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.
