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- [Voiceover] We left
off in the last video

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trying to solve for gamma.

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We set up this equation
and then we had the insight

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that look, we could
particular, we could pick

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a particular event that is
connected by light signal,

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and in that case x would be equal to c t,

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but also x prime would
be equal to c t prime.

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And if gamma's gonna hold
for any transformations

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between events, between x and
x prime and t and t prime,

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it should definitely hold
for this particular event,

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and so maybe we can use
this to substitute back in

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and solve for gamma, so
that's exactly what we're

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gonna do right now.

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So for all the xes I'm gonna
substitute it, with the c ts,

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so I'm gonna substitute with
a c t, so x, substitute ct.

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x, I'm gonna substitute c t.

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x, substitute c t,

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and that's it.

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And then all the x primes
I'm gonna substitute

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with a c t prime, with a c t prime.

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So x prime, c t prime.

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x prime, c t prime.

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And then I have an x prime here,
so it's gonna be c t prime.

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So let's simplify, and
now I'm gonna switch to

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a neutral color.

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So I'm gonna now have, I'm now gonna have

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c times t times t times t prime.

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So that's gonna be c
squared, actually let me just

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keep using the t, t primes, t, t primes.

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OK, I'll still do a little
bit of color coding.

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T prime is equal to gamma squared,

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gamma squared times, so
it's gonna be c squared

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c times c is c, let me do
that in the yellow color,

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so c squared, t times t prime,

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so t times

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t prime.

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and then we have plus

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c times v times t times t prime.

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plus c times v,

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I'll just write it this way.

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C times t,

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times v,

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times t prime.

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And then we have minus, minus, let's see,

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we're gonna have a c here.

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So minus c,

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minus c times t

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t

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times v, times t prime

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times v times t prime.

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I wrote this v in blue just
so it matches up with this.

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And we see something
interesting is about to happen.

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And then finally we have minus v squared.

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Minus v squared,

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times t times t prime.

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Times t times t prime.

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It doesn't look that much
simpler, but we're about

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to simplify it a good bit.

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And so we're gonna get
these two middle terms

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to cancel out.

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So plus c t v t prime,
minus c t v t prime.

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So those are going to cancel out.

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And then every other term has a

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t, t prime in it.

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So let's divide both sides of
this equation by t t prime.

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And so we're gonna get, if
we divide the left-hand side

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by t t prime, we're just
gonna be left with c squared,

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and then we're just gonna
divide everything by t t prime,

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and there our whole thing
has simplified quite nicely.

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Our equation is now, I'll
continue it over here.

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Our equation now is, c squared is

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equal to gamma squared,
is equal to gamma squared,

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times c squared,

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times c squared minus v squared.

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Minus v squared.

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Minus v squared.

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Close the parentheses.
And now we can divide

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both sides by c squared minus v squared,

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and we would get gamma squared,

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and I'm gonna swap the sides, too.

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So gamma squared is equal to

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c squared, c squared over,

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c squared, I'll write
it all in one color now.

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c squared minus v squared.

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Now if we like we can divide the numerator

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and the denominator by
c squared, in which case

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this will be equal to one over

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c squared divided by c squared is one,

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and then, v squared divided by c squared,

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and we are in the home stretch now,

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we can just take the
square root of both sides,

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and we get, we deserve a
little bit of a drum roll.

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Actually let me continue it up here where

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I have some real estate.

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We get gamma is equal to
the square root of this,

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well the square root of one is just one,

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over the square root of the denominator.

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square root of one minus v squared

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over c squared.

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So hopefully you found that as satisfying

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as I did, because all we
did, we just thought about

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the symmetry of x prime is
gonna be some scaling factor

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times the traditional
Galilean transformation,

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and x is going to be
some scaling factor times

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the traditional Galilean
transformation from

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the prime coordinates.

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We use that and it's important that we use

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one of the fundamental assumptions of

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special relativity,
that the speed of light

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is absolute in either frame of reference,

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that x divided by t is c, that
x prime divided by t prime

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is going to be equal to c,

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for some event that's associated with a,

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a light beam.

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We use that to substitute back in,

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and we were able to solve for gamma.

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So this looks pretty neat.

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And so some of y'all might be saying,

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well what about, what about,

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so we've been able to do the derivation

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for the x coordinates,

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but what about the, the
Lorentz transformation

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for the t and t prime coordinates?

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And I'll let you think
about how we do that.

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And I'll give you a clue,
it's just going to be

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a little bit more algebra,
and we're going to do that

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in the next video.