Past 3 of exploring the Lorentz Transformation,

here are equations back again, and I've changed them back to A and B for Bob and

Alice, because we're going to write a new version of them up here and

it'll be easier to see what's going on if we're thinking about Alice and Bob again.

But of course we can think about the rest frame, and the moving frame,

or a lab frame and rocket frame.

We want to think about it that way where Alice, in this case,

would be the rest frame or the lab frame, Bob would be in the rocket frame,

as moving in respect to the lab, with positive velocity v.

And the question is here, really, two part question, it's actually the same question,

just ask them two different ways.

First one is, what happens in terms of Bob looking at Alice?

Bob thinks he's stationary, that's his perception, and

it's Alice who's moving with constant velocity, but

in the opposite direction, in the negative X direction.

What would the equations look like in that case?

Another way to say that is, what if we just turned Bob around, and

Bob goes in the negative X direction at velocity v, then what would it be?

Let's ask, in terms of the first version, in terms, okay,

now let's look at it from Bob's perspective.

He's seeing Alice move with velocity negative v, essentially in the negative X

direction, or we can say with velocity v in the negative X direction.

And the key here is because, we could back and we have to do that derivation again,

and go through and change all any time the velocity appears,

turn it into a negative sign and then see what we, a negative v and

see what happens, and fortunately the answer is no.

Even if it was yes, we probably wouldn't go through it all again,

we'd just give you the result.

But here's our basic equations that were derived on the assumption that the rocket

frame, the moving frame, is going with velocity v in the pause of X direction.

If we have a velocity v in the negative X direction,

all we have to do is change the sign any place we see v appear in our equations,

because this is, v is assumed to be a positive quantity here, so for

just changing direction to the negative direction, it just becomes a negative v.

And again we'll assume the actual value of v here is positive, so

by putting a negative sign in, that will just change the direction.

And the other factor, we have remember is well, v also is in gamma.

So, we can use either though is gamma of course,

square of 1-v squared over c squared.

v squared saved us in this case, because, of course,

if you take a negative number and square it, it turns into a positive number.

So that whether v is negative or positive, gamma has the same value,

gamma does not change, so don't have to worry about gamma, and

the only changes that are made here are in these two plus signs.

Instead of having a plus v over c squared, and a plus v,

I have a negative v over c squared, and a negative v, or minus v in each case.

So we can write up like this,

from Bob's perspective we say,

okay, tB for Bob is gamma tA plus,

not plus, minus, minus v over

c squared XA and XB for Bob,

is gamma (XA-vtA), like that.

In other words, given X and t values in Alice's frame of reference,

if she measures them, then we can figure out what the values would be

in Bob's frame of reference, despite bringing minus signs in there.

So that gives us a sense of how we can go back and forth very easily, and

it's probably useful to memorize one form of these and perhaps if you like

to use the terminology lab and rocket because it's very visual that way.

So if you think, okay, I'm going to have a rocket moving to the right in the positive

X direction, and so therefore the Bs here would be the lab quantities and

the As would be, I'm sorry, the Bs here would be the rocket quantities and

the As would be the lab quantities, if you want to think about that, or

the rest frame quantities and the moving quantities.

This assumes motion to the right, positive X direction.

This version down here, essentially says, now the Bs are the lab frame of reference,

the rest frame in this sense, because now Bob sitting in his spaceship,

is assuming that, from his perspective, he's in the rest frame,

in the lab frame, and it's Alice moving, whether she's in the ship or not.

She's moving to the left there, and

therefore you get a negative signs in there otherwise it's the same form there.

So as we proceed, we will explore more of the implications of this.

This week we'll do a little bit more quantitative league,

with leading clocks lag, but then as we move forward we'll start actually

re-introducing space time diagrams and, not this week but next week, and then

also getting into some of the paradoxes involved with the special theory of

relativity that the concepts and equations we've build out so far leads us into.

And so we need to figure out some of those things, but

before we get into that, we have few more things to do for this week.