Let's now, look at the key things we did here in week 5.

So, we do a little summary, not quite in the order we did them but,

hitting all the high points,

clearly one of the big things of the week was the Lorentz transformation.

It was sort of the theme of space time switches.

In other words switching between one frame of reference to another frame of

reference and

having some nice equations such that we don't have to start from first principles.

All the time in doing that so the idea was of course that if

you have two frames of references, this is A and B, maybe Alice and Bob again, so

if you have Bob doing a measurement in his system he sees an event as x,

y, and z space coordinates, in a certain time on his lattice of clocks.

Then can we transform that into the coordinates in Alice's system.

And here were the equations that we came up with, yet

we won't go into all the details because we spent some time actually exploring them

a little bit to see what the implications were.

But for x sub a, and remember a and b here we're doing

b as the frame of reference, moving to the right, positive x direction of speed v.

So another way we've written these in the past is with a being the lag frame

of reference and b being r for rocket, the rocket frame of reference and sometimes

that's easier to visualize and remember which way it goes, as one will remember.

it's the rocket's going to the right for these equations here.

You may remember, if we have a situation where the rocket is going to the left,

then we have minus v and minus v over c squared in those cases.

So again advice is, memorize the basic situation, make sure you understand

which direction it's going and what's the rocket frame, what's the lab frame or

what's the rest frame, what's the moving frame, where here we're just doing a and

b for the rest frame and the moving frame in this case.

So this would be Alice watching Bob moving and he see's an event.

Inserting coordinates that involve x b and t b and therefore,

Alice's coordinate where that event occurs x sub a is given by this,

t sub a on her clocks is given by this.

Remember the y in z directions, we often don't deal with those, but

if we had to they are not transformed.

There's no difference in them because the things like the length contraction

effects, time dilation effects only apply in the direction of motion.

So we don't have to worry about those, typically,

at least in terms of the more complicated transformation equations.

So we explored those equations a little bit, and

then we used them to revisit The idea of leading clocks lag and

we found that the quantitative amount by which they lag is d v over c squared,

we v again is the velocity between the two frames of reference.

d is the distance between the two clocks in the frame that's moving so

again if you have a rocket ship with two clocks one at either end and

we see it moving past us, then the leading clock is going to lag as we've argued

qualitatively before this week.

The leading clock will lag the trailing clock as we watch it go by and

now we've discovered the exact amount it will lag by.

In this distance d is the distance in the frame of reference, in this case,

of the space ship.

So its not a contracted length or anything like that.

We've set the spaceship down, measure the distance between the clocks at rest,

in that frame of reference, say it was d and then as it was moving past us the lag

time between the two clocks would be d times the velocity between the frames of

reference divided by c squared, and note that in general

this is a very very small number and that's why we don't see it in our ordinary

everyday experience even with velocities up near the speed of light.

Point 9 c here, we still have this c squared down here.

So this does not get significant until both d and v are fairly large.

We start seeing that effect.

But, what was nice about it and one of the reasons we did it here, was that it helped

solve our star tours puzzle where we explored length contraction,

time dilatation a little bit more with Alice and

Bob Bob taking a trip to a nearby star.

And we looked at it from Alice's perspective and Bob's perspective and

the different things they would see on their clocks.

And what lengths they would measure from Earth to the star or

the start moving toward Bob and so on and so forth.

And we hit sort of a wall, a puzzle there.

And discovered that it was answered very nicely,

by remembering this leading clocks lag concept, which really is

a special case of the whole idea of the relativity of simultaneity that

clocks in one frame of reference will be synchronized, they're Bob's clocks.

Clocks in Alice's frame of reference will be synchronized.

Yet if one looks at the other's clocks, they're not synchronized.

And they're not synchronized by this factor between any two different clocks

as you observe them.

And that all goes back to remember Einsteins' principle

that he derived from his two postulate of the constancy of the speed of light.

The speed of light is the same to all observers.

And that's where we got this relativity of simultaneity effects,

as well as the time dilation and the length of contraction of x.

So leading clocks lag, we discovered by this amount, d v over c squared there.

And then another thing we looked at was combining velocities.

We've done this before in our Galilean transformation case of our everyday

experience case, when If you have a car moving along and

you shoot a basketball out from that car

at a certain velocity with respect to the car, the velocities just add.

If you have an observer observing what the velocity of the ball is.

It is like running forward and throwing something versus just throwing something.

Well because the speed of light is an ultimate speed limit,

which we also talked about this week, then you can't have

things if you have something going up 0.9 c and you try to add something at 0.7 c,

so if you have Bob in a spaceship going at 0.9 c by Alice and then

he shoots out an escape pod going up 0.7 c, whatever it is, with respect to him,