PBS Space Time

We're told there are four fundamental forces - the strong and weak nuclear forces, electromagnetism, and gravity.

Except maybe gravity is actually no more fundamental than the force of a stretched elastic band.

Maybe gravity is just an entropic byproductan emergent effect of the universes tendency to disorder.

So if you allow that entropy to keep you in your seat for a bit, Ill tell you all about it.

Gravity is the odd one out among the fundamental forces.

Its enormously weaker than the other three and it's also not a quantum forceat least, it's not in general relativity, our best current description of gravity.

And gravity has steadfastly resisted our century-long efforts to quantize it; to unify it with the other fundamental forces.

But what if thats because gravity isnt quantum?

In fact, what if its because gravity isnt even fundamental?

There are a number of proposals along this line, including the buzzy recent work on by Jonathan Oppenheim.

But were going to have to come back to that, because theres an idea that youve been asking us to cover for years now.

Thats the emergent gravity of Dutch physicist Erik Verlinde, who tells us that its not a fundamental force or the curvature of spacetime thats keeping you in your chair right now, but rather the rise of entropy on the boundary of the universe.

Today were going to lay out the basics of Verlindes entropic gravity as it was published back in 2010.

Well follow up with an episode on the evidence and the criticism and what the idea has to say about dark matter and dark energy.

This follows directly from our last episode where we explored how space can emerge as an inward projection from its infinitely distant boundary through the holographic principle.

A lot of what we talk about today will draw from that episode, so it might not be a bad idea to check it out.

But let me recap anyway to emphasize the most important stuff.

Im also going to sprinkle in a bit of math that well use to construct Verlindes idea.

But if the math isnt your cup of tea, relaxIll give you everything you need to follow the story with the normal human words.

So the holographic principle grew out of black hole thermodynamicsfrom the fact that the information that can fit inside a black hole is proportional to its surface area.

That information is hidden from the outside world, so this is also the black holes entropy, given by the Bekenstein-Hawking formula.

It turns out the same information limit applies to all space, so that its in principle possible to encode the contents of a universe on its boundary.

In fact, according to the holographic principle, the particles and fields and gravity and laws of physics that govern our universe also play out in lock step on its infinitely distant but infinitely compacted boundary.

On that lower-dimensional, gravity-free boundary, a very different set of laws encode everything that happens on the interior.

The boundary encodes what we call the bulk, and perhaps vice versa..

The only concrete mechanism for this thats currently known is Juan Maldacenas AdS/CFT correspondence, which unfortunately doesnt quite apply to our universe.

Still, its a strong lead that theres a version which does.

In AdS/CFT the interior spacethe bulkcontains a string theory, and the gravity emerges within that string theory.

But we dont need string theory to see how gravity might be a natural prediction of the holographic principle.

Last episode I showed you a scheme by which this extra dimension can be encoded in the scale of structures on the boundary, with larger structures on the boundary manifesting as structures closer to the center of the bulk.

Today were going to explore one idea for how gravity can also emerge within that space as a statistical side effect of the interplay of whatever it is thats happening out there on the boundary.

This is the entropic gravity of Eric Verlinde.

Now entropic gravity is by no means broadly accepted, but it is taken seriously by reasonable physicists.

After all, there is a fascinating and still mysterious connection between gravity and entropy, as Bekenstein and Hawking discovered.

And theres a fascinating neatness to Verlindes idea that seems like its telling us something, even if it isnt the whole picture, and maybe even if it's wrong.

To understand how gravity might arise entropically, lets think about a less out-there system we'll use the same thought experiment that Verlindeuses in his entropic gravity paper.

Imagine a long molecule that is free to move and fold in any direction.

We place it in a box of constant temperature, with one end fixed to the wall of the box.

If we ignore any possible external forces acting on the molecule, we might expect it to just curl up.

This is because, of all the ways the molecule could move, its far more likely to end up in a coiled configuration than remain straight.

This is one way to think about entropy.

The molecule will almost always take a more probable - higher entropy - configuration of being curled up because there are way more configurationsor what we call microstatesin which the molecule is curled compared to it being straight.

Being in one of the many random coiled states is a higher-entropy configuration than the very few straight states.

If we straighten the molecule we have to exert a force and expend some energy to do so.

If we let go itll curl up again because theres an effective force pulling it back.

I should add that there's nothing magical going on here.

The molecule shares its temperature with the air in the box.

Its atoms have randomly oriented vibrations and are being randomly smacked by air molecules, and these will pull and push the molecule towards random configurations, which are overwhelmingly the coiled ones.

Theres a simple relationship between the entropic force required to pull a molecule or that the molecule exerts on you: This is really saying the amount of energythe force times the distance pulledis equal to the temperature of the system times the change in entropy after that little motion happens.

Or that the force is equal to the temperature times the entropy gradient.

We call this an entropic force.

This is exactly what you experience when you pull on an elastic band.

In fact, any time you have the movement of matter in service of increasing entropy, theres an entropic force.

For example, if you force all the air in a room into a box, then release it, itll rush to fill the room and generate an enormous entropic force in doing so.

The proposal of Erik Verlinde is that gravity is also an entropic force.

At first glance that seems odd.

Gravity is a property of spacetime itselfeven empty spacetimeso what exactly is pushing or pulling in empty space?

Verlinde constructs his argument in the context of a holographic universe, in which at least one dimension of space is also emergent.

He argues that the entropy of the stuff on the holographic boundary must increase, and that rising entropy manifests as gravity in the interior.

To build up this idea were going to need to keep in mind dual picturessomethings happening on the boundary and somethings happening in the bulk and they encode the same thing, even if they sit very differently in our mental imagery.

Well flip between them as convenient, and even merge them slightly.

But remember that these are two distinct ways of describing the same system.

If all that gives you a headache then youre not alone.

OK, so, somewhere in the bulk of a holographic universe we have a star with some mass.

Lets see if we can figure out the gravitational force produced by the star without ever using any theory of gravityjust by visiting the boundary.

To derive such a law we want to know show much gravitational force is felt at different distances, so lets imagine a series of spherical surfaces around the star.

If the star is massive and compact enough then one of these surfaces would be an event horizon and wed have a black hole.

Then, the entropy of that surface would be the Bekenstein-Hawking entropybasically, it would represent the amount of information of everything that previously fell through that surface.

But even if this surface isnt an event horizon we can give it an entropy.

Its also the entropy of everything interior to the surface.

But that makes the most sense if we zip out to the boundary.

If this is a holographic universe, then the particles onthis surface map to the boundary.

In fact, the particles on surfaces of all sizes map to the same boundary and play out together, overlapping in this lower dimensional space.

We want the entropy of this one surface with respect to someone outside that surfacethat translates to how much information is hidden within the surface.

Lets start with the holographic boundary from which our bulk universe emerges.

Imagine that it emerges from the outside-in.

Thats not really the case, but it helps us represent this visually.

Were going to partially emerge our universe down to this one surface so we can depict a special subset of the boundary as actually lying on this surface.

This is the part of the boundary that corresponds to everything below this surface.

So from now on, when I say boundary Ill mean the component of the holographic boundary corresponding to the region of the bulk enclosed by this surface.

OK, hold on for a little bit of math.

We know that this surface contains a mass, so we can say that it also contains energy by Einsteins E=mc^2.

Thats the energy of the interior, but also has to be the energy of the corresponding holographic boundary.

We can also give that boundary a temperature, assuming the stuff on the boundary is in thermal equilibrium so that the energy is evenly spread over all possible states.

And that N thing is just the number of possible states on the boundary and the total number of arrangements of particles inside the volume that would give you particular values of energy, mass, and temperature.

But that also corresponds to the amount of hidden information within surfaceits entropy.

We know the maximum value for thisits the Bekenstein-Hawking entropy, so the number of Planck-length squares over that surface.

For our surface lets just assume that the entropy and N are still proportional to the surface area.

But presumably smaller than a black hole.

OK, one more step.

We want the gravitational force, so we need another particle to feel that force.

Lets add a tiny mass and move it close to surface from the above emerged part of space.

When that happens, the entropy of the boundary increases because the information from that object is lost from the external region.

The boundary gains the same amount of entropy as dropping something into a black hole event horizon.

This equation is just saying that the surface gains minimal entropy, equivalent roughly to a bit, when the particle merges with the surface, which we define as it getting within its own quantum wavelengthin this case the Compton wavelengthof the surface.

But just like we saw with the coiling molecule, this tiny increase in entropy should have a corresponding entropic force.

Whatever crazy interactions are happening on the boundary, they are statistically inclined to bring our particle closer to this surface because that motion increases entropy.

If we bring everything together - the \Delta S/\Delta x from dropping a particle through the horizon, and the temperature from the overall entropy of the surface, all the h-bars and the cs and the kds cancel out and we replace the area of thesphere with 4 times pi times its radius we see that the algebra shakes down to Newtons universal law of gravitation, within some constantand if that constant is one because we got our surface entropy formula right then we have Newton's exact equation.

Even though that last step is dubious, we sort of just derived Newtonian gravity with arguments that are entirely thermodynamic.

This is basically saying that if objects in the bulk move in such a way as to maximize entropy on the boundary, then that motion means falling towards other masses in the bulk.

Remember that Hawking and Bekenstein used gravitational theory and quantum mechanics to get black hole thermodynamics, but entropic gravity turns this on its headit starts with thermodynamics and finds that gravity falls out.

But lets not get ahead of ourselves.

Firstly, this is just Newtonian gravitycan entropic gravity reproduce Einsteins general relativity?

Well, in the 2010 paper, Verlinde argues that yes it can, although the derivation is a bit much for this episode.

In 2016, he published another paper that argued that dark matter can also be explained by this idea, and that its connected to dark energyalthough this requires some extra assumptions.

And speaking of assumptionsthe validity of this idea rests on the validity of the founding assumptions.

Not least of those is the requirement of a holographic dual to the gravitational universeand until we can find a version of AdS/CFT that works for our universe this feels like a big if.

The debate over entropically emergent gravity is real, and its taken seriously by serious physicists.

That means its worth doing another episode to pick this apart.

In the meantime, this has been pretty hard work, so why not have a little lie down.

I mean, can we really be expected to fight the rising entropy at the infinite boundary of our holographic space time.