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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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PROFESSOR: Welcome back to 8.20.

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In this section and
the following ones,

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we talk about paradoxes
in special relativity.

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A paradox is something which is
absurd or self-contradictory.

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So we have statements
which don't really make

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any sense when put together.

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The pole in the barn paradox
is rather interesting.

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And we will analyze this, and
at the end of the discussion,

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we hopefully agree that
there is no paradox here.

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It's a pseudo paradox.

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So the situation is as follows.

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We have Alice.

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She has a pole.

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The pole is 10 meters long
in her reference frame.

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And Bob is very proud
of his New England

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barn which is, in his
reference frame, 8 meters long.

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Alice, however, is moving
with a velocity of 0.6 times

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the speed of light, which gives
us a gamma factor of 1.25.

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Does the pole fit into the
barn is the question, or not?

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So stop here and think
about this for a second,

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and we will continue
with an analysis of this.

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So here is the analysis
for Alice's frame

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and the analysis
for Bob's frame.

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For Alice, the barn
in her reference frame

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is Lorentz contracted.

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It's 6.4 meters long, but
her pole is 10 meters long.

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So we should clearly
answer this question

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by saying it doesn't fit.

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In Bob's frame, the
barn is 8 meters long,

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and the pole is
Lorentz contracted--

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also 8 meters long.

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So Bob will say, yeah, it
fits-- it just barely fits.

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They're exactly the same
size, so yes, it all

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fits into the barn.

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And here is where
you might think

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this is an absurd statement.

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They cannot be both right.

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We will see they can.

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They can both be right.

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They just disagreed on
the fact that events

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happen simultaneously.

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What are the
crucial events here?

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When does the barn
hit the end-- does

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the pole hit the
end of the barn,

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and when does the back of the
pole hit the front of the barn?

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Those are the two things
we have to study in detail.

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But let's get to it.

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How can they, or why
can they disagree?

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So the idea is that you
draw space-time diagrams

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for the pole in
the barn, and show

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that there's no paradox by using
the world lines of the pole.

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Before we do this, we're going
to analyze this a little bit

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more.

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So assume that the front of the
pole enters the barn at time

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equals 0 for both Bob and Alice.

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Then Bob observes the
pole entering his barn,

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and it takes 44.4 nanoseconds--

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8 meters divided by 0.6
times the speed of light--

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for the front of the pole to
reach the back of the barn,

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and the back of the pole to
reach the front of the barn.

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So after 44 nanoseconds,
in Bob's reference frame,

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the pole is in the barn.

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Alice, however, sees the
barn Lorentz contracted.

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It's 6.4 meters long.

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She moves this 0.6 times
the speed of light.

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So for her, she reaches the
back after 35.6 nanoseconds,

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in which case, Bob's
clock only shows

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28.4 nanoseconds,
because Alice's clock

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time is Lorentz contracted.

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So we can clearly conclude
here that that's not

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enough time for Bob such that
the pole actually entered

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the barn for the full length.

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So the back of the
pole is still outside.

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So we want to consider
three different events.

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The first event is
after 44 nanoseconds,

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and in the space of 8 meters
in Bob's reference frame.

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For Alice-- this is the
situation we just analyzed--

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36.6-- 35.6 nanoseconds passed.

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And in her reference frame,
the front of the pole

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is at 0 meters.

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The second event is
then the other side

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of the barn in Bob's
reference frame

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after 44.4 nanoseconds 0 meters.

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He sees-- or she sees that
55 nanoseconds have passed.

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We use Lorentz
transformation here,

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but the position
is minus 10 meters.

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And the last point is 28.49
nanoseconds and 0 meters.

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That is the observation
when Alice sees the end

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of the-- front of the barn--

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the front of the pole
at the end of the barn.

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That translates
into Alice's frame

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a 35.6 nanoseconds,
and minus 6.4 meters.

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So the minus 6.4 meters
tells you very clearly

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what we just already said.

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The back of the pole
is still outside.

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So that's the quantitative or
numerical kind of evaluation.

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And we can also show
the very same thing

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in the space-time diagram.

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So we show the
space-time diagram here,

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and this is Bob's
reference frame.

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So the pole just touched
the front of his barn,

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and the barn is
located at 8 meters--

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the end of the barn
is located 8 meters.

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The front of the barn
is located at 0 meters.

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After 44 nanoseconds, there's
event number one and event

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number two.

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The pole is fully in the barn.

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But we can also show the
pole in event number three.

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So one-- where is
the end of the pole?

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We look at this diagram here.

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Where is the end of the pole
when the front of the pole

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hits the end of the barn?

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You see clearly there's
a piece sticking out.

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We saw that there's--

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in this event here,
6.6 meters in Alice's

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frame still seeking out.

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So we see that event number
three is located here,

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and not all of the pole is
actually contained within the.

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So Bob and Alice disagree
on whether the front

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and the back of the pole are
in the barn simultaneously.

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That's where the situation
becomes contradictory.

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They don't agree that two events
which happened at the same time

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in their reference frame--

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in Bob's reference frame
occurs at the same time

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in Alice's reference.