Uploaded by numberphile on 18.07.2012

Transcript:

BRADY HARAN: You got another number for us?

TONY PADILLA: I have, yeah.

Yeah.

It's the--

well, I'll just write it down, shall I?

That's probably the best thing to do.

So it is 10 to the 10 to the 10 to the 10 to the 10--

and then this is the strange bit-- to the 1.1.

OK?

This has been claimed to be the largest finite time that

has ever been calculated by a physicist

in a published paper.

This is the paper.

It's a bit of a weird paper.

It's not had a huge impact or anything.

It's about black hole information loss

and conscious beings.

We won't go down that route.

He actually calculates something called the Poincare

recurrence time for a certain type of universe within a

certain cosmological model.

And this is the number that he gets out.

So this is the one on I'm on about.

Let me just check I got the number of 10's right.

Yeah.

I did.

Here it is, equation 16.

So he's put Planck times, millennia, or whatever,

because basically, it don't really make any difference

whether you use seconds, Planck times, millennia, years

when the number is this big.

But there's other interesting numbers in here,

which is this one here.

You got three 10's to the 2.08.

That's the Poincare recurrence time for our universe.

BRADY HARAN: What's all this about, dude?

TONY PADILLA: OK.

So Poincare recurrence time.

This is something that's arisen

from statistical mechanics.

Very simply, if we had a pack of cards, and we've only got a

finite number of cards in that pack.

And let's say we keep dealing each other

hands of five cards.

Eventually, if we do it for long enough, you're going to

get a royal flush, Brady.

That's guaranteed to happen.

And if you wait long enough again you'll get another one.

And so on, and so on, And that's true because there's a

finite number of cards.

Now what Poincare realized is that if you take a gas, the

particles, you can put the gas in a box.

Now you can put all the particles in one corner of the

box, and then they'll disperse and then they'll move around.

But Poincare recurrence tells us is that after a very, very,

very, very long time, those particles will eventually

return to the corner of the box.

You always get a repetition.

And it's basically because the thing that controls the

evolution of that system only has a finite region of what we

call [INAUDIBLE] space, of solution space, that's

accessible to it.

And so eventually, you always come back to it arbitrarily

close to where you started.

The time scales are truly enormous before you start

expecting it to happen.

So you can apply this quantum mechanically as well.

So what you're doing in quantum mechanics, what you're

really talking about is the evolution of microstates

within the quantums, so these sort of quantum building

blocks of your system.

And they will eventually return--

they will evolve, but eventually return to their

initial states.

And what Don Page in this paper has tried to do is he's

actually applied that to various types of universe,

models of various types of universe.

It's a bit of a cheat, because a universe is what we call a

macroscopic object.

It's a large object.

It's not really a quantum micro state, in some sense.

What it is, is it an ensemble, an average, of all the

microstates.

So what he's tried to do is he's tried to say, OK, I'm

going to treat the universe as this average

of all these states.

But then I'm going to count up all the possible averages and

treat them as a sort of microstate in itself.

So it's a bit of a cheat, but he gets an extra exponential

out from that.

BRADY HARAN: Are we talking about, Jupiter's over there,

the Andromeda Galaxy's over there, I'm in this room

filming you.

TONY PADILLA: Yeah.

Yeah.

All that.

OK?

Now there's a very large number of possibilities that

you can have, but it's finite.

And so as the system evolves, it's only got to--

and it can only evolve, the system can only evolve through

a finite number of possibilities.

And eventually, it evolves back to where it started.

So actually, it's--

I've often heard it said that the universe, for example,

will evolve, will expand.

Eventually, everything will be spread out very far because of

the expansion of the universe.

That all objects will have collapsed to form black holes.

That then those black holes will evaporate from Hawking

radiation, and all you will have is this very sort of

bleak landscape of just radiation that's come out

these evaporated black holes, that's uniformly distributed,

and it'll be very boring.

OK?

But that isn't the end state of the universe.

The end state of the universe is that after these truly epic

time scales, you will eventually have a Poincare

recurrence, and you'll wind up back where you started from.

And it's quite easy to see how it might happen.

Imagine you have this sort of bleak universe, right?

Just have a little fluctuation.

That little fluctuation sort of gathers together, builds up

other things.

Eventually, it sort of forms a sort of galaxy, even.

From that galaxy, you get planets, stars.

And you keep going, you keep going.

Eventually, you'll get back to the situation where it looks

like it is today.

Now, what I think is fair--

I think it's a fair point--

is that there's no way of ever being aware of these

repetitions over these large times.

And the reason is, you could never build a device.

You could never be an observer that could measure this.

And that's because over these huge time scales, such a

device or such an observer would definitely thermalize,

would definitely become part of this recurrence itself.

And so there's never anybody or anything that

could measure it.

It's this sort of idea, this sort of notion within physics,

is that if you can't measure it, it's

irrelevant in some sense.

So.

BRADY HARAN: But according to this, in this number of--

in this time, in this number of years, you and I are going

to make this video again?

You and I are going to make this video again?

TONY PADILLA: Oh, I know this is--

less than that.

This is for a special type of universe that's

particularly large.

The number for us is at least less than this other number

that he's written down here, which is 10 to the 10 to the

10 to the 10 to the 2.08.

I'll just say years.

So this one is the one that applies to us in our physical

universe, what we call our causal patch.

This one applies to seeing what is the Poincare

recurrence time for a truly vast domain of universe that

you can get out of certain models of cosmology.

So they're all based, of course, on the same idea.

We can work through where these numbers come from.

So the Poincare recurrence time of any system is roughly

proportional to the number of states in that system.

Because we're applying this to the universe, this is really

the number of macrostates, the number of these averages of

microstates that you could talk about.

So let's call this Nmacro.

This, then, where we'd expect it to be about the exponential

of the number of microstates.

Now why is that?

Well, this doesn't really have to be an e.

It could be a 2 or whatever.

Basically, any microstate is either in or out of the

averaging with some weighting.

And so this is the number that you get out.

The number of microstates when you relate to the entropy is e

to the entropy.

Let's look at some volume of the universe,

of some radius r.

Then the entropy--

we've done this before.

This is the same arguments you have before--

the entropy that you could possibly have in this region

of space, basically it's proportional to--

I'll just be a bit sloppy with factors--

r squared over the Planck length squared.

So the next step is e to the e to the--

So now let's apply it.

So let's apply it to the really big

number that he does.

The question is, what's r?

Well, the radius of the universe is about 10 to the 26

meters, our visible universe, so far as we can see.

What we call our causal patch.

The Planck length is about 10 to the minus 34 meters.

So r over n l Planck squared is about 10 to the 120, which

is a number you often see in physics, actually.

This is sort of the number that's associated with

cosmological constant problems.

But anyway.

Well, 120 is about 10 to the 2.08.

That's where that is coming from.

I don't know why he's so precise about this 2.08,

because what he's going to do next is--

the number we have is e to the e to the 10 to

the 10 to the 2.08.

But what he does here is he just approximates

these e's as 10's.

Which is fine, really, in the broader scheme of things.

e, 10.

For the sake of cosmology, they're more or

less the same thing.

All the e's become 10's, and you're got 10 to the 10 to the

10 to the 2.08.

Which hopefully is what he's got there, and it is.

So that's where that number comes from.

So this is basically the Poincare recurrence time for

our visible universe, for our patch.

BRADY HARAN: Why is the other number bigger?

TONY PADILLA: Well, because the other number--

there, he's looking at--

he's trying to get a big number is what

I guess he's doing.

He's trying to get a bigger number.

What he's looking out there is a model of inflation.

Now, inflation is a model of the very early universe, where

the universe grew really quickly out of a very small

patch of the universe.

The amount by which it blows up depends on various

parameters in the model.

But basically, the thing that you get is you get to the size

of the universe.

So r, for that case, is of order e to

the 4 pi over M squared.

So this is actually r over l Planck.

He's done everything in Planck units.

But M here is the mass of some of this influx, what we call

the inflaton field.

It's just some field in the model

that causes the expansion.

1 over M squared is 10 to the 12.

I think this might be 13, actually.

10 to the 12.

So this is about 10 to the 13 overall,

because 13 is about 10.

Bear with me.

So this is e to the 10 to the 13, roughly.

The 13, well, is about 10 to the 1.1.

That's where the 1.1 comes from.

He's very precise about this 1.1, and yet he's very sloppy

with some of the other factors.

You get e to the e to another e to the 10 to

the 10 to the 1.1.

And then we do the same thing.

We turn all the e's into 10's.

BRADY HARAN: Help me understand, because obviously,

you have bit numbers there.

How long a time is this?

TONY PADILLA: OK.

So this is truly vast.

Like I said, there's no device, there's no observer,

anything that could survive this kind of

length of time scale.

In fact, you would probably say that the universe is more

likely to tunnel out of the current state before this

could happen, in some sense.

This is such a long time scale that one might say that

actually the probability of tunneling to a new phase of

the universe, completely different, is actually going

to dominate over this.

That that would occur first.

So maybe this is kind of an irrelevant point.

Yeah.

It's truly vast.

It's bigger than a googol, clearly.

Way bigger.

Is it bigger than a googolplex?

I think so.

So you know, let's just check that.

Yeah, clearly it is.

It's enormous.

It won't be as big as Graham's number, but you know, Graham's

number's the daddy, right?

So.

Well, I didn't actually know about this paper.

But yeah, I was just--

I just thought, oh, big numbers.

Let's see what's interesting about big numbers.

And then I stumbled across this paper.

What he claims-- in fact, he says it--

he claims, "So far as I know, these are the longest finite

times that have been explicitly calculated by any

physicist." So whether somebody has calculated a

longer one--

and I'm sure some of the viewers will try to calculate

a longer one and then claim that they've--

but this is in a published paper.

So you've got to get the paper published.

But the challenge is on, I guess, to find a longer one.

There probably has been, too, but he was the one that

pointed it out.

TONY PADILLA: I have, yeah.

Yeah.

It's the--

well, I'll just write it down, shall I?

That's probably the best thing to do.

So it is 10 to the 10 to the 10 to the 10 to the 10--

and then this is the strange bit-- to the 1.1.

OK?

This has been claimed to be the largest finite time that

has ever been calculated by a physicist

in a published paper.

This is the paper.

It's a bit of a weird paper.

It's not had a huge impact or anything.

It's about black hole information loss

and conscious beings.

We won't go down that route.

He actually calculates something called the Poincare

recurrence time for a certain type of universe within a

certain cosmological model.

And this is the number that he gets out.

So this is the one on I'm on about.

Let me just check I got the number of 10's right.

Yeah.

I did.

Here it is, equation 16.

So he's put Planck times, millennia, or whatever,

because basically, it don't really make any difference

whether you use seconds, Planck times, millennia, years

when the number is this big.

But there's other interesting numbers in here,

which is this one here.

You got three 10's to the 2.08.

That's the Poincare recurrence time for our universe.

BRADY HARAN: What's all this about, dude?

TONY PADILLA: OK.

So Poincare recurrence time.

This is something that's arisen

from statistical mechanics.

Very simply, if we had a pack of cards, and we've only got a

finite number of cards in that pack.

And let's say we keep dealing each other

hands of five cards.

Eventually, if we do it for long enough, you're going to

get a royal flush, Brady.

That's guaranteed to happen.

And if you wait long enough again you'll get another one.

And so on, and so on, And that's true because there's a

finite number of cards.

Now what Poincare realized is that if you take a gas, the

particles, you can put the gas in a box.

Now you can put all the particles in one corner of the

box, and then they'll disperse and then they'll move around.

But Poincare recurrence tells us is that after a very, very,

very, very long time, those particles will eventually

return to the corner of the box.

You always get a repetition.

And it's basically because the thing that controls the

evolution of that system only has a finite region of what we

call [INAUDIBLE] space, of solution space, that's

accessible to it.

And so eventually, you always come back to it arbitrarily

close to where you started.

The time scales are truly enormous before you start

expecting it to happen.

So you can apply this quantum mechanically as well.

So what you're doing in quantum mechanics, what you're

really talking about is the evolution of microstates

within the quantums, so these sort of quantum building

blocks of your system.

And they will eventually return--

they will evolve, but eventually return to their

initial states.

And what Don Page in this paper has tried to do is he's

actually applied that to various types of universe,

models of various types of universe.

It's a bit of a cheat, because a universe is what we call a

macroscopic object.

It's a large object.

It's not really a quantum micro state, in some sense.

What it is, is it an ensemble, an average, of all the

microstates.

So what he's tried to do is he's tried to say, OK, I'm

going to treat the universe as this average

of all these states.

But then I'm going to count up all the possible averages and

treat them as a sort of microstate in itself.

So it's a bit of a cheat, but he gets an extra exponential

out from that.

BRADY HARAN: Are we talking about, Jupiter's over there,

the Andromeda Galaxy's over there, I'm in this room

filming you.

TONY PADILLA: Yeah.

Yeah.

All that.

OK?

Now there's a very large number of possibilities that

you can have, but it's finite.

And so as the system evolves, it's only got to--

and it can only evolve, the system can only evolve through

a finite number of possibilities.

And eventually, it evolves back to where it started.

So actually, it's--

I've often heard it said that the universe, for example,

will evolve, will expand.

Eventually, everything will be spread out very far because of

the expansion of the universe.

That all objects will have collapsed to form black holes.

That then those black holes will evaporate from Hawking

radiation, and all you will have is this very sort of

bleak landscape of just radiation that's come out

these evaporated black holes, that's uniformly distributed,

and it'll be very boring.

OK?

But that isn't the end state of the universe.

The end state of the universe is that after these truly epic

time scales, you will eventually have a Poincare

recurrence, and you'll wind up back where you started from.

And it's quite easy to see how it might happen.

Imagine you have this sort of bleak universe, right?

Just have a little fluctuation.

That little fluctuation sort of gathers together, builds up

other things.

Eventually, it sort of forms a sort of galaxy, even.

From that galaxy, you get planets, stars.

And you keep going, you keep going.

Eventually, you'll get back to the situation where it looks

like it is today.

Now, what I think is fair--

I think it's a fair point--

is that there's no way of ever being aware of these

repetitions over these large times.

And the reason is, you could never build a device.

You could never be an observer that could measure this.

And that's because over these huge time scales, such a

device or such an observer would definitely thermalize,

would definitely become part of this recurrence itself.

And so there's never anybody or anything that

could measure it.

It's this sort of idea, this sort of notion within physics,

is that if you can't measure it, it's

irrelevant in some sense.

So.

BRADY HARAN: But according to this, in this number of--

in this time, in this number of years, you and I are going

to make this video again?

You and I are going to make this video again?

TONY PADILLA: Oh, I know this is--

less than that.

This is for a special type of universe that's

particularly large.

The number for us is at least less than this other number

that he's written down here, which is 10 to the 10 to the

10 to the 10 to the 2.08.

I'll just say years.

So this one is the one that applies to us in our physical

universe, what we call our causal patch.

This one applies to seeing what is the Poincare

recurrence time for a truly vast domain of universe that

you can get out of certain models of cosmology.

So they're all based, of course, on the same idea.

We can work through where these numbers come from.

So the Poincare recurrence time of any system is roughly

proportional to the number of states in that system.

Because we're applying this to the universe, this is really

the number of macrostates, the number of these averages of

microstates that you could talk about.

So let's call this Nmacro.

This, then, where we'd expect it to be about the exponential

of the number of microstates.

Now why is that?

Well, this doesn't really have to be an e.

It could be a 2 or whatever.

Basically, any microstate is either in or out of the

averaging with some weighting.

And so this is the number that you get out.

The number of microstates when you relate to the entropy is e

to the entropy.

Let's look at some volume of the universe,

of some radius r.

Then the entropy--

we've done this before.

This is the same arguments you have before--

the entropy that you could possibly have in this region

of space, basically it's proportional to--

I'll just be a bit sloppy with factors--

r squared over the Planck length squared.

So the next step is e to the e to the--

So now let's apply it.

So let's apply it to the really big

number that he does.

The question is, what's r?

Well, the radius of the universe is about 10 to the 26

meters, our visible universe, so far as we can see.

What we call our causal patch.

The Planck length is about 10 to the minus 34 meters.

So r over n l Planck squared is about 10 to the 120, which

is a number you often see in physics, actually.

This is sort of the number that's associated with

cosmological constant problems.

But anyway.

Well, 120 is about 10 to the 2.08.

That's where that is coming from.

I don't know why he's so precise about this 2.08,

because what he's going to do next is--

the number we have is e to the e to the 10 to

the 10 to the 2.08.

But what he does here is he just approximates

these e's as 10's.

Which is fine, really, in the broader scheme of things.

e, 10.

For the sake of cosmology, they're more or

less the same thing.

All the e's become 10's, and you're got 10 to the 10 to the

10 to the 2.08.

Which hopefully is what he's got there, and it is.

So that's where that number comes from.

So this is basically the Poincare recurrence time for

our visible universe, for our patch.

BRADY HARAN: Why is the other number bigger?

TONY PADILLA: Well, because the other number--

there, he's looking at--

he's trying to get a big number is what

I guess he's doing.

He's trying to get a bigger number.

What he's looking out there is a model of inflation.

Now, inflation is a model of the very early universe, where

the universe grew really quickly out of a very small

patch of the universe.

The amount by which it blows up depends on various

parameters in the model.

But basically, the thing that you get is you get to the size

of the universe.

So r, for that case, is of order e to

the 4 pi over M squared.

So this is actually r over l Planck.

He's done everything in Planck units.

But M here is the mass of some of this influx, what we call

the inflaton field.

It's just some field in the model

that causes the expansion.

1 over M squared is 10 to the 12.

I think this might be 13, actually.

10 to the 12.

So this is about 10 to the 13 overall,

because 13 is about 10.

Bear with me.

So this is e to the 10 to the 13, roughly.

The 13, well, is about 10 to the 1.1.

That's where the 1.1 comes from.

He's very precise about this 1.1, and yet he's very sloppy

with some of the other factors.

You get e to the e to another e to the 10 to

the 10 to the 1.1.

And then we do the same thing.

We turn all the e's into 10's.

BRADY HARAN: Help me understand, because obviously,

you have bit numbers there.

How long a time is this?

TONY PADILLA: OK.

So this is truly vast.

Like I said, there's no device, there's no observer,

anything that could survive this kind of

length of time scale.

In fact, you would probably say that the universe is more

likely to tunnel out of the current state before this

could happen, in some sense.

This is such a long time scale that one might say that

actually the probability of tunneling to a new phase of

the universe, completely different, is actually going

to dominate over this.

That that would occur first.

So maybe this is kind of an irrelevant point.

Yeah.

It's truly vast.

It's bigger than a googol, clearly.

Way bigger.

Is it bigger than a googolplex?

I think so.

So you know, let's just check that.

Yeah, clearly it is.

It's enormous.

It won't be as big as Graham's number, but you know, Graham's

number's the daddy, right?

So.

Well, I didn't actually know about this paper.

But yeah, I was just--

I just thought, oh, big numbers.

Let's see what's interesting about big numbers.

And then I stumbled across this paper.

What he claims-- in fact, he says it--

he claims, "So far as I know, these are the longest finite

times that have been explicitly calculated by any

physicist." So whether somebody has calculated a

longer one--

and I'm sure some of the viewers will try to calculate

a longer one and then claim that they've--

but this is in a published paper.

So you've got to get the paper published.

But the challenge is on, I guess, to find a longer one.

There probably has been, too, but he was the one that

pointed it out.