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MARKUS KLUTE:
Welcome back to 8.20.

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In this short video, we want
to discuss length contraction.

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We're going to actually
derive length contraction.

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And we do this
with an experiment.

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The question is how long
is Bob's spacecraft?

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The experiment is conducted
by taking two pictures.

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So let's [? read ?] it here.

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The first picture is when
Alice's and Bob's spacecrafts

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just start to meet.

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Bob's relative to Alice's,
is moving with a velocity, v.

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And we take the
second picture when

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the back of Bob's spacecraft is
meeting the front of Alice's,

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just like it's shown here.

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And the pictures-- take
pictures of clocks.

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So in the first picture,
we see Alice's clock

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showed a T A1 and Bob's T B1.

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And for the second equivalently
we see T A2 and T B2.

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So your task is now to express
the length of Bob's spacecraft

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as Alice sees this
from these pictures

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and as Bob now see it
from his own pictures

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by comparing the time
and the velocity.

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Now pause the video and
try to work this out.

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So I did this here for you.

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We calculate the length.

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We can do this with
the velocity and simply

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the product of the velocity,
and the time difference,

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and the pictures as shown
[? of ?] those two clocks.

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We see this for Alice,
and we see this for Bob.

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And now we can start to compare.

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For Bob, Alice now is moving.

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So the time difference
in Alice's clock

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will be smaller by 1 over
gamma compared to what

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Bob sees on his own clocks.

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Bob says, "Your clock is slow."

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So we can use this.

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We can then calculate the
length of the spacecraft

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as Alice sees it equal
to v times delta T A.

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And we'll just use the
time dilation here.

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In this equation, we find
that the lengths are actually

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not the same.

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The length as Alice sees
this is 1 over gamma times

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the length as Bob sees this
of the very same spacecraft.

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So the length of the spacecrafts
are not the same as seen

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by Alice and seen by Bob.

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Now here you have to see that,
in this example, what I just

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did is I changed around who's
moving and who's resting.

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So here the observation of
Alice of Bob's spacecraft

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is that of a moving spacecraft.

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So Alice sees a
moving spacecraft,

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which is shorter than the
spacecraft itself at rest.

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So how can we resolve this?

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Alice will argue that
the two clocks Bob used

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are actually not synchronized.

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And if you paid a
lot of attention,

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you'll see that I'm actually
not looking at the same clock.

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I'm [? having ?] a
look at the clock

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at the beginning
of the spacecraft

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and at the end of
the spacecraft.

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And while they're
synchronized for Bob,

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they're not
synchronized for Alice.

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Who's right?

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Both are right.

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They're just observing
events or sequences

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of events as two different
[? reference ?] points.