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On Emergent Gravity

We need to talk about gravity, people. Why ? Because there might be a lot more to it than meets the eye !

As we all know, gravity is well described locally by Einstein’s General Relativity, in the sense that the model gives predictions which match the observational evidence to an extremely high degree. This holds true locally, up to an intra-galactic scale; on galactic scales and above, the model can still be made to work, but requires the addition of dark matter and dark energy to match observational data. This is not inherently a problem as such, if it weren’t for the fact that on microscopic scales, we have neither a good model nor any observational candidates as to what dark matter actually is. Thus far, all searches for a dark matter particle have come back empty handed, and each failed search further restricts the set of possibilities; it becomes harder and harder to hypothesise as to the nature of these particles, and how to fit them into the Standard Model.

Here is where Prof Erik Verlinde’s recent proposal on “Emergent Gravity” comes in, and the purpose of this article is to present an overview of his paper on the subject [1], aimed at interested lay people. As such, I will endeavour to keep the maths to a minimum; anyone who would like to delve deeper into this subject should consult the original paper, as well as all further sources referenced therein.

The central idea of Verlinde’s paper ( as well as his previous work ) is that – unlike the electroweak and strong interactions – gravity ( and hence spacetime itself ) is not a fundamental force at all, but an emergent phenomenon which arises from the dynamics of an underlying system.  This sets gravity apart from the other three fundamental forces of the Standard Model, and is ultimately the reason why our attempts at quantising it using the traditional tools of quantum field theory have failed so miserably. You can’t open a can using a spoon, just like you can’t quantise gravity using quantum field theory; it’s the wrong tool for the job, which is why the results are of no physical value. In fact, given Verlinde’s ideas, in my mind there is a major question mark as to whether it is even meaningful to speak of such a thing as “quantum gravity” – but I will leave this topic to another article, since the author of [1] does not address this subject.

In Emergent Gravity ( henceforth abbreviated as “EG” ), the basic building block which forms the primary constituent of smooth, continuous spacetime with its geometry as described by GR is a unit of quantum information, a qubit. Verlinde makes no attempt at specifying just what it is that physically carries this information – he merely describes the dynamics which arise from it, not the exact nature of whatever it is that is represented; EG is a model of how spacetime and its geometry arises from an underlying system of degrees of freedom, not an attempt at explaining that system itself, just like you can describe the  dynamics of water without having a good model of atoms. That particular ( albeit very interesting ! ) question is left open. For now, we deal with events, their relationships, and the information needed to describe them; it’s a first start, not yet a complete model.

Of central importance in EG is a form of the holographic principle – the idea that information about “the bulk” is encoded in its boundary. This concept was originally discovered from String Theory, and finds a natural home within EG.

It all starts with something that Verlinde calls the “dark energy medium”, or the matrix. This can be thought of as an ensemble of metastable quantum states ( again, no attempt is made at specifying just what it is that exhibits these states ), which form an as yet unstructured system that can be described in terms of thermodynamics. Spacetime and gravity arise directly as the dynamics of the underlying matrix.

The crucial step which turns the generic medium into the smooth spacetime we are used to is to add entanglement relationships between its constituent degrees of freedom. In EG, entanglement comes in two types – there is firstly short-range entanglement between neighbouring ( =local ) degrees of freedom, and this type of entanglement obeys a strict area law, in the sense that the area of any boundary defines the entropy content of the bulk it encloses. If that bulk ( or volume ) contains exactly the limit of its storage capacity as determined by the area of its boundary, then we are dealing with a de Sitter spacetime, which is essentially what we observe in the universe around us. The boundary is given locally by event horizons, and globally by cosmological horizons; I should also think that any boundary we care to define around any bulk should work for this, but I am not sure.

The second type of entanglement is extremely long-range entanglement between some degrees of freedom, with other very distant degrees of freedom; this type of entanglement obeys a volume law, in the sense that the entropy associated with it scales with volume, and not just area.

Classical spacetime and its geometry arises due to entanglement of degrees of freedom of the underlying matrix, just as water with all its properties arises from the interactions of the underlying molecules and atoms of which it is made up. It contains both short range ( area law ) and long range ( volume law ) contributions. In small local regions, it is the area law of entropy which wins out, and that leads to the usual model of General Relativity ( it turns out that GR requires strict area law entropy as a necessary condition for its validity ), which, as we know, is well borne out by observational evidence. However, on larger scales ( galactic and above ) and long time scales, the long-range volume law entanglement entropy of spacetime starts to make a non-negligible contribution, which leads to deviations of the law of gravity from standard GR. These contributions make it appear as if gravity is stronger than it ought to be based on visible matter distributions and GR, which is what led us initially to postulate the existence of dark matter; however, in EG no dark matter exists. Rather, what happens is that gravity across large distances and long time scales is simply no longer adequately described by GR, because Einstein’s model does not account for the volume law contributions of entanglement entropy.

In EG, “gravity” is best understood as changes in entropy caused by the presence of energy-momentum. Short distance entanglement leads to entanglement entropy, and long distance entanglement to de Sitter entropy, and together they make up spacetime, which now behaves like a glassy, elastic medium. This medium is described in the language of thermodynamics and the material sciences, meaning it is subject to stresses and strains, and has an elastic response to these; it is precisely this elastic response which is dark energy on long scales. Local laws of physics do not describe this elastic response, since its origin is in the long-range entanglement of distant degrees of freedom, and hence a delocalised, emergent phenomenon. This is why, locally, GR describes gravity so well without the inclusion of the cosmological constant term.

Verlinde argues that the presence of energy-momentum leads to a displacement ( reduction ) of entropy density around that region, which implies changes in entanglement relationships; this is precisely what the Einstein equations describe. Because the dynamics of the matrix are glassy rather than crystalline, this displacement can “relax” only slowly, leaving behind residual strains and stresses in the matrix. These stresses and strains in turn back-react on the matter, and this back reaction is exactly what we observe as excess gravity, currently explained as dark matter. However, in EG there is no dark matter or dark energy, rather, the additional gravitational influence is a property of the matrix itself; the laws of gravity as given by GR are modified. This is why we haven’t found any trace of a dark matter particle, or dark energy – according to EG, they quite simply don’t exist, we have just been describing the properties of spacetime incorrectly. Dark matter is an elastic memory effect within the matrix, not a particle at all.

In this context, it should be made clear though that EG is not the same as MoND, since these two models work on the back of completely different physics. EG also has nothing to do with “aether”, since the underlying matrix is one of entanglement relationships, and not a medium of a mechanical nature.

In his paper, Verlinde makes all of this mathematically precise; he formulates an exact correspondence between gravitational and thermodynamic quantities, and arrives at the following expression for the amount of de Sitter entropy contained in a volume B, around a point x and bounded by a surface A [2] :

(1)   \begin{equation*} \displaystyle{S(\ss )=\frac{1}{V_0} \int_{\ss}dV=\oint_{\partial \ss}\frac{x_i}{L} \frac{dA_i}{4G \hbar}} \end{equation*}

This is essentially just a restatement of Stoke’s theorem in physical terms. The presence of matter removes the following amount of entropy from a region B, whose volume is proportional to mass M [3] :

(2)   \begin{equation*} \displaystyle{S_M(\ss )=-\frac{1}{V_0} \int_{\ss}dV=\frac{1}{V_{0}^{*}} \oint_{\partial \ss}u_idA_i} \end{equation*}

From these relations, and general considerations about strain in media, Verlinde than derives an expression for how baryonic mass and inferred dark matter mass should be related, which is valid for spherical distributions [4] :

(3)   \begin{equation*} \displaystyle{\int_{0}^{r}\frac{GM_{D}^{2}(r')}{r'^2}dr'=\frac{M_B(r)a_0 r}{6}} \end{equation*}

which is very close to what we empirically find in observational data. Take careful note that the above expression contains only G and a as dimensionful constants, and makes no reference whatsoever to any other fundamental constant – this is in fact true for all of EG ! This in itself is really quite remarkable – the entire model is based on fundamental principles, and does not require any ad-hoc parameters at all.

In summary, EG postulates that spacetime and gravity are emergent phenomena from the dynamics of an underlying system of degrees of freedom, which Verlinde calls “the matrix”. Spacetime arises as a combination of short-range ( area law ) and long-range ( volume law ) entanglements of these degrees of freedom; on small scales and short times, the area law wins over, which is described by General Relativity. On large scales and long times, there is a volume law contribution as well, which leads to deviations from GR. These deviations are described thermodynamically as memory effects in a glassy medium; dark matter is hence not a particle, but rather an intrinsic property of the medium itself. On very large scales, the dynamics are described almost exclusively by volume law entanglement, which leads to the appearance of dark energy as a driving force. In this model, neither spacetime nor gravity are fundamental in themselves, but emergent properties of an underlying system, the nature of which Verlinde does not at this time speculate on.

I personally think this to be an exciting and promising proposal – at present, [1] is just a first starting point, and much more research is to be done. Specifically, Verlinde does not explain in detail under what conditions the model is valid, and he also does not attempt to apply it to the evolution of the universe as a whole, which is probably the crucial test. Nonetheless, being able to derive the correct magnitude of dark matter from first principles, using no other inputs than two constants, is a truly remarkable achievement.

I, for one, shall keep a very close eye on further developments in Emergent Gravity.



[2], eqn. 7.22

[3], eqn. 7.23

[4], eqn. 7.40



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