0:04

All right, before we actually move on and consider the Galilean transformation,

whatever that happens to be.

We'll get to that next video clip.

I want to take the opportunity to actually say a few more words on worldlines and

look at two specific examples that are going to be important for

us later on in terms of trying to understand

a couple of the paradoxes that arise in the theory of special relativity.

And one of the paradoxes is the so-called pole in the barn paradox, and

another, perhaps the most famous paradox is called the twin paradox.

And so a couple diagrams we're going to look at here,

we'll see variations of them later on when we're considering these paradoxes.

So I wanted to take the opportunity now just to introduce the ideas.

So, again, let's imagine a situation.

We've got Bob in his spaceship traveling along at velocity v here.

Here's Alice.

So Alice is doing the measuring, this is her lattice of clocks.

Her measuring stick, as it were.

We've just chosen different units here, minus 10, 0,10, 20, 30, 40,

on her measuring lattice there.

So here comes Bob along, and if we imagine that at time t equals zero, he's here.

So, Alice records his position on her clocks,

the front of his ship is at 10, his cockpit is positioned at 0, and

the back of his ship is positioned at negative 10.

Maybe, a second later, he's moved ahead, so now his front of his ship is at 20.

He'd be at 10 in the cockpit and

then the back of the ship would be at 0 and so on and so forth.

It will move 10 to 30 and 20 to 40 and so on and so forth like that.

So, that's the idea.

In other words, before when we're looking at these situations,

we had just imaging Bob was a point, his ship was a point and so we plot that.

Now, lets actually plot that as a extended object here.

And so this is the physical situation, this is what Alice would actually see.

This is what Bob would see as he's traveling around.

Now let's add the space-time diagram.

So we'll use the x axis here for Alice and

we'll add her time axis on there.

And what would Bob's ship look like, this extended object,

what would that look like on her space-time diagram?

More clearly, we'll use our red again, at the beginning if we were to plot just Bob

in his cockpit at time t equals 0.

It's right there, then we'll imagine that,

let's put some time marks here, the tick marks for time, so

at the next time we'll imagine that the ship is moved over 10.

So now he's at position 10 and so he'd be right there, and

then move again and he'd be at position 20 and so on and so forth.

So he would looks something like that and so his worldline,

of Bob in the cockpit, would look like that.

3:07

What about the front of the ship?

Let's do the front of the ship here.

So, at time t equals 0, it's at 10.

So it would be there and then, at the next time, it would be at 20.

So it would be here.

And then 30 and 40 and so on and so forth at each time.

So that's 30 and so the front of the ship, it's worldline looks like that.

So this is the cockpit where Bob is sitting, this is the front of the ship.

3:34

The worldline of the front of the ship and

the back of the ship is similar, of course.

It starts off at equals 0 here, then as the ship moves over one second or

whatever the unit of time is later, it's going to be here and then it's going to

be here and so on and so forth and we get that as well more or less.

So this is the back of the ship.

So in terms of the ship's worldline, it's sort of a world,

I don't know if you'd call it a line but a blob there, so it moves like this.

So at any given instant, if this represents the ship,

we'll do it like this.

So there's the ship, then it, Alice sees it here and here and here.

Actually, of course, physically she's seeing it here and then here and

then here and then here, so on and so forth.

She was representing that in the space-time diagram

as moving along this way.

So the worldline at the front of the ship, worldline of the cockpit,

worldline of the back.

And so things that move in parallel.

We see these are diagonal lines, of course, but

they're all parallel to each other, represent a moving object,

an extended object in Alice's frame of reference on her space-time diagram.

5:40

So, minus 10, 0, 10, 20.

And this is x Bob and t Bob.

His frame of reference, of course, in his frame of reference he's stationary.

The ship is stationary.

It's not moving as far as he is concerned.

He always stays at position 0.

So his worldline, as we mentioned in the previous video clip.

If something is stationary in a frame of reference, the worldline is just vertical.

So he's just staying here in the cockpit.

In any given time, whatever time t we're talking about, he is at 0.

And the front of the ship stays at, get this right here, stays at 10.

The back of the ship stays at, not 10, but minus 10, here.

So the back of the ship is here.

6:24

So on and so forth.

So, here's the front of the ship worldline.

Here's Bob's worldline in the cockpit.

And here is the worldline of the back of the ship so as far as he's concerned,

the world path as it were of his ship, the space-time path,

the path in the space-time diagram, not the actual physical path.

The physical path is just moving along, well he's stationary,

to Alice he's moving along like this, is just straight up and down like that.

Okay so, that's that example.

Let's look at a different one here.

We'll just call this, in general, x and t.

So, it's going to be some observer.

It could be Alice, could be Bob, could be any observer and their frame of reference.

Meaning they're stationary,

they've got their lattice of clocks they're measuring things with.

And, they analyze the motion of an object.

And they get something that looks like this on their space-time diagram.

It moves out to here.

7:22

That's the space-time diagram.

Here's a question for you,

what does this represent in terms of the physical motion of an object, all right?

If you want, you can pause the video for a minute here, ponder that, figure out,

okay, what's actually going on?

Remember this is for the x-axis, motion along the x-axis, we have our space-time

diagram that represents the time, position in time, of a given event.

But what's going on if we were to translate this back into the physical

situation of motion along the x-axis, what does this represent?

So again, if you really want, pause or ponder a few minutes on that.

Let's think about this then.

8:02

Okay, so what this is saying, we're starting at t equals 0 here.

And it's moving out this way, straight line.

So it's going to be constant velocity motion.

And so time is going on here.

Time 0, 1, 2, 3, 4.

So what this is telling us up to that point,

it's moving in the positive x direction through time.

So up to this point here, it's moved out maybe, well, let's,

we'll say 1, 2, 3, something like that.

So it's at position 3 at some given time.

8:39

All right, so we'll go 1, 2, 3, 4, 5, 6, 7, 8.

Okay, so at time t equals 4, four seconds, four minutes, four years,

whatever it happens to be it's at position 3.

It's moved in the positive x direction and if it kept going,

it'd just keep moving in a positive x direction.

But evidently it stops and changes direction at that point.

And now at time t equals 5, it's back, so

here's the t equals 5 point right there, it's back someplace in here.

Sorry my diagram's all messed up there.

9:13

And then at t equals 6, it's here, at t equals 7, it's here.

It's moving back this direction, though it's moving to the left on the x-axis.

And at t equals 8, where is it?

It's back where it started from.

So a worldline that looks like this, where it moves out like that and

then back like this, represents an object that has moved out to position three or

wherever it is, and then has moved back to position zero again.

Out to position 3, then as time continues going on, it moves back to position not

position 8, that's time equals 8, but position 0 where it started from here.

You could also have something like this,

a worldline like this where it moves out here, moves back there.

So this time, we're starting at 0, start at position 5 or

6 out here, moved out to some further position, and then moved back there.

So again, along the x-axis, moving out and then back again, moving out and back in.

Or if you want to go the other direction, you can move this way.

And then to some position like that, so there it's moving to the left and

then back to the right again.

So it's helpful to sort of think about these things to get a more

intuitive understanding by able to look at a space-time diagram and understand what's

actually going on, what does it represent in terms of the motion going on.

And as I mentioned this example here we did will be useful when we consider

the pole in the barn paradox.

This example will be very useful when we consider the twin paradox later on.

So that's a few more words on worldlines, and then in the next video

clip we'll talk about this so called Galilean transformation.