In 2010, an unconventional paper from a string theorist made headlines. Erik Verlinde proposed that gravity is an emergent phenomenon that could be understood as an entropic force caused by changes in the information associated with the positions of material bodies. An entropic force, which does not require a force field, is an effective macroscopic force that originates in a system with many degrees of freedom by the statistical tendency to increase its entropy.
Verlinde derives Newton’s law of gravity from classical statistical mechanics applying the holographic principle, i.e., that the description of a volume of space can be entirely encoded on a lower-dimensional boundary to that region. This boundary is not accessible to us living in the volume, it’s another description of the volume itself. He starts with a closed spherical boundary surface, with N number of used bits of binary information encoded on the boundary. This information of the volume is evenly distributed over the area, and the number of bits is proportional to it. Now he wants to find the origin of the entropic force, so he needs to know where the temperature comes from. If the total energy is divided into bits, the statistical equipartition rule relates the energy to the N number of degrees of freedom and the temperature. Then with E=mc^2 and massaging all these equations, he reaches Newton’s law of gravity, by which masses move in the volume maximizing entropy in the boundary.
He also derives Newton’s second law of inertia as another entropic force from the change of entropy associated with the information on the boundary and Unruh radiation associated with acceleration.
Verlinde’s claim that gravity is derived from first principles is not exempt of assumptions, the main one being evidently the holographic scenario. This goes back to Bekenstein’s idea of the entropy of a black hole. Another assumption is that only a finite number of degrees of freedom are associated with a given spatial volume, relating every bit of information to the existence of a Planck area.
Verlinde later argued in 2016 that his approach would give rise to a modification of Newtonian laws similar to that required to explain galaxy rotation curves without dark matter, which is Milgrom’s Modified Newtonian Dynamics. We explained MOND in our previous post. He also claimed that dark energy could be derived from his ideas, but I will focus on dark matter here. Notice how Verlinde directly compares his result with that of MOND instead of providing tests of his equations in real galaxy rotation curves to prove that such a modification really removes the need for dark matter in galaxies. This should tell you how strong MOND is in addressing dark matter in galaxies.
One of the key differences with MOND is that the interpolating function of Verlinde contains the radial derivative of the Newtonian gravitational acceleration. It is also not clear whether if it has an external field effect, something quantized inertia doesn’t have. The external field effect is considered crucial in MOND in order to fit observations.
And what was the response from the MOND community? There is an answer from Milgrom himself and Robert Sanders, in which they state that Verlinde’s dark matter explanation is, and I quote, “a formula – of rather limited applicability, and with no road to generalization in sight – for the effective gravitational field of a spherical, isolated, static baryonic system. [It] cannot be used to calculate the gravitational field inside, or near, disk galaxies, with their rich MOND phenomenology. Notably, it cannot predict their rotation curves, except asymptotically. It does not apply to the few- or many-body problem. [… It] cannot be applied to the internal dynamics of a system embedded in an external field, where MOND predicts important consequences.”. But most remarkably, they state that the way Verlinde arrives to MOND’s introduced constant, the acceleration scale constant a0 is not a consequence of established principles or theories, but it is due to the freedom of choosing numerical factors along his derivation.
There is also a study from Benoit Famaey and others which claim that emergent gravity’s modification á la MOND is inconsistent with solar system constraints due to its interpolating function, which was directly given by Verlinde. And another big name in MOND discussions, Stacy McGaugh, also pointed out that emergent gravity is different from MOND when you have an extended mass and not the idealized point mass case, and that the data from galaxy rotation curves is challenging for Verlinde’s model.
So no, it doesn’t look like Verlinde has derived the acceleration scale constant. There are several other candidates for a0, such as the size of a cosmic Rindler horizon, a property of the vacuum related to the cosmological constant, the relative acceleration of most matter in the observable universe due to cosmic expansion, of the inverse squared mass contributions of the large scale. More about the last one soon.
But as Milgrom and Sanders say in their paper, these problems of emergent gravity are only there to stress that much more work is needed to be done in this line of research. We don’t raise these problems to put an end to his idea, but to encourage further work on it, because we think its promising and at least, is a novel and original idea in theoretical physics.
References:
Erik Verlinde, On the Origin of Gravity and the Laws of Newton, 2010
Erik Verlinde, Emergent Gravity and the Dark Universe, 2016
Benoit Famaey and others, Emergent gravity in galaxies and in the Solar System, 2017