Is gravity an entropic force after all?
This question has introduced me to the whole "entropic force" area which has several papers during 2010. I see that there are references to "entropic force" explanations for Coulomb's law and other areas too. Here is a link to a simple introduction to these ideas.
The Verlinde paper and others however are deriving Newtons Law, Einstein's GR etc as classical theories. The underlying formulation of course being a stochastic behaviour of unknown microstates. Despite the presence of $\hbar$ and the motivation from the Black Hole area formulae the Verlinde paper does not introduce an explicit link with quantum mechanics. Thus there is no derivation of Schrodinger's equation and no introduction of $\Psi$.
The Kobakhidze paper says "One starts with a "holographic screen” S which contains macroscopically large number of microscopic states which we denote as $\left|i(z)\right\rangle$, $i(z) = 1, 2, ...,N(E(z), z).$ The screen is then described by the mixed state
$$\rho(z)=\sum p_{i(z)}\left|i(z)\rangle \langle i(z)\right|$$
However Verlinde does not explicitly introduce microstates as quantum states, with density matrices etc, although this is a tempting extension.
Now it might be that this is the only sensible quantum development of the stochastic basis of the "entropic idea", but Verlinde has not taken it. So what is disproved is a theory that Verlinde has not written down.
Having said this, there is a resemblance between "entropy" and the idea of introducing "stochastics" into quantum theory. One such attempt is known as "Stochastic Electrodynamics" (link to Wikipedia). As you will see from the summary this has had successes with e.g. the Unruh effect, but problems modelling genuine quantum phenomena.
I dont know whether anyone has considered combining the two areas directly.
Dear Vagelford, you're totally right. Gravity cannot be an entropic force because
- its phenomena would be irreversible
- the degeneracy of the states coming from the entropy would destroy the interference patterns that have been measured e.g. by neutron interferometry.
More than a year ago, this was also explained on my weblog
http://motls.blogspot.com/2010/01/erik-verlinde-comments-about-entropic.html
and Erik Verlinde much like some of his junior Dutch colleagues tried to react but as far as I can say, none of their reactions has ever made any sense.
The neutron interferometry experiments are pretty impressive. They not only show that the interference pattern survives the action by the force of gravity. But it is exactly as shifted as the equivalence principle implies.
And in fact, the changes of the phases have been measured so accurately that the experimenters may deduce not only the zeroth order gravitational acceleration but also the higher-order corrections to it such as the tidal forces. All of these effects preserve the interference pattern - which wouldn't be possible if there were many states representing a macroscopic configuration - and this pattern exactly moves and behaves according to general relativity.
For these two and other reasons, gravity cannot be entropic. We also know it from the explicit models in the AdS/CFT correspondence and elsewhere: only event horizons may produce a large entropy of this order. A cold binary star doesn't carry any entropy associated with the gravitational attraction, certainly not an entropy comparable to the black hole entropy which is what Erik Verlinde claims.
But a multi-million euro grant has already paid by some European politicians to endorse this "research" so it may be unreasonable to expect that too many people aside from the two of us will say these obvious things too comprehensibly and loudly. After all, many people can be bought very easily and inexpensively.
One additional disclaimer: If you originally encountered the proofs that gravity can't be entropic on my blog, you shouldn't treat this answer as an independent confirmation of my previous claims. ;-)
All the best, LM
I think there has been some confusion over this matter. Of course if makes little sense to think a trajectory around a black hole will exhibit an entropy increase. Verlinde proposed an entropy force of gravity from which Newton's law of gravity may be derived. This is a thermodynamic principle for the entropy variation of a holographic screen $$ \Delta S~=~2\pi k_B(mc/\hbar)\Delta x $$ where $\Delta x$ is the distance between the holographic screen and a test particle of mass $m$. The entropy is then in increments of $2\pi k_B$ according to displacements of the screen equal to the Compton wavelength $\lambda~=~\hbar/mc$. The standard entropy formula $S~=~k_BA/4L_p^2$ indicates a proportionality with respect to area. For the radius of the screen adjusted $S_0~\rightarrow~S~=$ $S_0~+~\Delta S$ by the radial change in the screen $r~=~r_0~+~\Delta r$ then $$ ΔS~=~k_B/4L_p^2(A~–~A_0)~=~(2\pi k_B/L_p^2)r\Delta r~=~(2\pi k_Bc^3/G\hbar)r\Delta r, $$ which is linear in the radial displacement. By equating $\hbar/mc~=~G\hbar/rc^3$ gives a radius $r~=~Gm/c^2$, which is appropriate for the Newtonian result, but is half the Schwarzschild result.
The entropy for an orbit of a test mass is constant, and this entropy is a measure of the holographic screen. So if you place a Gaussian surface around a gravitating radially symmetric body that cloaks the configuration of the body, the entropy of the screen is the maximum entropy of the system. For a particle orbiting the body the entropy is constant, or $\Delta S~=~0$, for the screen remains constant.