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.