- [Voiceover] In the last video,
we started to construct a space-time diagram
for my frame of reference
and I'm just drifting through space
and I'm assuming that I'm in an intertial frame of reference
which means I'm moving at a constant velocity
relative to all other frames of reference
and we set up a situation where
I emitted a photon right at time zero,
so after one second, it would have moved
3 times 10 to the 8th meters.
After two seconds, it would have moved
6 times 10 to the 8th meters,
and then we had a,
we added a little bit of flavor
to our little scenario
where right at time zero, a friend passes me up
in her spaceship and she is traveling
relative to me in the positive-x direction
at half the speed of light,
and so we plotted her path right at time zero.
Her spaceship is right there.
I could draw the spaceship.
It's at the origin.
Then after one,
after one second,
she would have traveled
1 1/2 times 10 to the 8th meters.
After two seconds, she would have traveled
3 times 10 to the 8th meters,
so this blue line in the last video
was her path.
But now I want to make it even more interesting.
Let's assume that we have actually a whole train
of spaceships, all traveling in the positive-x direction
at the same velocity as her spaceship.
So this is her spaceship right over here,
at time equals zero,
she is exactly where I am,
but let's say that 3 times 10 to the 8th meters in front
of her, there's another spaceship,
traveling at exactly
1 1/2 times 10 to the 8th meters per second,
so they're traveling at the same relative velocity to me,
but if you think about it, from each of their point of view,
they would seem to be stationary,
because the distance between them,
in this x-direction is going to stay the same.
So this person, at time zero, is going to be
3 times 10 to the 8th meters in the positive-x direction
from me.
Now, if I were to wait two seconds,
they are going to be,
so if I were to wait two seconds,
they are going to be,
they are going to be 6 times 10 to the 8th meters
away from me, because they're going half
the speed of light,
and so I could draw their path
and I'm going to do it in a slightly more muted,
thinner color, so that we,
what I'm essentially going to be doing here
is I'm setting up gridlines for
my friend's alternate frame of reference.
I'm going to put that on top of my frame of reference
and to be clear,
I'm not assuming special relativity.
I'm assuming a Newtonian world,
classical mechanics.
Just to get familiar with these ideas
and to see where the Newtonian world is going to break down
but let's say that's not the only ship,
let's say there was another ship,
that at time equals zero is 6 times 10 to the 8th meters
away from me in the positive x-direction.
Well, where are they going to be?
Where are they going to be after
two seconds?
Well, after,
let's see, after one second,
they're going to be 1 1/2 times 10 to the 8th meters
further.
Then after two seconds, they're going to be
3 times 10 to the 8th meters further,
and then after three seconds,
after three seconds, so they're going to be
1 1/2 times 10 to the 8th meters further than that,
so their path is going to look like,
so let me draw this.
I'm really just doing these to show what stationary
objects in my friend's frame of reference,
how they actually look to me,
and these should be parallel lines.
These should be parallel lines,
so that's a pretty good attempt,
and so I'll,
let me, and so let me actually just set up
that alternate frame of reference,
so this is going to be not only my x-axis,
I'm also going to call this my x-prime axis
and I'll do it in a second of thinking about
how we can read this.
So, I can color this in with the blue color.
Actually, let me do it in that light blue color
that we're going to be using for my friend's
frame of reference,
so this is also,
this is also going to be our x-prime axis overlaid,
and this one, let me call this the t-prime axis,
so my friend's frame of reference,
I'm going to call that the prime frame of reference,
or if I called,
if I called my frame of reference the S frame of reference,
I can call my friend's frame of reference
the S-prime frame of reference,
and so, let me draw horizontal lines to show the
passage of each second,
so, that's one second,
two seconds,
three seconds,
so on and so forth.
So let's make sure we're comfortable with reading
what's going on here.
So, a point,
let's say this point on our diagram.
Let me just see.
This point on our diagram,
from my frame of reference, in the S frame of reference,
this is the point time is,
or space,
or my distance,
my distance from the origin in the positive-x direction
is going to be,
I just drop a,
I just go parallel to my vertical axis,
so I'm just going to do that,
and I get 4.5 times 10 to the 8th meters,
so 4.5 times 10 to the 8th meters,
and what is our,
so I can write the units there if I want,
and our time is at one second,
one second.
So this is our coordinates from,
this is the coordinates from my frame of reference,
from the S frame of reference,
I guess S for Sal,
and so, what would be the frame?
What would be those same coordinates
from my friend's frame of reference?
Let's say her name is Sally, or S-prime.
Well, from her frame of reference,
our time is still one second.
We see the horizontal line there.
We're just going parallel to our horizontal axis
or our x-axis
so the time is still one second,
but her,
the distance that the,
or the x-coordinate is 3 times 10 to the 8th meters,
3 times 10 to the 8th meters,
and I really want you to sit and think about this
and even take out a ruler and draw this on your
own paper and feel good about this.
Now, why does this make sense?
Well, this, you could view this point as where
that second spaceship was
after a second, and from Sally's point of view,
from the S-prime frame of reference,
at time equals zero,
that person, that ship was 3 times 10 to the 8th meters
in front of her, and after a second, that spaceship is still
3 times 10 to the 8th meters in front of her.
After two seconds, it's still
3 times 10 to the 8th meters in front of her,
and so from her point of view and her frame of reference,
it is stationary,
so the way to read this for any point,
if you want my frame of reference,
you, the time is easy,
you just go horizontally to either one,
so on her axis, we've essentially just taken the coordinate
axis and sheared it and skewed it.
This is still one second,
this is two seconds, this is three seconds,
but it turned all of these kind of a grid
of squares if I were to draw a grid for my coordinates
into a bunch of,
into a bunch of parallelograms
which we see right over here,
and so for any given point in,
remember we're thinking of Newtonian space
and time,
to read it, we go horizontally for time,
in either case,
but for me, you would go straight down
because my time axis is vertical.
You go straight down and you read where it intersects
the x-axis to figure out our position
but from her frame of reference
we move parallel to her t-axis,
or her t-prime axis, I should say,
to get what her x,
I guess we could say what her x-prime coordinates
are going to be, so this is x-prime, t-prime.
This is x and this is t.
Now, what's clear, at any given point,
at any given point,
t is going to be equal to t-prime,
or we could say t-prime is equal to t,
either way,
but what's the relationship between x-prime and x?
Well, let's think about that for a second,
so when x is 4.5 times 10 to the 8th meters,
x-prime is 3 times 10 to the 8th meters,
and so it looks like,
it looks like x-prime is going to be equal to
x minus, well what's it going to be minus?
It's going to be minus
1.5 times 10 to the 8th meters
but that difference is going to grow as time goes on,
meters per second times,
times the amount of time that has passed by,
and you should see that
because after one second,
after one second, the difference is 1.5 times 10 to the 8th
meters, after two seconds,
after two seconds, so let's look at this point
right over here,
in my frame of reference, that person is sitting at
6 times 10 to the 8th meters in the positive-x direction,
but in Sally's frame of reference,
in the S-prime frame of reference,
they're still at 3 times 10 to the 8th meters
in the positive-x direction,
so we'd multiply two seconds
times 1.5 times 10 to the 8th
so we have 3 times 10 to the 8th meters,
so we take 6 times 10 to the 8th
which would be that.
We subtract out 3 times 10 to the 8th
and we get the coordinates in S-prime,
so this might get a little bit confusing
and the key here is really try to draw this yourself
and really try to plot points
and think about how they are different coordinates
in the different frames of reference,
or different time-space coordinates in the different
frames of reference.