In the next several videos, we're going to revisit space time diagrams.

In particular, we want to

extend them to the cases involving the special theory of relativity.

And we're doing this,

because we'll find it's particularly useful next week in week seven.

When we talk about some of the paradoxes, such as the twin paradox and

the pole in the barn paradox.

So first here, we just want to remind ourselves of a couple things about

spacetime diagrams, and then we're going to look at it also in

the case of a light beam, how we actually plot that on a space time diagram.

So remember, we've got in our basic configuration.

We've got the x axis horizontally, the t axis vertically.

And we can plot the path of an object through space and time on the diagram.

So, this line right here we can see at T equals 1.

We'll just assume it's 1, 2, 3, 4 and x2 1, 2, 3, 4.

-1, -2, -3, and so on and so forth.

So at t = 0, x= 0, this object is

at the origin, basically, and then one second later its here.

Its moved two positions to the right along the x axis.

And then at two seconds it's four positions to the right and so

on and so forth.

And remember all our cases here we're just going to be doing constant velocity motion

that remember, if you get a curve line that means the velocity is changing and

you have accelerations Involved.

So this is space time diagram or world line is what we call it.

The world line of an object moving in the positive x direction

through time equals 1, equals 2, equals 3, so on and so forth.

This is the world line of an object moving with actually a little bit slower here,

because if you think about it, this object here moves in one second

it moves over two, x equals two, and then one second this one moves roughly

one half there perhaps, and so it's velocity is going to be slower.

In fact, as you've already probably read here, I've got a reminder that velocity

is the change in distance, the displacement divided by the elapse time

delta x over delta t, or we also call the run over the rise.

The run is the elapse distance covered in the x direction arises

how far up on our diagram you go in time.

And that also is one over the slope,

because remember the slope of the line by definition is a rise over the run.

It's the vertical distance covered divided by delta

whatever this X is here, the horizontal distance covered.

So, that if you have a line with a low slope,

it's not sloping very much, that means a high velocity.

It covering a lot of X in a short amount of time.

A line of this slope, a higher slope

It means that's covering shorter distances in a given amount of time.

And so this is a high velocity here, this is a lower velocity.

Also note of course that we can have things going backwards, we can have things

going in the negative x direction, so that's what this line here represents.

Okay, so that's just a quick reminder of some of the basics of spacetime diagrams.

We'll come back to this in a minute or two.

Let's move over to this diagram though and talk about lines of same location and

lines of same time, or the latter one here is sometimes called lines of simultaneity.

And essentially what we have for

lines of same location, that means x equals a constant.

In other words, and I've drawn in green here several of them,

this is the x equals 1 line, and remember what that means.

If this is the world line of an object in space and

time, it means it's just sitting there at the x equals one position.

As time goes on, it's always at the x equals one position.

This one's always at the x equals 2 position.

So, this is the line of same location for x equals 2.

This is the line of same location for x equals 3, and certainly for

negative x equals negative 1, x equals negative 2.

So again, if this one here was the world line of an auditor,

it would mean it's sitting at x equals negative 2 motionless

in this frame of reference of course and so on and so forth for each of those.

So, those are lines of same location.

Lines of same time, or more commonly we say lines of simultaneity,

that's where t is a constant.

And so here we have the t equals 1 line.

This is the line, everything that happens at t equals 1,

any play on any x here is on this line, the line of simultaneity.

So if have something that occurs, you know, right here, and

also something maybe that occurs over there,

those two events are simultaneous in this frame of reference.

Those two events are simultaneous in this frame of reference because they both occur

at t equals one.

And here's the t equals two line of simultaneity equals three.

And of course you can have any line in between those as well.

So in our diagrams, horizontal lines here are lines of simultaneity.

Anything on that horizontal line, any event on that horizontal line means that

happened at the same time, they were simultaneous with each other.

Of course, that's why we call it lines of simultaneity.

And that's with t equals constant.

And of course, the x-axis itself is t=0.

So it is a line of simultaneity.

And I forgot to mention the t-axis, the vertical axis is the x=0.

So if you have something sitting at the origin in our frame of reference and

not moving at all as time goes on.

It's real line would just be a vertical line right along the t-axis there.

Okay, so line's the same location, lines of simultaneity.

Let's also do a couple of other things on here.

Remind ourselves of

one more thing here that we can have things change direction of course.

And so something like this, if it's a world line going out this way,

and then maybe back that way.

Okay, so this is an object.

Think about this a minute.

What's going on here.

Remember these spacetime diagrams are not the actual physical situation.

Right? They're the representation of

what's going on.

Everything is going on along the x axis here.

This is not an object shooting up at some angle.

And then another angle another direction.

It's just moving along the x axis.

So this line here says during the first two seconds,

one two here, it's moving over here to about this point.

And so, it's moving along the x-axis to there, and then, it changes direction.

It stops and changes direction and starts moving back this way at,

actually, remember a little faster velocity here.

Because lines of lower slope mean a faster velocity.

And then at this point maybe it stays right there.

So now it's stopped for the next few seconds, and then maybe, you know,

it goes off that way again.

So this is an object that moves here to the x.

Over here x equals maybe 2.3 or something, and then moves back at a faster

velocity over here, and then stops there for a few seconds, and then starts moving

to the right again at a certain velocity given by the slope of that line.

Again, lower slopes mean higher velocity so this slope here is

closer to the horizontal, so it's a higher velocity than this slope or that slope.

So just from looking at the diagram,

you can get a sense of what that all trip is doing,

again this is only along the X axis here, that's what the diagram is showing us.

And the vertical axis is what is happening through out time.

So that is just a reminder of some of the basics of space time diagrams,

pretty much everything, not everything we covered before but

some of the main points we've covered.

Before, actually I don't remember if we introduced these concepts or not but

they're pretty basic here.

We didn't really need them until now.

We're going to use those in the next couple of video clips

as we do a few things.

But, one other thing we want to look at here is what if you shine a light beam?

Say you have a flashlight at the origin here.

You shine that light beam along the x-axis.

So again, I'm just going to shoot it that direction,

laser beam or whatever, how would we plot it on the spacetime diagram?

What would its worldline look like?

Well, clearly it has a velocity c.

And now we have to think about units a little bit, because often,

at least previously, we had maybe units x equals meters, time equals seconds.

Let's just think about that.

If x here was in meters, and time was in

seconds, what would the whirl line of that laser beam going off in that direction.

What would that look like?

Well, we know it travels 300 million meters per second or

300,000 kilometers per second.

So if this is one second here,

it's location along the x axis at that point would be 300 million.

Clearly, I'm going to have a problem fitting that

on my white board here, right?

It's going to be way out there.

Roughly, I'm just guessing,

probably three million meters that way, at least, maybe even more than that.

So, it would be a long ways away.

And so, we have a line.

If we actually had a planet on here, it's really going to look something like that.

You wouldn't be able to plot it very well using these

units for time and space, the x distance.

So what we typically do is we adjust the units.

Get rid of that a minute, and so instead of meters,

we might say well we could do kilometers, but

then it's 300 thousand kilometers per second, so again it's way way out there.

So what we might have to do instead here is,

if we've got units here we might say this is 100 thousand kilometers.

This is two hundred thousand kilometers.

This is three hundred thousand kilometers.

The guy will do that.

So, just imagine that's there.

So then, in one second the light beam would travel 300,000 kilometers.

And this is where it'd be, right there.

And in another second, it would choose another 300,000.

1-2-3, so it'd be out here someplace.

Just very roughly there and so, that'd be the worldline of our light beam.

Assuming x then is 100,000 kilometers, 200,000 300,000 400,00 and so

on and so forth.

Now that works, but it's a little awkward to have these huge units.

On the x-axis, you could certainly, another way you could do it,

let's do it like this.

Okay, so that's one option.

Another option, let's say we like x in meters.

It's fairly intuitive.

We have a sense of how long one meter is,

or one yard roughly, three feet if you want to think about it in those terms.

So what if we wanted to keep meters here, how can we change our time?

Well, we could have seconds, we could do nanoseconds.

We have nanoseconds, a billionth of a second,

going with the American usage there.

So nanoseconds.

And therefore, you may remember that in one nanosecond

light travels approximately one foot, approximately one third of a meter.

So in one seconds here, the light beam would be one third of a meter, so

it would be roughly right there, and then in one In two seconds here,

it would be, actually let's go up to three seconds, because in three seconds roughly

it would travel to a meter at that point, and so our line looks something like that.

So you can see we get different drawings here just depending on how we change

the units of course.

So this is meters and nanoseconds.

That also is not too convenient.

We don't typically think in terms of nanoseconds,

especially in maybe if you're doing table top experiments in a laboratory.

But often, we're thinking about astronomical distances.

And so we think about this a minute, and we say remember what we were talking about

last week, and that is that useful units for

the speed of light are something like light-years per year or

light-seconds per second because then c just becomes 1.

So it's 1 light-years per year, because by definition,

of course, a light year is the distance light travels in one year.

Or light seconds per second, or light days per day, or light months per month, or

whatever unit you want to use there.

Just remember light years is a unit of distance not a unit of time.

So what if we did something like that,

because then as we will see our diagram becomes very nice.

So let's just say x is in light years.

And then whatever we choose for x that determines our time unit here,

doesn't have to technically, we could put seconds

if we wanted It doesn't make much sense diagrammatically to do it that way.

So light years and years, c is one light year per year.

So that means in one year, it goes one light year.

So if we're putting the speed of light,

our laser beam shooting off here, in one year its gone one light year.

In two years, here its gone two light years.

And so on and so forth, three years, three light years.

And so we get a nice line at 45 degrees.

Nice symmetry to that, and therefore this is what we're going to choose for our

space time diagrams when we're using them with the special theory of relativity.

We're going to have light years per year typically, or it could be in light seconds

if we want to do seconds for time and so on and so forth with that.

Another way sometimes you will see this, by the way,

if you read other books on the special theory of relativity.

And later on in the course I will give you a list of some nice books for those of you

want to go beyond the course, sort of next steps you might take for books out there.

Sometimes you'll see this as as CT on the vertical axis there, okay.

We won't put that in.

We're just assuming that C is in light-years per year or

light seconds per second and so on and so forth, but if you see that,

that essentially is the same thing we've got going on, going on here.

Because if c is 300 million meters per second and time is in seconds,

c is essentially in 300 million, you put x in the same units as well.

And essentially get the same idea here.

So, just a reminder, we're going to take c as one light years per year typically.

And that means the slope of R, our beam of light along the X axis,

remember it's along the X axis is not traveling at some angle here it's just

going along the X axis at an angle of 45 degrees.

So that is actually a slop of one, because you have a rise over run,

the rise here is one, the run is one, or the rise if you go over here, it's two

here, rise is two, run is two, and so on and so forth, and that means the velocity

is one as well it is one over the slope velocity is one light year per year.

Okay, so that is a brief review of spacetime diagrams, and

then also how we are going to put something going the speed of light, so

this is the world line of that light beam going along the x axis here.

And then were going to expand on this in the next couple of video clips, and

see what happens when we introduce some of the relativistic concepts into this.

Spacetime diagrams revisited (part 1a)

Understanding Einstein: The Special Theory of Relativity

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