Is Verlinde’s Emergent Gravity compatible with General Relativity?

Dark matter filaments, Millenium Simulation
Image: Volker Springel

A few months ago, Erik Verlinde published an update of his 2010 idea that gravity might originate in the entropy of so-far undetected microscopic constituents of space-time. Gravity, then, would not be fundamental but emergent.

With the new formalism, he derived an equation for a modified gravitational law that, on galactic scales, results in an effect similar to dark matter.

Verlinde’s emergent gravity builds on the idea that gravity can be reformulated as a thermodynamic theory, that is as if it was caused by the dynamics of a large number of small entities whose exact identity is unknown and also unnecessary to describe their bulk behavior.

If one wants to get back usual general relativity from the thermodynamic approach, one uses an entropy that scales with the surface area of a volume. Verlinde postulates there is another contribution to the entropy which scales with the volume itself. It’s this additional entropy that causes the deviations from general relativity.

However, in the vicinity of matter the volume-scaling entropy decreases until it’s entirely gone. Then, one is left with only the area-scaling part and gets normal general relativity. That’s why on scales where the average density is high – high compared to galaxies or galaxy clusters – the equation which Verlinde derives doesn’t apply. This would be the case, for example, near stars.

The idea quickly attracted attention in the astrophysics community, where a number of papers have since appeared which confront said equation with data. Not all of these papers are correct. Two of them seemed to have missed entirely that the equation which they are using doesn’t apply on solar-system scales. Of the remaining papers, three are fairly neutral in the conclusions, while one – by Lelli et al – is critical. The authors find that Verlinde’s equation – which assumes spherical symmetry – is a worse fit to the data than particle dark matter.

There has not, however, so far been much response from theoretical physicists. I’m not sure why that is. I spoke with science writer Anil Ananthaswamy some weeks ago and he told me he didn’t have an easy time finding a theorist willing to do as much as comment on Verlinde’s paper. In a recent Nautilus article, Anil speculates on why that might be:

“A handful of theorists that I contacted declined to comment, saying they hadn’t read the paper; in physics, this silent treatment can sometimes be a polite way to reject an idea, although, in fairness, Verlinde’s paper is not an easy read even for physicists.”

Verlinde’s paper is indeed not an easy read. I spent some time trying to make sense of it and originally didn’t get very far. The whole framework that he uses – dealing with an elastic medium and a strain-tensor and all that – isn’t only unfamiliar but also doesn’t fit together with general relativity.

The basic tenet of general relativity is coordinate invariance, and it’s absolutely not clear how it’s respected in Verlinde’s framework. So, I tried to see whether there is a way to make Verlinde’s approach generally covariant. The answer is yes, it’s possible. And it actually works better than I expected. I’ve written up my findings in a paper which just appeared on the arxiv:

A Covariant Version of Verlinde's Emergent Gravity
S. Hossenfelder
arXiv:1703.01415 [gr-qc]

It took some trying around, but I finally managed to guess a covariant Lagrangian that would produce the equations in Verlinde’s paper when one makes the same approximations. Without these approximations, the equations are fully compatible with general relativity. They are however – as so often in general relativity – hideously difficult to solve.

Making some simplifying assumptions allows one to at least find an approximate solution. It turns out however, that even if one makes the same approximations as in Verlinde’s paper, the equation one obtains is not exactly the same that he has – it has an additional integration constant.

My first impulse was to set that constant to zero, but upon closer inspection that didn’t make sense: The constant has to be determined by a boundary condition that ensures the gravitational field of a galaxy (or galaxy cluster) asymptotes to Friedmann-Robertson-Walker space filled with normal matter and a cosmological constant. Unfortunately, I haven’t been able to find the solution that one should get in the asymptotic limit, hence wasn’t able to fix the integration constant.

This means, importantly, that the data fits which assume the additional constant is zero do not actually constrain Verlinde’s model.

With the Lagrangian approach that I have tried, the interpretation of Verlinde’s model is very different – I dare to say far less outlandish. There’s an additional vector-field which permeates space-time and which interacts with normal matter. It’s a strange vector field both because it’s not – as the other vector-fields we know of – a gauge-boson, and has a different kinetic energy term. In addition, the kinetic term also appears in a way one doesn’t commonly have in particle physics but instead in condensed matter physics.

Interestingly, if you look at what this field would do if there was no other matter, it would behave exactly like a cosmological constant.

This, however, isn’t to say I’m sold on the idea. What I am missing is, most importantly, some clue that would tell me the additional field actually behaves like matter on cosmological scales, or at least sufficiently similar to reproduce other observables, like eg baryon acoustic oscillation. This should be possible to find out with the equations in my paper – if one manages to actually solve them.

Finding solutions to Einstein’s field equations is a specialized discipline and I’m not familiar with all the relevant techniques. I will admit that my primary method of solving the equations – to the big frustration of my reviewers – is to guess solutions. It works until it doesn’t. In the case of Friedmann-Robertson-Walker with two coupled fluids, one of which is the new vector field, it hasn’t worked. At least not so far. But the equations are in the paper and maybe someone else will be able to find a solution.

In summary, Verlinde’s emergent gravity has withstood the first-line bullshit test. Yes, it’s compatible with general relativity.