- [Voiceover] Things are starting to get interesting.
In the first video, we set up a space-time diagram
from my frame of reference and started to plot things
past in that space-time diagram.
And then we thought about our friend, Sally, who
right at time equals zero is at x equals zero,
but she's passing me up at a relative velocity
of half the speed of light in the positive x direction.
So after one second, she has gone 1 1/2 times 10
to the 8th meters in the positive x direction.
After two seconds, she's gone three times 10 to the 8th.
And then we assume that she was part of a train
of spaceships.
So three times 10 to the 8th meters in front of her
was another spaceship and it's moving with the same
relative velocity relative to me, or in my frame
of reference.
So at time equals zero is three times 10 to the 8th meters
in front of me.
But then at time equals one, if we take this point
in space-time, in my frame of reference,
that ship is now, it now has,
so t, let me write this.
If we look at my space-time coordinates,
t is equal to one second
and x is equal to 4.5
times 10 to the 8th meters.
That's if you look at my space-time coordinates.
But what about Sally's space-time coordinates?
Well, to figure out Sally's space-time coordinates,
we go parallel to the x prime axis and see where we
intersect the t prime axis, but that's still t
is equal to one second, this is t is equal to two seconds,
this is t is equal to three seconds.
So t, or I should say t prime is equal to one second,
t prime is two seconds, t prime is three seconds.
So t prime is equal to one second still.
And so in general, we can say that t prime
is going to be equal to t, but what is x prime
going to be equal to?
So x prime, well, to figure out x prime,
you go horizontal to the t prime axis
and see where you intersect the x prime axis,
and you see that it's three times 10
to the 8th meters.
And hopefully, this makes intuitive sense.
If it doesn't, pause the video and really think about it
because in her frame of reference, that spaceship
looks stationary because it's moving with the exact same
relative velocity to me.
It's going to continue to stay three times 10
to the 8th meters in front of her, which is exactly
what we see.
So that's why its x prime coordinates stay
three times 10 to the 8th meters.
From my point of view, it's getting further and further
away from me at the relative velocity,
at 1.5 times 10 to the 8th meters per second.
So how do we translate between our,
between our x coordinates?
Let me do this in a different color.
Between our x coordinates and our x prime coordinates?
Well, you see for these examples,
we see x prime is going to be less than x,
and that should also make sense because
especially for this case, it's stationary
from the s prime point of view,
but its x is continuously increasing as time passes on
from my frame of reference, from Sally's frame of reference,
from the S frame of reference.
So if we start with x, we should subtract something,
and the difference between the two,
the discrepancy between the two is going to be
the relative velocity times time.
And so for this particular example,
we saw that three times 10 to the 8th meters
was equal to 4.5 times 10 to the 8th,
minus the relative velocity, 1.5 times 10
to the 8th meters per second times time,
which was one second.
And I didn't write the units but if you write the units,
it all works out and you get exactly this.
So hopefully that's starting to get you comfortable
with having these two coordinate planes
or two space-time diagrams over on top of each other.
And the reason why the blue one is distorted
is because it's on top, they're moving with a relative
velocity relative to what I'm considering
to be a stationary frame of reference, which is mine.
Obviously, there's no such thing as an absolute
stationary frame of reference, and we'll talk more
about that in the future.
But what I now want to focus on is that photon
that I emitted at time equal zero.
Because we saw it moves with the speed of light
in my frame of reference.
After one second, it has moved,
its x coordinate is three times 10 to the 8th meters.
After two seconds, after two seconds,
the photon is at six times 10 to the 8th meters.
But let's see what that photon looks like
from the s prime frame of reference, from Sally's
frame of reference.
Well, from Sally's frame of reference,
let's think about that photon after two seconds.
So the photon is right over there.
So t prime is equal to two seconds, two seconds,
but what is x prime?
What is x prime going to be equal to?
Well, x prime, we go parallel to the t prime axis,
is three times 10 to the 8th meters.
Three times 10 to the 8th meters.
So in her frame of reference, it took that photon of light
two seconds to go three times 10 to the 8th meters,
or it looks like the velocity of that photon
is one and a half times 10 to the 8th meters per second
in the positive x direction.
And this should hopefully makes sense from a Newtonian
point of view, or a Galilean point of view.
These are called Galilean transformations
because if I'm in a car
and there's another car and you see this on the highway
all the time, if I'm in a car going 60 miles per hour,
there's another car going 65 miles per hour,
from my point of view, it looks like it's only
moving forward at five miles per hour.
So that photon will look slower to Sally.
Similarly, if we assume this Newtonian, this Galilean world,
if she had a flashlight,
if she had a flashlight right over here
and right at time equals zero she turned it on,
and that first photon we were to plot it on her
frame of reference, well, it should go the speed of light
in her frame of reference.
So it starts here at the origin.
And then after one second, in the s prime,
in the s prime coordinates, it should have gone
three times 10 to the 8th meters.
After two seconds, it should've gone six times 10
to the 8th meters.
And so it's path on her space-time diagram
should look like that.
That's her photon, that first photon that was emitted
from it.
So you might be noticing something interesting.
That photon from my point of view is going faster
than the speed of light.
After one second, its x coordinate is 4.5
times 10 to the 8th meters.
It's going 4.5 times 10 to the 8th meters per second.
It's going faster than the speed of light.
It's going faster than my photon,
and that might make intuitive sense
except it's not what we actually observe in nature.
And anytime we try to make a prediction
that's not what's observed in nature, it means that
our understanding of the universe is not complete
because it turns out that regardless of which
inertial reference frame we are in,
the speed of light, regardless of the speed
or the relative velocity of the source of that light,
is always going three times 10 to the 8th meters per second.
So we know from observations of the universe
that Sally, when she looked at my photon,
she wouldn't see it going half the speed of light,
she would see it going three times 10 to the 8th meters
per second.
And we know from observations of the universe
that Sally's photon, I would not observe it
as moving at 4.5 times 10 to the 8th meters per second,
that it would actually still be moving
at three times 10 to the 8th meters per second.
So something has got to give.
This is breaking down our classical, our Newtonian,
our Galilean views of the world.
It's very exciting.
We need to think of some other way to conceptualize things,
some other way to visualize these space-time diagrams
for the different frames of reference.