air” for awhile going back at least until 1984, if not further. Like many things in modern theoretical physics, it started with the black hole.

In 1972 Jacob Berkenstein argued on largely information-theoretic grounds that:

… black-hole entropy is equal to the ratio of the black-hole area to the square of the Planck length times a dimensionless constant of order unity.

Black holes, already with the reputation that everything “checks in” but nothing “checks out”, were already a bit of an oddity. That they couldn’t exist in thermodynamic equilibrium with other ‘normal’ objects wasn’t much of a stretch for something that could swallow entire galaxies. Thermodynamics, usually considered an emergent property, just didn’t contain in its ‘domain of applicability’ something as exotic as a black hole.

Nevertheless, in 1975 Hawking published a paper describing what has come to be known as “Hawking radiation“. As is well known, stuff can come out of black holes. And importantly the radiation has the frequency distribution of blackbody radiation. Hawking’s calculation was a pretty straight forward application of quantum field theory in curved spacetime where he took into account the boundaries associated with black holes.

The next year, Unruh with the general relativity equivalence principal in mind,  published “Notes on Black Hole Evaporation” which is actually better remember for what is variously called the “Unruh effect”, “Unruh temperature” or “Unruh temperature”. The take-away from that article was:

The behavior of particle detectors under acceleration is investigated where it is shown that an accelerated detector even in flat spacetime will detect particles in the vacuum.

The equation is linked here. He also wrote (remember this for later):

The similarity of this case with the behavior of a detector near the black hole is brought out, and it is shown that a geodesic detector near the horizon will not see the Hawking flux of particles.

So far in this story there are a lot of things going on:

• [1972] Berkenstein thinks black holes should have entropy because in many respects the area of event horizon acts like an entropy measure.
• [1975] Hawking does some quantum field calculations for the curved space around a black hole and finds black holes are thermodynamic objects.
• [1976] Unruh goes further and claims “empty space” will have a temperature relative to an accelerated observer.

There is an important and interesting backdrop to these results. Berkenstein’s original proposal that black holes had entropy faced an objection from Geroch. Here is a description of the problem from Stanford’s philosophy website:

From the time that Bekenstein first proposed that the area of a black hole could be a measure of its entropy, it was know to face difficulties that appeared insurmountable. Geroch (1971) proposed a scenario that seems to allow a violation of the generalized second law. If we have a box full of energetic radiation with a high entropy, that box will have a certain weight as it is attracted by the gravitational force of a black hole. One can use this weight to drive an engine to produce energy (e.g., to produce electricity) while slowly lowering the box towards the event horizon of the black hole. This process extracts energy, but not entropy, from the radiation in the box; once the box reaches the event horizon itself, it can have an arbitrarily small amount of energy remaining. If one then opens the box to let the radiation fall into the black hole, the size of the event horizon will not increase any appreciable amount (because the mass-energy of the black hole has barely been increased), but the thermodynamic entropy outside the black hole has decreased. Thus we seem to have violated the generalized second law.

“… the generalized second law” meaning the second law of thermodynamics.  Not a good thing. Perpetual motion machines and all that. In 1982 Unruh and Wald satisfactorily resolved the apparent ‘paradox’. By applying the ‘Unruh effect’ to the situation of the box they showed the change in the global conservation equations was exactly the amount to eliminate the problem. Exactly the amount.

Something is obviously going on here. Gravity, quantum field theory and the laws of thermodynamics fit together perfectly consistently. That Jacobson was able in 1995 to derive Einstein’s equation from the Unruh temperature, the proportionality of entropy to event horizon surface area and a generalized second law of thermodynamics says something about the components of that relationship.

To be continued…