MARKUS KLUTE: Welcome back to Special Relativity, 8.20.
In this section, we're going to talk about proper velocity.
We have seen already concepts of proper time and proper length
as the time and the space as seen
in the object's own reference frame.
So now we want to try to find something similar
for velocities, as we have seen the Lorentz
transformation applied to velocities
of a quite difficult form.
As a reminder, velocity is given as a change
in space of a change in time.
We have seen the Lorentz transformation of a velocity x
in a reference frame which is boosted in the same direction
x.
And you see that this new velocity x prime
is given by new x minus v. So there's
a velocity addition going on, which
is corrected then by this factor 1 minus uxv over c squared.
You've also seen that even so, the boost is in x direction.
There's also modification of the velocity in y direction
and in z direction.
OK?
So you see that basically, there is this--
the velocity itself is corrected with this new factor.
Note here that there is a special case
in which the direct-- the velocity in x direction
is equal to 0.
Think about this object being in its own rest frame,
again, where the velocity in the booth direction is 0.
You see that both equations simplify
for ux prime that would be simply equal to minus v,
where uy prime would be uy prime and uv prime would be uv.
So let's try to get at it.
Let's try to express velocities in terms of the proper time,
at the time as it ticks in the object reference frame.
So we have seen that the time is given by gamma times
the time in the rest frame or time in the proper--
times gamma times the proper time.
Note here that we have two different gamma
factors to play with.
One is a gamma factor of the Lorentz transformation.
And this gamma here is the gamma using
the speed of the object in a specific reference frame.
So this is the gamma which is a gamma factor which
is a gamma of v of the velocity or the speed of the object.
And now we can just simply define proper velocity
if I use this vector eta here, which is a four vector, which
is the derivative of the spatial component with the proper time.
And when we do this, you find this relatively simple solution
of gamma times c for the 0's component, gamma times ux,
gamma times uy, and gamma times uz for the last component.
So the question now is, we defined this new velocity
of an object where the time of the object
ticks in its own reference frame using
this property as proper time.
OK?
If we now try a Lorentz transformation on this,
we can simply apply the matrix for Lorentz transformation
on this four vector and find the solution here.
You can see that this is consistent by doing
this with the original components in this proper time
as well.
You see that those actually are a consistent answer.
It makes a lot of sense.