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.