WEBVTT

00:00:07.010 --> 00:00:10.220
MARKUS KLUTE: Welcome back
to Special Relativity, 8.20.

00:00:10.220 --> 00:00:13.250
In this section, we're going
to talk about proper velocity.

00:00:13.250 --> 00:00:17.330
We have seen already concepts
of proper time and proper length

00:00:17.330 --> 00:00:20.930
as the time and
the space as seen

00:00:20.930 --> 00:00:24.170
in the object's own
reference frame.

00:00:24.170 --> 00:00:26.600
So now we want to try to
find something similar

00:00:26.600 --> 00:00:29.420
for velocities, as we
have seen the Lorentz

00:00:29.420 --> 00:00:32.150
transformation
applied to velocities

00:00:32.150 --> 00:00:34.370
of a quite difficult form.

00:00:34.370 --> 00:00:37.220
As a reminder, velocity
is given as a change

00:00:37.220 --> 00:00:39.830
in space of a change in time.

00:00:39.830 --> 00:00:44.542
We have seen the Lorentz
transformation of a velocity x

00:00:44.542 --> 00:00:47.000
in a reference frame which is
boosted in the same direction

00:00:47.000 --> 00:00:48.050
x.

00:00:48.050 --> 00:00:52.620
And you see that this
new velocity x prime

00:00:52.620 --> 00:00:55.350
is given by new x
minus v. So there's

00:00:55.350 --> 00:00:57.740
a velocity addition
going on, which

00:00:57.740 --> 00:01:03.290
is corrected then by this factor
1 minus uxv over c squared.

00:01:03.290 --> 00:01:08.540
You've also seen that even so,
the boost is in x direction.

00:01:08.540 --> 00:01:12.140
There's also modification of
the velocity in y direction

00:01:12.140 --> 00:01:13.890
and in z direction.

00:01:13.890 --> 00:01:14.390
OK?

00:01:14.390 --> 00:01:17.270
So you see that
basically, there is this--

00:01:19.810 --> 00:01:23.570
the velocity itself is
corrected with this new factor.

00:01:23.570 --> 00:01:26.390
Note here that there
is a special case

00:01:26.390 --> 00:01:29.300
in which the direct-- the
velocity in x direction

00:01:29.300 --> 00:01:30.680
is equal to 0.

00:01:30.680 --> 00:01:33.620
Think about this object
being in its own rest frame,

00:01:33.620 --> 00:01:38.310
again, where the velocity
in the booth direction is 0.

00:01:38.310 --> 00:01:40.710
You see that both
equations simplify

00:01:40.710 --> 00:01:46.470
for ux prime that would be
simply equal to minus v,

00:01:46.470 --> 00:01:54.420
where uy prime would be uy
prime and uv prime would be uv.

00:01:54.420 --> 00:01:58.170
So let's try to get at it.

00:01:58.170 --> 00:02:02.850
Let's try to express velocities
in terms of the proper time,

00:02:02.850 --> 00:02:07.360
at the time as it ticks in
the object reference frame.

00:02:07.360 --> 00:02:10.979
So we have seen that the
time is given by gamma times

00:02:10.979 --> 00:02:13.440
the time in the rest frame
or time in the proper--

00:02:13.440 --> 00:02:15.750
times gamma times
the proper time.

00:02:15.750 --> 00:02:19.350
Note here that we have
two different gamma

00:02:19.350 --> 00:02:20.760
factors to play with.

00:02:20.760 --> 00:02:24.840
One is a gamma factor of
the Lorentz transformation.

00:02:24.840 --> 00:02:28.980
And this gamma here
is the gamma using

00:02:28.980 --> 00:02:32.440
the speed of the object in
a specific reference frame.

00:02:32.440 --> 00:02:36.270
So this is the gamma which
is a gamma factor which

00:02:36.270 --> 00:02:41.660
is a gamma of v of the velocity
or the speed of the object.

00:02:41.660 --> 00:02:44.680
And now we can just simply
define proper velocity

00:02:44.680 --> 00:02:49.640
if I use this vector eta here,
which is a four vector, which

00:02:49.640 --> 00:02:54.880
is the derivative of the spatial
component with the proper time.

00:02:54.880 --> 00:02:58.320
And when we do this, you find
this relatively simple solution

00:02:58.320 --> 00:03:02.880
of gamma times c for the 0's
component, gamma times ux,

00:03:02.880 --> 00:03:11.570
gamma times uy, and gamma times
uz for the last component.

00:03:11.570 --> 00:03:16.460
So the question now is, we
defined this new velocity

00:03:16.460 --> 00:03:19.800
of an object where
the time of the object

00:03:19.800 --> 00:03:22.880
ticks in its own
reference frame using

00:03:22.880 --> 00:03:25.440
this property as proper time.

00:03:25.440 --> 00:03:26.490
OK?

00:03:26.490 --> 00:03:28.860
If we now try a Lorentz
transformation on this,

00:03:28.860 --> 00:03:32.910
we can simply apply the matrix
for Lorentz transformation

00:03:32.910 --> 00:03:35.370
on this four vector and
find the solution here.

00:03:38.900 --> 00:03:43.160
You can see that this
is consistent by doing

00:03:43.160 --> 00:03:48.680
this with the original
components in this proper time

00:03:48.680 --> 00:03:49.180
as well.

00:03:49.180 --> 00:03:52.440
You see that those actually
are a consistent answer.

00:03:52.440 --> 00:03:54.590
It makes a lot of sense.