0:03

So now we come to the pole in the barn paradox,

one of the famous paradoxes coming out of the special theory of relativity.

You can see we've already written a number of things on the board here so

we can save a little time as we go along.

But we'll take it one step at a time, make sure we understand where we start and

where we end up and in between, where we go with this.

So here's the situation, back to Alice and Bob again.

We assume Alice has some pole vaulting ability.

So she has a long pole, that is 10 meters in her frame of reference.

Okay, so Alice's pole here.

10 meters in her frame of reference, and she is moving.

She is a very fast pole vaulter.

She's moving at a velocity of 0.6 times the speed of light.

That gives us a gamma factor, a Lorentz factor of 1.25.

And then we have Bob over here.

Bob owns a barn.

It's Bob's barn.

And, his barn is 8 meters long, okay?

Not a huge barn, but 8 meters long there.

And that's in his frame of reference.

Again, Alice's 10 meter long pole is in her frame of reference,

in terms of how she measures it.

It's stationery with respect to her.

She gets out her tape measure, it's 10 meters long.

When Bob is stationed in his barn,

he gets out his tape measure and it's 8 meters long for him.

They both have accurate tape measures.

1:25

And Bob also has, at the barn, has a door in the front and

in the back which we haven't shown here, but you can imagine that, and clocks,

two clocks synchronized, one at the front door and one at the backdoor.

And now let's just write down some of the key numbers here.

In Alice's reference frame, first for

pole, L sub pole sub Alice, A for Alice, of course, is 10 meters long.

1:49

But in her reference frame, Bob's barn

is Lorentz contracted, so it's a 1 over gamma factor here.

And the proper length of the barn is 8 meters, and so

we've got 1 over gamma, 1 over 1.25 times 8, 8 divided by 1.25 you get 6.4 meters.

So from Alice's perspective, from her frame of reference,

she sees Bob's barn as 6.4 meters long and her poll as 10 meters long.

Meanwhile, Bob, in his reference frame, he sees his barn of course has been 8

meters long and he's observing Alice's pole she's coming along here.

And, again, Lorentz contracted.

And so, 1 over gamma, the 1.25 factor here, times 10 meters for

her pole, gives us 8 meters.

And so what Bob says to Alice, Alice, hey, why don't you run through the barn here,

I'll open the doors.

And, in fact, do a little test here.

Just as you get inside the barn, okay, or not just as you get inside, but

as you get all the way inside the barn, so

your pole reaches the back door I will close both doors simultaneously.

And clearly there's enough room, if I do it quickly enough, very, very quickly.

My barn is 8 meters long, your pole is 8 meters long.

Clearly there's enough room to close both doors simultaneously,

you'll fit this nicely.

And don't worry, I'll help them again really quickly so

you can keep running out there.

Of course in actual fact you couldn't do that and

poor Alice would splat against the back door probably.

But to make this situation a little more realistic,

perhaps, we won't have the doors close.

We'll just have Bob take two photographs, okay.

So, as Alice come through here, Bob again will see the barn is 8 meters long.

Alice's pole has also 8 meters long.

And just when Alice's pole reaches the back door here,

he'll take a photograph of the front of the pole and at the same time according to

his perspective here, the back of the pole should be right at the front door and

it'll take a photograph then simultaneous.

And therefore he'll see the front and

back of the pole in both photographs and the two clocks will be synchronized.

They'll both read the same thing.

4:02

Alice, meanwhile, is just thinking Bob is crazy here.

There is no way my 10 meter long pole is going to fit inside his barn

there because it's not even 8 meters, its 6.4 meters there.

And so she thinks he's crazy.

So let's now analyze this and

see what happens if they had actually do something like this and we'll assume

that the front of the pole is Alice runs to the barn, enters the barn at 0.

Okay, so Alice's clock reads 0, Bob's clock reads 0.

We can certainly do that for one spacetime point, we can synchronize clocks.

We can't assume that other clocks of course are synchronized in terms of

4:43

I should say Alice's clocks will synchronized to her.

Bob's clocks are synchronized to him.

But we can't assume that just because Alice and

Bob synchronized one pair of clocks at a certain baseline point.

It means L sub Bob's clocks,

of course, are synchronized and in other instances as well.

So, let's see what happens.

Bob observation is pretty straightforward, assuming

we think about just the fact that he's got 8 meters here.

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And sees that Alice is coming toward him at 0.6c, meters per second.

And we want to first just figure out,

how long does it take the pole to reach the back door.

So again, the pole enters right at t = 0 here, traveling along,

meets the back door.

Well, as we go 8 meters from here to here, as far as Bob's concerned.

It's at 0.6c meters per second.

When you do all the math there and get the units right,

you come out with 44.4 nanoseconds, billionths of a second there.

So that's what Bob reads on clock number 2 here,

in other words the front of the pole reaches the back door in 44.4 nanoseconds.

And actually, the rear of the pole, the back of the pole reaches the front door

in that same time because again it's 8 meters long so when the front is here,

the back of the pole has to be here as far as Bob is concerned so

both clocks will read 44.4 nanoseconds at that point in time.

And so Bob says hey, no problem at all here.

What's the deal Alice?

Don't worry about anything.

Well let's look at what Alice observes then.

Because clearly from her perspective it seems how can Bob close both doors or

take both photos and show that the front of the pole is right there and

the back of the pole is right there.

It's way too big here, 10 meters versus 6.4 meters.

So, let's break down Alice's observation step by step.

So, remember, from Alice's perspective it's not, she who is running toward

the barn, the barn is actually approaching her, is coming towards her.

In her frame of reference, she is stationary.

And so she sees, again, we start at t = 0, right, when her pole gets

to the barn here, well, I should say, the barn gets to her at the front of her pole.

And that's a t = 0.

And then the question is okay,

how long does it take the back door to get to where the front of the pole is.

Well, from her perspective, 6.4 meters, that's how long the barn is.

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So 6.4 meters, the barn is coming towards her at 0.6 times the speed of light.

Do the simple arithmetic there, you get 35.6 nanoseconds,

billionths of a second, to reach the front of the pole.

So the rear of the barn takes, from the time the pole just enters the barn,

or really the barn just gets to the pole.

It takes another 35.6 nanoseconds to from the back of

the barn to get to the front of the pole.

And then you look at that element you say, that how can that be because,

Bob's clock is 44.4 nanoseconds, how can Alice begin at 35.6 nanoseconds.

Well we have to remember a couple of things here,

we have done things like this before actually, so now this is Alice's clock.

If she is observing Bob's clocks,

she observes them running slowly, right, moving clocks run slow.

As far as she's concerned, Bob's clocks are approaching her at 0.6c,

they're running slowly.

And so we get the 1 over gamma factor.

So in actual fact, this doesn't even seem to help though because she says okay,

hey, Bob's clocks are ticking off between the time the Barn reached the front of

the pole and then moved back, so the rear of the Barn got to the front of the pole,

and on my clock and Alice's clock, 35.6 nanoseconds.

But if she's looking at Bob's, clock they're ticking more slowly compared to

her by the 1 over gamma factor, so it's actually 28.4 nanoseconds.

It seems like we're going in the wrong direction here.

We're trying to get the match 44.4 nanoseconds because remember,

there was a photograph taken right at that instant in time.

Just when the pole, the front of the pole,

reached the back door, Bob took a photograph.

Everything was there, all Alice's clocks.

She has clocks on her pole.

And 44.4 nanoseconds, and, yet,

Alice is saying Bob only reached 28.4 nanoseconds?

Well, one other factor as you've probably read ahead here, you can see here.I didn't

write it down, but you should recognize this, leading clock's lag, right?

So relativity of simultaneity, many of these relativistic paradoxes

in terms of the special theory of relativity go back to, have at the heart,

their foundation, this idea of the concept of relativity is simultaneity,

that what is synchronized in one frame of reference is not in another.

So remember the formula for that.

Again, so Alice is observing Bob's clocks coming toward her.

She's looking at two of Bob's clocks here at the front and the back.

Leading clocks lag.

This clock, clock number 1 is going to lag clock number 2.

Let's focus on clock number 2 for a moment here.

10:11

Bob's clock number 2, as Alice observed it, she had her clocks and

took another photograph right at that point would be reading

Dv over c squared ahead of this clock.

Both Alice and Bob's clocks, remember, were synchronized at zero at that point.

So at zero at this point, and if Alice snuck a photograph of clock

number 2 at that instant in time, in her frame of reference, she would see clock

number 2 for Bob, not to be synchronized with this one but to be running ahead.

Remember leading clocks lags, so

this one is running ahead, as the Dv over c squared factor.

One of the reasons we did that calculation awhile back so we could do things like

this and actually make it consistent, make it come out, so what is that?

Well, remember the D here is the difference between the two clocks

in their frame of reference.

And the distance is 8 meters in Bob's frame of reference.

So we have an 8 for D.

V, of course, is 0.6c, the relative velocity between the frames of reference.

And, c squared on the bottom, do a little bit of algebra, you get 16 nanoseconds.

So, look what happens,

when the Barn reaches the front of Alice's pole,

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But this clock number two is already ahead by 16 nanoseconds.

So what we have here is it's running 60 nanoseconds ahead,

plus then it ticks off the 28.4 as the barn moves

to that point where the rear of the barn reaches the front of the pole.

And add those together and what do you get?

16, it was running ahead by 16 nanoseconds.

24.8 nanoseconds ticked off while the Barn, the rear of the Barn moved

to where the front of the pole was, as far as Alice is observing.

And you get 44.4 nanoseconds.

So, when that photograph actually occurs, yes, both Bob and

Alice agree Bob's clock should read 44.4 nanoseconds but for different reasons.

Bob for fairly straightforward reasons,

just 8 meters divided by the speed of the pole going through.

Alice, a little bit more complicated calculation because we have to take into

account these relativistic effects.

But yet she can agree that it reads 44.4 nanoseconds.

She doesn't disagree with the photographic evidence,

though she does tell Bob your clocks are really messed Bob.

You need to get that fixed.

Okay, so that takes care of the backdoor, in terms of 44.4

nanoseconds at that point, but remember going back here,

Alice thinks the Barn is 6.4 meters long, her pole is 10 meters long.

That's not a thing, she observes those to be that way.

So when the rear of the Barn reaches the front of the pole,

her pole is still sticking out.

And so that's our next question we have to do.

What about the front door?

What's going on with the front door in terms of the photograph that's taken then?

Because remember Bob had two photographs taken, one at the front, one in the back.

And both of his clocks were at 44.4 nanoseconds.

We've matched up the backdoor situation, we understand that.

And now we need to look at the front door situation.

So let's give it a little room here with that.

14:08

Okay, let's try a little diagram what's going on at this point, again,

from Alice's perspective now.

From Bob's perspective, straightforward, just two synchronized photographs,

front and back, 44.4 nanoseconds, pole's right there.

Boom, we're done.

Okay. Now though, let's look at, so

let me just emphasize this, okay?

So this is the key figure, the 44.4 nanoseconds there and

remember that Alice got there first.

She saw her clock tick off 35.6 nanoseconds but

then it actually means Bob's clocks take off 28.4

nanoseconds because at time dilation as she is observing Bob's clocks.

And then we've got the leading clocks leg factor

from the clock number 1 versus clock number 2 and that's 16 nanoseconds.

And then you put those together, the 16 plus 28.4, and we get

the 44.4 nanoseconds which Bob also saw on his clock and which was in the photograph.

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Okay, where the barn has been rushing towards her and

we're just to the point where the rear of the barn reaches the front of the poll.

And then Alice is there and get the 44.4 seconds.

But, remember she sees the barn not as 8 but

6.4 meters because it's length contracted.

So for her, she got 6.4 meters of the pole inside the barn at this instance and

another 3.6 outside because the pole is 10 meters for her.

Okay, so what's going on?

Well, again,

remember we need to have a situation such that when this end gets to there.

When, really, from Alice's perspective, when the front door gets to the rear of

her pole, this clock needs to read 44.4, because that's the photograph Bob took.

Well again, remember the leading clocks lag effect.

At this instant in time, okay, when this photograph is taken we know that

clock is reading 44.4 nanoseconds.

In Alice's frame of reference, this clock is ahead of this one.

And by a factor of 16 nanoseconds.

So in fact from Alice's perspective when this reads 44.4 the front one

16:49

here is going to read 44.4 minus 16.

Which we've already calculated actually it's 28.4, so 28.4 nanoseconds.

Okay, so, now imagine Alice.

In very, very slow motion here she's watching the barn go by, and

she's watching Bob's clocks tick off.

And so, the barn is going to move now from square meter is along green here sobarn.

From the barn is still moving expect her and it's going to move to that point,

right there and that's when the photograph would be taken.

Bob's photograph showing this cloth and so, let's analyze then

how long that take's would suspect to Alice's frame of references, right?

Well, again, this is just v here, 0.6c.

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So she has to watch this, so it's 28.4 nanoseconds,

60 nanoseconds to go on Bob's clocks here until it get to 44.4.

So the question is how long does it take the front of the barn here

to get to the end of the pole?

Does it take that extra 16 nanoseconds?

On Bob's clocks, yes, but what about Alice's perspective?

Once again we have to remember time dilation.

From Alice's perspective, she's seeing Bob's clocks tick slowly.

18:14

And we used it over here.

And this case we know that

Bob's clocks are going to tick 60 nanoseconds from here to here.

Because again, we know the photograph was taken at this point and

at that point clock number one reads 44.4.

And right now it reached 28.4 from Alice's perspective, difference of 16 nanoseconds.

So Alice is going to see Bob's clock tick off 16 nanoseconds

while the barn goes from there to there.

But it's not 16 nanoseconds for Alice.

Bob's clocks are running slowly.

So, for Alice, it's more time.

In fact, it's gamma times 16

nanoseconds equals 20 nanoseconds.

Okay, during the time Bob's clock ticked off 16 nanoseconds while

the front of the barn is traveling toward the end of the pole,

as Alice is observing it Bob's clocks tick off 16 nanoseconds because again we know,

we got 20.4 here, we know the photograph is taken at 44.4 nanoseconds

according to this clock here, so we got the 16 nanosecond difference.

Drawing that 16 nanosecond on Bob's clocks,

Alice's clocks tick off 20 nanoseconds.

19:32

Okay, so from Alice's perspective, she's waiting 20 more nanoseconds for

the front of the barn to get to the rear of the pole.

And how long, what distance does the barn cover during that period?

Well, we just do 20 nanoseconds,

that's how long we have on Alice's clocks and speed 0.6c,

that's how fast the barn is traveling along here, from her perspective.

If you do the math here, 20 nanoseconds times 0.6c, you get 3.6 meters.

In other words the exact distance the front of the barn needs to travel to

get to that point according to Alice's clocks.

So somewhat incredibly it might seem everything checks out,

everything is consistent.

Except, of course, Alice and Bob disagree on whether things are synchronized or not.

Bob is saying, hey my clocks are synchronized, okay?

And with length contraction of your pole, clearly your pole fit in my barn.

Okay, because I have photographic evidence of it.

The front of the pole was at the rear door at 44.4 nanoseconds, and the front door

was at the rear of the pole, and fit just inside, also at 44.4 nanoseconds.

What is there to debate here?

Meanwhile, Alice is saying, well, somehow I made it through here,

but clearly your clocks are off and clearly at this instant in time,

when your rear clock was reading 44.4 nanoseconds and we took the photograph or

you took the photograph there I agree with that, yes, it is 44.4.

But, the pole was still sticking up by 3.6 meters,

and it's only because your clocks are so unsynchronized that,

once the barn actually got to this end that it registered 44.4 nanoseconds.

So yes I agree with the photographic evidence there,

but I disagree that the pole actually could fit inside the barn, just because

if we're actually closing doors you'd have to close the door quickly here and

open again so that the pole could get out.

But again it goes back to the relativity of simultaneity which in turn goes back to

the fundamental postulate of Einstein

that the speed of light is constant in all reference frames.

Once we have that and

the principle of relativity then we start to building these principles.

On top of that, and

got to things like the relativity of simultaneity like time dilation,

like length contraction, and then the Lorentz transformation after that.

And now we're working out some of the consequences in examples like this.

Again, we don't see this typically in real life, because even though

technically it does occur in real life but the velocities involved are so small,

that the time differences are so small and length contraction is so small that we

just don't observe it unless you're using very precise physical experiments.