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- [Voiceover] Things are
starting to get interesting.

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In the first video, we set
up a space-time diagram

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from my frame of reference
and started to plot things

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past in that space-time diagram.

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And then we thought about
our friend, Sally, who

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right at time equals
zero is at x equals zero,

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but she's passing me up
at a relative velocity

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of half the speed of light
in the positive x direction.

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So after one second, she
has gone 1 1/2 times 10

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to the 8th meters in the
positive x direction.

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After two seconds, she's gone
three times 10 to the 8th.

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And then we assume that
she was part of a train

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of spaceships.

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So three times 10 to the
8th meters in front of her

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was another spaceship and
it's moving with the same

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relative velocity relative
to me, or in my frame

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of reference.

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So at time equals zero is three
times 10 to the 8th meters

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in front of me.

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But then at time equals
one, if we take this point

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in space-time, in my frame of reference,

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that ship is now, it now has,

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so t, let me write this.

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If we look at my space-time coordinates,

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t is equal to one second

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and x is equal to 4.5

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times 10 to the 8th meters.

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That's if you look at my
space-time coordinates.

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But what about Sally's
space-time coordinates?

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Well, to figure out Sally's
space-time coordinates,

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we go parallel to the x
prime axis and see where we

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intersect the t prime
axis, but that's still t

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is equal to one second, this
is t is equal to two seconds,

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this is t is equal to three seconds.

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So t, or I should say t
prime is equal to one second,

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t prime is two seconds,
t prime is three seconds.

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So t prime is equal to one second still.

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And so in general, we can say that t prime

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is going to be equal to
t, but what is x prime

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going to be equal to?

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So x prime, well, to figure out x prime,

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you go horizontal to the t prime axis

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and see where you
intersect the x prime axis,

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and you see that it's three times 10

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to the 8th meters.

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And hopefully, this makes intuitive sense.

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If it doesn't, pause the video
and really think about it

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because in her frame of
reference, that spaceship

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looks stationary because it's
moving with the exact same

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relative velocity to me.

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It's going to continue
to stay three times 10

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to the 8th meters in front
of her, which is exactly

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what we see.

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So that's why its x prime coordinates stay

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three times 10 to the 8th meters.

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From my point of view, it's
getting further and further

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away from me at the relative velocity,

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at 1.5 times 10 to the
8th meters per second.

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So how do we translate between our,

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between our x coordinates?

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Let me do this in a different color.

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Between our x coordinates
and our x prime coordinates?

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Well, you see for these examples,

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we see x prime is going to be less than x,

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and that should also make sense because

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especially for this case, it's stationary

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from the s prime point of view,

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but its x is continuously
increasing as time passes on

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from my frame of reference,
from Sally's frame of reference,

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from the S frame of reference.

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So if we start with x, we
should subtract something,

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and the difference between the two,

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the discrepancy between
the two is going to be

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the relative velocity times time.

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And so for this particular example,

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we saw that three times
10 to the 8th meters

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was equal to 4.5 times 10 to the 8th,

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minus the relative velocity, 1.5 times 10

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to the 8th meters per second times time,

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which was one second.

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And I didn't write the units
but if you write the units,

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it all works out and you get exactly this.

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So hopefully that's starting
to get you comfortable

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with having these two coordinate planes

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or two space-time diagrams
over on top of each other.

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And the reason why the
blue one is distorted

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is because it's on top,
they're moving with a relative

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velocity relative to what I'm considering

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to be a stationary frame of
reference, which is mine.

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Obviously, there's no
such thing as an absolute

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stationary frame of
reference, and we'll talk more

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about that in the future.

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But what I now want to
focus on is that photon

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that I emitted at time equal zero.

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Because we saw it moves
with the speed of light

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in my frame of reference.

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After one second, it has moved,

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its x coordinate is three
times 10 to the 8th meters.

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After two seconds, after two seconds,

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the photon is at six times
10 to the 8th meters.

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But let's see what that photon looks like

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from the s prime frame of
reference, from Sally's

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frame of reference.

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Well, from Sally's frame of reference,

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let's think about that
photon after two seconds.

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So the photon is right over there.

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So t prime is equal to
two seconds, two seconds,

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but what is x prime?

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What is x prime going to be equal to?

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Well, x prime, we go
parallel to the t prime axis,

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is three times 10 to the 8th meters.

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Three times 10 to the 8th meters.

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So in her frame of reference,
it took that photon of light

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two seconds to go three
times 10 to the 8th meters,

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or it looks like the
velocity of that photon

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is one and a half times 10
to the 8th meters per second

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in the positive x direction.

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And this should hopefully
makes sense from a Newtonian

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point of view, or a
Galilean point of view.

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These are called Galilean transformations

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because if I'm in a car

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and there's another car and
you see this on the highway

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all the time, if I'm in a
car going 60 miles per hour,

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there's another car
going 65 miles per hour,

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from my point of view,
it looks like it's only

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moving forward at five miles per hour.

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So that photon will look slower to Sally.

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Similarly, if we assume this
Newtonian, this Galilean world,

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if she had a flashlight,

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if she had a flashlight right over here

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and right at time equals
zero she turned it on,

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and that first photon we
were to plot it on her

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frame of reference, well, it
should go the speed of light

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in her frame of reference.

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So it starts here at the origin.

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And then after one second, in the s prime,

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in the s prime coordinates,
it should have gone

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three times 10 to the 8th meters.

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After two seconds, it
should've gone six times 10

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to the 8th meters.

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And so it's path on her space-time diagram

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should look like that.

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That's her photon, that
first photon that was emitted

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from it.

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So you might be noticing
something interesting.

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That photon from my point
of view is going faster

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than the speed of light.

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After one second, its x coordinate is 4.5

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times 10 to the 8th meters.

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It's going 4.5 times 10 to
the 8th meters per second.

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It's going faster than the speed of light.

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It's going faster than my photon,

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and that might make intuitive sense

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except it's not what we
actually observe in nature.

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And anytime we try to make a prediction

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that's not what's observed
in nature, it means that

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our understanding of the
universe is not complete

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because it turns out
that regardless of which

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inertial reference frame we are in,

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the speed of light,
regardless of the speed

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or the relative velocity of
the source of that light,

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is always going three times 10
to the 8th meters per second.

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So we know from
observations of the universe

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that Sally, when she looked at my photon,

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she wouldn't see it going
half the speed of light,

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she would see it going three
times 10 to the 8th meters

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per second.

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And we know from
observations of the universe

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that Sally's photon,
I would not observe it

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as moving at 4.5 times 10 to
the 8th meters per second,

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that it would actually still be moving

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at three times 10 to the
8th meters per second.

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So something has got to give.

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This is breaking down our
classical, our Newtonian,

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our Galilean views of the world.

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It's very exciting.

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We need to think of some other
way to conceptualize things,

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some other way to visualize
these space-time diagrams

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for the different frames of reference.