Archil Kobakhidze of the University of Melbourne (where my uncle has taught maths since the late 1960s) released his new paper on the arXiv,

Once more: gravity is not an entropic forceHe repeats and refines his arguments from his September 2010 paper, Gravity is not an entropic force (Phys Rev D 83, 2011), and he kindly cites your humble correspondent's January 2010 blog entry, Why gravity can't be entropic, as the first source of the argument that an entropic force predicts the disappearance of interference patterns that are however beautifully seen in neutron interferometry (see e.g. Nesvizhevsky et al. in Nature 2002: PDF free for a recent paper in this research direction that began in the mid 1970s: PDF) - a point that Kobakhidze (as far as I know) realized independently of me.

What motivated him to write another paper was, among other things, an April 2011 preprint, On gravity as an entropic force, by Masud Chaichian, Markku Oksanen, and Anca Tureanu. This paper contains some invalid criticism of Kobakhidze's paper which is why Kobakhidze decided to explain the flaw in a new preprint.

The authors of the flawed preprint used at least two (but related) invalid arguments in their attempts to resuscitate Erik Verlinde's theory. One of them was the claim that Verlinde's theory produces the "right classical limit". When this classical limit is quantized, one obtains the right quantum theory, including the neutron interference. However, this argument incorrectly assumes that quantum physics is uniquely determined by a classical limit. It's not. If you take the classical limit C of a quantum theory Q and "quantize" C again, you don't necessarily get Q.

In particular, when we talk about the distance-dependent entropy, it's a feature of a physical theory that holds both in the quantum theory Q and in the classical limit C. And in the quantum theory, it automatically destroys the interference patterns because there exists no one-to-one way how to link microstates at different separations (because their numbers differ). So there can't exist any quantum theory that preserves the interference but that still produces a classical limit with a distance-dependent entropy.

Kobakhidze looks at a particular method used by the three authors to mask the error in their paper: they try to hide the distance-dependent character of the number of microstates - which is the very defining feature of Verlinde's proposal - by using different conventions for coarse-graining at different distances. Of course, this is just a trick to fool themselves (as well as insufficiently careful readers of their paper). If one uses a self-consistent description and a fixed definition of the entropy, the entropy simply does depend on the distance between the gravitating bodies (such as the Earth and the neutrons) and the interference pattern disappears.

In my opinion, Kobakhidze also tries to present Verlinde's proposal in a maximally comprehensible, no-nonsense way. Instead of accepting the usual potential force as found by Newton,

\[ \vec F = -m \vec \nabla \Phi,\] one must adopt an entropic force that depends on a temperature and an entropy,

\[ \vec F = T \vec \nabla S. \] The temperature should be associated with the Unruh temperature which depends on the gravitational acceleration. This assumption is very problematic and probably inconsistent by itself because different pairs of bodies would have different temperatures which means that they couldn't be in a thermal equilibrium.

Moreover, everyone knows that the actual temperature of the Sun - and the temperature of all degrees of freedom on the Surface of the Sun - is 6000 °C and has nothing to do with tiny temperatures associated with the Sun by Verlinde.

Fine. Ignore that the temperatures in the real situations would be highly non-constant which would lead to a huge heat transfer and irreversibility. Kobakhidze writes down what the formula for entropy \(S = S(r)\) as a function of the distance has to be,

\[ S = 2\pi m \log(r/r_{\rm Verlinde}). \] I've integrated his formula for \(\vec \nabla S\). An unknown distance scale, \(r_{\rm Verlinde}\), really has to be added as well which is almost certainly another inconsistency but that's not the way how Kobakhidze shows that Verlinde's theory is falsified by the observations.

*Lisa Randall's new book on particle physics and philosophy of science, Knocking on Heaven's Door, will be released on September 20th. Pre-order now: I've read it thrice, it's great.*

Kobakhidze realizes that the number of microstates depends on the distance. But in agreement with his previous paper, he still assumes that the neutron may be associated with a wave function. If you try to do so, however, the momentum operator in quantum mechanics inevitably contains a non-Hermitian piece which takes care of the separation of the wave function among many microstates when the number of microstates goes up:

\[ \hat p = -i \frac{\partial}{\partial r} - 2\pi i m. \] Well, this is too optimistic because the relative phases between all the microstates would be undetermined and would evolve chaotically because of small differences in the energy between the macroscopically indistinguishable microstates - which would eliminate any trace of quantum coherence. But even if you assume that the quantum coherence is preserved, you get totally wrong new and very large terms in the Schrödinger equation that will obviously predict that the neutron interferometry experiments should see something completely different than what they do see.

The author describes some wrong predictions of Verlinde's theory for the neutron in the gravitational field in some detail. At any rate, the conclusion is that the neutron interferometry experiments in the Earth's gravitational field falsify all forms of "gravity as an entropic force" hypotheses.

**Challenge**

If you think that Verlinde's proposal has not been falsified, just explain the neutron interferometry experiments with his theory! Define some microstates so that the entropy depends on the distance according to Verlinde's description. And then try to derive the usual Schrödinger equation for the neutron in the external potential - which has been experimentally observed to govern the neutrons - from your original Hamiltonian for the neutron-Earth system by defining the state \( |h\rangle \) for the neutron at height \(h\) as a linear combination of your many microstates (whose density depends on the height \(h\)).

In advance, I can assure you that your attempts will fail. The very meaning of the entropic force is that the coherence is lost whenever the force acts, so the information about the relative phases of the neutron's wave function will disappear. Try it and if you fail - and you will surely fail - please stop with the nonsensical suggestions that Verlinde's proposal could still be OK in some way. Its very fundamental assumptions contradict some experimentally established pillars of modern physics.

If you couldn't have figured this simple thing out for more than 1 year, it's pretty painful, but if you will fail to do so for 2 or more years, it will be even more painful. ;-)

## snail feedback (3) :

This reminds me of Garrett Lisi's E8 theory to some extent. I noticed that Woit has on entry on his "Not Even Wrong" blog to the effect that Verlinde has been awarded some big grant money to continue his work:

"$6.5 Million for Entropic Gravity"http://www.math.columbia.edu/~woit/wordpress/?p=3781

So it seems that even if experts can be confident that he is wrong, Verlinde has a short-term victory.

Why in the world would one even suggest to associate the temperatures of the atoms of gravitating objects with the "temperature" of the microscopic degrees of freedom that cause gravity in the case of entropic gravity???

The universe is clearly far, far away from equilibrium with regard to the effective temperature of the real vacuum. By the time it will get to that equilibrium, all notions of space and time will be gone and there will be no observers left to amuse themselves about physics.

As far as Kobakhidze's derivation of the (in his eyes) correct Schroedinger equation is concerned... Schroedinger quantum mechanics is not even a self-consistent theory of quantum systems. It has nothing to say about fields and it does not treat gravity as a field. It treats gravity as a CLASSICAL potential. Kobakhidze's argument is basically nonsense in, nonsense out (in that he pretends that one can say anything about gravitation by using non-relativistic single particle quantum mechanics) and nature tells him so by agreeing completely with Verlinde's hand-waving and disagreeing with his.

Now, if someone would be so nice to show me a quantum field theoretical derivation of neutron wave functions formulated in a quantum field theory of gravity that incorporates Verlinde's argument and that still predicts the wrong outcome, then, maybe, I would be convinced. Until then I go with the experimental proof, which clearly states that Verlinde's hands wave far more successfully than Kobakhidze's.

And why entropic forces would immediately destroy coherence is also not clear to me. I talk a lot to people, which means that the entropic forces which transport the sound waves from my mouth to their ears work quite well without destroying short term coherence in sound waves. They retain both their amplitude and phase over quite some distance. And anybody who has done atomic spectroscopy knows that random em fields acting on atoms do not automatically destroy line spectra, they merely modify them, leading to line broadening and changes in transition probabilities. How much coherence gets destroyed is all a matter of scales. The effective scale of a neutron scattering experiment is far, far, far away from the scale on which entropic gravity emerges from microscopic degrees on freedom holographic screens... and by the time the neutrons get to react to them, the collective sum of all those interactions behaves like a Newtonian potential.

Sorry, your comment is pure crackpottery.

Whenever some degrees of freedom do interact with the rest so that this interaction matters - and it surely matters if one wants to use this behavior to explain huge effects such as the gravitational attraction - then they inevitably do converge to equilibrium.

It's pure crackpottery for you to suggest that they don't thermalize and/or that their temperature isn't really temperature and can't be measured. The temperature is always the same thing. The entropy is always the same thing, too. You can't have it both ways. You either try to use these temperatures and entropy to explain something - but then you also have to accept that these crazy new terms in the temperature and entropy also have other consequences that instantly falsify the model - or you preserve your sanity in realizing that there's no extra temperature or entropy coming from a nonzero gravitational potential.

I won't even discuss your utterly idiotic comments about Schrodinger's equations being inapplicable or bad or whatever rubbish you are writing. I wouldn't be doing anything else if I had to respond to every paragraph written by every crank of your type. Schrodinger's equation works perfectly, even in the presence of the gravitational potential as has been tested since the 1970s by neutron interferometry etc.

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