Translator: Tijana Mihajlović Reviewer: Rhonda Jacobs
This is a map.
It happens to be a map of Cleveland
because that's where I'm talking to you today,
but it could have been a map of Chicago, or Toronto, or New York,
wherever you're from or going to.
And if the mapmakers did their job,
then you can use this map to navigate around the city,
to explore the city, without concern about getting lost.
The important feature I want you to notice about this map is so obvious,
you probably paid it no attention.
This map is flat.
Being a flat map is wonderfully convenient.
For one thing, if you put it on a piece of paper,
you can fold it up and put it into your pocket.
You can carry it in your purse or in your backpack.
You can print it in a book or hang it on a wall.
And it doesn't run down the batteries of your cell phone.
But if you want to, you can put this flat map on any screen of any device,
and it won't get distorted.
Being a flat map, it possesses certain properties
that we've known for over 2,300 years,
since Euclid taught them to us.
Let me show you some.
On a flat surface, I can draw a triangle.
And the sum of the angles of that triangle will be 180 degrees.
It doesn't matter where I put the triangle on the flat sheet of paper,
it doesn't matter how big the triangle is,
it doesn't matter what shape triangle I make,
the sum of the angles of the triangle will always be 180 degrees.
I can also draw two parallel lines
across this flat sheet of paper.
And if I do that,
those lines stay the same distance apart across the entire sheet.
Now, the earth is not flat. It's spheroidal.
If you try to make a flat map of the earth, then you have two choices:
you either distort the shapes of the continents and the seas,
or you cut the map into all sorts of inconvenient, annoying shapes.
If you don't need to pull the map of the earth out of your pocket,
or if you don't need to put it into your pocket and then pull it out,
you're much better off making a spheroidal map,
a spherical map of the earth.
Soon enough we will have holographic displays
that allow us to do that,
but in the meantime, I have something simpler,
it's called a globe.
(Laughter)
So, one of the nice properties of a globe
is that it is different than a flat sheet of paper
in ways that completely contradict Euclid.
So for example, I can start here, on the equator,
I can head west along the equator, a quarter of the way around the earth,
and then I take a 90-degree turn and head north, to the North Pole.
And then I can take another 90-degree right turn and head south,
exactly to where I started,
and I will get another triangle.
And that triangle has three 90-degree angles.
And 90 plus 90 plus 90 is 270,
which is more than 180.
So, the sum of the angles of the triangle on this sphere is more than 180.
No matter what triangle I draw on this sphere,
the sum of the angles of the triangle will be more than 180.
If I started two planes flying
from any points on the equator of this globe, heading north,
they would start out on parallel paths,
but as they got closer and closer to the North Pole,
they would get closer until they met at the North Pole.
So parallel lines don't stay the same distance apart.
Apparently, we're learning that there are surfaces of different curvature.
This sphere is not flat. It's positively curved.
There are also surfaces of negative curvature.
The most important property of the globe
that I'd like you to understand, actually, is not that.
It's one other property which is that if I start anywhere on this globe
and head off in any direction,
and keep going straight, without turning,
I will come back to exactly where I started.
I can head any direction.
I could head off to this way, and I can go around,
and I don't have to turn to come back to where I started.
That's something that's completely different
than this sheet of paper.
And this is the property that I'd like to focus on
for the rest of this talk,
the ways in which in a space we can head off in some direction
and come back to where we started
because those describe the topology of the space,
the ways in which the space is connected.
It's not obvious that we can do that also on a space where Euclid is right,
on a flat space.
In other words, on a space
where the sum of the angles of a triangle are 180 degrees.
You can't do it on this flat piece of paper,
because if I head off in some direction, I'd fall of the edge.
Well, I could make the paper bigger,
but no matter what, I'm going to eventually fall off the edge.
However, I could take the piece of paper and I could wrap it up into a cylinder.
And if I have a cylinder, then I can head off in some direction,
and by going straight, come right back to where I started,
just like on the sphere.
Now, I can also take my cylinder and I can wrap it up -
I can glue its ends together,
and I can get a doughnut-like shape like this.
And on this shape, there are two different types of loops
which are straight.
I can start here and head around and through the hole,
or I can start here and head straight along this torus,
this doughnut shape,
and come back to where I started by going around the hole.
So on the torus, there are two different straight paths
that both come back to where they started,
without any turning.
Now, you might say, "Wait, you're trying to trick me."
After all, the cylinder doesn't look very flat,
this torus certainly doesn't look very flat.
But think about what I did
to make a cylinder out of a flat sheet of paper;
I rolled it up.
And so, if I was living in the sheet of paper,
as opposed to looking at it from the outside,
then when I went and measured the angles of that triangle,
they would still add up to 180 degrees.
And if that's what you mean by flat,
then we have a surface in which it's both flat,
and I can go around it by heading straight.
So the lesson is that spaces can have the same geometry,
the same curvature,
but different topologies, different ways of being connected.
On the sheet there are no ways of going straight
and coming back without turning.
On the cylinder there's one, by going around it,
and on the torus there are two:
I can go through the hole or around the hole.
Now, this is supposed to be a talk about the topology of the universe,
not the topology of pool noodles.
About general relativity or GR, as we've heard for short,
Einstein's theory of gravity.
And GR is a theory about space and time.
Einstein's field equations,
which he announced a hundred years ago, in 1915,
tell us how matter affects both space and time,
and how space and time affect matter.
But Einstein's equations for GR are what we call local.
They tell us how matter here affects space and time here.
They tell us also how matter here responds to the curvature of space and time here,
whereas topology is about the whole thing,
about all of space.
It's an important property of space.
It can even affect how objects in the space behave.
Einstein and his colleagues argued at the beginning
about what was the proper topology to choose,
to guess for space, for the universe,
which makes you understand that general relativity doesn't predict
the topology of the universe.
We have to go out and observe it.
But how?
Well, and you might ask, "Why?"
After all, who cares what the topology of the universe is.
Well, my personal view is that it's really the ultimate question
or one of those ultimate questions.
It's like trying to put your arms around the whole universe,
around the whole thing,
seeing all of reality,
and eventually finding your place in it.
It's like picking up a globe and noticing the Arctic ice sheets,
and noticing the great sand deserts of the Sahara,
or how small Western Europe is and how vast the Russian Steppes are,
and then eventually being able to look around
and say, "I live here."
That's what we're trying to do for the whole universe.
Well, back to work.
So Magellan, or at least his crew,
(Laughter)
circumnavigated the earth.
How are we going to do the same thing in the universe,
where distances are measured in billions of light years?
Well, clearly we can't.
But what we can do is we can look for something
that has made that journey for us.
This is a picture of the cosmic microwave background.
When the universe was young, long ago,
it was very hot and full of a glowing plasma.
That plasma was opaque.
The photons, the particles of light in the universe, bounced around;
they couldn't go very far.
But as the universe grew older, it expanded and grew cooler.
And when the universe was about 400,000 years old,
it was only about half the temperature of the surface of the sun.
And at that point, that plasma became a transparent gas.
The light was released
and has been traveling through the universe
for 14 billion years since that time.
So when we look out at the universe, we can detect this light,
stretched by the expansion of the universe,
and it's not visible radiation anymore,
but microwaves.
So we don't look at it and see it with our eyes,
but we see it with microwave antennas,
the cosmic microwave background radiation or CMB.
After 14 billion years, it is just a dull glow -
slightly hotter in some directions, slightly cooler in some other directions,
but just a dull glow.
But if you go out on the beach, you are basking not just in the sunlight,
but in the glow of the Big Bang.
Okay, so, suppose I want to make a measurement
of the microwave background.
So here I am, making a measurement of the microwave background
with my microwave antenna.
And the light is coming to me from a sphere,
a vast sphere 14 billion light-years in radius,
centered on me.
And my friend over here, he's doing the same thing,
he's making a measurement of the microwave background,
he has a microwave antenna.
And the light is coming to him from a vast sphere
14 billion light years in radius, centered on him.
Well ...
If I'm making one from over there, and he's making one from over there,
then if we're separated, our two spheres will intersect.
The intersection of two spheres is a circle.
That's a circle that I see off in this direction,
and he sees off in this direction.
Now, that's not very interesting
if we're just separated by the width of the stage.
It's also not very interesting
if we're just separated by the width of the solar system.
But if we're separated by the width of the universe,
that would be fascinating.
Because if I start over here and cross the universe,
and arrive back to where I started
because I've followed a loop around the universe,
that means that this me over here
can look out and see that circle off in this direction,
and this me over here can look out and see that circle in this direction,
but, of course, it's the same me.
And that means that there will be a circle on the sky
that I see over here,
and that I also see over here,
which is the same place in space
and therefore has the exact same pattern
of hot and cold microwave background radiation on it.
Well, when my colleagues Neil Cornish, David Spergel and I realized this,
we were very excited
because it meant we had a way to look for topology of the universe
that didn't depend on exactly which topology we were trying to find.
No matter what topology there was,
as long as there were two paths around the universe,
one from each side of that circle to me,
I would be able to find these pairs of matching circles,
I'd be able to see that circle in space from two sides.
Now ...
we looked.
We looked. We compared.
We compared the data from the Wilkinson Microwave Anisotropy Probe,
and we looked at the data from the Planck satellite
of the European Space Agency.
We compared billions and billions of circles to each other,
looking to see if any of them were matched.
And ...
nothing.
(Laughter)
The circles were no more matched than they would be just by accident,
when you compare that many billions of circles.
And of course that meant we were a little bit disappointed,
but we had learned something really, really important.
We had learned that the universe is big.
Really big.
A shortcut around the universe,
a loop around the universe has to be at least 84 billion light years long.
That's a long way.
If you want to make a trip around the universe,
that's probably too far.
And it also sounds discouraging
if you're looking for the topology of the universe.
Well ...
Fortunately, we don't believe that all is lost.
We think there's one last chance, and it has to do -
We may not have seen the topology of the universe,
but we believe we can hear the topology of the universe.
And it works like this.
The patterns in the CMB that you see
are actually the impressions of sound waves,
of the sound of the Big Bang echoing in the universe.
As acoustic waves traveled through the universe,
they compressed the gas here and they rarefied it there.
Compressed it here and rarefied it there.
General relativity tells us that as the light left those dense places,
it had to climb towards us and lose energy.
As it left the less dense places,
it had to climb less and it lost less energy.
And that means that the dense places should appear a little cooler,
and the less dense places should appear a little warmer.
So like a light show at a rock concert,
we should be able to see the sounds
of the cosmic Big Bang in microwave.
Now ...
what does that have to do with topology?
Well, to show you, I have here two drums.
And they're two drums of different shape.
And when I hit them,
they make different sounds.
If you listen very carefully,
you can tell the difference between almost any pair of drums.
The shape of the drum imprints itself on the sound.
Different drums, different sounds.
Listen carefully and tell the shape of the drum.
So too the shape of the universe.
The topology of the universe is like the rim of the drum,
imprinting itself on the sounds that are in the universe.
And those sounds are turned into the hot and cold temperatures
of the cosmic microwave background.
The question remains: how?
How do we do this?
How do we like great cosmic bats use the echoes of the Big Bang
to find our place in the vast universe?
All that I can say is: stay tuned.
Thank you.
(Applause)