0:03
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
2:40
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.
4:22
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.
9:00
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,
10:25
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.
11:23
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.
12:02
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.
13:12
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.
15:05
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.