Another thing, just to mention here, and as because some of you

are probably still thinking, well, It just seems, aren't they symmetrical?

In other words, Bob watches Alice go away and come back in.

Alice and her spaceship, from her frame reference, watches Bob go that way,

the star came this way, and then back again.

Just think about it, it seems very symmetrical.

But we made the point that only Alice has acceleration involved,

and Bob actually does not feel any acceleration.

We mentioned of course, if you're in a car that's accelerating, you can feel being

pushed back in the seat, or decelerating you're thrown forward a little bit, or

even moving side to side around around the curve.

Another, maybe, more physics way of doing that is to do a simple experiment.

Drop a ball, and if you're moving at constant velocity motion, and there's no

wind or anything like that, if you drop a ball, it will drop right down at your

feet, because the ball participates, again we're talking about slower speeds here.

We obviously did the analysis with relativistic speeds,

but the ball participates in the motion of the object as you're moving.

If you're accelerating,

however, what happens is the ball falls behind you for large enough acceleration.

For smaller acceleration, it actually does,

but you just can't notice it as much.

And so, essentially, what happens is, as you drop it,

the ball has the velocity at that point, so it would be going forward with you, but

you sort of accelerate out from underneath it and therefor, it falls behind you.

Or if you decelerate, it falls ahead of you, because it still has that initial

velocity, you've slowed down now, and so it moves ahead of you.

So Bob could do that dropping ball experiment all he wants,

and during this whole time, it would just fall at his feet.

He would not undergo acceleration, whereas Alice would

when she was at the turn around point, decelerating and accelerating.

So if that helps a little bit to see what's going on,

that even though it seems symmetrical, it actually is not.

And of course, the acceleration or

deceleration acceleration at the turnaround point is the key factor,

because that's what changes the frames of reference for Alice.

And then you get the jump going on there,

the jump in time as Alice observes Bob's clock.

So in one sense, we've talked about how the special theory of relativity is all

about inertial frames of reference.

It only applies to that, not In the case of acceleration and deceleration, but

you really don't need, say, the general theory of relativity here which deals with

acceleration and deceleration because you just assume the acceleration and

deceleration occurs over a short enough time

that you can essentially analyze it using the special theory.

And it's really the change in reference frame,

which is covered by the special theory, that gives us the results there.

So, yes, acceleration deceleration is involved, but

we can analyze the results and get the correct results using the special theory.

So, all that being said, what about the Lorentz transformation?

What we want to show is that you can get the same numbers using Lorentz

transformation, perhaps a little faster as well.

The downside is, first of all, you've got to use the correct formulas of course,

the correct versions of the formulas, so you've got to be careful about that.

But the downside is, you don't get as much feel for

what's going on as you do when you break it down via time dilation,

length contraction, and leading clocks lag, the relativity of simultaneity.

I do want to show that you can do this.

Again, our basic equation that we suggested you memorize,

if you're in the mood for that and want to use this a lot, is this one here

where we have the rocket moving away in a positive x direction l 4 lab And

one way to memorize whether or remember whether it's a plus or minus here.

Is just think about, okay.

The rock is moving away from me If I'm in the lab.

That mean's any rocket value for X would be a bigger value for me.

because it's moved away and so, If they measure something ten meters ahead of them

that's going to be maybe thirty meters, for me it's a twenty meters In that time.

So, if you can remember that basic situation, rocket moving away,

it's going to be a plus sign here in the basic forms of the equation.

Then from that you can derive pretty easily just in your head

that the other forms of the equation here.

So let's start over here with A.

Calculating Alice's results using Bob's Values and

so Bob's values are the lab frame values and

to Alice here Bob is moving in the negative x direction to the left.

And that's why we got minus signs in this versions.