Have you ever asked “what is beyond the
edge of the universe?” And have you ever been told that an infinite
universe that has no edge? You were told wrong. In a sense. We can define a boundary to an infinite universe,
at least mathematically. And it turns out that boundary may be as real
or even more real than the universe it contains. Our universe may be infinite. In order to wrap our puny human minds around
such a notion we like to come up with boundaries. For example we have the “observable universe”
– that patch that we can see, and beyond which light has not yet had time to reach
us. It’s boundary is called the particle horizon. Beyond it there exists at a minimum of thousands
and possibly infinitely more regions just as large. Our observable universe is like a tiny patch
of land in a vast plain. We define its horizon like we might build
a little picket fence around our little patch – meaningless from the point of view of
the plain, but it makes our patch feel more homey and us less crushingly insignificant. We visited these cosmic horizons in one of
the early Space Time episodes. To review: the particle horizon defines the
limit of the visible past, and there’s also cosmic event horizon defining the limit of
the visible future. But these correspond to actual spherical boundaries
in space, whose distances can be calculated. How mundane. There’s another way to define the boundary
of the universe that isn’t so shy in the face of an infinite cosmos. In fact, if we twist our human intuition and
our mathematics to its limit we can build our picket fence around an infinite universe. A mathematical boundary at infinity turns
out to be not just useful for doing calculations in physics, but may be as real as the physical
universe it contains. It may encode that universe as a hologram
on its surface. In today’s episode we’re going to talk
about two ways to define an infinite boundary, and this will set us up finally for laying
open the holographic principle. I promise! Let’s start with a quick review of types
of universe. At least the three basic types described by
Einstein’s general theory of relativity. The only type of non-infinite universe – or
closed universe – is the one that curves back on itself – like a 3-D analog of the
2-D surface of a sphere. We call that positive curvature. Travel far enough and you get back where you
started. Geometry is a bit broken in such a universe
– for example, two parallel lines will eventually converge and cross each other. In general relativity we call a universe with
this geometry de Sitter space, after Dutch astronomer Willem de Sitter. Then there’s the flat universe – classic,
straightforward geometry – parallel lines stay parallel – and it goes on forever. In GR this is Minkowski space, after Hermann
Minkowski, teacher and colleague of Albert Einstein’s. Finally there’s the universe with negative
curvature, and the 2-D analog of that is the hyperbolic surface, like an infinite saddle
or pringle. In this geometry parallel lines actually diverge
from each other. In GR this is anti-de Sitter space because
it’s the “opposite” of the positive curvature de Sitter space. Terrible name, so we abbreviate it AdS space. OK, so 2 out of 3 possible universes are infinite. On the other hand, assuming all these types
exist, there should be infinitely more people in infinite universes compared to people in
non-infinite universes. You’re probably one of the former, so let’s
ignore puny de Sitter space and for today assume we’re in one of the infinite ones. How do we put bounds on infinity? This all got going in the early 60s when physicists
tried to find ways to map infinite spacetime –to the edge of an infinite universe or
across the event horizon of a black hole. Regular coordinates of space and time are
useless there – they blow up to infinities. Physicists found mathematical ways to fuse
space and time into new coordinates that suppressed the infinities. We call this process compactification. The first efforts were designed to allow physicists
to cross the event horizon of black holes – mathematically. We have Kruskal–Szekeres coordinates, Eddington-Finkelstein
coordinates, and others. But these coordinates only defeated the artificial
infinity – the coordinate singularity - of the event horizon – also something we’ve
discussed (the phantom singularity) It was Roger Penrose who defeated the true
infinity of an infinitely large universe. In the early 60s he developed his Penrose
coordinates and Penrose diagrams – also known as Penrose-Carter diagrams, for Brandon
Carter who came up with them around the same time. We’ve used these before to understand black
hole event horizons, but these were originally conceived to understand the boundaries of
the universe. As a quick review: start with a graph of space
versus time – a spacetime diagram – then compactify. These horizontal-ish contours are our old
time ticks - moments of constant time across the universe, while the vertical-ish lines
are set locations in space in only one spatial dimension. The contours bunch up towards the boundaries
so that every step on the map covers more and more space and time. The boundaries themselves represent infinite
distance and infinite past and future. One amazing thing about the Penrose diagram
is that the transformation preserves all internal angles – all angles between intersecting
lines relative to each other stay the same. We call such a transformation “conformal”. Remember that word – it’s going to be
very, very important. This particular conformal compactification
is designed to ensure that the path of every ray of light remains at 45 degrees across
the map. That means only lightspeed paths can hope
to reach these boundaries ahead, and only light speed paths can originate from these
boundaries behind. Any sub-lightspeed paths, which means anything
with mass, will be swept along with the contours of space. All matter must originate at this point representing
all of space in the infinite past, and must also converge to this point which represents
all of space in the infinite future. Only lightspeed paths – or in the language
of quantum field theory “massless fields” can access these diagonal boundaries. If we write the equations of these fields
in Penrose’s compactified coordinates then we can do something that seems impossible
– we can track a quantum field to infinite distance and calculate its behavior there. That has a very particular use. Penrose diagrams represent a universe that
is “asymptotically flat” – it may have some local curvature due to gravity of massive
objects inside, but at its boundaries the simple rules of non-curved, Minkowski spacetime
apply. That’s handy because flat space is the only
space where quantum mechanics is fully solvable. The most famous use of this is by Steven Hawking,
as we saw in our Hawking radiation episode. He connected a quantum field between two points
at infinite distance – past and future - where he could define the state of the quantum vacuum
in solvable flat space. Then he placed a black hole in between these
points and calculated how it perturbed the balance of a quantum field traced between
them. He found that two “infinitely distant”
regions could not both be in a perfect vacuum state if a black hole lay between them. He concluded that the black hole must generate
particles – Hawking radiation. Which, by the way, was a key discovery on
the path to the holographic principle, as we’ve discussed before and which I’ll
. review again - but not today. Today we’re just talking about boundaries
– and we need a very different infinite boundary to give us our hologram. Penrose diagrams define the infinite boundary
of a flat universe as a useful tool in calculation. For the holographic principle we need the
infinite boundary of a negatively-curved universe – an anti-de Sitter, AdS universe. Let’s start with another map. Just like the Penrose diagram we’ll do a
conformal transformation – all internal angles preserved, and will compactify layers
towards the edge. But this time we’re not mapping space versus
time – we’ll just map two dimensions of hyperbolic space. In fact, let’s use the most famous conformal
compactification of hyperbolic space. That’s right, there is a most famous conformal
compactification of hyperbolic space, and you’ve probably seen it. This is M.C. Escher’s Circle Limit IV – the final in
a series of woodcuts inspired by a projection of hyperbolic plane called the Poincaré or
conformal disk. The basic construction is straightforward
enough – start with a circle. Now fill it with a set of circle segments
that all intersect the circumference at right angles. Those circle segments represent the straight
lines of a hyperbolic geometry projected onto the disk. We can see the hyperbolic behavior these arcs
– they are geodesics – the straightest-possible paths in the geometry. Any two paths that are parallel at one point
will diverge from each other in either direction. This is a conformal transformation of a hyperbolic
surface because the angles of intersection of these lines are preserved, and it’s compactified
because an infinite hyperbolic surface fits on a finite disk, with lengths represented
by shorter and shorter arcs towards the rim. The other feature of a conformal mapping is
that shapes are preserved – at least locally. We see that when we fill the circle with a
regular choice of circle segments. They define a set of enclosed shapes that
vary in size but not in shape. This is an example of tessellation – tiling
a space with regular repeated shapes. Hyperbolic space is fascinating because there
are literally infinite ways it can be tessellated with regular polygons, while spheres and flat
space each have only a small finite number of possible tessellations. So this disk can represent an infinite anti-de
Sitter universe with 2 spatial dimensions at a single instant in time. Each tile represent the same size region of
space. The boundary is infinitely far away and looks
the same no matter where we travel. If we wanted to represent a 3-D AdS universe
we could use a Poincaré ball instead. But the real power of AdS space isn’t the
cool art you can do with it. At least to physicists. It’s the nature of the infinite boundary. Here’s a mouthful: the boundary of a conformally-compactified
anti-de Sitter space is itself a conformally-compactified Minkowski space with one fewer dimension. Got it? Cool, we’d done here. OK, let’s unpack that. In fact let’s add the dimension of time
to our hyperbolic projection. Stack a bunch of Poincare disks – each representing
an instant in time. They give you a cylinder and representing
an AdS spacetime with 2 spatial and one temporal dimensions – let’s call that 2+1 dimension. On the other hand the surface of the disk
has only one dimension of space – the circumference – and the same one-D of time – 1+1. Right. So it turns out that the surface of the cylinder
– which exists only in the compactified coordinates of the interior volume - is mathematically
exactly a flat, Minkowski space. You can extrapolate to any number of extra
dimensions – say a 3+1 dimensional - Poincare ball. Compactify it so you the infinite boundary
becomes a surface – that surface is a 2+1 Minkowski plain. And the crazy thing is that you can treat
that surface space and the interior space – also called the “bulk” - as separate
spacetimes with their own physics. But they are connected. Every point on the flat surface maps to a
set of paths through the hyperbolic interior – remember those circle arcs? Patterns on the surface define the structure
of interior. In 1997 Argentinian physicist Juan Maldacena
found an incredible correspondence between these spaces. He realized that if you define a conformal
quantum field theory in a 3+1-dimensional Minkowski space, that corresponded to an interesting
mathematical structure in the enclosed 4+1-D AdS space. That structure looked exactly like a string
theory with gravity and everything. This is the AdS/CFT correspondence. Quantum mechanics in the form of a conformal
field theory in one space is a theory of quantum gravity in a space with one higher dimension. The hologram part is because the lower dimensional
space can be thought of as the infinitely distant boundary of the higher dimensional
space. Every particle, every gravitational effect
in the bulk is represented by quantum fields on an infinitely distant surface. OK, I’m going to have to cut us off here. The deeply abstract relationship between these
two spaces needs an entire episode. Stay tuned for the final installment of the
holographic principle in not-so-infinitely-distant future of spacetime. Last week we enjoyed another potential end
of the universe when we talked about the Big Rip - in which space tears itself to shreds
on subatomic scales due to runaway increase in dark energy. Your excitement at our possible doom really
showed in your comments. Many of you asked what happens to black holes
in the big rip. That's... a great question. So I thought that the answer was that black
holes would be eroded into nothing. After all, if space is expanding faster than
light at the event horizon, that should counter the light-speed flow of space into the event
horizon, causing the event horizon to shrink and the black hole to dissolve. Some internet physicists agree with my intuition. But I also read an interesting take by Alan
Rominger on physics stackexchange, in which he suggests that the shrinking cosmological
horizon could merge with the event horizon to produce a global state where everything
just looks like an inflationary spacetime. Chris Hanline wisely notes that dark energy
appears to break the conservation of energy. That's sort of true - except that the law
of conservation of energy has a very clear range of validity - it's valid in systems
that are time symmetric - systems where the global properties of the spacetime don't evolve
over time. In fact conservation of energy comes from
this symmetry, as revealed by Noether's theorem. However our universe on its largest scales
is not time symmetric - it's expanding, so the past looks very different to the future. In its most familiar form, conservation of
energy doesn't apply and so dark energy CAN be created from nothing. Some have argued that energy is conserved
and that dark energy is created from the increasing negative potential energy of the cosmic gravitational
field, but I think at that level this is all just different interpretations of the math. Swole Kot asks if the last months in the big
rip scenario would be a painful and horrible experience for any sentient life still around
at that point. Well it wouldn't be great, that's for sure. After some millions of years watching the
galaxies fall apart, the last phase of the destruction of the solar system would happen
pretty fast. There's going to be an unpleasant period between
the destruction of planet sized things and the destruction of atom sized things where
you have the destruction of people-sized things. But I guess the painful part would be when
expansion is fast enough that it starts to disrupt the way chemistry works, without actually
ripping matter apart. Our bodies are pretty dependent on chemistry
working normally, so there would be some minutes to hours of bad times as our molecules start
to betray us. To make matters worse, the Earth would be
falling apart at the same time. It's like 2012 meets World War Z meets Chronicles
of Riddick. Guys, I think we just sold a movie.
TIL: the boundary of a conformally-compactified anti-de sitter space is itself a conformally-compactified minkowski space with one fewer dimension!
Gotta be honest, this one was hard to follow for me, my math and physics University courses are starting to be a long time ago. I'm only 31 and I barely remember any math past highschool xD
Still love the video though !
Finally, I’ve been waiting for pbsspacetime to cover ads/cft for over a year now and I can say it was worth the wait. The animations in this one were on point and very well done. I hope this is the start of a new series from them. To be quite honest, it makes me happy knowing there are enough curious people out there that a resource such as pbsspacetime can exist.