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- [Voiceover] Let's now
dig a little bit deeper

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into the Lorentz Transformation.

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In particular, let's
put some numbers here,

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so that we're, we get a little bit more

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familiar manipulating and

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then we'll start to get a
little bit more intuition

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on how this transformation
or sometimes it's

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spoken of in the plural,
the transformations behave.

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So let's pick the scenario in
which our friend passes us by,

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and this is the same scenario
that we've been doing

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in previous videos, with
a relative velocity,

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from my frame of reference,
at half the speed of light.

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So the magnitude of her velocity
is half the speed of light.

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She is moving in the
positive x direction and

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our space-time diagrams,
they coincide at the origin.

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And so let's pick an event in space-time.

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And so let's say in my coordinate system,

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in my frame of reference,
this event that we focused on

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in the last video, let's say that is at

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x is equal to one meter...

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Let's use the same color...

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X is equal to one meter and let's say

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that time or ct is also
equal to one meter.

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And like we said in, I think
it was several videos ago,

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we could do this as a light meter,

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the time it takes for
light to go one meter.

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So we will also say this is one meter.

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So in my coordinate system,
in my frame of reference,

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this would be the point one comma one.

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One meter in the x direction,
one meter in the ct direction.

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Now, based on that, think about what

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would be the prime coordinates.

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What would be the coordinates
in her frame of reference?

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And I encourage you to pause the video,

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evaluate the Lorentz Factor using v and c,

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and then evaluate what x
prime and ct prime would be.

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All right, I'm assuming
you've had a go at it.

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Now let's work through this together.

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So first let's figure out
what the Lorentz Factor...

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Actually, let's first figure
out what beta would be.

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That will simplify everything.

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So, beta, we'll do it in the blue color,

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beta in this case is going to
be equal to zero point five c.

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That's her relative velocity
in my frame of reference.

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The ratio between that
and the speed of light,

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so that's just going to be
equal to zero point five.

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You could just view beta as

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what fraction of the speed of light

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is that person traveling
in my frame of reference

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since we're using that as kind of the

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non-primed frame of reference.

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And so let's now think about
what gamma is going to be,

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the Lorentz Factor.

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The Lorentz Factor is going to be...

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We'll do it in that reddish
color not the magenta...

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it is going to be, so gamma is going to be

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one over the square root
of one minus beta squared.

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Beta squared is, zero point five squared

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is going to be zero point two five.

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Actually let me just write
zero point five squared

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just so you can see what I'm doing.

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So this is zero point five squared

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and if we were to evaluate that...

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Let's see this is going to be
one minus zero point two five.

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So that's going to be point seven five.

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This is going to be equal
to one over the square root

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of point seven five,
one over the square root

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of zero point seven five.

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Let me get my calculator out.

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We can at least approximate it.

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So point seven five, let's
take the square root.

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And I'll just take the reciprocal
of that, so approximately

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one point one five.

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So our Lorentz Factor is
approximately one point one five.

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And now using that we can figure what

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x prime and ct prime are going to be.

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X prime is going to be
equal to my Lorentz Factor

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which is approximately one point one five.

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So maybe I'll write it as approximately

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going to be equal to one,
we'll do that same color,

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it's going to be one point,
I'm having trouble switching

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colors today, one point one five.

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One point one five times, now we're saying

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x is one meter, so x is
one meter minus beta,

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beta is zero point five,
so zero point five.

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And ct we're also saying is one meter,

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so times one.

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And then t prime or I should say ct prime,

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ct prime is going to be
approximately the Lorentz Factor

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one point one, I always have
trouble switching colors

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for the Lorentz Factor,
it's going to be approximate

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to one point one five times...

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Well, ct is one.

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I think you see a little
bit of symmetry here.

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This one in particular because

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it had the same x and ct coordinates.

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So one minus beta, one minus beta,

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so zero point five times
x, which is once again one.

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So times one.

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So in this particular
case, it simplifies to half

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of the Lorentz Factor because
this one minus zero point five

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times one that's just going
to be zero point five.

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And same thing over here, zero point five.

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So these things are going
to be approximately equal to

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zero point five times the Lorentz Factor.

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We already had the Lorentz
Factor in my calculator,

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so let me just multiply
by zero point five.

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And I get, it's approximately
point five, I'll just say

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point five eight.

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So, zero point five eight,
zero point five eight.

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And once again the units are in meters.

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So even though this is one meter and

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one meter, this over here x
prime is zero point five eight,

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zero point five eight meters and

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ct prime is also zero
point five eight meters.

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So this is also equal to
zero point five eight meters.

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So one way to think about
it, if right when our two,

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right as she was passing
me at x equals zero,

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time equals zero in my frame of reference,

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If I were to shoot my lazer gun or

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I were to turn my flashlight on and

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that very first photon starts traveling

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and so I could think about
it's path through space-time,

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it would look, that very first photon

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would look something like that.

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When I think that a, when
I think that that photon

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has traveled one meter in
the positive x direction

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and one light meter of time has passed,

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from my friend's frame of reference,

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she would say, "No, no, no, no."

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At exactly that moment,
let's say it hits an asteroid

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at that moment, it lights up an asteroid.

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She would say, "No, no, no, no, no."

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That happened zero point
five eight light meters

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after she passed me up.

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And it happened zero
point five eight meters

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in the positive x direction.

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So something very, very,
very interesting is going on.

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And I encourage you to think
about what's actually going on

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with these different parts of
the Lorentz Transformations.

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The most interesting is
what's going on, well,

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actually it's all interesting.

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In fact, the symmetry's interesting.

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But the Lorentz Factor, think
about what's happening here.

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Think about what's happening
here for low velocities when

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v is a very, very, very small fraction

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of the speed of light.

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Well then beta is going to
be pretty close to zero,

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and then the Lorentz Factor

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is going to be pretty close to one.

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And think about what happens when

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v approaches the speed of light.

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Well then this thing just booms.

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This thing gets larger and
larger and larger as we see

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this denominator getting
smaller and smaller and smaller.

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If v were actually equal
to the speed of light,

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well then you're going
to be dividing by zeros.

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Well, you know, that's when

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all sorts of silliness starts to happen.

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So I really encourage you to
try out different numbers.

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We tried very high relative velocity,

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half the speed of light,

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incredibly, incredibly high velocity.

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Try it out for something more mundane

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like the speed of a bullet
or something like that.

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But definitely get very
familiar with this.

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And also manipulate it algebraically.

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In fact, maybe in the next
video I'll manipulate this

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a little bit algebraically
so that you can reconcile

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the way I've written the
Lorentz Transformation or

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the Lorentz Transformations
with the way that

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you might see it in your
textbook or other resources.