In this video clip, we're going to take one more look

at this whole idea of leading clocks, lagging a trailing clock.

So this is take two, you may remember last time we

did this derivation in terms of just looking at Bob.

If he has a machine that gives off two photons,

now Bob is on a space ship as far as he's concerned, he's at rest.

As far as Alice's concerned down here,

which we'll get to in a minute, she's going at velocity v from left to right.

So on Bob's space ship,

there's a little device that emits two photons or light waves.

And one goes toward his front clock, and one goes toward the back clock.

And we'll say to Alice's perspective, as Bob moves this way,

this is the leading clock and this would be the trailing clock,

the front clock or the rear clock, however you want to specify them.

And for both perspective, remember both of these light ways are going to hit his

clocks at the same time assuming that the device is right and exact center of

his spaceship and the spaceship has length L as far as he's concerned.

And therefore, when the light beams hit each clock, they trigger a flash photo

take the picture and therefore you see these blocks is having the same time.

Alice however remember as Bob goes by here,

the key fact is that she certainly sees the light means go off in each difference.

But because the speed of light is constant in frames of reference,

it means that she sees this light being traveling at the speed of light C and

she sees this one traveling at the speed of light C.

But Bob's leading clock is moving away from that light beam and

his trailing clock is moving toward that light beam.

And therefore it takes longer for the light beam going toward the leader,

leading clock to reach that and trigger the flash photo versus the one

going backwards to the rear clock and triggering that flash photo.

So on Bob's clocks, when the flash photos are triggered both of them show,

both these photos, one here and one here, show his clocks having identical time.

But in Alice's clocks down here again, remember that the lattice of clocks so

that we always have to imagine in any frame.

Bob also would have a whole lattice of clocks, but we only need two for

this case.

So imagine Alice's lattice of clocks all synchronized in her frame,

just as Bob's clocks are synchronized in his frame.

And we can see just qualitatively, we argue this awhile back that again,

as far as Alice is concerned, even though Bob's 2 o'clock,

the photo indicates flashes hit at the same time.

As far as she's concerned,

they aren't synchronized because she sees this flash occur after that one.

And so Alice's perspective, she sees this flash occurs,

records the time there and she can compare it to her clock here.

And take this same flash photo of both clocks at that instant in space and time.

And then a little while after that,

she would see the light beam hit Bob's front clock, the leading clock.

And she'd record a time for that as well so

that this clock, to Alice's perspective is ahead of this clock.

Or this clock lags this clock because Alice,

as far as she's concerned, her clocks are synchronized.

It's Bob who has a problem with the synchronization of his clocks.

And so we capture that idea in the phrase Leading Clocks Lag,

that this clock on the leading edge of Bob's spaceship lags behind this clock

as far as Alice's observations are concerned.

And so last time we sort of worked our way through derivation of that,

because we're only doing that quantitatively.

We wanted to actually get a result that we might be able to use in certain cases.

And so we found that result was to Alice now,

that the difference in Bob's two clocks was in at

general at distance D, which here was the length L.

But in general is going to be at distance D between the two clocks we're talking

about, the relative velocity between the two frames Alice and Bob here and

then the speed of light squared.

So that was the result we came away with after a little bit of calculation.

By this point in time, especially after the last few video clips,

you might be thinking that these are getting a little long here in terms of

the calculations we have to work through.

But first of all, remember that it's just a little basic algebra.

Once we sort of get this situation set up and analyze it, then we just work our

way through the end and if it helps there's a nice quotation from Einstein.

The story goes he was at a dinner honoring him as a national academy scientist

dinner.

And sort of halfway through listening to the speeches honoring him,

he turned to his companion and said, I now have a new theory of eternity,

obviously referring to the long winded speeches that had been going on.

So maybe you feel like that a little bit as well, going through some of our

analysis here, that it seems to take an eternal time to get to the end.

But once we get to the end and again the analysis, we're not doing anything fancy,

too fancy in terms of the math involved.

But just some basic assumptions, principles that Einstein started off with.

And then we're following them through in certain instances to see what

the conclusions are.

And one of the startling conclusions or the weird conclusions,

we get is that when a spaceship like this is traveling by,

it's got two clocks, one on either end or just any moving clocks.

They could have a clock way out there and a clock way over here.

In terms of Bob's lattice of clocks, Alice would see it in terms of her synchronized

clocks as a time difference Dv over c squared.

Now, note here that this is going to be a very small number in practice.

And it's why we don't see it in common everyday life, because we've got

a C squared on the bottom, a huge number and only a v on top here and the D.

And normally, the distances involved are much smaller, even when you multiply

distance times the velocity, much smaller than c squared here.

So in general this is a very small number, but

when you get up to speed near the speed of light and large distances,

then this can have some interesting effects that can be observed.

And we'll talk about that a little bit later on in our course.

What we're going to do now though, in terms of Take 2, is use our

Lorentz transformation to get this result, much more quickly than we did before.

As an example of the power of the Lorentz transformation,

remember where we got it from we're using variance of the interval and

time dilation came in along the way and so on and so forth.

But once we get to the end of it, based on those basic principles

then we have this tool we can apply in situations like this.

So we don't always have to go back sort of first principle in basic situations to get

the answer that we might want.

So let's see if we can apply the Lorentz transformation to this situation, and

get our Dv over C squared result or Lv,

if you're using the length L for the spaceship here.

So let's just modify this slightly.

So we're not going to use the light beams anymore here or

the photons, if you want to call them that.