- [Voiceover] We left off in the last video
trying to solve for gamma.
We set up this equation and then we had the insight
that look, we could particular, we could pick
a particular event that is connected by light signal,
and in that case x would be equal to c t,
but also x prime would be equal to c t prime.
And if gamma's gonna hold for any transformations
between events, between x and x prime and t and t prime,
it should definitely hold for this particular event,
and so maybe we can use this to substitute back in
and solve for gamma, so that's exactly what we're
gonna do right now.
So for all the xes I'm gonna substitute it, with the c ts,
so I'm gonna substitute with a c t, so x, substitute ct.
x, I'm gonna substitute c t.
x, substitute c t,
and that's it.
And then all the x primes I'm gonna substitute
with a c t prime, with a c t prime.
So x prime, c t prime.
x prime, c t prime.
And then I have an x prime here, so it's gonna be c t prime.
So let's simplify, and now I'm gonna switch to
a neutral color.
So I'm gonna now have, I'm now gonna have
c times t times t times t prime.
So that's gonna be c squared, actually let me just
keep using the t, t primes, t, t primes.
OK, I'll still do a little bit of color coding.
T prime is equal to gamma squared,
gamma squared times, so it's gonna be c squared
c times c is c, let me do that in the yellow color,
so c squared, t times t prime,
so t times
t prime.
and then we have plus
c times v times t times t prime.
plus c times v,
I'll just write it this way.
C times t,
times v,
times t prime.
And then we have minus, minus, let's see,
we're gonna have a c here.
So minus c,
minus c times t
t
times v, times t prime
times v times t prime.
I wrote this v in blue just so it matches up with this.
And we see something interesting is about to happen.
And then finally we have minus v squared.
Minus v squared,
times t times t prime.
Times t times t prime.
It doesn't look that much simpler, but we're about
to simplify it a good bit.
And so we're gonna get these two middle terms
to cancel out.
So plus c t v t prime, minus c t v t prime.
So those are going to cancel out.
And then every other term has a
t, t prime in it.
So let's divide both sides of this equation by t t prime.
And so we're gonna get, if we divide the left-hand side
by t t prime, we're just gonna be left with c squared,
and then we're just gonna divide everything by t t prime,
and there our whole thing has simplified quite nicely.
Our equation is now, I'll continue it over here.
Our equation now is, c squared is
equal to gamma squared, is equal to gamma squared,
times c squared,
times c squared minus v squared.
Minus v squared.
Close the parentheses. And now we can divide
both sides by c squared minus v squared,
and we would get gamma squared,
and I'm gonna swap the sides, too.
So gamma squared is equal to
c squared, c squared over,
c squared, I'll write it all in one color now.
c squared minus v squared.
Now if we like we can divide the numerator
and the denominator by c squared, in which case
this will be equal to one over
c squared divided by c squared is one,
and then, v squared divided by c squared,
and we are in the home stretch now,
we can just take the square root of both sides,
and we get, we deserve a little bit of a drum roll.
Actually let me continue it up here where
I have some real estate.
We get gamma is equal to the square root of this,
well the square root of one is just one,
over the square root of the denominator.
square root of one minus v squared
over c squared.
So hopefully you found that as satisfying
as I did, because all we did, we just thought about
the symmetry of x prime is gonna be some scaling factor
times the traditional Galilean transformation,
and x is going to be some scaling factor times
the traditional Galilean transformation from
the prime coordinates.
We use that and it's important that we use
one of the fundamental assumptions of
special relativity, that the speed of light
is absolute in either frame of reference,
that x divided by t is c, that x prime divided by t prime
is going to be equal to c,
for some event that's associated with a,
a light beam.
We use that to substitute back in,
and we were able to solve for gamma.
So this looks pretty neat.
And so some of y'all might be saying,
well what about, what about,
so we've been able to do the derivation
for the x coordinates,
but what about the, the Lorentz transformation
for the t and t prime coordinates?
And I'll let you think about how we do that.
And I'll give you a clue, it's just going to be
a little bit more algebra, and we're going to do that
in the next video.