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MARKUS KLUTE: Welcome back
to Special Relativity, 8.20.

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In this section, we're going
to talk about proper velocity.

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We have seen already concepts
of proper time and proper length

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as the time and
the space as seen

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in the object's own
reference frame.

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So now we want to try to
find something similar

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for velocities, as we
have seen the Lorentz

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transformation
applied to velocities

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of a quite difficult form.

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As a reminder, velocity
is given as a change

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in space of a change in time.

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We have seen the Lorentz
transformation of a velocity x

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in a reference frame which is
boosted in the same direction

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x.

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And you see that this
new velocity x prime

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is given by new x
minus v. So there's

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a velocity addition
going on, which

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is corrected then by this factor
1 minus uxv over c squared.

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You've also seen that even so,
the boost is in x direction.

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There's also modification of
the velocity in y direction

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and in z direction.

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OK?

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So you see that
basically, there is this--

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the velocity itself is
corrected with this new factor.

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Note here that there
is a special case

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in which the direct-- the
velocity in x direction

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is equal to 0.

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Think about this object
being in its own rest frame,

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again, where the velocity
in the booth direction is 0.

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You see that both
equations simplify

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for ux prime that would be
simply equal to minus v,

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where uy prime would be uy
prime and uv prime would be uv.

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So let's try to get at it.

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Let's try to express velocities
in terms of the proper time,

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at the time as it ticks in
the object reference frame.

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So we have seen that the
time is given by gamma times

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the time in the rest frame
or time in the proper--

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times gamma times
the proper time.

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Note here that we have
two different gamma

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factors to play with.

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One is a gamma factor of
the Lorentz transformation.

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And this gamma here
is the gamma using

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the speed of the object in
a specific reference frame.

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So this is the gamma which
is a gamma factor which

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is a gamma of v of the velocity
or the speed of the object.

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And now we can just simply
define proper velocity

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if I use this vector eta here,
which is a four vector, which

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is the derivative of the spatial
component with the proper time.

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And when we do this, you find
this relatively simple solution

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of gamma times c for the 0's
component, gamma times ux,

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gamma times uy, and gamma times
uz for the last component.

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So the question now is, we
defined this new velocity

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of an object where
the time of the object

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ticks in its own
reference frame using

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this property as proper time.

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OK?

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If we now try a Lorentz
transformation on this,

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we can simply apply the matrix
for Lorentz transformation

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on this four vector and
find the solution here.

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You can see that this
is consistent by doing

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this with the original
components in this proper time

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as well.

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You see that those actually
are a consistent answer.

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It makes a lot of sense.