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Now that we've learned quantitatively about this whole idea of

leading clocks lag, let's return to our Star Tours example,

our trip to a star five light years away.

So let's remind ourselves what was going on there, so

I've added a few things to the original diagram.

But essentially Earth, Alice on Earth, star five light years away,

according to Alice's lattice system of measurement and clocks.

Bob traveling along at velocity v, we're going to say v is 0.943c,

mainly so that we can say gamma is 3,

and we did it from Alice's perspective, the Earth observer perspective.

We said, okay, five light year distance, we see Bob floating along at 0.943c.

The time it takes him, then, will be 5.3 years.

In other words, and what I've drawn in here is this is the Alice's,

the Earth Star system, lattice of clocks there, the imaginary lattice of clocks.

And so at, when Bob passes by, remember when Bob passes by

Earth on his trip here, Bob's clock will read 0.

Alice's clock, on the Earth system, will read 0 right at that point, so

they both are measuring time from that instant on their clocks.

They just set their clocks so they're 0 at that point.

So then as far as Alice is concerned, Bob moves along.

When he gets to the star,

another photograph is taken to compare the two clocks at that time.

And Alice on her clock in the star system clock, Earth Star system, will

read 5.3 years, because 5 light years, 0.943c takes 5.3 years to get there.

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the photograph of Bob's clock as well, what does she read?

Well, to her, Bob is travelling along at speed v.

There's a time dilation effect.

She sees Bob's moving clock running slow.

And for gamma factor of 3, what that means is that her clocks tick off 5.3 years.

His clock, from her perspective, as she's watching the moving clock go by,

ticks off 5.3 divided by gamma, which is 3, 5.3 divided by 3, 1.77 years.

So that's the photograph and

that photograph there that's taken, that's what Bob's clock says, 1.77 years.

Now, Bob can't disagree with that.

He says, yeah, obviously, it's 1.77 years.

I see, it's in the photograph.

I looked when I went by, too.

It was 1.77 years and

Alice's result is 5.3 years and that's the part you may remember it was,

when we first did this, we really didn't have the capability to understand that.

And that's what we're going to do now in a few minutes, but let's continue on here.

What's Bob's perspective?

He's the rocket observer.

He observes, in his frame of reference, their star system moving to the left

and therefore he sees a star rushing toward him at 0.943c.

And therefore this five light year distance, five light years to Alice,

to him is only 1.67 years, as a contracted distance.

The distance for the Earth Star system from Bob's perspective,

the rocket perspective, is 5 light years divided by gamma,

gamma being 3, and that's 1.67 light years.

So to Bob, when he just passes Earth, he sees the star at 1.67

light years away and travellng toward him at 0.943c.

Therefore, the time for the star to actually reach Bob and then have

the photograph taken is 1.67 light years, the distance divided by 0.943c.

We're taking c to be one light year per year, and

therefore it comes out to be 1.77 years.

And he says, yes, absolutely, that's what my clock should say.

That's what the photograph shows and

I agree absolutely because I traveled for 1.77 years.

The distance was 1.67 light years.

The speed, actually I should backup, he didn't travel for 1.77 years.

The star travel toward him for 1.77 years,

is a distance of 1.67 light years, the speed 0.943c.

It all works out to him.

But then we ask the question, and we held off on this,

we said how does Bob explain this 5.3 years business?

Especially when we took the next step and said okay,

shouldn't time dilation also apply to Bob observing Alice's clocks?

It applies for Alice observing Bob's clock here, or system of clocks.

Bob, of course, has his own lattice of clocks, which we didn't draw in here.

But doesn't it apply the other way as well?

And the answer is, absolutely, it does.

So we said Bob's observation of the elapsed time on Alice's clock, so

he's sitting there, he's watching Alice's clocks go by.

He compares the timekeeping on her clocks to his clock, and

sees the time dilation factor, such that,

I squeezed it in down here, it's 1.77 years.

That's the elapsed time he sees on his clock,

as the star rushes towards him, and divided by the gamma factor.

So he sees elapsed time on Alice's clocks,

from when Earth was next to him to when the star goes by, as 0.59 years.

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And that's the big puzzle then or was a big puzzle because we say 0.59 years,

that's what Bob sees Alice's elapsed time being.

Alice is seeing 5.3 years.

In fact, the photographic evidence shows that when,

say this clock right here, okay?

So here's that clock that's going to get photographed,

when it gets over here to Bob, when that clock reaches there,

the photographic evidence shows that it's reading 5.3 years.

And yet Bob is saying it should only be 0.59 years.

Well, as I mentioned when we first did this,

this is a very nice example because it shows that in these types of problems,

typically, you have to think about time dilation, it comes into play.

You often have to also think about length contraction, it comes into play.

And often people just think about those things and

they forget one of the key things

about the special theory of relativity that comes into play in these problems.

A third thing that comes into play and that's a relativity of simultaneity.

And this is where you get into trouble where you get these weird paradoxes

sometimes that don't seem to makes sense and you say,

gee, maybe the special theory of relativity is wrong after all.

You have to remember the relativity of simultaneity.

And, that the relativity of simultaneity,

one aspect of that is the fact that leading clocks lag,

as we showed qualitatively a while back.

And now quantitatively we know that the factor by

which they lag by is Dv over c squared, so

this is where you're in a frame of reference and

you're observing clocks going by you with some velocity, v, okay?

So they're in a separate frame of reference, they're moving by you.

So in this case, Bob is in his frame of reference,

sitting in his ship, he's observing Alice's clocks move by him.

And pick any two clocks in that lattice, and the distance between

them here is the distance in the moving frame of reference, okay?

So, it's the distance that Alice sees between the two clocks,

not the distance Bob sees between the two clocks.

That's important because Bob's distance is contracted, as we saw.

But, it's the distance between that Alice sees

in the clock's frame of reference, okay?

v is the velocity of the two frames of reference and, of course, c squared here.

So let's apply it to our situation here, okay?

So photograph is taken right here, clocks, both Bob's clock and Alice's clock read 0.

The star, from Bob's perspective, travels to him.

Another photograph is taken at that point, from Alice's perspective,

Bob travels to the star, photograph taken.

And, again, we've said, to Bob's perspective,

the elapsed time while he waits for

the star to get to him between the 0 time here and the is 0.59 years, okay?

And yet clearly this clock, the photographic evidence shows,

it reads 5.3 years.

How do we understand that?

Well, it's the leading clock's lag factor.

It's the relativity of simultaneity because, at the beginning here, okay,

right when Bob is next to Earth,

the star is moving toward him, these clocks, they both read 0.

Well, at that point, for Bob, if he had his whole lattice of clocks here, and took

photographs, so Bob has his own lattice of clocks, let's find my green pen here.

So Bob, I guess we'll make that green there, but

Bob has his own lattice of clocks.

Imaginary, of course, but useful for making measurements.

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This clock here we know that instant in time is t equals 0, it's reading 0.

What does this clock read according to Bob,

according to his synchronized lattice of clocks?

Well, that clock is the clock in the rear.

This clock here, and note that here is the distance that you make this clear here,

that's the distance D that Alice sees between those two clocks.

Her distance D, that's this distance.

Bob sees a lag between that clock and

the clock at the star here that's eventually going to get to him.

That leg is going to be Dv over c squared.

Now is if this is 0, leading clocks lag, so this clock,

as it's moving this way, lags this clock.

This clock is ahead.

Clocks in the rear are running ahead of clocks in the front, leading,

or the way we've been saying it, leading clocks lag.

The clocks in front lag behind a clock in the rear,

if you have clocks moving past you.

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C squared, we're using light years, so c squared,

c is just one light year per year squared.

Just one squared.

Do the math and what do you get here?

You got 5 times 0.943,

you get this right, 4.71.

Again, I just semi-memorized that.

4.71 years.

Let's make sure we understand that.

In other words, at the instant when Earth and Bob are lined up here and

both the clocks here read 0 from Bob's frame of reference.

If he took a photograph of the star clock out there that is moving toward him,

it would read 4.71 years.

Alice's clock here reads 0 and it lags.

This clock here by 4.71 years, which is based on the distance between the two

clocks in Alice's frame of reference and the relative velocity between them.

If you took a clock halfway along the way, if Bob looked at this clock, then 4.71

years, it'd be what, about 2.36, something like that, 2.355 whatever, years distant.

So all these clocks, the farther away you get here, the more the lag time.

Or the more really these clocks are ahead of the leading clock.

And we're guessing that this is the leading clock, okay?

But, you say, well wait a minute, though.

We're trying to get 5.3 years on this clock.

When this clock reaches here, the photograph evidence shows,

it reads 5.3 years.

4.71 years is not 5.3 years.

But remember, the elapsed time between this instant,

when Earth is here, and then when the star gets there is

0.59 years, according to Bob's clocks.

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So, I should say not according to Bob's clocks, that's the elapsed time he

sees on Alice's clocks, they're running slow compared to him.

So, as they're going by, he's watching them tick off and

he sees them tick off 0.95 years, and it reaches here.

But this clock started 4.71 years ahead,

and so 4.71 plus 0.59 years,

4.71 plus 0.59, 5.3, and it all hangs together.

It's all consistent, as long as you remember to put everything in there

in terms of the special theory of relativity.

We've got time dilation involved in this, both ways to Bob and Alice.

We've got length contraction, as well.

The way we were analyzing it, it's really Bob viewing the length contraction there.

But, you can reverse things and also do Alice's perspective and

length contraction for Bob.

If Bob had two space ships here, one here and one here, then Alice would see

a length contraction for Bob's distance between them and so on and so forth.

But the point here is that when this clock reaches here, and the photographs

are taken, this clock says 5.3 years, Alice understands that very easily.

Just 5 light years divided by 0.943c, sure, it's gotta be 5.3 years.

My clocks are all synchronized.

It started out at 0, took 5.3 years, yes, it's going to read that.

And Bob's clock reads 1.77 years.

And again, Alice understands that.

She says yes, because your clocks are running slow, Bob.

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So they only ticked off 1.77 years.

And meanwhile, Bob says, my clock's just fine, but

the fact that it's 1.77 years is because I didn't have to travel as far.

It's your distance that's messed up, Alice.

It's not 5 light years.

It's 1.67 light years.

So I traveled for, according to my clock, I didn't travel.

The star traveled toward me for 1.77 years,

it's a distance of 1.67 light years, 0.943c.

It all makes sense to me.

And then, the final thing that Bob has to explain to make sense is,

how does Alice read this 5.3 years on the star clock when it reaches Bob and

they take the photograph?

And the answer is, from Bob's perspective again,

that he sees Alice's clocks time dilated, that his clocks tick off 1.77 years,

he sees her clocks tick off 0.59 years.

But, this clock back here at the star, when it starts off here and

Earth is right next to Bob, is already 4.71 years ahead because of

the leading clock's lag factor, the relativity of simultaneity.

So that it's reading 4.71 years right here, according to Bob's perspective.

And then Bob watches it tick off as it gets closer and closer, and

when it reaches here, it has ticked off an additional 0.59 years.

0.59 years plus the starting value of 4.71 years gives us the 5.3 years and

Bob says, yes, of course your clock says 5.3 years.

Not because you're right.

Your clocks are all messed up, they're all unsynchronized.

But yes, we can agree that that's what should show, 5.3 years.

And they both agree that yes, the 1.77 years should show on Bob's clock,

but for different reasons there.

So, again, something to ponder here to sort of work through in terms of how

time dilation works into it, how length contraction works into it.

And then, sort of the missing factor that people often forget about

is the relativity of simultaneity that you get this

desynchronization in a sense that Bob's clocks are synchronized to him.

Alice's clocks are synchronized to her in her Earth Star system, but

when one looks at the other's clocks, they are not synchronized.

And the leading clock's lag factor is this Dv over c squared, D being the distance

in the moving frame, you're thinking about times the velocity divided by c squared.