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MARKUS KLUTE: Welcome back to 8.20, Special Relativity.
So in this section, we want to look again
at collisions and study momentum conservation.
And so we have this scenario here
in which we have two balls colliding with velocities uA
and uB, mass mA and mB.
And after the collisions there, the mass
is changed to mC and mD.
And the velocities are uC and uD.
Momentum conservation tells you that the product
of the mass and velocities and the sum of the two
particles before and after the collisions is the same.
But now what happens if you boost the system?
If we look at the very same system with the
boosted reference frame, and we can just simplify the case here
by just considering the x direction.
So the question is if momentum is conserved
in a frame s, like the one we're looking here in this picture,
is the momentum also conserved in a moving reference
frame with relative velocity v?
And so you can show this quite easily, that this is actually
not the case, right?
So you write down the velocities.
The momentum equation is the same as before
with boosted velocities.
And you find that this is not the case.
You can easily show this by, for example,
setting the right part of this equation to 0,
and see whether or not this equation will
hold true in general.
And it doesn't.
But in the last section, we introduced the new concept
of proper velocity.
So how about redefining momentum through proper velocity
and just saying mA times proper velocity A plus mB times
proper velocity B is equal to mC times proper velocity C,
and so on, and see whether or not we can learn something
from this equation?
So why don't we write down this very same equation
and with proper velocity and see whether or not
it's invariant in the Lorentz transformation?
OK.
Again, a good moment to stop the video
and just work out the math by yourself, on your own.
So I did this here.
And I'm just doing this for the x component.
So we have our proper velocity vector,
which is gamma times C, gamma times uX, gamma times uY,
and sometimes uZ.
And so the x component is the first component.
And in the boosted reference frame,
the proper velocity of the x component in our boosted frame
is gamma times the proper velocity of the particle
A, first component, minus beta, upper velocity
of A0's component.
And then you can write your equation.
And this should be our momentum energy, momentum conservation
equation.
OK.
And the Lorentz transformation?
We find this one here.
And so all we need to do now in order
to show that this is always true, or true in general,
is to reassign, relabel, reorder the individual terms.
And I did this here to make this visible.
You see everything behind, in this bracket behind the beta,
is equal to the equation we had before with the minus sign
in between, which means it's 0.
And everything we see on the top of the first component,
it also be 0.
That's for the boosted reference frame.
So we see here, and it just shows this for the x component.
You can show this for all components
that momentum is conserved.