Emergent Gravity in the Solar System

In a prior post I outlined Erik Verlinde’s recent proposal for Emergent Gravity that may obviate the need for dark matter.

Emergent gravity is a statistical, thermodynamic phenomenon that emerges from the underlying quantum entanglement of micro states found in dark energy and in ordinary matter. Most of the entropy is in the dark energy, but the presence of ordinary baryonic matter can displace entropy in its neighborhood and the dark energy exerts a restoring force that is an additional contribution to gravity.

Emergent gravity yields both an area entropy term that reproduces general relativity (and Newtonian dynamics) and a volume entropy term that provides extra gravity. The interesting point is that this is coupled to the cosmological parameters, basically the dark energy term which now dominates our de Sitter-like universe and which acts like a cosmological constant Λ.

In a paper that appeared in arxiv.org last month, a trio of astronomers Hees, Famaey and Bertone claim that emergent gravity fails by seven orders of magnitude in the solar system. They look at the advance of the perihelion for six planets out through Saturn and claim that Verlinde’s formula predicts perihelion advances seven orders of magnitude larger than should be seen.

No emergent gravity needed here. Image credit: NASA GSFC

But his formula does not apply in the solar system.

“..the authors claiming that they have ruled out the model by seven orders of magnitude using solar system data. But they seem not to have taken into account that the equation they are using does not apply on solar system scales. Their conclusion, therefore, is invalid.” – Sabine Hossenfelder, theoretical physicist (quantum gravity) Forbes blog 

Why is this the case? Verlinde makes 3 main assumptions: (1) a spherically symmetric, isolated system, (2) a system that is quasi-static, and (3) a de Sitter spacetime. Well, check for (1) and check for (2) in the case of the Solar System. However, the Solar System is manifestly not a dark energy-dominated de Sitter space.

It is overwhelmingly dominated by ordinary matter. In our Milky Way galaxy the average density of ordinary matter is some 45,000 times larger than the dark energy density (which corresponds to only about 4 protons per cubic meter). And in our Solar System it is concentrated in the Sun, but on average out to the orbit of Saturn is a whopping times the dark energy density.

The whole derivation of the Verlinde formula comes from looking at the incremental entropy (contained in the dark energy) that is displaced by ordinary matter. Well with over 17 orders of magnitude more energy density, one can be assured that all of the dark energy entropy was long ago displaced within the Solar System, and one is well outside of the domain of Verlinde’s formula, which only becomes relevant when acceleration drops near to or below  c * H. The Verlinde acceleration parameter takes the value of  centimeters/second/second for the observed value of the Hubble parameter. The Newtonian acceleration at Saturn is .006 centimeters/second/second or 50,000 times larger.

The conditions where dark energy is being displaced only occur when the gravity has dropped to much smaller values; his approximation is not simply a second order term that can be applied in a domain where dark energy is of no consequence.

There is no entropy left to displace, and thus the Verlinde formula is irrelevant at the orbit of Saturn, or at the orbit of Pluto, for that matter. The authors have not disproven Verlinde’s proposal for emergent gravity.