We live in a universe with 3 dimensions of
space and one of time. Up, down, left, right, forward, back, past,
future. 3+1 dimensions. Or so our primitive Pleistocene-evolved brains
find it useful to believe. And we cling to this intuition, even as physics
shows us that this view of reality may be only a very narrow perception. One of the most startling possibilities is
that our 3+1 dimensional universe may better described as resulting from a spacetime one
dimension lower – like a hologram projected from a surface infinitely far away. The holographic principle emerged from many
subtle clues – clues discovered over decades of theoretical exploration of the universe. Over the past several months on Space Time,
we’ve seen those close clues, and we’ve built a the foundations needed to glimpse
the true meaning of the holographic principle. We’ve moved from quantum field theory to
black hole thermodynamics to string theory. We’ve made a background playlist if you
want to start from scratch, and I especially recommend catching last week’s episode. But this is tough material, so let’s do
a review. The story started with black holes, and with
Jacob Bekenstein, who derived an equation to describe their entropy. A black hole’s entropy represents the amount
of quantum information of everything that ever fell into it. This Bekenstein bound represents maximum possible
entropy-slash-information of any volume of space. Oddly, that maximum in is proportional to
the surface area of that space, not its volume. That was surprising – surely the information
in a volume of space depends on the volume – like, 1 bit per infinitesimal voxel – not
one bit per pixel on its surface. Steven Hawking confirmed the Bekenstein bound
by calculating the amount of information leaked by a black hole as it evaporated in Hawking
radiation. His discovery of Hawking radiation led to
the black hole information paradox, because this radiation was expected to erase the quantum
information of everything that fell into the black hole. But destroying quantum information would break
the foundations of quantum mechanics. Hence the paradox. This conundrum inspired Gerard t’ Hooft
to show that the information of all material that fell into the black hole could be imprinted
on that outgoing Hawking radiation. And while it’s waiting to be radiated, that
information should be encoded on the event horizon. Nice solution, but new paradox. Things that fall into a black hole do actually
experience crossing the event horizon and being inside the black hole. So the interior of the black hole has a dual
existence. From the point of view of outside observers,
its contents is smeared into 2-D on the surface, but from the PoV of anyone falling in they
are definitely inside the black hole, plummeting to their doom in full 3-D glory. This is the first glimpse of a holographic
spacetime: a 2-D surface that encodes the properties of the 3-D interior. ‘t Hooft along with Leonard Susskind extrapolated
this to proposed that not only is any surface sufficient to fully describe the locations
of all particles in its volume, but also the full machinery of the volume can exist on
its surface – all degrees of freedom needed to describe the behavior of everything within. But it’s one thing for this stuff to FIT
on the surface – but how is it actually encoded? How does the 2-D surface store information
about that extra dimension? And how do interactions on that surface correspond
to interactions in the volume? Leonard Susskind laid out the first steps
towards how this could be achieved using string theory, but ultimately it was Juan Maldacena
who figured out a concrete string theoretic realization of the holographic principle with
AdS/CFT correspondence. But I’m getting ahead of myself. Let’s ignore string theory for the moment
and just think about how to create an extra dimension. Let’s say we start with a plane – a flat,
2-D spacetime. Now grid it up into a lattice of cells and
make a set of rules about how those cells interact with each other. Those rules are a field theory, the lattice
itself is the field, and the cells are some elementary component of the field. But perhaps not the smallest possible component. For now let’s just say the size of those
cells depends on how we’re looking at the grid. For example, the resolution of our microscope
or the power of our particle collider. Probably the rules between cells – the field
theory – depends on this scale. Focus on a very small scale and we see a very
fine grid that interacts according to one set of rules, zoom out and we see a courser
grid – with cells that are the average of smaller cells, and which presumably interact
via different rules. Or so you’d think, but we’re going to
add something weird. We’re going to say our field theory is scale-invariant. We’ll say the rules are the same for the
small pixels or big pixels. We see scale invariance in fractal patterns,
where the rules defining the structures repeat to infinitely large or small scales. We also see it in string theory, which I’ll
come back to. A field theory with this property is called
a conformal field theory. In the last episode I said that a conformal
transformation is one that leaves all internal angles unchanged. A conformal field theory has this property. For example, you can change the scale at every
point on the grid separately and not change the internal angles or the shapes of the pixels,
which corresponds to not changing the rules of interaction. By making this a conformal field theory we’ve
added a symmetry –invariance under local changes in scale – also known as Weyl invariance. This adds a degree of freedom everywhere – like
a new infinite number line at each of the 2-D grid points. Objects on the 2-D grid also have values on
that number line - they exist at a certain scale. If objects at different scales don’t tend
to interact with each other then this new degree of freedom behaves just like another
dimension. Our 2-D grid behaves like a 3-D volume, and
we can treat it like one – mathematically. You might say it’s not a real 3-D volume
because the 3rd dimension is fake. But is it? What is a dimension but a number-line of possible
values which a) exists alongside the other dimensions but is independent of them; b)
over which the rules of physics stay the same, and c) imposes some kind of locality – for
example, elements of that number line need to be next to each other to interact. Crudely, this is how an extra dimension can
be coded in a holographic universe. But for the details we need string theory. Even from the beginning string theory had
hints of this scale invariance and dimensional weirdness. The first iteration of the theory, around
1970, tried to model the strong force between pairs of quarks – mesons – as this strand
of gluons that behaves like a vibrating string. A nice feature is that the changing the length
of the strand – which defines the energy in the bond – doesn’t change the basic
physics. That means you can pretend string length-slash-energy
is a separate dimension as a calculation trick. The weird thing is that when you write the
quantum wave equation for the gluon strand with length expressed as a separate dimension
you get the wave equation for a graviton – the quantum particle of gravity. Which is ridiculous given the puny energy
scale of a meson - gravitons shouldn’t exist there. This and other glitches led to string theory
being abandoned as a model for the strong force. But it was quickly rejigged to make it a theory
of quantum gravity, and the scale invariance of the strings becoming a central feature
of string theory. Fast forward a couple of decades to the 90s. We now have a several versions string theory
that try to explain how vibrating strings can lead to the familiar particles of this
universe. These were tentatively united by Ed Witten’s
M-theory, which showed that different types of string and string theories were all related
by dualities. A duality is when two seemingly different
theories prove to represent the same underlying physical reality. These arose from the way string size and energy
scales could be rescaled. But the strangest string duality was still
to come with AdS/CFT correspondence, proposed by Argentinean physicist Juan Maldecena in
1997. Strange because it provided the first concrete
description of a holographic universe. Maldecena imagined a set of string theory
objects called branes. These are like multidimensional strings that
can serve as start and end points for strings, but also as spaces embedded within higher-dimensions. Maldacena considered geometrically flat 3-D
branes. These branes are extremely close together
– basically overlapping. The strings connected to these branes are
scale invariant, so their length and energy can vary without changing the physics. Under certain assumptions he found that the
resulting braney structure looked just like a Minkowski spacetime of 3+1 dimensions on
which their lived a field theory that arose from interactions between branes. In itself that field theory wasn’t stringy–
rather it was a quantum field theory like the ones that gives us our standard model
of particle physics – a Yang-Mills theory, but with supersymmetry added in. It was also a conformal field theory – a
CFT - so it was invariant to the scaling of grid sizes. This quality fcame from the energy-scale-invariance
of the strings embedded in the construction of this space. In good string-theorist style, Maldacena defined
incorporated that scale factor into be a new spatial dimension. The 3-D space became a 4-D space. While the original space was flat, the new
space had negative curvature – it was a hyperbolic, anti-de Sitter or AdS space. The conformal field theory in the original
space included no gravity, but in the higher-dimensional space it became a full quantum theory of gravity. This is AdS/CFT duality. As with the other dualities in string theory,
this one was extremely useful for calculations. When interactions in the lower dimensional
field theory are extremely strong – we would say the fields are strongly coupled - then
the corresponding higher-dimensional gravitational structures in the higher dimensional space
would be weak and solvable. Conversely, strong gravitational fields in
the higher dimensional space – like in black holes – look like a solvable configuration
of particles in the low-D space. Among other things, this provided a new resolution
to the black hole information paradox: the information lost in a black hole persists
perfectly comfortably in the lower-dimensional space. The techniques of AdS/CFT correspondence are
even extended to disparate fields like nuclear and condensed matter physics. But the more startling implication of AdS/CFT
is that it’s the first concrete realization of a holographic universe. The lower dimensional CFT space is the surface
of the AdS space because the field theory exists where the new dimension becomes infinite. That’s tough to imagine – so let’s go
back to our depiction of an infinite hyperbolic space from the last episode. Represent a 2-D hyperbolic space as a compactified
map and it has an edge – at least a mathematical one. Anyone inside the hyperbolic space still has
to travel infinitely far to get to that edge. Now stack many maps to represent slices in
time. The resulting column has a geometrically flat
and finite surface that is a spacetime all on its own. The rules of interactions between cells on
the surface is a quantum field theory. But those rules translate to interactions
in the volume – in the bulk – and there they are a theory of gravity. AdS/CFT is a hint that we may live in a holographic
universe. It doesn’t represent THIS universe, because
our universe doesn’t appear to be negatively curved AdS space, nor does it have 4 spatial
dimensions as in Maldecena’s calculation. But there are efforts to generalize this to
a universe more like our own. The question we now wrestle with is this:
a series of mathematical clues indicate that our universe may be holographic – or at
least have a dual representation in a lower dimension. Can these just be crazy mathematical coincidences? Maybe, but perhaps our familiar 3+1 universe
has an alternative – perhaps a more true representation out there. An abstract mathematical surface infinitely
far from our location and from our intuition, projecting inwards our familiar holographic
spacetime. Before we jump into comments, we wanted to
let everyone know there's new merch in the merch store. Including the return of our Game of Thrones
inspired shirt the heat death of the universe is coming. It's a great way to support us as is joining
us on Patreon. Links in the description. Last week was the warm-up to today's episode,
in which we looked out how an infinite spacetime can have a finite boundary. First up: no, no psychadelics were involved
in making that episode. The universe is just that weird. A few of you asked whether our percieved universe
is just the surface of a higher dimensional space. So that's actually the opposite of the proposition
behind the holographic principle, which suggests that our percieved universe is the volume,
but it can be encoded on its lower dimensional surface. In AdS/CFT correspondence, the volume exhibits
gravity via a type of string theory, while the surface exhibits no gravity - only a quantum
field theory similar to the field theory behind the standard model. Part of the confusion comes from the fact
that Maldacena's derivation is for a volume with 4 spatial dimensions, which would have
a 3-D surface. So that obviously doesn't directly correspond
to our universe. But there's work to generalize it to our case
of a 3-D volume with a 2-D surface. Related to that, Musical Ways asks whether,
according to Ads/CFT correspondence, can we say there would be no gravity on the surface
of the (2+1)Minkowski spacetime. So first - the "surface" in current AdS/CFT
spacetime is 3+1. 3 spatial, one temporal dimensions. That surface contains only a conformal field
theory and no gravity. The strange miracle of AdS/CFT is that gravity
arises naturally when you add an extra spatial dimension, which ends up looking like the
volume contained by the "3-D" surface. KI9 asks whether the things we learn from
AdS/CFT are applicable to the universe we live in given that our universe doesn't have
negative curvature. Well we don't know for sure that it doesn't
have negative curvature - just that any curvature - negative or positive - is very weak compared
to our current ability to measure it. Measurements of the geometry of the universe
indicate flatness, but we may never know whether it's truly flat, or just flat as far as we
can see. Several people were offended that I dissed
Chronicals of Riddick. I want to go on record as saying Pitch Black
was an artistic masterpiece. Realmaml summarizes my position well: Chronicals
is the third best Riddick movie but it is still better than any Marvel movie. And I'm sure saying this will cause no further
comments. Cuallito notes that it's looking more and
more like Roger Penrose might literally be a timelord. In a separate comment Midplanewanderer states
that Sir Roger Penrose is an unsung wizard. So apparently we can't agree on what genre
Roger Penrose belongs to. Personally, I'd always thought of him as a
Jedi master - especially with all that dubious quantum consciousness stuff. Midichlorians, microtubuals, potato potahto. Anyway, perhaps we need to accept that Penrose
is beyond genre - like if Gandalf had a TARDIS and a lightsaber. By the way, if anyone feels like drawing Roger
Penrose as dressed as Gandalf with a lightsaber and a TARDIS, you would win the internet.
Sooo good
A professor lent me the book to read. It's fucking rad.
i understood some of those words.
Must watch this. Thanks!!
I still dont understand
When the spooky music kicks in
Far out!
Whattt
.