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- What I hope to do in
this video is get even more

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algebraically familiar with
the Lorentz transformation,

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so that we can recognize
it in its different forms

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and start to build our
intuition for how it behaves.

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So let's just write down
the Lorentz transformation,

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or at least the way
that I like to write it.

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So let's just remind ourselves,

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if I'm in my frame of reference,

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I am floating through
space, so I could say s

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for Sal's frame of reference.

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Some event in space time,
from my point of view,

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is going to have some x-coordinate,

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I will do that in green,
and it's going to have

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some ct coordinate, I
can do that in orange.

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So the Lorentz transformations
are going to go from

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coordinates in my frame of reference,

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space-time coordinates for an event,

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to my friend's frame of reference,

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so we can say that's the
s-prime frame of reference,

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and her frame of reference, the event,

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will have space-time coordinates x-prime,

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let me write it this way: x-prime,

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comma, ct-prime,

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I'm really having trouble
switching colors today,

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ct-prime, so let's just
write it down the way

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I've written it down
in the previous videos,

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and then I'll do a little bit
of algebraic manipulations,

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so we can recognize its different forms.

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So if we want to get x-prime,

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we see that it's going to be based on

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the Lorentz factor times x

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minus a scaled version of ct,

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and the scaling factor is beta,

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and we will redefine beta in a second,

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so beta times ct,

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where our Lorentz factor,

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let me write it over here,

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the Lorentz factor is
one over the square root

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of one minus v-squared over c-squared,

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or we could write it as
one over the square root

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of one minus beta-squared,

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where beta is equal to

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v over c. So there you go.

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That's how we get x-prime
and it's gonna be based on

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the Lorentz factors dependent on v

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and of course the rest
of this is going to be

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dependent on x and ct.

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And so how do we get ct-prime?

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Well ct-prime is going to be equal to,

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I'll just write it right over here,

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ct-prime is going to be equal to

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the Lorentz factor times,

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and once again this is going to be

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the nice symmetry we talked about,

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times ct, we'll do that in orange,

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so ct minus

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beta times x.

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And like I said before, I
like to write it this way.

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I find it easier to remember.

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I find it easier to
remember because it has this

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nice, beautiful symmetry to it.

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When I'm trying to solve for x-prime,

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it's x minus beta times ct.

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When I'm solving for ct-prime,
it's ct minus beta times x.

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And in both cases, I'm
scaling by the Lorentz factor.

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Well let's manipulate this a little bit,

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just to understand a little better,

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and reconcile with what you might see

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with other sources,
including, say, your textbook.

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Well we know that beta
is equal to v over c,

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and this is v over c,

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and so we can write this
as v over c times c,

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this c's going to cancel with that c.

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And so we could rewrite this as

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x-prime is equal to the Lorentz factor,

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gamma times x minus vt.

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V times t.

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Now this is really
interesting right over here,

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'cause if you'd ignored gamma,
or if gamma was one here,

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this is essentially the
Galilean transformation.

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If it was just x minus vt, that was just

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our intuition about our everyday life,

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but Newtonian physics
would actually tell us,

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and so when you view it in this form,

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you really think, okay, you're
just going to scale that

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by this Lorentz transformation,

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which has this interesting behavior

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that if v is much, much smaller
than the speed of light,

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well then, this whole factor

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is going to be pretty close to one,

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and that's why the Galilean
transformation's worked for us,

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for kind of everyday velocities.

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But then if v starts to
approach the speed of light,

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this thing blows up and we
get a very different result

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than with our traditional
Galilean transformation.

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Well let's think about
what happens over here,

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and over here, instead
of staying in ct-prime,

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I'm gonna also divide both sides by c,

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so we're just solving for time

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as we normally associate it.

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Just the t-prime variables
as opposed to ct-prime.

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So let's also divide both sides by c,

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so you divide by c there, and
you can divide by c there,

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and you can divide by c there,

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those c's cancel, those c's cancel,

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and so we're left with t-prime is equal to

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the Lorentz factor,

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times t times t minus,

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now you're going to get v times x,

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and you're going to divide by
c twice, so over c-squared.

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So vx,

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v times x over c-squared.

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And this is actually a more
typical way, both of these,

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of seeing the Lorentz transformation.

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The reason why I don't
like this form as much,

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even though this does
have the neat, kind of,

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when you look at it, it looks
like you're just scaling up

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the Galilean transformation,

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is it you no longer
see the symmetry there,

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and you should see the symmetry there,

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because we're talking about space-time.

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We're not talking about this
independence of space and time.

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We saw how the angles in
the Minkowski diagram,

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how those were symmetrical,
the angles between the regular,

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or the unprimed axes, and the primed axes.

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And so what I don't like about this

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is you no longer see the symmetry,

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while you did see it the
first way that I wrote it.

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And frankly I find this
harder to remember.

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But let's just think
about what happens here

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when v is a very small
fraction of the speed of light.

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Well as we've already
said, our Lorentz factor

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is going to be pretty close to one,

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and if v is a very small fraction of c,

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well then this second
term right over here,

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is going to be pretty close to zero.

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And so if v is a small fraction of c,

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then this thing is going to be get pretty,

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just let me just write this down,

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so if v is much less than c,

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then this is going to reduce to

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t-prime being approximately equal to,

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because our Lorentz factor

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is going to be pretty close to one,

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this is going to be pretty close to zero.

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So this is going to be pretty close to t.

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Likewise, for v

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much lower than c over here,

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our Lorentz factor is going
to be pretty close to one,

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and so x-prime is going to be

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approximately equal to x minus vt.

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So for small v's,

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and small can even be
the speed of a bullet,

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or even the speed of the space shuttle,

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or things that are much, much smaller

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than the speed of light.

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Three times 10 to the
eighth meters per second,

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that's why the Galilean transformations

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are pretty good approximations.

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So hopefully this starts to give you

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a little bit of intuition.

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Start evaluating this,
evaluate this for v's

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in our everyday life,
and then see what happens

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when v starts to approach
.8 times the speed of light.

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.9 c, .99 c, think about what
happens to the Lorentz factor.

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And hopefully you'll
get an appreciation for

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how this whole thing behaves.