Could gravity be an emergent property of nature? Sorry if this question is naive. It is just a curiosity that I have.
Are there theoretical or experimental reasons why gravity should not be an emergent property of nature?
Assume a standard model view of the world in the very small. Is it possible that gravity only applies to systems with a scale large enough to encompass very large numbers of particles as an emergent property?
After all: the standard model works very well without gravity; general relativity (and gravity in general) has only been measured at distances on the millimeter scale.
How could gravity emerge? For example, it could be that space-time only gets curved by systems which have measurable properties, or only gets curved by average values. In other words that the stress-energy tensor has a minimum scale by which it varies.

Edit to explain a bit better what I'm thinking of.


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*We would not have a proper quantum gravity as such. I.e. no unified theory that contains QM and GR at the same time. 

*We could have a "small" (possibly semi-classical) glue theory that only needs to explain how the two theories cross over: 

*

*the conditions and mechanism of wave packet reduction (or the other corresponding phenomena in other QM interpretations, like universe branching or decoherence or whatnot)

*how this is correlated to curvature - how GM phenomena arise at this transition point.




Are there theoretical or experimental reasons why such a reasoning is fundamentally incorrect?

 A: I'm not an expert in gravity, however, this is what I know.
There's a hypothesis about gravity being an entropic property. The paper from Verlinde is available at arXiv. That said, I would be surprised for this to be true. The reason is simple. As you probably know, entropy is an emergent property out of statistical probability. If you have non-interacting, adimensional particles into one half of a box, with the other half empty and separated by a valve, it's probability, thus entropy, that drives the transformation. If you look at it from the energetic point of view, the energy is exactly the same before and after the transformation. This works nicely for statistical distribution, but when you have to explain why things are attracted to each other statistically, it's much harder. From the probabilistic point of view, it would be the opposite: the more degrees of freedom your particles have, the more entropy they have. A clump has less degrees of freedom, hence has less entropy, meaning that, in a closed system, the existence of gravity is baffling. This is out of my speculation, and I think I am wrong. The paper seems to be a pleasure to read, but I haven't had the chance to go through it.
A: You might want to look up the Weinberg-Witten theorem which shows that's not possible given certain assumptions. If the original model from which quantum gravity is supposed to emerge is an ordinary Poincaré covariant quantum field theory over flat nondynamical Minkowski space, they showed it's not possible for massless helicity $\pm 2$ particles to emerge. As a theory of quantum gravity ought to contain gravitons, this appears to rule out such models.
Of course, these assumptions are questionable. For instance, the theory from which gravity emerges might not be a quantum field theory. This is the case for superstring theory. 
Another possibility might be the "fundamental" model isn't Lorentz covariant. However, we still need the low energy effective theory to be approximately Lorentz covariant. In typical condensed matter analog models, different quasiparticles couple to different metrics, and there is no universality to the gravitational couplings, or the speed of light. Unless all the quasiparticles co-emerge together, I don't see any way around this problem.
It might be a bit hard to come up with the positive energy theorem in an emergent theory of gravity. The positive energy theorem states that the ADM energy of an asymptotically flat spacetime always has to be nonnegative. In an emergent theory, the ADM energy could just as easily be negative for some states. To see this, note first that the ADM energy can be defined locally as the limit as we go to spatial infinity of a locally defined integral over an enclosing spatial surface with one spatial and one time codimension. If we assume the "fundamental" theory is local, this means the now emergent ADM flux also has to be defined locally in terms of the more fundamental fields. As the enclosing boundary becomes larger and larger, its extrinsic curvature becomes closer and closer to zero. If we have a positive ADM flux passing through a plane — as defined with respect to a choice of normal vector orientation — a reflection by the plane will give us another state where this ADM flux is now negative. So, we can certainly imagine performing some sort of approximate reflection about the enclosing submanifold on a local patchwork basis, at least for the regions at or around the enclosing surface. We then need to find an interpolation of the resulting state far into the interior, which of course, might not look anything like a reflection at all. But if the fundamental theory also satisfies local independence, that ought to be possible. But the end result of all this construction is a state with negative emergent ADM energy. I know this argument is very handwavy and nonrigorous, but it sounds plausible. But there might be some loopholes. For instance, the fundamental theory might be local, but the emergent large scale excitations — and hence emergent spacetime — might be delocalized with respect to the underlying background spacetime. Or the underlying fundamental theory might be inherently nonlocal.
A: Isn't the answer to the question of the title widely believed to be "yes"?
If you believe that searching for what high-energy theorists call a "theory of everything" is a valuable and worthwhile enterprise, then you probably also believe that gravity as we currently understand it (General Relativity, say) "emerges" from some deeper theory (in the effective field theory sense) which unifies it with all other known fundamental forces.  
Of course nobody yet knows for sure what that theory is, but I'm told certain flavors of string theory are the most viable candidates as of 2010.  You can find some indication towards how gravity emerges from string theory in the first few sentences of this answer by Eric Zaslow.  
Perhaps Eric Zaslow or some other expert can give more details at the level of saying, for instance, how Einstein's equations arise from string theory (I would ask this as a question on this site, except that I know I could find the answer in any book on string theory if I cared enough to look).  I'm told that it has something to do with the renormalization group equations of the conformal field theory on the worldsheet, but I'm afraid I can't reproduce or explain that argument any further for you here.
A: In December, Carlo Rovelli summarized the last twenty years of the research agenda of a group of researchers in Loop Quantum Gravity theory.  In a nutshell, LQG argues that gravity is a property of space-time rather than a quantum field theory mediated by a boson, and that space-time is fundamentally discrete with point-like locations in space-time connected by a network of connections to each other.  In this approach, described by three main equations, the number of dimensions in space-time itself is emergent and neither locality nor the number of dimensions of space-time are well defined concepts at the most fine grained level.  You are at point A and connected to points B, C and D, related by the equations, which when repeated ad infinitum are well approximated by a continuous, four dimensions space that may satisfy the properties of GR in classical approximation.  As he sums up the research agenda:
"There is substantial circumstantial evidence that the large distance limit of the theory is correctly general relativity, from asymptotic analysis and from large distance calculations of n-point functions and in spinfoam cosmology; and there are open directions of investigations to reinforce this evidence. The degrees of freedom are correct and the theory is generally covariant: the low-energy limit is not likely to be much else than general relativity. But there is no solid proof yet." 
A: The question is fairly old, but new ideas have appeared in literature in the meantime.
An interesting possibility is that gravity is an emergent property of a more fundamental quantum theory. In particular, space and time are emergent from the entanglement of quantum fields and gravitation can be inferred from quantum information constraints. It's a very fascinating proposal and there is a great deal of calculations supporting this idea. 
These theories makes significant use of the AdS/CFT correspondence, that is indeed a duality between a gravitational theory and a quantum theory without gravity. 
A pedagogical reference
A: Despite all I wrote in my other answer, there's a very interesting attempt by Xiao-Gang Wen to come up with emergent models of gravity starting from quantum lattice models with no gravity, and only nearest neighbor interactions. His work can be found at gr-qc/0606100 and arXiv:0907.1203. He managed to show that quasiparticles with no energy gap and a helicity of $\pm 2$ can emerge without being accompanied by helicity $\pm 1$ or $0$ quasiparticles. Whether or not this model can be considered a model of gravity though is another matter.
