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Current time:0:00Total duration:8:04

- [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.