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.