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Moving on to part two of the Lorentz transformation.

We've brought back one of our props here, taped it up on the board.

We'll see what we're going to do with that in a minute.

But in this part we want to focus on the time equation,

the time transformation equation.

In the last real clip we were focusing on the x location,

the x transformation equation.

So we have t rest equals gamma in a rest frame and moving flame here.

And we're measuring Alice is in the rest frame, Bob is in the moving frame,

moving to the right velocity, being in the positive x direction.

So gamma times time value in the moving frame

plus gamma v over c squared times an x value in the moving frame.

In other words, x and t, the moving frame values are measurements made by Bob,

we want to transform that into a time value for Alice in this case.

So just as we did with last equation, let's explore this a little bit.

Let's just say, let's let x moving here, in other words

Bob's measurement of some event in the moving frame let that equal zero.

Okay, very simple and then of course this term becomes zero and we are left with

t in the rest frame equals gamma times t in the moving frame, which should remind

ourselves of the time dilation equation, because that's exactly what it is.

Does this make sense?

Well, remember when we derived the time dilation equation, we used a light clock.

And the light clock in the frame here in Bob's frame okay?

Is just a beam of light bouncing up and

down like that, and Bob's frame is carrying along with him.

And Alice of course sees it at a diagonal and

that is where of course we see the dilation effect but as far as Bob is

concern, if he is carrying the light clock with him as he goes along.

That bouncing up and down is happening in his moving frame.

Well, we're labeling it a moving frame from Alice's perspective.

But in Bob's frame at x equals zero.

Okay because it's in his cockpit with him and

that's his definition of the origin right.

As far as he's concerned he's not Moving.

And so essentially for that light clock example, we had the x value for

Bob in the moving frame be equal to zero and

we see it falls right out of the Lorentz transformation equation as is should.

We just get t in the rest frame, time measure in the rest frame is gamma

times the time measured in the moving frame.

Gamma is greater than one, greater or equal to one of course.

And so the way we wrote it previously was something like this.

We said time in the moving frame equals one over gamma times time at rest.

And obviously you can write it either way there.

But this emphasizes for us at least the time dilation effect,

gamma being greater than or equal to one.

So the elapsed time in the moving frame is slower than in the rest frame.

Moving clocks run slow is how we term that.

So Lorentz transformation equation for that, it checks out.

It gives us our basic time dilation results that we derived earlier.

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Now let's say, what if we just let x moving be anything here.

So we are going to come back to Alice here.

So we can't simplify our equation at all.

We're assuming that v is some value.

I guess we could say certainly if v is zero here as we did before,

then gamma is one, gamma is one and

you just get time in the rest frame equals time in the moving frame.

In other words both Alice and Bob are sitting there with their clocks.

They're not moving with respect to each other and they measure the same time.

So that's a trivial case there.

But what about the general case where Bob is measuring some time in his moving frame

and of course he's not considering himself to be moving.

This is from Alice's perspective in his frame and an x value in the frame.

Well let's imagine the situation that Alice is here.

We will put her in a spaceship but we'll assume she's at rest.

Let's bring back Bob in his spaceship with their lattice and

clocks each synchronized in their own frame of reference and

as we drawn here of course Bob is hold like this.

And so let's imagine right about, let me get this lined up here a little bit.

Right about here, obviously it probably would've traveled much farther than that,

but that Bob observes a certain event.

Or actually, let's not even do it that way.

Let's say that Alice at an instant in time on her clocks,

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sets off photographs all along her lattice of clocks and

she photographs not only her clocks but the close one in clocks of Bob

where we happen to be at that instant in time, okay?

And let's just say it's ten seconds so on Alice's clock's.

So ten seconds, Bob is flying by here at when they're right next to each other

both their times say t equals zero, tA equals zero, tB equals zero.

And then later on, at some point way down the line of course, but

we won't show that, we'll just imagine about right here ten seconds have gone by,

Alice takes photographs at all her clocks at that instance in time.

The question is, what are the readings on Bob's clocks?

And we're not going to put numbers in here,

we just want to get a qualitative sense of this at this point.

So let's just imagine it's something like that there.

And note one thing about our equation here,

is that let's say we've got a time right here.

These are all ten seconds reading on Alice's clock.

So we've got ten seconds on that one, what is the value of the corresponding

clock like here going to be on Bob's clock on that incident in time?

Well, according to Lorentz transformation equation here, if this is ten

seconds over here according to Alice, I've got something plus something over here.

And assuming both those terms are both positive terms here,

we're assuming nothing is zero.

because v is not zero again obviously, x is not zero in other words.

The x value is maybe one, two the third clock out there, the second clock or

something like that.

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So what that essentially means is if this is ten

than this term here has to be less than ten.

And in particular t moving has to be less than ten.

In other words the time on Bob's clock,

whichever one you point to out there has to be less than ten.

And that, again, is an indication of a time dilation effect.

That the moving clocks are going to run slow.

They're perfectly, at least the clocks right here at t equals zero, said fine,

everything was set up, not all the clocks but just the clocks at t equals zero.

Where down here as it went on,

we find that Alice took the photo that the time on.

I've just been pointing to the second one there.

Which ever one I've been pointing to time has to be less than ten because gamma is

greater than one.

This term also is positive and therefor the value of

time on Bob's clock in the moving frame, has to be less than ten.

Okay, so that sort of interesting result just reminds us of time dilation.

But now let's also think about something else because remember

Alice took pictures along the whole line here, of her clocks.

And so, ten seconds on this clock, and

we can read off the value on Bob's clock at that instant in time.

And then what about the next clock over?

Also ten seconds for Alice.

So again, this is ten.

But note what happens as we move over one clock?

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The x value here in Bob's frame also gets larger by whatever amount it is.

We could use our other equation to get that.

We're interested more in the qualitative argument here.

So as we move this way so ten seconds on this clock.

Some value on Bob's clock, that's less than ten, as we move this way.

Let's just say it's eight, give us a number here.

So there was eight seconds on Bob's clock for that one.

As we move this way, this value is getting bigger.

If we move from that clock to that clock therefore, for

the next clock over this has be the less if this was eight for this clock.

And when we move it one over this term is getting a little bit bigger,

this term has get a little smaller.

So this has be also smaller than eight now.

If we move over one more still ten seconds in our Alice's clock.

So this is still ten but this term's getting a little big bigger even, and

therefore, this has to get a little smaller.

So as you move in this direction.

Again, all Alice's clocks are ten seconds.

But Bob's clocks, the time as Alice observes them through her photos here

that she's taken, the time on these things is getting smaller.

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It's really prior in time.

Again, if this was eight seconds, this maybe has to be seven point six.

And then the next one over is seven point two or something like that as it goes on.

And what that means is this is another verification of

something we talked about before and that is leading clocks lag.

Okay, in other words leading clocks lag is

built to Lorentz transformation equation through this term right here.

because otherwise, put this down for a minute.

Otherwise this sort of looks strange because the Galilean transformation,

we just had tA equals tB, t rest equals t moving, there was no change.

And so we expect, okay special theory of relativity, gamma's going to be involved.

So we might expect something like t rest equals gamma t sub moving,

without this term here.

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And we can sort of get a handle on that.

But then this term is just weird.

We say, how does that work in terms of not only an effect on time here but

you're adding some distance in here as it were, in a sense,

a distance factor that's affecting the time observed in

Alice's frame of reference compared to Bob's frame of reference.

And what this term essentially is doing

that another way to put it is leading clocks lag.

As we're just trying to argue here is that again,

Alice takes all the photographs at t equals ten seconds.

So all of her clocks say ten.

But maybe Bob's block here, because I've got two terms here, they're both positive.

If this is ten, that has to be less than ten, we said maybe it's eight.

But then as you move over one,

this is getting bigger, moving in more positive x direction there.

And therefore if this is eight, that's gotta be 7.6 or something like that.

Move over again, got to be down to 7.2 now,

because this term keeps getting bigger, this term has to keep getting smaller.

Again, we don't see these effects, at least not in every day real life.

Because look at the factor here, not only gamma of course, but v over c squared.

This is a very, very small number.

So it does show that leading clocks lag.

In fact, in a later video clip, we're going to look at that in a more

quantitative fashion to figure out exactly how much do leading clocks lag by.

But this isn't showing therefore that the time equation as part of the Lorentz

transformation equations has this built in.

Not only the time dilation effect, which we got from our light clock.

But also the leading clock's lag effect as well.

So that's part two of exploring Lorentz transformation.

In part three we're going to look at the case where,

where about from Bob's perspective looking at Alice or in general.

What if we have some frame of reference moving to the left instead of to

the right.