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# Lorentz transformation derivation part 1

Video transcript

- [Voiceover] So in all of our
videos on special relativity so far, we've had this
little thought experiment, where I'm floating in space,
and right at time equals zero, a friend passes by in her spaceship. She's traveling in the
positive x direction, velocity is equal to v,
and we draw space-time diagrams for both of us. First, I draw my space-time
diagram in white, and then I overlay her space-time diagram. And the angle that is formed between the time axes and the position
axes, right over there, that's going to be dictated by how fast v is, how fast
she is actually traveling. And we give the space-time diagram from her frame of reference, we
see with the little primes right over there. Now, one thing that you might
have been thinking about throughout this entire series, is well, if I perceive her traveling
with a velocity of v in the positive x direction,
if we took her point of view, and that's what I have right over here. If she views herself as
just floating in space, what she will see me as
right at time equals zero, actually I'd say right at
t prime is equal to zero. We're saying that t prime and
t equals zero are coinciding. Right at that moment,
she will see me fly by at negative v, going in
the negative x direction. Once again, there is no
absolute frame of reference. These frames of reference
are all absolute... I'm sorry, these frames of
reference are all relative. And so you could imagine
what it would look like, 'cause if we drew her
space-time diagram where her ct prime axis and x
prime axes that they are perpendicular to each other
and then based on that, my space-time diagram
would be at an angle. It's at an angle like this. You can kind of see the positive ct axis is in the second quadrant here because I'm traveling with the
velocity of negative v, but these angles are going to be the same. This is going to be alpha
and that is going to be, let me write this, is going to be alpha
and this is going to be, and this right over here
is going to be alpha. Now what I want to do in this video is use this symmetry, use these two ideas to give us a derivation of
the Lorentz Transformation or the Lorentz Transformations. And the way we might
start, and this is actually a reasonable way that the
Lorentz Transformations were stumbled upon, is to say, all right, we could start with
the Galilean Transformation, where we could say, all right, the Galilean Transformation
would be x prime is equal to, is going to be equal to x minus v times t. V times t. Now we already know that if you just use the Galilean Transformation, then the speed of light
would not be absolute, it would not be the same in
every frame of reference. And so we had to let go of the constraints that time and space are absolute, and so there's going to be some
type of scaling factor involved. And so we can call that
scaling factor, gamma. So, we could say all
right, let's just postulate that x prime, if we assume the
speed of light is absolute, is going to be some scaling
factor, gamma times x minus vt. Well, you could make the same
argument the other way around. If you view it from
her frame of reference, and you're trying to translate
it to my coordinates, you could say, well, x,
instead of just using the Galilean Transformation that
x is going to be equal to, x is going to be equal to x prime, and now instead of a v, we
have a negative v, right? So if you subtract a negative v, in fact, let me just write it that way. X minus negative v times t prime, that would be the Galilean Transformation, but whatever scaling factor we used here, there's a symmetry here. I shouldn't have to use a
different scaling factor if I assume a different, kind of, if I'm in a different frame of reference. So, if we assume the absoluteness
of the speed of light, we're going to have some
other scaling factor just like that or we
could rewrite this as x, let me do that same color, we could rewrite it as x is equal to this scaling factor. I'm really having trouble
changing colors today. It's gonna be equal to that scaling factor times x prime, subtract the negative plus vt prime. And if you ignore the scaling
factor right over here, this is the Galilean Transformation from the primed frame of reference to the non-primed frame of reference. So an interesting thing is
what is this scaling factor? How do we figure out what that scaling factor is going to be? And so we can do a little bit
of interesting algebra here. What we could do is, Actually, let me just
write what I just wrote, let me write it right below here. So we could say that x... Once again changing colors is difficult. We could write that x is
equal to our scaling factor, gamma times x prime, times x prime plus vt prime. And now what I'm going to
do in order to have myself an equation that involves all
of the interesting variables, I'm gonna multiply both
sides of this equation by, one way to think about
it, I'm going to multiply both sides of this top equation by x. So if I multiply the left-hand side by x, I'm going to have x times x prime, x times x prime. And then the right-hand
side of the equation, I could multiply by x,
but x is the same thing. I'm saying it's the same thing as gamma times all of this business. So, I'm just going to
multiply the left-hand sides of the equation and I'm
going to multiply the right-hand sides of the equation. So if I multiply the right-hand
sides of the equation, I am going to get gamma squared times, and I'm gonna have a big expression here, and so just really applying the
distributive property twice, x times x prime, x times x prime, and then x times positive vt, so, that prime doesn't look like a prime, x times x prime plus x times positive vt plus x times, actually
positive vt prime I should say, gotta be careful here. X times positive vt prime and then I have negative vt times x prime. So it's gonna be negative vt times x prime, times x prime. And then finally, I'll
have negative vt times positive vt prime. So that's going to be, I could
write that as a negative. Let's write that as vt squared, I'm sorry, negative v squared. And actually, let me
delete this parentheses. I don't wanna force myself
to squeeze for no reason. So I'm gonna have negative v times v, so that's negative v squared times t, times t prime. Times t prime. And now let me place my parentheses. So how can I use all
of this craziness here to actually solve for gamma? And here we're going to go back to one of the fundamental postulates, one of the assumptions
of special relativity and that's the speed of light is absolute. You're going to measure it to be the same in any frame of reference. And to think about that,
let's imagine an event that is connected with the
origin with a light beam. So let's say right at time
and t prime is equal to zero, I were to shoot my flashlight and let's say it hits something at some point. We look at some event right over there and they're connected by a
light beam, by photons. So let me connect them. So let me connect them. And so if you say, and once again this could be me turning on my flashlight and the photon at some future, at some forward distance
and some forward time, it could just be at some position or maybe it hits something, it
triggers some type of reaction. Who know what it does, but
we're going to talk about this event right over there. That event in my frame of reference, its coordinates are going to be x and ct. And since we know the
speed of light is absolute and the way that we've
set up these diagrams, any path of light is always
going to be at a 45 degree or a negative 45 degree angle, we know that x is going to be equal to ct. X is going to be equal to
ct for this particular case. I could draw it on this
diagram as well, if I'd like. Just to show that I can,
so let me draw that. So it would look like this. It would look like this and we would once again have x equaling ct. How would you read that? Well, to get the x coordinate, you go parallel to the ct axis. So that would be the x
coordinate on this diagram. And then the ct coordinate,
you go parallel to the x axis. So, just like that. But, once again, x is
going to be equal to ct, and, similarly, because the speed of light is going to be absolute
in any frame of reference, if we look at x prime, x
prime is going to be the same, is going to need to be
equal to ct prime, ct prime. If we look at over here, x prime is going to be equal to ct prime. Once again, because
light, this is going to be at a 45 degree angle. So x prime is equal to ct prime. They're connected by light events. So if you take your change in x divided by your change of time is going
to be the speed of light. So what we can do is use this information, for this particular event, if gamma's going to be true
for all transformations, it definitely should be true
for this particular event. I can use this information
to substitute back in and then solve for gamma. And that's exactly what I'm
gonna do in the next video, although, I encourage
you to try it on your own before you watch the next video.