# Emergent Gravity

Erik Verlinde (Wikipedia) has expanded on earlier work pointing out that gravity can be understood as an entropic force, arising from the information content of the region of space in which it's happening. I've only, so far, had a cursory look at the work (I haven't worked out what it looks like when expressed in my own notation), so the following is just my provisional impression of what Verlinde is saying; for the real deal, follow the links below.

First, starting in general relativity's description of the universe as a smooth manifold, take a volume of space-time on whose boundary: the Newtonian gravitational potential, or the gravitational red-shift relative to observers far from all massive bodies, is constant. The boundary of that region, at each moment of time, has a definite area. Conjecture that the information content of the interior is proportional to the area of the boundary and that this information content has the entropy it naturally would given the temperature that would arise from the energy (including the energy-equivalent of mass) within the volume being distributed thermodynamically among the bits of information. Assume that a massive body passing through the boundary makes the transition between inside and outside during movement through its Compton wavelength. (These are essentially the assumptions used to infer the thermodynamic properties of black holes, IIUC.) Verlinde applies a thermodynamic argument to then show that the system's tendency to increase energy is sufficient to explain the forces conventionally explained as gravity.

Now, Verlinde is at pains to point out that you don't need to presume, as I just did, the smooth manifold of general relativity: all he actually needs is an information-storage system with a well-defined (and variable) number of bits for holding information, time variation, a temperature associated with a total energy and a formal parameter, associated with parcels of that energy entering or leaving the system described by the information-content, for which the rate of change of the system's entropy with that parameter, as the packet leaves or enters, is proportional to the energy in the packet. Once you reason your way through this, interpreting the number of bits as proportional to an area (in fact, IIUC, as the measure of the boundary of a volume in a space potentially with higher dimension than we're used to; a factor of (1 +1/(dim−3))/2 is needed, in space-time of dimension dim (including the time dimension; so we're used to dim=4, where this factor is 1), in the number of bits per unit measure), the parameter naturally being understood as a spatial parameter perpendicular to the area, you end up inducing the notion of space-time and the laws of general relativity. As such, the smooth manifold of general relativity and the dynamical equation controlling it emerge from this information-theoretic analysis.

It is worth nothing that, in general relativity, gravity is already an emergent effect, arising from the fact that space-time is a smooth manifold, whose curvature is influenced by matter via a dynamical equation, on which matter travels along geodesics (the appropriate notion of straight lines for use on a curved space). Masses cause curvature in space-time and that curvature tends to cause the geodesics followed by nearby masses to approach one another; our perception of a force of gravity arises from our habit of looking at the universe from an accelerating frame of reference, arising because the ground beneath our feet is exerting forces on us that prevent us from following geodesics.

Now, Verlinde tells us nothing about how information is stored on his holagraphic screens, nor about how it does the implied information processing to model the system it models: but he does show that no matter how that happens, the laws of general relativity are bound to emerge. So it's a very neat result, albeit from an incomplete theory. A lot of Verlinde's explanation takes the emerging rather literally as an on-going process, with space-time outside the surface having already emerged and that inside not yet having done so; I find this peculiar and unnecessary to the analysis, at least in so far as one can apply it to regions of space-time for which we have a perfectly clear sense of space and time both within and without. None of this is a problem: indeed the the incompleteness may even be a strength, as we don't need to pin down the rest of the details to infer a model sufficient for our needs.

We presently have

• a quantum-mechanical theory that all work nicely together as an explanation of the dynamics of systems within regions of space-time small enough to permit a single chart with respect to which space-time's metric takes its canonical diagonal form (with only ±1 on the diagonal); it can deal with all the known types of matter.
• a geometric theory of space-time that works nicely for explaining the dynamics of matter in large enough clumps that quantum-mechanical effects are ignorable and, in particular, the only properties of the matter we have to consider are charge, energy-momentum-stress and total angular momentum.

Potentially, we might be able to extend either (or each) to embrace the subject-matter we presently only know how to describe in one of them, along with the interface between the two. If we ever do so for both, we can reasonably suppose there shall be some transformation that maps each of these two descriptions onto the other, thereby showing that they're the same theory, in the sense that they make the same predictions. My take on Verlinde is that he's taking the first stesps towards a third such model, based on information theory, which would (once completed) be identifiable with the others via similar transformations. He already has, in effect, the transformation he needs to map between his model and general relativity. Combine that with how it transforms onto the standard model of quantum mechanics and he'll have a complete theory, which may indeed serve to help us work out how to complete each of the others.  Written by Eddy.