0:03

On to part two of our revisitation of spacetime diagrams.

And the subject here is "cool" plots, "don't be a square".

I don't know how to translate it to other languages or something.

But, the idea is If you want to be cool, you want to be with it,

then you don't want to be a square.

What does that have to do with what we're doing here?

Well as you'll see, maybe this you can see from the diagrams already.

We're doing some sort of strange type of plotting here.

So let's see what we've got.

Non-square plotting, to bring it back to don't be a square.

If you don't like that reference you can ignore it, it's not important for

our purposes here.

So, here's our normal, we're just doing x, y plots to start off with here, okay.

So we've got two dimensions, x axis, y axis.

You could imagine maybe, we have a field or a white board and

we want to plot points.

In the field or surveyor maybe we want to mark locations of various things within

a field, or just mark locations of various things on a flat surface.

And clearly, we have our scale on here so this is (x1,y1), this set of axes.

This point here is x1 = 2, so 2 over for x1, y1 = 3, the point (2,3) there.

Nothing surprising about that.

But you could say, why do we have to have our axes at right angles?

This is the way we've always done it and maybe it makes sense in some ways but

1:58

x2 = 2, y2 = 3.

So, note how we plot this.

So we go again over x2, but then we'll go up parallel to the y axis, the y2 axis.

So that's 1, 2, 3 there.

So that's our point x2 = 2, y2 = 3.

And we're going to come back and

do a comparison of these two situations in just a second.

But we could go one step further and

say hey, let's get really wild here, why not put the x axis at an angle as well?

And so that's what we've done in our third example here where we've

tilted up the x axis at some angle.

We've got the y axis tilted at some angle as well.

And so in this case with the x3 and y3 axes, the point x3 = 2,

y3 = 3, again we go 2 on the x3 axis and then

parallel to the y-axis up 1, 2, 3, and so that's where the point is on that plot.

2:53

I hope if you think about it a minute, you can see that if we're plotting a field,

surveying a field, or just points on a flat surface like the white board here,

any one of these would work as long as we're consistent about it.

Again, it'd be a little strange to do it in these second two cases but

we could certainly do it.

We could say let's define the x and y axis like this.

And then for any given point on the board here, we could say okay, that has an x3

value, so maybe over here, it will have a negative x3 value over here and

a sort of a negative y value, too, y3 value, and so on and so forth.

So any given point could be plotted with the (x3,y3) cordinates or the (x2,y2)

coordinates, or the normal, quote unquote, normal x1 and y1 coordinates.

3:55

So we took this origin here, put that on top here, and took this one and

put this on top here.

I won't draw it because it gets really messy, it's hard to draw it.

But if we superimposed all three of them,

especially in this case you can see clearly the (x3 = 2,

y3 = 3) point is would be way up there, so if we moved that over here and

superimposed it here, then that point would probably be,

just roughly speaking, be some place up here, right.

It would be different from the (x1 = 2, y1 = 3) point.

In effect for this one too, again, we didn't draw it quite to scale.

But it clearly would be at least over this direction more,

some place, might be up here some place or something like that.

So the point being is the coordinates of our three systems are different.

Okay, so if I do (2,3) here, it's a different point, for

(2,3) here is a different point for (2,3) there.

Again, assuming I have all three of them superimposed

with the origin at the same point.

Another way to think about that is if I have, let's do the most obvious example.

If I have this point here, okay, and I am measuring it in those coordinates and

I get okay, (x3 = 2, y3 = 3) for that, what would those

coordinates be if I transformed them into the (x1,y1) coordinates?

because it's is a different point, okay.

This (2,3) point in this coordinate system is a different point than the (2,3) point

in this coordinate system.

In fact, as we've drawn it in here, just roughly speaking again,

by eyeballing it, let's say it's up here.

So, this is the (x3 = 2,

y3 = 3) point.

If we superimpose the two, that point we're talking about is right there.

And we can ask the question, what's the value of that?

So let's just get rid of this for a moment here.

What is the value of this point in the (x1,y1) coordinate system?

Like in just sort of eyeballing, we can say okay it's something like this,

like that and maybe 1, 2, 3, 4, so maybe has a value x1 = 4 and y1,

1, 2, 3, 4, maybe 5, 4, 5, or something like that.

Okay.

I hope that reminds you of something.

It should remind you of things like the Galilean transformation and

the Lorentz transformation,

because in those cases we're transforming between coordinate systems.

We did a lot of examples with the Lorentz transformation with Alice and Bob.

Alice having her frame of reference, Bob having his frame of reference.

And Bob has a certain coordinate system in his frame of reference.

And we, using the Lorentz transformation, if he has any given point, spacetime point

in his frame of reference, we transform it into Alice's frame of reference,

her coordinate system in space and time, using the Lorenz transformation.

7:05

Same exact idea here.

We could have this coordinate system here for whatever reason, maybe Bob chose this

crazy-looking coordinate system, Alice chose the more regular, normal system.

And they would wonder if they were both surveying the same field

using the two different coordinate systems,

they would have to have a transformation between them.

So if Bob measures something and say hey, I got that well over there that we're

surveying for in our field is at point (2,3), (x = 2, y = 3),

would not be Alice's coordinates for the well, (x = 2, y = 3).

It'd be something else, maybe as we said maybe 4, 5, there.

Okay, so that's why we're introducing this idea, because as we will see in the next

video clip or so, is that this type of thing is going to be

useful because what we like to do is put both Bob and Alice

on one plot using two coordinate systems, so two coordinate systems in one plot.

So the idea is something like this, we're going to superimpose it on our regular

type plot, so we can get an idea of what's going on in this Lorentz transformation,

back and forth between the two frames of reference.

9:07

So that's the X equals two line of the same location.

Anything that happens at that location at any given time will be

on that vertical line, and then lines of simultaneity,

lines of same time are horizontal lines on this plot.

So they say any event that happens at the same time that is simultaneous with

another event, both those events will be on the same horizontal line,

that same value.

Now, see I was using generic values for y here.

So now we're talking about, let's change this now to our t values because

9:58

Okay, so now we've turned them into spacetime diagrams, same idea applies.

Any of these are valid, just different ways of doing it.

But just because we measured (2,3) in one position here, actually then,

let's change all these too, be consistent here.

So this is t2, t3, and we've got a t3 here, and so on and so forth.

Just because we measure an (x,t) point, the spacetime point there with certain

coordinates, it's going to have different coordinates in another coordinate system.

And really if you think about it,

that's what's going on with Lorentz transformation.

10:53

Vertical and horizontal here, note that they're parallel to the respective axes.

And same thing when we get these skewed plots, as it were,

these non-square plots, the lines of same location here are going to be like this,

they're going to be parallel to the t axis because the t axis is the line,

the same location for x = 0.

Oops, that's not a t1 there, sorry about that.

It should be a t2.

But anyways, same lines of same, oops, curved line there, but.

So lines of same location on this plot are parallel to the time axis.

Lines of same time, because the x axis is horizontal here,

are still going to be the horizontal lines there.

So that is where the lines of same location, same time, lines of simultaneity

would look like, within the horizontal lines being lines of simultaneity.

11:47

On our third plot here though, both lines are skewed.

And again, they're always, the lines of location, same location, same time,

are always parallel to the appropriate axis.

So that in this case, you've got lines like this, actually

not crooked of course, but nice straight lines as much as possible, like that.

Those are the lines of what?

Well, those are the lines of constant x values, so

constant location, same location.

Those are the lines parallel to the t axis.

Lines of simultaneity are the lines parallel to the x axis,

so that it goes, they go like this.

12:30

Okay.

So it's important to sort of get that concept down because we're going to

be using that in the next video clip or two, or three.

It'll come up very shortly here.

Okay, so, again, cool plots, don't be a square.

So non-square plots, perfectly legitimate, we usually don't use those, but

we will find that a plot actually like this,

we're going to superimpose it on a plot like this.

Because we'll find that that's what the Lorenz transformation is telling us to do

when we're going between one frame of reference to another frame of reference.

So, that's coming up.