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that took the principle of relativity to be true,

the principle of light constancy to be true.

When you put them together, the conclusion is that the speed of light is constant for

all observers.

That no matter how a flashlight, a light source will be moving, whether you're

moving toward it, it's moving toward you, at some angle or whatever.

Going through the whole proof theorem.

But essentially argue that based on Einstein's two postulates,

speed of light is always constant.

So this week then we've been working out some of the implications of that and

we started with Einstein's famous phrase, the idea that time is suspect,

which led us to the idea that simultaneity is relative.

If you have two inertial frames of reference,

inertial being constant velocity with respect to each other, and

you have two lattices of clocks in each one, say Alice and

Bob each have their lattice of clocks, their system of clocks and measurement.

Everything is synchronized for Alice.

Everything is synchronized for Bob.

The fact that the speed of light is the same for

the both of them leads to some weird results.

Just remind ourselves of the basic value RAM we used.

Obviously we won't go into all the details.

You can go back and watch the video lecture for that.

But theme of relativity of simultaneity and

then also this idea that leading clocks lag.

So let's just quickly remind ourselves about that.

We start off with this situation here, an imaginary, say,

Alice's ship with a clock at each end, and

some sort of device that could shoot out either paint balls, or light pulses.

If Alice's ship is stationary, then- and

she shoots out whether it's paint balls or light pulses.

Clearly, if this little device is right in the center of her ship,

she will see the two paint balls, light pulses, hit the clocks at the same time.

So assuming her two clocks are synchronized and

when these light pulses hit the clocks, the clocks stop or you take a photograph

at that point, then both of these clocks will be the same thing.

And Bob, down here, using his lattice of clocks, we get the same result.

Everything is stationary here.

Nothing is moving.

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to the right, positive x direction with a velocity v, speed v.

And then we analyzed what happens in that case

with the paint balls, one paint ball going that way, one paint ball going that way.

The velocity of the ship gets added to this paint ball.

So even though this clock, once the paint ball's released here,

it's sort of halfway in-between here on its way over there.

This clock is moving away from it.

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the case of light pulses, then the light pulses in-between there Travel at speed c.

The velocity of the ship does not add on to the light pulses.

This is the whole idea of the constancy of light, of the velocity of light,

of the speed of light.

That, it's always c, no matter what's happening.

And so, even though it seems like, intuitively,

because our everyday experience, if Alice's ship was going that way,

the velocity of this light pulse going that way should be c + v, it's not.

It's simply c, and this one is not c- v, it's simply c going that way.

So what that means from Bob's perspective now here,

let's take a look at Bob's perspective.

He's watching Alice's ship go that way.

This clock is receding away from this light pulse,

obviously the light pulse is going faster so eventually it will catch it, but

it has to sort of run ahead to catch it up here.

Whereas this clock is heading toward the leftward going light pulse,

the result being Bob will see the rightward going

light pulse hit this clock later than this light pulse hits this clock.

Or, say, perhaps a better way, is that this light pulse hits first, and

then a little bit later, that light pulse hits that clock.

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Well, what's so strange about that?

Well, consider from Alice's perspective, her frame of reference.

She's at rest as far as she's concerned.

She shoots one light pulse out one way, one light pulse out the other way.

They're both with velocity of c, identical velocities.

She will see this light pulse hit the clock.

Let's just say it hits at t sub a here on her clock.

And this we'll also hit at t sub a.

And she's going hey, no big deal.

Bob on the other hand is saying wait a minute,

Alice, something's wrong with your clocks.

Because Alice says hey, my clocks are synchronized.

They're in fine working order.

Bob is saying though, as I observe it I see this light pulse,

actually let's do the left one, hit this clock first and

then a little bit later this light pulse hits that clock.

Take a photo.

They both take photos at that instant.

Clearly, they have to agree that when this light pulse hits that clock it's t sub a.

When this light pulse hits that clock it's t sub a..

They agree on that but Bob's saying,

according to my clocks down here, my lattice of clocks, this one hits first and

then this one hits second which means, Alice, your clocks are not synchronized.

They are out of whack.

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The point being is, what seems like should simultaneous events,

these two light pulses hitting these two clocks,

Alice thinks they're simultaneous, Bob thinks they're not simultaneous.

Assuming obviously that v here,

the velocity is high enough such that you start to see these effects.

The effects occur no matter how big the velocity is here,

it's just that the speed of light is so great compared to our normal velocity,

the actual effects are so small we don't see them in everyday life but

get up near the velocity of the light and you'll start seeing these effects.

So we talked about the relativity of simultaneity.

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We also talked about this idea of leading clocks lag, because think about this

a minute that as Bob sees Alice's ship going by here, this is the leading clock.

She's going that way, Bob observing it and this is the rearward clock,

the lagging [INAUDIBLE].

So, as we just argued, Bob sees this one hit this clock first and sees.

Hey, that's t sub a.

And then a little bit later, he sees this one hit that clock.

And then that clock reads t sub a.

In other words, this clock, as far as Bob is concerned, reaches tA first,

and then the leading clock reaches it a little bit after that.

In other words, from Bob's perspective, from his frame of reference observing

Alice's ship, the leading clock lags, a trailing clock in that direction.

And they're on actually next week, I think it is, we'll look at that quantitatively.

But for now, we can see qualitatively, leading clocks lag,

it's all based on the fact that speed of light is the same for all observers.

So you get relativity of simultaneity that what's simultaneous

in one frame of reference, inertial frame of reference, is not necessarily, is not

in general in another frame of reference in this whole idea of leading clocks lag.

So, that was time as suspect.

Then we looked at this case of the light clock and

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discovered that there's this time dilation effect.

So let's just briefly remind ourselves.

We have a stationary clock here, the light clock, at two meters.

We bounce c up, a beam of light up and a beam of light back down again, so that's

one tick of the clock, up and down, or tick tock, if you want to do it that way.

And then we said, okay, let's have two identical light clocks, and

we'll keep one beside us here.

Maybe Bob has that one.

And then we'll give another one to Alice.

And side by side, together, they're in perfect sync, perfect working order.

And then we give one to Alice, she gets on her spaceship and

travels by Bob at some velocity v.

And so Bob then can see his stationary clock right here.

He's at rest with respect to his frame of reference, and

he watches Alice's clock go by and takes three snapshots.

So here's snapshot 1 of the clock, snapshot 2, snapshot 3.

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And he sees, clearly he has to see, the beam of light hit the top mirror,

all right, because Alice,

in her frame of reference, in her spaceship, as she's travelling along.

But as far as she's concerned, the clock is right here, it's just going up and

down, up and down, hitting the top here, bottom here, top here, bottom here.

So clearly, Bob has to see essentially the same thing, in other words,

he has to see it hit the top here.

And so the only way it can do that is if, from his perspective,

the beam of light travels a diagonal path up to the top mirror, and

then third snapshot, he sees it back down there at him.

And clearly then, one tick of the stationary clock,

Bob's stationary clock next to him, is faster than one tick of the moving clock,

or the way we often say it is the moving clock runs slow.

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And maybe if you can get close to it,

time'll even stop if you get to the speed of light.

Can't quite get there, but say close enough, time essentially stops.

That's not quite true because Alice, who is in the spaceship, going say,

near the speed of light, her light clock is working just fine.

She notices nothing, right?

But it's Bob observing her clock that he says,

hey Alice, if they got back together again, your clock ran more slowly.

And this will be one of the key points,

when we look at the twin paradox later on in the course, okay?

And we did a little quantitative analysis, and

we found that the elapsed time on a moving clock, delta T on a moving clock,

elapsed time is 1/gamma(delta T) on the clock at rest, the stationary clock.

Where gamma, the so-called Lorentz factor, is 1/square root of 1-v squared/c squared.

And we also explore that a little bit more on one of the video lectures, the short

one, just plugging in various values of v to see what gamma would be like.

And we found that v has to get pretty big, up around half the speed of light,

maybe 0.3, 0.4 the speed of light, you see some noticeable effects,

but not until then, does gamma really get much bigger than 1.

And we know that gamma's always greater than or equal to 1.

If v is 0 here, then both our clocks would be just stationary next to each other, and

gamma would be 1.

And delta T of the moving clock, elapsed time on the moving clock,

would be the same on the stationary clock, they'd both be stationary.

But put one into motion, and the elapsed time on the moving clock

will be less than the elapsed time on the clock at rest.

Because gamma is always greater than or equal to 1 here, as for

any positive v, any v at all really, any value of v here, this,

in the denominator, as long as v is not 0, will be less than 1.

So 1 divided by something less than 1 gives you a quantity for

gamma greater than 1.

So again, moving clocks run slow.

The elapsed time on a moving clock is less than the elapsed time on an identical

clock at rest because gamma is always going to be greater or

equal to the 1 here, so like 1/3 of gamma is 3, that would be 1/3.

The moving clock for every, say, the clock at rest ticks through 9 seconds,

the moving clock, if gamma's 3, will tick through 3 seconds.

So, that's substantial difference there.

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Okay, so that was the idea of time dilation, time slows down for a moving

clock, but it's not the person who's actually doing the moving who observes it.

It's the person who is observing the person moving that their clock runs slow.

And it's in common language when people talk about this they often get

that mixed up.

They assume that moving clocks run slow.

So the person's moving if you get near the speed of light will actually feel

like they're aging more slowly.

Their clocks will run more slowly.

In actual fact, it doesn't work quite like that.

Then we moved on and

saw that well, there's something weird about time, time is suspect.

We also learned that length is suspect,

we talked about the length contraction effect, did an analysis like this.

Just to remind you of the picture that we drew here, we had Alice and

just one clock for Alice.

And say, here's spaceship, we're just sitting there, right?

And here comes Bob by at some velocity, v.

And he has his lattice of clocks going along there.

Alice, of course, has hers, too,

but we just need to focus on one of her clocks for this analysis, and

the idea is Alice is going to measure the length of Bob's spaceship.

So, he's coming by at velocity v, and when the front of his spaceship gets right

to her clock, she takes a photograph and notes the time on her clock.

And then waits until the end of Bob's spaceship gets here, takes

another photograph, and notes the time on her clock on that photograph as well, and

then they also note the time on Bob's clock.

So, photograph at the beginning on this clock of Bob's, and then

as this one comes along, take a photograph of that one and note the time on that.

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And so he's saying, well, that's the problem, Alice.

Your clock isn't working right.

My clocks are in fine working order.

Compared to my clocks, your clock is running slow.

And that's why you got the wrong length for my ship.

Meanwhile, Alice, her perspective, of course,

is Bob is coming by her with this velocity and she sees it slightly differently.

She actually is, for Bob, he's using two different clocks to measure the length.

He knows the time here and then he knows the velocity, v,

he knows the time when he gets to there.

And therefore the difference in times, times the velocity, gives the length.

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Remember, leading clocks lag.

She says, your leading clock here is lagging this clock.

They're not synchronized.

No wonder you're getting the wrong answer.

So they get two different answers and they explained it in different ways, but

ways that are consistent with our results from the special theory of relativity.

And quantitatively, then, we show that the result is that

the length of a moving object, length in the direction of motion,

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is, if you observe that moving object and you measure the lengths,

observe like Alice measuring Bob's moving spaceship here, it will be 1 over gamma,

gamma factor again, greater than 0, times the length at rest.

In other words, Alice says, I want to figure out Bob's ship here,

what its dimensions are, and so Bob comes in, they sit down.

They both measure it at rest, they get one value.

But then when he's going flying by with some velocity and

Alice takes a measurement again, she would get a shorter result for that length.

So gamma might be 3, so it would be one-third times the length at rest.

So length contraction, the length of a moving object when you

measure it is less than you'd measure it when it's at rest.

And when it's at rest, sometimes we call that the proper length, or

just the rest length.

Proper length, again, as I mentioned in the video lecture,

it's not a misnomer in a sense, but it's a little confusing because it then seems

like other lengths are not proper, improper lengths.

But it's just an idea that the length at rest versus the length when an object

is moving, measured when it's moving.

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Just remind ourselves a couple things about that.

So what is not suspect?

Because we have, well, time is suspect, length seems to be suspect here.

What else is suspect or not suspect?

And so we had an argument to say that transverse dimensions are not suspect.

We looked at the train car with the train tracks and we looked at both Alice and

Bob's perspective, and we saw that if the width of something moving along this

way contracted, we get contradictory results.

One of them, we'd see the width getting narrower of the train car.

One of them, we'd see, so say Alice would see, if it was width contraction,

as Bob went by in the train car, Alice would see the width getting narrower and

therefore the train would fall through the tracks.

From Bob's perspective,

sitting in the train car, he'd see the tracks coming toward him.

So if there's a width contraction effect in effect, then he'd see the tracks

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getting narrower, and so the train car wouldn't fall through the tracks, so

the wheels would sort of go off the ends.

So we have two contradictory effects there,

the conclusion being that there is no width contraction.

And then we did a similar argument for the height of the train car versus the tunnel.

Again, concluded no width contraction, no height contraction, or anything like that.

It's only in the direction of motion that you get the length contraction effect.

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Took a little more doing, maybe not too much doing, to show this result.

But the idea here is if you have two events and you measure the location and

time of those two events in different frames, even though,

say, Bob and Alice and we had Chris as well, they all disagreed

exactly where the two events occurred and the time between the two events.

We use flashes of light and light clock.

But this quantity, if we did c squared

21:23

And it actually has to do, we still have the little light clock here,

as you may remember, it has to do with the height of the light clock was the same for

Alice and Chris and Bob.

The transverse dimension does not get contracted or expanded or

anything like that.

It's constant and that's really where this result comes from.

So what that means is, for

Alice, say, speed of light, c, is the same for all of them.

Alice would have a certain time, t sub A, between two events and

a certain distance between two events measured in her frame of reference.

Chris could take similar measurements and she would also get c squared

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tk squared minus xk squared.

She'd have different values for the time between the events and

the distance between events than Alice would, but you get the same result.

When you do this calculation, you'd have the same answers there.

And the same thing for Bob, if we did a version for Bob.

x sub B squared = constant, the same constant as well.

So all these things are the same.

The example we actually did had Bob being the distance between the events was 0.

So sometimes you just get that.

But in general, time and distance between two events, between various initial frames

of reference, this value, this quantity will always be a constant.

We call it the invariant interval and it'll be useful for

us later on with the Lorentz transformation.

And then finally we did an example with a muon,

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So subatomic particle, what happens is, as cosmic rays come into the atmosphere,

they hit some atomic nuclei up there in the atmosphere.

And it caused them to raise very high energy particles coming in,

probably generated in supernova explosions elsewhere in the galaxy.

And so we're always getting cosmic ray particles coming through.

Some of them hit things up in the atmosphere, create muons and

other particles.

Some of them actually make it down to earth.

Maybe you've been at a science museum before where they had a cosmic ray

detector, a sort of spark gap type thing.

And the cosmic ray going through would generate a spark, so

you can actually see their paths go through.

Anyway, that's a side point there.

What about the muon?

Well, As these cosmic rays come down.

Remember you get these muons being generated at about 10 kilometers

high in the atmosphere.

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And it turns out the speed of the muons generated is very, very high,

about 0.998 times the speed of light.

It's very close to the speed of light.

And at that speed, turns out that gamma, the gamma factor,

Lorentz factor, is about 15.

And so the question is, okay,

this is the lifetime, you can say the proper lifetime, the rest lifetime.

In other words, if a muon is at rest, it lasts 2.2 microseconds.

At this speed, 0.998c, if it lasts for

2.2 microseconds, it can only go about 600 to 700 meters.

Okay? So if it's generated 10 kilometers high in

the atmosphere, it shouldn't be able to get down to Earth.

It can go about less than one-tenth of that way, not even one kilometer.

And yet we see the muons down here at the surface of the Earth.

So somehow they get from 10 kilometers high,

we know that's when they're generated, a lot of them, and they get down to earth.

So what's going on here?

Well we can explain it and

get the correct result only using special theory of relativity.

The idea is this.

We're observing the muons.

Its lifetime is 2.2 microseconds in the laboratory, when it's at rest essentially,

but when it's traveling at a high velocity, what do we see?

We observe its internal clock, as it were, undergoing time dilation.

And the time dilation with gamma 15, that means it's about a 15 times factor for

the time dilation effect.

In other words, we observe the internal clock of the muon as it comes screaming

down towards us at 0.998c, that its lifetime lengthens by 15 times.

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This is average lifetime there.

That explains how it gets down to Earth.

But then also, what about from the perspective of the muon?

Because you could say, well, let's ride along with the muon, if we could,

and it has its sort of internal clock, ticking away.

There's no time dilation effect in that case.

It's in its rest frame of reference.

It's going to live for 2.2 microseconds.

How in the world, from this perspective,

does it get down to earth ten kilometers away from the surface of the Earth?

Well, here's sort of the ingenious part, or

the amazing part of the special theory of relativity.

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It doesn't have time valuation, but it does have length contraction going on,

not for itself, but it's seen the surface of the Earth rushing up towards it

in its rest frame of reference because the surface of the Earth is coming towards

it at 0.998c, Gamma factor 15.

It sees that length to get down to the surface of the Earth,

not to be ten kilometers, but to be one-fifteenth of that.

In other words, about 600 or 700 meters or so.

So it only, in its frame of reference, it only survives for 2.2 microseconds, but

it doesn't have as far to go to get down to the surface of the Earth.

Whereas, from our perspective, our perspective on the outside of the muon,

it has to go ten kilometers, because that's what we see in our rest frame of

reference, but it lasts 15 times longer, and therefore is able to get there.

So that's a quick, relatively quick maybe,

summary of Week Four, where we saw some of the weirdness begin,

and these are things that we're constructing mental models here.

And mental models are very different from how our ordinary experience goes.

And therefore it takes some time to sink in, to play around with these things.

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From my own personal experience learning this, it takes time to sink in and

become familiar with it, and comfortable with it enough so

that, yeah, you sort of understand that there.

Remember again, I don't know if we mentioned it recently, but we started off

with the course with one of Einsteins' quotes about the struggle to understand.

And certainly, at times, there can be a struggle to understand some of these

things, but if you persevere, work through it, and hopefully gain some insight

in that, as Einstein put it, it can be a even a noble and enriching experience.

I hope we're getting a little bit of that as we get into some of these weird things

with the special theory of relativity.