What is "entropic gravity"?

The story (and math) behind Erik Verlinde's finding...

This is a story about Erik Verlinde. However, to talk about him and how he changed how some view the very nature of what gravity is, we need to set the stage…

In 1995, Ted Jacobson showed you could derive Einstein’s field equations from thermodynamics. What this means concretely is if you start with dQ = T dS applied to local Rindler horizons, then assuming the entropy-area relation…

\(S = A/4l_P^2 \ \ \text{gives you} \ \ G_{\mu\nu} = 8\pi T_{\mu\nu}\)

…The result was extremely interesting and unexpected (at least to me), but didn’t change the field of physics. In fact, I spoke to Ted Jacobson below about this exact topic and it went viral:

Fifteen years later, string theorist Erik Verlinde tried to push this relationship even further. The reaction was… let’s say, complicated. Like your ex. But with tensors.

I’m writing these as some public notes in my preparation for my interview with Verlinde. If you click here you’ll see many of my other writings on physics / consciousness / philosophy / reality that I often write to sort my thoughts out as preparing for guests often takes months of reading papers, talking with colleagues, and testing myself. Peer into that process here. Errors are mine.

So in 2010, Verlinde published a paper On the Origin of Gravity and the Laws of Newton claiming gravity isn’t a fundamental interaction / force / entity at all. It’s emergent. Much like temperature’s emergent and not fundamental. Air pressure also isn’t fundamental but comes from collisions at an emergent level, so this isn’t a foreign concept. Some people mistake the idea that if something isn’t fundamental, it doesn’t mean it’s “not real.”

Verlinde says that gravitational attraction is just entropy gradients on holographic surfaces. Literally the same as. Not “is analogous to.”

Here’s the core logic. Take the holographic principle seriously: information in a volume is encoded on its boundary, with one bit per Planck area. Now consider a particle of mass m approaching a spatial “holographic screen” enclosing mass M. When I say “spatial” I mean we’re not in 4D spacetime but rather 3D space, thus a spatial screen enclosing something is a 2D shell of a sphere (a “closed” sphere, technically). Importantly, the screen is not a physical object but the M inside it is, and the m moving toward the screen (from the outside of it) is.

Verlinde says there’s going to be an entropy change… Here’s where I was confused because if the screen is supposed to encode everything inside, then why the heck does it matter that a little mass m moves toward it from the outside, if nothing about the inside M is changing?

Verlinde’s implicit answer is that the screen encodes everything inside holographically. As m approaches, the screen’s bits must rearrange (or whatever) to eventually incorporate m’s information. The entropy change is the screen preparing (or whatever) to absorb new data.

Anyhow, Verlinde postulates the entropy change:

\( \Delta S = 2\pi k_B \frac{mc}{\hbar} \Delta x \)

Combine this with the temperature seen by an accelerating observer:

\( T = \frac{\hbar a}{2\pi k_B c} \)

Use the entropic force relation FΔx = TΔS, and you get… Newton’s law:

\( F = \frac{GmM}{r^2} \)

The derivation is super short. Lee Smolin called it “remarkable in that we all felt so stupid for not having seen it before… The mathematics involved is just high school algebra.” (Wikipedia does the derivation in half a page)

But here’s where it gets uncomfortable. Jacobson himself (whose work Verlinde explicitly builds on) said he “couldn’t make sense of it.” Thanu Padmanabhan, another pioneer in gravity‑thermodynamics connections, said he could “see little difference” between the papers and that “the new element of an entropic force lacked mathematical rigor.” Padmanabhan wrote, “I doubt whether these ideas will stand the test of time.” At a workshop in Texas, Raphael Bousso of Berkeley led a discussion on the paper, saying “the end result was that everyone else didn’t understand it either, including people who initially thought that did make some sense to them.”

Where does the factor of 2π in Verlinde’s entropy formula come from? Jacobson’s derivation uses local Rindler horizons (mathematically well-defined objects in general relativity). Verlinde uses “holographic screens,” whose physical status is unclear. Are they observer‑dependent? Fundamental? Both?

I’ve been researching for many weeks now, so forgive for random name / year dropping but it helps paint the picture… Zhi‑Wei Wang and Samuel Braunstein in 2018 showed that, yes, stretched horizons near black holes do obey thermodynamic‑like first laws, however ordinary spacetime surfaces (likethe holographic screens central to Verlinde’s program) generally don’t.

In other words, thermodynamic first law for surfaces requires something like:

\(dE=TdS+ \text{work terms}\)

As far as I can tell, the difference may be interpretative. Jacobson’s result is that Einstein’s equations have thermodynamic structure. It’s compatible with gravity being fundamental (you can still quantize it, still treat it as a real force). Jacobson himself said, now jumping forward to 2003, that “condensed matter physics abounds with examples of collective modes that become meaningless at short length scales, and which are nevertheless accurately treated as quantum fields within the appropriate domain.” The thermodynamic structure doesn’t necessarily mean gravity is merely emergent.

Verlinde goes a bit further. “For me gravity doesn’t exist,” he told the New York Times.

The above is a much stronger claim.

Actually, Verlinde himself was cautious about overclaiming: “This is not the basis of a theory,” he said. “I don’t pretend this to be a theory. People should read the words I am saying opposed to the details of equations.”

However, in 2016, it seemed like Verlinde doubled down. His paper Emergent Gravity and the Dark Universe extends this thermodynamic gravity idea to de Sitter space which better models an accelerating universe.

By the way, I was going to say “better models our accelerating universe,” but my recent interview with Subir Sarkar has appreciably changed my credence in our universe actually accelerating to the degree we think.

Anyhow, in Verlinde’s picture, there’s entropy associated with the cosmological horizon and it’s distributed throughout the volume, contributing to a sort of “volume law” term that becomes important at large scales.

Interestingly, when you add matter to this picture, you’re removing entropy from the dark energy and localizing it. The universe “remembers” this (it creates an elastic response).

Verlinde demonstrates how this elastic response will produce an additional gravitational force but one that becomes pronounced only at galactic scales. One that looks exactly like what we attribute to dark matter.

If right now you’re scratching your head wondering “Mate, what are you on about elastic responses when, first, you were talking about a holographic screen? It’s like, mate, you’re all of a sudden talking about viscosity and conductivity of an imaginary surface.” Well, what’s odd is that (other than me turning British for a moment), when I start researching black‑hole thermodynamics, people who think about certain surfaces around certain objects and then start studying the entropy of those enclosures (even if they’re non‑physical) actually inherit properties that we would ordinarily think of as belonging to ordinary matter (like a balloon‑stretched surface, such as viscosity, elasticity, conductivity, tension). The reasons for this are beyond the scope of this particular Substack article, but of course you can feel free to subscribe, as I’ll talk about these more and more later.

Note that I’m not trying to imply that Verlinde’s surface (the “holographic screen”) is one such surface that inherits these transport properties (viscosity, tension, etc.). This is what’s under question by other physicists!

Anyhow, Verlinde’s prediction is specific. The apparent dark matter contribution relates the baryonic gravitational acceleration g_B to the Hubble constant H_0:

\( g_D \approx \sqrt{\frac{g_B \cdot cH_0}{6}} \)

This resembles something called Modified Newtonian Dynamics phenomenologically but derives from different principles. Verlinde predicts that dark matter effects depend on the cosmological horizon. MOND but different. Physics remix culture.

This means that the expansion rate of the universe directly affects galaxy dynamics. This is super interesting. Dirac also noticed that some constants which appear local seem to have something to do with the entire universe’s structure. Verlinde’s results aren’t the same in text but they are the same in spirit.

Some observations support Verlinde. For instance, Margot Brouwer measured gravitational lensing around 33,000+ galaxies and found the predictions from emergent gravity were “in good agreement” with observations. The upside is that it does this matching without free parameters. People love everything free, except their parameters.

The counter‑side to Brouwer’s finding comes from Kris Pardo, who in 2017 showed that Verlinde’s predictions are inconsistent with dwarf‑galaxy rotation curves. Also, the Bullet Cluster (which is always the dark‑matterists’ “but, actually…” UNO card) remains a tension where dark matter and baryonic matter appear spatially separated after a collision.

So far my assessment is that Verlinde has proposed something attractive and freaky. Much like myself.

The idea that gravity comes from information dynamics obviously has some connection to other approaches like AdS/CFT and tensor network models of spacetime. His 2016 predictions are near-term falsifiable, which is more than you can say for most quantum gravity proposals / pictures / programs (whatever you want to call them).

But, as far as I can tell, his microscopic degrees of freedom are never specified. And the person whose work most directly preceded Verlinde’s (Jacobson) doesn’t exactly buy it. It doesn’t mean Jacobson is sending it back to the kitchen for reheating, but more like it’s a bit too raw for his taste. The espresso needs a bit of milk to increase its entropy.

Verlinde may just retort that steak tartare’s an acquired taste. And that real physicists like their coffee like they like their spacetime holes… black.

—Curt Jaimungal