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- [Voiceover] So in all of our
videos on special relativity

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so far, we've had this
little thought experiment,

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where I'm floating in space,
and right at time equals zero,

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a friend passes by in her spaceship.

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She's traveling in the
positive x direction,

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velocity is equal to v,
and we draw space-time

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diagrams for both of us.

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First, I draw my space-time
diagram in white,

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and then I overlay her space-time diagram.

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And the angle that is formed between

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the time axes and the position
axes, right over there,

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that's going to be dictated by

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how fast v is, how fast
she is actually traveling.

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And we give the space-time diagram from

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her frame of reference, we
see with the little primes

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right over there.

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Now, one thing that you might
have been thinking about

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throughout this entire series, is well,

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if I perceive her traveling
with a velocity of v

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in the positive x direction,
if we took her point of view,

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and that's what I have right over here.

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If she views herself as
just floating in space,

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what she will see me as
right at time equals zero,

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actually I'd say right at
t prime is equal to zero.

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We're saying that t prime and
t equals zero are coinciding.

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Right at that moment,
she will see me fly by

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at negative v, going in
the negative x direction.

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Once again, there is no
absolute frame of reference.

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These frames of reference
are all absolute...

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I'm sorry, these frames of
reference are all relative.

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And so you could imagine
what it would look like,

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'cause if we drew her
space-time diagram where her

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ct prime axis and x
prime axes that they are

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perpendicular to each other
and then based on that,

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my space-time diagram
would be at an angle.

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It's at an angle like this.

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You can kind of see the positive ct axis

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is in the second quadrant here because

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I'm traveling with the
velocity of negative v,

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but these angles are going to be the same.

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This is going to be alpha
and that is going to be,

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let me write this,

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is going to be alpha
and this is going to be,

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and this right over here
is going to be alpha.

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Now what I want to do in this video is

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use this symmetry, use these two ideas to

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give us a derivation of
the Lorentz Transformation

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or the Lorentz Transformations.

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And the way we might
start, and this is actually

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a reasonable way that the
Lorentz Transformations

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were stumbled upon, is to say,

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all right, we could start with
the Galilean Transformation,

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where we could say, all right,

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the Galilean Transformation
would be x prime is equal to,

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is going to be equal to

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x minus v times t.

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V times t.

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Now we already know that if you just use

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the Galilean Transformation,

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then the speed of light
would not be absolute,

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it would not be the same in
every frame of reference.

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And so we had to let go of the constraints

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that time and space are absolute, and

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so there's going to be some
type of scaling factor involved.

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And so we can call that
scaling factor, gamma.

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So, we could say all
right, let's just postulate

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that x prime, if we assume the
speed of light is absolute,

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is going to be some scaling
factor, gamma times x minus vt.

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Well, you could make the same
argument the other way around.

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If you view it from
her frame of reference,

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and you're trying to translate
it to my coordinates,

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you could say, well, x,
instead of just using the

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Galilean Transformation that
x is going to be equal to,

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x is going to be equal to x prime,

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and now instead of a v, we
have a negative v, right?

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So if you subtract a negative v, in fact,

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let me just write it that way.

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X minus negative v times t prime,

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that would be the Galilean Transformation,

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but whatever scaling factor we used here,

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there's a symmetry here.

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I shouldn't have to use a
different scaling factor

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if I assume a different, kind of,

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if I'm in a different frame of reference.

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So, if we assume the absoluteness
of the speed of light,

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we're going to have some
other scaling factor

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just like that or we
could rewrite this as x,

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let me do that same color,

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we could rewrite it as x is equal to

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this scaling factor.

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I'm really having trouble
changing colors today.

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It's gonna be equal to that scaling factor

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times x prime,

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subtract the negative plus vt prime.

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And if you ignore the scaling
factor right over here,

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this is the Galilean Transformation

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from the primed frame of reference

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to the non-primed frame of reference.

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So an interesting thing is
what is this scaling factor?

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How do we figure out

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what that scaling factor is going to be?

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And so we can do a little bit
of interesting algebra here.

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What we could do is,

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Actually, let me just
write what I just wrote,

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let me write it right below here.

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So we could say that x...

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Once again changing colors is difficult.

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We could write that x is
equal to our scaling factor,

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gamma times x prime,

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times x prime

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plus vt prime.

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And now what I'm going to
do in order to have myself

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an equation that involves all
of the interesting variables,

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I'm gonna multiply both
sides of this equation by,

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one way to think about
it, I'm going to multiply

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both sides of this top equation by x.

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So if I multiply the left-hand side by x,

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I'm going to have x times x prime,

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x times x prime.

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And then the right-hand
side of the equation,

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I could multiply by x,
but x is the same thing.

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I'm saying it's the same thing as

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gamma times all of this business.

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So, I'm just going to
multiply the left-hand sides

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of the equation and I'm
going to multiply the

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right-hand sides of the equation.

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So if I multiply the right-hand
sides of the equation,

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I am going to get gamma squared times,

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and I'm gonna have a big expression here,

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and so just really applying the
distributive property twice,

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x times x prime,

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x times x prime,

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and then x times positive vt, so,

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that prime doesn't look like a prime,

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x times x prime

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plus x times positive vt

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plus x times, actually
positive vt prime I should say,

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gotta be careful here.

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X times positive vt prime and then

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I have negative vt times x prime.

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So it's gonna be negative vt

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times x prime, times x prime.

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And then finally, I'll
have negative vt times

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positive vt prime.

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So that's going to be, I could
write that as a negative.

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Let's write that as vt squared,

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I'm sorry, negative v squared.

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And actually, let me
delete this parentheses.

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I don't wanna force myself
to squeeze for no reason.

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So I'm gonna have negative v times v,

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so that's negative v squared times t,

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times t prime.

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Times t prime.

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And now let me place my parentheses.

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So how can I use all
of this craziness here

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to actually solve for gamma?

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And here we're going to go back

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to one of the fundamental postulates,

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one of the assumptions
of special relativity

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and that's the speed of light is absolute.

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You're going to measure it to be

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the same in any frame of reference.

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And to think about that,
let's imagine an event

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that is connected with the
origin with a light beam.

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So let's say right at time
and t prime is equal to zero,

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I were to shoot my flashlight and

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let's say it hits something at some point.

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We look at some event right over there and

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they're connected by a
light beam, by photons.

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So let me connect them.

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So let me connect them.

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And so if you say, and once again

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this could be me turning on my flashlight

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and the photon at some future,

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at some forward distance
and some forward time,

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it could just be at some position or

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maybe it hits something, it
triggers some type of reaction.

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Who know what it does, but
we're going to talk about

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this event right over there.

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That event in my frame of reference,

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its coordinates are going to be x and ct.

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And since we know the
speed of light is absolute

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and the way that we've
set up these diagrams,

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any path of light is always
going to be at a 45 degree

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or a negative 45 degree angle,

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we know that x is going to be equal to ct.

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X is going to be equal to
ct for this particular case.

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I could draw it on this
diagram as well, if I'd like.

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Just to show that I can,
so let me draw that.

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So it would look like this.

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It would look like this and

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we would once again have x equaling ct.

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How would you read that?

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Well, to get the x coordinate,

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you go parallel to the ct axis.

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So that would be the x
coordinate on this diagram.

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And then the ct coordinate,
you go parallel to the x axis.

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So, just like that.

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But, once again, x is
going to be equal to ct,

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and, similarly, because the speed of light

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is going to be absolute
in any frame of reference,

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if we look at x prime, x
prime is going to be the same,

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is going to need to be
equal to ct prime, ct prime.

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If we look at over here, x prime is going

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to be equal to ct prime.

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Once again, because
light, this is going to be

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at a 45 degree angle.

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So x prime is equal to ct prime.

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They're connected by light events.

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So if you take your change in x divided by

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your change of time is going
to be the speed of light.

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So what we can do is use this information,

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for this particular event,

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if gamma's going to be true
for all transformations,

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it definitely should be true
for this particular event.

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I can use this information
to substitute back in

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and then solve for gamma.

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And that's exactly what I'm
gonna do in the next video,

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although, I encourage
you to try it on your own

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before you watch the next video.