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   byproduct—an emergent effect of the  universe’s tendency to disorder.

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

Gravity is the odd one out among the fundamental  forces.

It’s enormously weaker than the other   three and it's also not a quantum force—at  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 that’s   because gravity isn’t quantum?

In fact, what  if it’s because gravity isn’t even fundamental?

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

But we’re going to have to come back  to that, because there’s an idea that you’ve   been asking us to cover for years now.

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

Today we’re going to lay out the basics  of Verlinde’s entropic gravity as it was   published back in 2010.

We’ll 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.

I’m also going to sprinkle   in a bit of math that we’ll use to construct  Verlinde’s idea.

But if the math isn’t your cup   of tea, relax—I’ll give you everything you need  to follow the story with the normal human words.

So the holographic principle grew out  of black hole thermodynamics—from 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 hole’s entropy,  given by the Bekenstein-Hawking formula.

It turns out the same information limit applies to  all space, so that it’s 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 that’s  currently known is Juan Maldacena’s AdS/CFT   correspondence, which unfortunately doesn’t quite  apply to our universe.

Still, it’s a strong lead   that there’s a version which does.

In AdS/CFT the  interior space—the bulk—contains a string theory,   and the gravity emerges within that string theory.

But we don’t 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 we’re 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 that’s 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 there’s a  fascinating neatness to Verlinde’s idea that seems   like it’s telling us something, even if it isn’t  the whole picture, and maybe even if it's wrong.

To understand how gravity might arise  entropically, let’s think about a less out-there   system— we'll use the same thought experiment that Verlinde uses 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, it’s 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 configurations—or what we call  microstates—in 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 it’ll curl up again  because there’s 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.

There’s 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 energy—the  force times the distance pulled—is 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,   there’s an entropic force.

For example, if  you force all the air in a room into a box,   then release it, it’ll 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 itself—even empty spacetime—so   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 we’re going to need to keep  in mind dual pictures—something’s happening on   the boundary and something’s happening in  the bulk and they encode the same thing,   even if they sit very differently in our mental  imagery.

We’ll 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 you’re not alone.

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

Let’s   see if we can figure out the gravitational  force produced by the star without ever using   any theory of gravity—just by visiting  the boundary.

To derive such a law we   want to know show much gravitational  force is felt at different distances,   so let’s 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 we’d have  a black hole.

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

But even if this surface isn’t an event horizon we  can give it an entropy.

It’s 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 on this 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 surface—that translates to how much  information is hidden within the surface.

Let’s   start with the holographic boundary from which  our bulk universe emerges.

Imagine that it emerges   from the outside-in.

That’s not really the case,  but it helps us represent this visually.

We’re   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” I’ll 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 Einstein’s E=mc^2.

That’s 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 surface—its entropy.

We know the maximum value for this—it’s   the Bekenstein-Hawking entropy, so the number  of Planck-length squares over that surface.

For our surface let’s 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.

Let’s 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 wavelength—in  this case the Compton wavelength—of 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 the sphere with 4 times pi times its radius … we   see that the algebra shakes down to … Newton’s  universal law of gravitation, within some   constant—and 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   head—it starts with thermodynamics  and finds that gravity falls out.

But let’s not get ahead of ourselves.

Firstly,  this is just Newtonian gravity—can entropic   gravity reproduce Einstein’s 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 it’s connected to dark energy—although  this requires some extra assumptions.

And speaking   of assumptions—the 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 universe—and 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 it’s taken seriously by serious   physicists.

That means it’s 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.