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Now that we've seen some of the results of the special theory of relativity

in terms of measuring time, more specifically time dilation,

the time dilation effect Let's turn now to measuring length and see what

we can find out about that, especially measuring length of a moving object.

And so here's our situation.

We've got Alice and Bob again.

Here is Alice.

She is stationary.

This is her ship or just some platform or something.

We're just going to focus on one clock of hers.

For this example and then here's Bob on his spaceship going by at some velocity v.

We've drawn a number of clocks for him.

Just to remind ourselves that actually in all of the things we're doing

we always have that lattice or grid of clocks all synchronized,

so In Bob or Alice's frame of reference,

so all Bob's clocks are synchronized, he thinks they're in good working order.

Alice, we're just showing one clock for her, but

she'd have a whole of clocks as well, and they would be synchronized for her.

So, but remember of course, as we showed earlier.

Alice would not see Bob's clock synchronize and the other way around.

So, that'll be a key point here in a few minutes.

But let's get back to the example.

So, Bob's going to fly by here, add some velocity v and

Alice is going to measure the length of Bob's ship as he goes by.

And so, to do that, when the front of his ship reaches her clock here,

She'll take a photograph, it's our photograph principal again, and

she'll see the time on her clock, we'll call that t sub a1,

in other words, Alice's clock time one, and

the photograph will also include the time on Bob's leading clock here, tb1.

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Okay? Her clock at time two, and

Bob's clock will have some tb2 on it as well, time t b number two there.

So, how then does Alice get the result for the length of Bob's ship?

Well, it's simply We know that it's going to be essentially the distance

traveled here, we know Bob is traveling at a constant velocity, V,

that's what Alice sees him doing.

He'll travel a certain distance in that amount of time,

represented by essentially the end of her ship coming to this point here.

And so that would be the length of his ship as start the clock here,

wait until you see the end of the ship go by.

Stop the clock as we did right here in our second snapshot.

So tA to there.

And it's very simply You take the velocity of the ship and

multiply by the elapsed time, between the beginning, the time that you see

the beginning of the ship, the front of the ship, and the back end of the ship.

So I also will say, okay, I've got a result.

And the result is the velocity times the time difference,

t sub a 2 minus t sub a 1.

Just elapsed time between those two Points on her clock.

So t sub a two.

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And the Bob says, well, you know, I can also get the length of my ship from our

little experiment here because I've recorded the photograph record t b 1 and

t b 2, so that's my elapsed time in terms of my clocks.

I know I'm going at a velocity v or really You could think of it from

his perspective as he could imagine he's at rest and Alice is going by.

So he measures when Alice is at the front of his ship,

TB1, and how long it takes Alice to get to the back of his ship, at TB2.

He's going with velocity, it'd be this way.

If he thinks he's at rest and Alice clearly is going velocity that way.

And therefore again the Length of his ship is going to be V,

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And they get back together again, and they say, they compare results, and

Bob says, hey, you didn't get the same length as I did.

What's going on here?

You screwed up, right, right?

Why did you mess up your result?

Well as Bob considers it He says, hey you know what Alice,

your clock is running slow.

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My clock is in perfect working order.

We have identical clocks.

But remember from our time dilation effect moving clocks run slow.

And so from Bob's perspective, he's at rest,

Alice is actually moving that way compared to him.

And he compares the ticking of her clock to the ticking of her clock,

in lapse time he says Alice your time is off.

I'm sorry to say that, but your clock is off.

That's why you're getting the wrong answer here and my answer is correct.

That, you know, whatever the number you could put in v times delta t, t sub b.

And obviously we are assuming v is fairly close to the speed of light to get any

effects there as we shall see.

So that is what Bob is saying.

Bob is saying hey, my answer is right.

You got the wrong answer Alice because your clock is not in working order, it's

running too slowly compared to my standard clocks which are in perfect working order.

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Well, what's Alice's response to that,

and actually a few minutes ago we hinted at that, so

You might want to think about in a minute from Alice's perspective, what's going on.

What does she point to in terms of Bob's measurement of the system,

where she says, it's not my fault, Bob.

You're the one who messed this up.

So think about that in a minute.

If you want it,you can pause.

And puzzle that in over a minute.

But here's the answer.

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If she were to look at all of Bob's clocks at the same instance.

So say at this instant in time, we not only take a photograph here, but

we have another photographing device out here.

Remember we imagined Alice having a whole lattice of clocks so

we can take a photograph either place along that we want.

And compare times with other clocks and locations.

So, Take a photograph.

When this photograph here goes off,

still take another photograph simultaneously here.

What will she see on this clock of Bob's?

Well, Bob's clocks are moving towards her like this, past her like this.

As we showed previously leading clocks lag she

will see this clock behind this clock here.

And, therefore, she'll say Bob your clocks aren't synchronized.

Notice that in Bob's measurement, he has to use two clocks,

Alice is only using one clock.

Bob is passing by her like that to measure his own length,

Bob has to use this clock and that clock for

the length of his spaceship, and Alice is saying,

sorry to say Bob but your front clock here is not synchronized with your back clock.

In fact it lags by a certain amount.

So that when this clock here, when this photograph is taken this clock is actually

further on in time at that point.

And that's why you got the wrong answer.

My answer is correct.

So what's going on here more quantitatively, okay.

So both of them are complaining about the other's results, and

basically in a sense they're both right.

But again, it's the relativity of simultaneity in terms of synchronized

closed, the relativity of time keeping with time dilation.

Which we'll get to here in just a second and

as we'll see the relativity of measuring length, when things are moving.

So here are the two results we got.

Okay, Alice is getting this result, Bob is getting this result.

They would turn out to be different answers if we looked at them.

And the reason they're getting different answers, we just explained.

Again Bob is saying, hey Alice your clock is running too slow.

And Alice is saying, Bob, your clocks aren't even synchronized.

So why even bothering to make these measurements?

So let's look at these results a little bit further, because we now can apply

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and, well let's see where we're going to do this.

Let's just remind ourselves up here for a second.

Remember our time dilation result, and our time dilation result,

from Bob's perspective now, he would say an elapsed time for

Alice is one over gamma his elapsed time.

Because from his perspective, so for Bob, From his perspective, Alice is moving.

So let's start with that.

So as he analyzes Alice's result here, I guess we will need some more room here.

So you have the pictures I forgot to mention, but it should be obvious.

The handout has this diagram on that, so you don't have to re-do it.

In this case, we just have this one, or

really two diagrams here, not too bad to reproduce those.

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So this becomes v times 1 over gamma delta t b.

In other words, remember Bob says Alice your clock is running too slowly,

therefore the elapsed time will be less remember gamma is greater than one.

So from Bob's perspective this is his explanation for

why Alice's clock is running too slowly.

It's running too slowly by a factor of one over gamma, so

we put that in here, but look what we get from this.

We just rearrange the terms, we'll have one over gamma in front.

Let's put that in front and move the v over this way, v delta

T Being that v

delta TB is the length of Bob's ship from Bob's perspective.

So this becomes 1 over gamma (LB)Bob.

And now we have a nice little equation.

Let me just write this down here, LB.

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Alice though, as she sees Bob's ship flying by and

trying to do the measuring with her clock and her knowledge of Bob's velocity.

Gets a result that's one over gamma, Bob's result.

Now remember gamma's greater than one here, so

her length is going to be less than Bob's length,

earlier we didn't specify that because we couldn't necessarily tell just from

the results there, which one was less and which one was more.

We could see they were different.

Go to analyze that.

But here's the quantitatively how they're different,

that when Alice measures Bob's moving spaceship, the length of it.

She'll get a result that's 1 over gamma times the length of whatever Bob gets just

from being his own spaceship at rest, as far as he's concerned.

And note that we can actually see similarity

here between our time dilation equation and

this is known as length contraction.

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So in other words, you have to get pretty close to the speed of light.

But if you're going that fast, and so if Bob's going by that fast and

Alice measures the length of his ship.

It'd be about 1/2 of what he would get for that measurement.

So let's write our two equations

in general terms here because we've been using Alice and Bob with them.

So let's do it like this.

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Because we have a few more points to make with this.

So this is the length contraction, the so called length contraction equation.

So called time dilation equation.

Einstein actually wasn't the first to come up with the thesis.

I think we've mentioned before.

People like Hendrik Lorentz and Henri Poincaré were also coming up with similar

results like this but they were getting them in a very different fashion.

And they almost seemed ad hoc results.

They didn't seem to be coming out of the actual analysis.

You sort of had to put it into their theory.

Einstein is getting his results just from his two basic postulates.

The principle of relativity, the principle of light constancy, put those together and

you discover that the speed of light is a constant.

Same value for all observers,

no matter how you're moving with respect to the speed of light.

And then you started analyzing things like the light clock example, and

start getting these somewhat weird results here.

That depending on how you're moving with respect to something, time,

the elapsed time that you will measure will be different, and

the same thing with lengths as well.

A little bit of terminology here.

When we talk about, let's start with the length here first, the length at rest and

that was if Bob was just sitting there on his ship he's at

rest as far as he's concerned, he's measuring his length of the ship.

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In German, the German terminology for it,

of course, which Einstein was using,

he called I believe his own length, just own length.

I think my German is really rusty, but I think it's eigen, I'm not sure now.

Anyway, own length in English, I know that much.

In other words, it's Bob's own length at rest as he measures his spaceship, or

whatever he happens to be measuring.

And proper length has connotations of, okay,

if this is proper length, the other length we might be talking about is improper.

And that's not the case at all.

So it's the length measured at rest.

And same thing for time.

We could talk about proper time.

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Proper time.

Or sometimes rest time, so

that is the time measured on a well running clock at rest.

Sometimes some people like to call to call it wristwatch time.

It's the time you would measure on the watch on your wrist.

It would be a different time if you saw somebody's watch flying by and

you were able to measure that as we've been analyzing,

there would be the one over gamma difference.

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Is this real?

What, as Bob's ship goes by, what does Alice actually see, right?

Well, here's where we have to go back to the distinction we made early on in terms

of observing something versus perhaps seeing something.

And we define observation using our photo principle.

That an observation meant we had this imaginary grid or lattice of clocks.

And at any given point we want

to fix something that something happened at that incident in time and space.

We could take a photograph there.

The clock would show it.

All the clocks are synchronized in our frame,

and we'd be able to specify the time and location in our frame of reference, and

that's what we meant by observing something.

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The reason it's important is that if you talk about seeing something,

then technically you'd have to think about okay,

if it's happening over there some place.

Even though I have my grid of clocks and I can take a photograph out there and

get the time, look at the photograph later, look what the time was and

the location was.

To see that actual event, I have to have light coming from, say there's a flash of

light over there, I have to have light coming from it back to my eyes of course.

There is a delay involved, and different things can happen if you have that delay.

Especially if you have an object moving past you, and

involves an additional layer of complexity.

It is possible to do that analysis, but it also gets a little confusing in terms of

the basic concepts that we are trying to get across here.

So, these concepts are correct and in terms of our analysis, and

they've also ported experimentally.

And if we talk about an observation being made in terms of our lattice of clocks and

our photo principle, yes, it is absolutely true that a clock at rest will

keep different time compared to a clock that you see moving along.

And the length of something at rest, if you put that in motion and

then watch it go by, you will actually measure a different length for that.

A little later on in the course, maybe almost the last week, next to last week or

the last week, we'll look at this in a little more detail because again you

can ask, do things really compress like that?

If Bob is riding in his spaceship,

does he sort of feel all of a sudden all sort of compressed?

And you could also ask, but what if they're multiple observers watching Bob,

who maybe one of them is at rest so he's going by at v, and

the other one is going at half v, so he sees Bob at different velocity.

Does Bob feel different types of length contraction?

And the answer, of course, is no.

Bob, as far as he's concerned, is at rest.

And he doesn't feel any of this.

It's the observations of other people making various determinations of

the length that will depend on the relative velocity between them.

But for now, we just have to focus on what we've got so far,

our two big equations here, time dilation and length contraction.

Again, elapsed time on a moving clock will be less than the elapsed time

on a identical clock at rest by this one over gamma factor.

And the measurement of a length will be less if you measure it while it's moving

by the 1 over gamma factor versus if you measure it when you're at rest.