Before we get started, just letting you know we have some limited time only Solar Eclipse
merch for sale at the merch store. I’ll tell you know more at the end of the episode.
Space seems pretty fundamental. To build a universe, surely you need something to build it on or
in. Many, maybe most physicists now think that the fabric of space emerges from something
deeper. And perhaps the most existentially disturbing such proposal is that our 3-D universe
is just the inward projection of an infinitely distant boundary. A hologram, or sorts. Let’s
see how that can actually work, and what the holographic principle really says about the
“realness” of this universe.
You could argue that the job of physics is to drill down and find the most fundamental
elements of reality. The most basic building blocks—the “ontological primitives”--from
which everything else emerges. A little while ago we asked whether the fabric of space deserved
that status. Isaac Newton certainly thought so, but many pretty smart people have wondered
if space might actually be an emergent phenomenon.
We’ve even seen some examples of this in past episodes—from the early ideas of Leibnitz
to modern loop quantum gravity. We’ve also seen how at least one dimension of space can
emerge as the inward projection from its infinitely distant boundary. This is the holographic
principle—which we spent many episodes building towards back in the day—culminating in a
pretty intense coverage of this wild idea. Well, today we’re travelling back to the
holographic boundary to try to understand better how the fabric of space can emerge
this way. This will set us up to do a couple of things in upcoming episodes: we’ll be
able to understand how gravity can emerge from entropic forces on a holographic boundary,
and how the fabric of space may be knit together by the quantum entanglement of entities that
themselves don’t exist in our familiar spacetime.
Feel free to watch the old holographic principle stuff later if you haven’t. I’m going
to summarize the core stuff for your convenience. So the whole argument started in a surprising
place—with Steven Hawking and Jacob Bekenstein thinking about the thermodynamics of black
holes. Normally we think of thermodynamics as the study of the statistical behavior of
many particles. It’s the study of emergent properties like temperature and pressure of,
say, a room full of air molecules. These are properties that the individual particles do
not have.
Another key thermodynamic property is entropy. We can think of this as a measure of the amount
of hidden information in a system—information about the stuff that makes up the system that
isn’t reflected in the observable, macroscopic properties. This ro-om full of air molecules
has a high entropy because all of the particles’ locations and speeds are hidden behind just
a few thermodynamic properties that I can sense like the temperature and pressure.
We wouldn’t necessarily expect an individual black hole to have thermodynamic properties.
After all, black holes are practically particle-like. They are the result of the ultimate gravitational
collapse of matter and are surrounded b y an event horizon which prevents anything exiting
the black hole from the inside. That includes any information about what exactly collapsed
to form it. Jacob Bekenstein realized that black holes viewed from the exterior possess
three properties—mass, charge, and spin—almost like an elementary particle. But it’s that
exact fact that gets us to black hole thermodynamics, and to emergent space.
The difference between a black hole and an elementary particle is that the former is
made by the extreme clumping together of the latter. That means we can define the entropy
of a black hole—and the number is huge because black holes hide almost all of the information
of the stuff that fell in. Bekenstein and Hawking figured out the entropy of black holes.
They found that there’s around one bit of hidden information per 4 square Planck length units
on the surface.
This simple formula was really surprising—and that surprise led physicists down the path
to the holographic principle. See it’s actually quite weird that the entropy should be proportional
to the surface area. Intuitively we might expect the amount of information that can
be hidden in any region of space is proportional to its volume—like, one bit per tiny block
in that volume, rather than a bit per tiny square on its surface. In fact, black holes
represent the maximum information density of any volume of space—this is the Bekenstein
bound. And that applies to the whole universe—the information about every particle in its 3-D
volume can, in principle, be encoded on its 2-D surface.
But it’s one thing to say that all the information describing a universe fits on its surface,
but the holographic principle goes way further. It suggests that we can think of our universe
as really playing out on that surface. More precisely, that surface is its own lower-dimensional
spacetime with its own particles and fields and a different set of laws of physics. But
encoded in the patterns that play out on that lower-dimensional spacetime is an entire universe
one dimension higher. It’s as though the volume of the universe—what we call the
“bulk”--emerges from the dynamics on its surface—the boundary.
The journey from black hole entropy to the Holographic Principle was led by Gerard t’Hooft
and Leonard Susskind. But it was Juan Maldacena who gave us the first concrete example of
how this could work in detail with his AdS/CFT correspondence. I talked in a lot of detail
about the nature of the AdS/CFT proposal in our previous episode—including the fact
AdS/CFT is valid for a negatively curved universe with a negative cosmological constant—so-called
anti-de Sitter or AdS space—quite different to our own de Sitter universe with its positive
cosmological constant.
But AdS/CFT is still a powerful tool for exploring holographic emergence. And we’re going to
use just one part of this proposal—the CFT part. CFT is for conformal field theory. It’s
the laws of physics that play out on the boundary—that encode the universe in the bulk. A
conformal field theory is one in which the laws governing the field theory don’t care
about the size-scale of the objects in the theory. So you could have little wiggles in
the field that interact with each other in a particular way, or gigantic wiggles that
interact in exactly the same way—as long as the wiggles are the same shape.
This conformal-ness gives us a way for the bulk to emerge from the boundary. Imagine
the boundary consists of many Planck-length squares, each able to hold a single bit—0
or 1—empty or full. You can have small-scale patterns of these bits, where each square
is independent. Or you could have larger-scale patterns that average over many squares—they
are, as we say, course-grained. Because it’s a conformal field theory these large patterns
experience the same physics as the smaller patterns. And these patterns could appear
on many scales, all behaving basically the same.
And let’s add another fact—a system of patterns on one scale only interacts with
patterns of the same scale. They don’t interact with patterns on larger or smaller scales.
It’s like having multiple sizes of ripples on a pond that just wash over or under each
other.
But there is something that’s different for the course-grained patterns vs the fine patterns —the amount
of information needed to represent them over the surface is lower. You’re effectively
lumping the Planck-length squares together. Let’s say we’re looking at just the patterns
that have twice the scale-length of the smallest-scale patterns. We can represent them using averaged
square sizes that each contain 4 of our original Planck-length squares. We can also imagine
a surface containing only this pattern scale—it has ¼ of the number of effective squares
as our original surface. That means we can also think of it as having an effective radius
of ½ that of the original.
The larger the pattern size, the more Planck squares you fit into the effective pixel scale of that pattern size, and that means
the smaller the effective radius of that scale will be.
So here's another way to represent this 2-D surface with its many pattern scales—and
as a series of nested spheres, each sphere representing one of those scales.
The spheres near the center represent the largest scales on the boundary where the patterns
are so big they have relatively few effective pixels to play with. That’s what makes the
inner spheres effectively “small”, while the outer spheres are occupied by the boundary
patterns that are finer or smaller in scale—we need higher resolution to represent them, so these spheres
are made of many effective pixels, and so are effectively large.
But because the boundary theory is a conformal field theory, the physics on each sphere is
exactly the same. And if the separation between these spheres is small enough—for example,
if it's also a Planck length—then we’ve produced a 3-D space out of a 2-D one.
This is a fairly cartoonish sketch of how a non-spatial degree of freedom in a lower
dimensional space can lead to an emergent space with one more dimension.
The real picture is likely to be way more complicated—for example, the emergence may
not come from simple scale invariance but rather from quantum entanglement across many
scales—and that’s something we’ll come back to another time.
But the story I just told is still a useful picture to get the basic gist of what a holographically
emergent space looks like. Because this new degree of freedom translates to a radial direction,
that emergent space is in a sense “inside” the original space. That’s why we can talk
about the higher dimensional space—the bulk—emerging from its boundary. Although really the bulk
and boundary are two distinct spacetimes. It may be that each completely contains all
of the information of the other and the two evolve over time in lock-step with each other.
But they evolve by very different laws of physics. There’s a boundary theory and a
bulk theory and on the surface they look like they shouldn’t predict the same thing, yet
they do; they both predict the same universe.
This is actually a recurring thing in physics. There are a number of places where two wildly
different mathematical descriptions describe the same system or phenomenon—and often
the different descriptions seem mutually incompatible. These are called dualities, and to me they’re
one of the most mysterious features of reality.
Now for a true duality, there is no preferred side—neither of the two theories are more
fundamental than the other. For example, in quantum mechanics you can represent a particle
as a sum of waves in momentum space OR in position space. Those are equally true, so
this is a true duality. So is the holographic duality a true one? If so then the boundary
and the bulk are equivalently real—it wouldn’t be quite right to say the interior emerges
from the boundary. Instead there’s a so-called ontological democracy. Perhaps both emerge
from something more fundamental, but neither is primary to the other.
On the other hand, there are also such things as approximate dualities, in which one side
really is more fundamental—one system “causes” the other system in a meaningful sense. Now if
the holographic duality is like this then it’s much more reasonable to describe one
sidie as emergent—and the emergent side in this case is most likely the bulk—but
only because in the most concrete version of a holographic duality—AdS/CFT—the boundary
theory is currently better defined than the bulk theory.
So which is the case? Well even if our universe possesses some sort of holographic duality,
we don’t know whether it’d be a true or approximate duality. In fact, we don’t even
know which type the AdS/CFT duality is. That’s something we’re going to have to spend a
lot more time on—and I mean this show and the theoretical physicists.
Ok, so we’re up to speed on the holographic emergence of space. In upcoming episodes we’ll
see how gravity can enter the picture—first through the entropic gravity proposal of Eric
Verlinde. We’ll also see the modern thinking for the emergence of space through quantum
entanglement, and what the prospects are for finding a concrete holographic description
for our universe. As a teaser, the holographic boundary may not even be at infinite distance,
but rather in the far future. And we’ll ultimately get to the big question: is our
universe really emergent from a more fundamental, lower dimensional reality? Or are we just
one side of the coin of a true duality—the boundary and the bulk—each an equally real
and mutually emergent spacetime.
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