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this is a way to visualize how events occur.

And we'll find it's very useful later on when we're dealing with

some of the concepts of the special theory of relativity.

But let's go back to the key concept from the last couple of video clips, and

that is a grid or lattice of clocks, that is our observer's system here.

So that, in this case, again, we're just doing the x axis, one dimension,

so we have a clock at every single point along the x axis that we're interested in.

And we'll assume that all those clocks are synchronized and

using a method like we talked about in the previous video clip.

And here's our observer again at the origin.

So what we want to do now is really think about a series of events that occur.

So we might, again, imagine some flashes of light,

little strobe lights go off or something like that.

And what we've done here is specify the x and t coordinates of these flashes.

So we'll just use time 0, 1, 2, 3, we'll assume seconds.

So I have 0 seconds, 1 second, 2 seconds, 3 seconds.

1:22

We're going to have, in this case, four flashes.

At time t equals 0, we'll have a flash that occurs at x equals 0.

At time t equals 1 we'll have a flash that occurs at x equals 1.

Time t equals 2, it occurs at x equals 2, time t equals 3 it occurs at x equals 3.

So if we were creating a little movie or animation of that,

it might look something like this.

Where at time t equals 0, according to these clocks,

which are all running just fine here.

So we have a flash of light there, one second later we'd

have a flash of light here, and then a flash of light here one second after that,

and a flash of light here one second after that.

So we get a series of flashes.

Each flash is recorded on the clock that's at that point, so

again, our photo clock principle or the clock principle.

That's what it means to observe an event.

That we take a photograph at that point and we know what the location of that

clock is, and then we know what the time on that clock is.

And we say can be very specific in terms of when that event occurs,

that flash of light in this case.

So physically, again, we're just looking at the x axis here.

We're just looking at events along the x axis.

Technically, you could say these flashes actually occur right on the x axis, but

we're sort of expanding it a little bit here.

So that's what our observer would see.

They could go back and check the photos generated by the clocks when those flashes

of lights occurred, and could say hey, at time t equals 0 there's flash at

x equals 0, time t equals 1 a flash at x equals 1, and so on and so forth.

Now, here's where we get into the new visual representation,

the so-called spacetime diagram.

So what we're going to do is, draw it down here first,

eventually we'll bring it up here, it's a little easier to work with, but I want to

keep this on here to remind ourselves this is physically what's going on,

we're just looking at motion in one dimension along the x axis.

We could expand this to three dimensions if we wanted to, but

it gets a little messy that way to draw off certainly and

also to see what's going on, so one dimension.

And I'm going to reproduce the x axis down here.

So something like this.

4:40

Time there, and do 1, 2, 3, we'll do three on it, 2, 3, okay?

So essentially we're creating a plot here.

All right, so remember back, maybe dredge your mathematical memory

a little bit if it's been awhile since you've done these things.

And we're just going to plot these events on this

t versus x plot, time versus its x location.

So that each flash here.

We'll use orange for our dots say to represent the green flashes here

just to make sure we realize this is not the physical representation that

we'll see here but it's representing the physical reality up here.

So at time t equals 0 and location x equals 0 we have a flash.

So that would be right here.

We'll a little circle there,

I don't want to get too much black ink on my nice orange marker here.

And then at t equals 1, at time equals 1,

I have an event at x equals 1.

So at t equals 1 here, x equals 1.

There's another event at time t equals 2, this flash right here,

the third flash occurs and that occurs at x equals 2.

So at t equals 2, I've got x equals 2 right here.

6:02

And finally, I have the fourth flash occur at x equals 3 and time t equals 3.

And so it occurs right about there, right?

And this one, we could fill in a little bit.

So those are the four points on our spacetime diagram.

It's called a spacetime diagram because we have space represented along the x axis.

Time represented on the vertical axis here, the t axis.

And so it allows us to do spacetime events, spacetime plots, as it were, okay?

because I had four events there that occurred here, here, here,

here in succession.

On the spacetime diagram, this is what it looks like.

At t equals 0, it occurred at 0.

t equals 1, I had one there.

t equals 2, I had one there.

t equals 3, I had one at that location.

Very important not to get this mixed up.

We're going to use these a fair amount, and sometimes when you start looking at

these you imagine, this means the flashes are going up at some angle.

In actual fact, no,

the flashes are just going along the x axis here, this is the x axis.

This is the time at which the flash occurs.

And, in fact, sometimes it's useful to imagine a little stop motion animation or

something that maybe you've done once upon a time.

And each time slice here at time t equals 0,

this is a photograph of the x axis.

And at time t equals 0, there's an event, a flash that occurs right there.

And then imagine at time t equals 1.

In fact, let me use my, so at time t equals 0,

if we could take a photograph along the whole axis that's what it'd see.

And then a little bit later at time t equals 1 it'd see a flash at x equals

1 there, right there.

And then a little bit later in time you see a flash at x equals 2.

And a little bit later in time you'd see that flash at at x equals 3.

So you can imagine a sort of strobe light type thing.

A strobe there and then one second, two seconds, three seconds.

So again, the actual flashes are not going up at some angle.

They're going just along the x axis.

But when we do the spacetime diagram, then we represent it as this upward going line.

And in fact we could, if we wanted to be nice about it,

we could even draw sort of a line through there like that to represent

that series of events, especially if we had a whole bunch

of other events in between there that followed that line.

Another way to think about this going back here, is what if we had a car or

a spaceship or something traveling in one dimension along the x axis, and

as it travels along it gives off flashes of light as it's going on.

So, as it's traveling along, when it hits time to equals 0 here,

flash of light there and then a little bit later, flash of light here.

All right?

And then a little bit later, flash of light there.

And a little bit later, flash of light there.

So really, the flashes of light record the progress of our spaceship or

car or whatever it is along the x axis.

And again, this then represents the line itself here, when I connect the dots,

represents the path of that spaceship or car through space and time.

So again, a spacetime diagram.

And what I'm going to do now is just do a few others here so

we can start getting a sense of using these spacetime diagrams.

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So let's just change up our numbers over here.

We'll use t equals 0 1 2 3 as before, but

let's just change the x locations of these flashes.

So, now we'll assume there's a car or spaceship going along and

giving off those flashes of light.

And so let's put some new ones in here, we'll just do, say,

0 2 4 6.

Okay, so again, we'll leave these here for a moment, this was case one.

Now case two at time t equals 0 I have x equals 0.

So again, a flash of light occurs there.

So okay, flash of light, t equals 0, x equals 0.

Next flash of light which occurs at t equals 1 occurs at x equals 2.

And then there's one at x equals 4, and one at x equals 6.

Okay, so at time t equals 1,

that flash occurs at x equals 2, so it's way over here.

So I've got a flash of light here now.

And then at time t equals 2, it's going to be x equals 4.

So about right there.

10:49

Maybe not, a little bit higher there.

Sorry, messy point.

And then at time t equals 3, it occurs at 6.

So 3, over here is 6, something like that.

I'll clean that up a little bit, I don't know.

Anyway, okay, so in this case again, if this represents a car or

spaceship going along, it flashes there, there, there, there,

assuming it's moving at a constant velocity here, so

it's not changing its speed, its velocity or anything.

Then we could draw a line here more or less through those points, okay.

And that would represent the spacetime progress of that car.

12:23

So, for this one let's do say -2, -1, 0, 1, okay.

So now, and you might actually pause it and do that plot and

see what you get here and then come back and see how we plot it here.

So we've got t equals 0, a flash at -2.

So here's our first flash.

Second flash at -1, third flash at 0, and fourth flash at 1.

And so if we were to plot those now the x coordinate at t equals 0, -2.

So it's actually over here, again,

I don't want to mess up with black on the nice orange marker again.

So there's t equals 0, it's at -2.

And t equals 1, it's at -1.

So here's time t equals one.

And it's -1 there, more or less, and at time t equals 0,

it's at 2 right here again.

And at time t equals 3, it's at 1, so right there.

And so then its progress looks like this for that case, all right?

Note that these two lines are actually parallel to each other, but

just moved over here.

In other words, this one, this series of flashes,

started at 0, at x equals 0, at time t equals 0.

This series of flashes started at -2,

over here, but its progress was the same,

it moved ahead one segment on the x axis each time, okay?

And actually let's do one more to demonstrate something.

14:13

So in this case, let's do 0, 1,

4, 9, for our flashes, okay.

So, this is the case where the four flashes and

sequence occurred at -2, -1, 0 and 1.

Again, that's the physical reality.

This is the spacetime diagram.

This is the world line, if that was a car or

spaceship traveling along with those flashes there.

14:59

And, at time t equals 3, it's at 9.

And so,8, 9, so it'd be out here someplace.

Roughly speaking, all right?

Note that if we are imagining a car or spaceship there traveling along,

it seems to be accelerating.

In fact, it is accelerating.

It's going faster and faster.

Our flashes are not equally spaced here.

It's slow, and then this time, in the next second, it jumps to here, from 1 to 4.

And the next second after that, it jumps from 4 to 9, so clearly,

it is actually accelerating.

And if we were to plot that on our plot here, I'm going to get rid of this one so

that it doesn't get too messy here.

15:39

Okay, so let's plot that.

Again, go ahead and pause if you like and do your own little plot,

it shouldn't be that difficult to see but good thing for

the brain, just let it sink in there a little bit.

And so at time t equals 0, x equals 0, so we're here again.

Time t equals 1, x equals 1, so it's really this point right here.

Time t equals 2, x equals 4.

Okay, so time t equals 2, we're at 4, actually this one.

So in fact, just to distinguish them here,

let's draw green circles around to indicate this case.

So, there's that one, of course it was at 0, and

then at t equals 2, it's at x equals 4.

And at t equals 3, it's at x equals 9, which would be you know here at some

place, so right in the middle of our world line thing, something like that.

And if we were to draw a line through that,

it would look something like this, roughly speaking, okay?

17:06

That things like this where we have a curved line,

where something is actually accelerating.

So when you see a curved line on a spacetime diagram,

whether it's curving that way or curving up, it's an accelerating situation.

And as we'll learn, the special theory of relativity does not apply

to accelerating situations, it only applies to constant velocity situations.

So just wanted to mention that because you might ask well,

why are we doing examples where everything's straight?

Well, the reason we're doing examples where everything's straight is that's

where the special theory of relativity will apply.

But this is the concept then, of spacetime diagrams.