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MARKUS KLUTE: Welcome back
to 8.20, Special Relativity.

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So in this section,
we're going to review

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the content of the
material, but underlined

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with a few questions and
examples, very similar

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to the previous section.

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It's just this one is
interleaved with activities.

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So let me start by bringing
back two of those Einstein

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quotes, which nicely relate
to each other in a sense

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that they feed off each other.

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The first one is, "I
have no special talent.

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I'm only passionately curious."

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Albert Einstein.

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And the second one, "It is
a miracle that curiosity

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survives formal education."

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And I [LAUGHS] sincerely
hope that I didn't stop

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your curiosity
with this lecture--

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quite the opposite.

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Like we discussed here,
it's just the starting point

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of a wider discussion
of general relativity

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in your education at MIT
in the Physics Department.

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You could learn about quantum
mechanics, quantum field

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theories.

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And a lot of physics
is out there,

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which is super exciting
and interesting.

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So it basically
needs your curiosity

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in order to tackle challenging
questions in physics today.

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We started the
discussion looking

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at the background
which led Einstein

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to make his discoveries.

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And specifically the
year 1905, in which

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he was able to come out with
five papers, all breakthrough

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papers, including the theory
of special relativity.

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His career didn't stop there.

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He developed the general
theory of relativity

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and published a paper
on this in 1915.

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And his fame as a
physicist really

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comes out of the predictions
he made at that time.

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So we set the context
of this class,

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and we started with a question
of Galilean transformation and

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whether or not you can
tell in a moving train car

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whether or not this is
actually moving or stationary.

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And we demonstrated that
time and acceleration

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is invariant in the
Galilean transformation.

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And then, therefore, since you
cannot distinguish the strength

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of a force in two reference
frames which move to each other

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with moderate velocity, you
cannot tell whether or not

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the train car is moving or not.

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Very important are the
development of clocks and times

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and signal processing
of the time.

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And we discussed this in the
context of trains and train

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lines, but also in Einstein
living in the city of Bern,

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with a large number
of clock towers

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which needed synchronization.

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And even working in
the patent office

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certainly confronted him with
those questions all the time.

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So time is suspect.

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That is really the
key to moving out

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from Galilean transformation
into Newtonian mechanics

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to special relativity.

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When we classify or when
we look at specific series,

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we have to understand that
they live within a context,

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within a range of validity.

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And so classical
mechanics is not wrong

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because it breaks down
at large velocities.

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It's just only correct in
the frame of slow velocities.

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And special relativity
also has its limitations,

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as in it only describes
reference frames or scenarios

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in which there is no
acceleration between two

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reference frames.

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Going back to the question
of the time, Michelson--

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and a number of other
experiments leading to the very

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same direction--

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was trying to
establish that there

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is an ether wind, a medium
in which light is moving.

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And his experiment,
at the time, failed

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to demonstrate that
ether actually exists.

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His experiment used a light
source and a couple of mirrors

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in order to show
interference patterns.

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And those interference patterns
didn't manifest themselves.

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So he thought for a long time
that his experiment is limited

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or that he has made a mistake.

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But it turns out that
ether indeed doesn't exist.

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Einstein tackled this problem
by making two postulates.

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The first one is the
principle of relativity,

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that there is no preferred
reference frame if you want;

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and the second, that the
speed of light is constant--

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and constant and the same
in all reference frames.

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And in this class we use
those two postulates in order

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to derive everything we know
about special relativity.

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So we looked at
the implications.

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And the implications
started from time dilation,

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length contractions.

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We were able to derive the
Lorentz factor and Lorentz

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transformation.

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And we did this by showing
this light clock here,

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where you observe a ticking
clock in which there

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is two mirrors and
light bouncing back.

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And each time there
is a bounce, we

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count this as one
tick of the clock.

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So if the clock is
moving, the light

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has to travel a longer distance.

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And hence time is delayed.

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And from just the
geometry of this problem,

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we were able to derive
this gamma factor here--

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1 over square root of
1 minus beta squared.

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The beta is velocity
over the speed of light.

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So as the first activity
in today's class,

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I want you to think
about a clock which

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is moving with a
photon, a clock which is

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moving with the speed of light.

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And also discuss why isn't
it possible to go faster

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than the speed of light.

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And why can you not
just keep accelerating?

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So think about
this a little bit.

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In the live class we
will have a discussion.

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But just come up with
some sort of answer

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of why this is the case.

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So here we have, in
the class, actually

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showed that there
is a real speed

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limit, that if you
try to go faster,

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past [INAUDIBLE]
velocity, you run

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against a boundary,
a real speed limit.

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If you think about keeping
accelerating, giving

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more energy, you
find that the amount

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of energy you need in order to
go faster and faster, faster,

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doesn't get you to velocities
which are faster and faster.

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And again, you enter a speed
limit-- the speed of light.

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An important topic
in understanding

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some of the paradoxes
in special relativity,

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and some of the
confusion, is the concept

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of the relativity
of simultaneity.

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Now, it can be illustrated quite
nicely in this example here,

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where you have a carriage train
car with light being emitted

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and clocks which record those
events at each end of the train

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car.

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For the stationary person,
those clocks will tick in sync.

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They will always show the
same tick and the same time.

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But for somebody who's observing
this train from a platform,

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or somebody who's moving
with a relative velocity

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towards this train car, you
will see those clocks not

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ticking at the same time.

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So the clear evidence
is given in this picture

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again, where you see that
the light's being emitted

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in the center, but one side
of the train car is hit first

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and the second side
is hit afterwards.

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So you see that the
leading clock lags.

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The leading clock in
this example lags behind.

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And so what you find
here is that events which

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are observed simultaneously
for one observer--

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in this case, the person
inside the carriage--

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they will not be simultaneous
for an observer who's moving

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with a relative velocity.

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And that led us to the
understanding of the pole

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in the barn paradox, where,
in one example, the event

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of the front of the pole
hitting the back of the barn

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and the event of the
back of the pole hitting

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the front of the barn,
they are simultaneously

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for the barn owner.

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But they're not simultaneously
happening for the person

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who's carrying the pole.

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In this case, the event of
hitting the back of the barn

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is simultaneous to an event
where the back of the pole

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is still sticking
out of the barn.

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So there is a
clear disagreement.

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But the disagreement
can be resolved

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by understanding that
simultaneous events are not

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necessarily simultaneous
to two observers.

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Then we moved on to a
variety of other paradoxes

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in special relativity.

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And the most famous likely
is the twin paradox,

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where we discussed that a person
moving away and then returning

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is younger than the person
who actually stayed at rest.

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And we discussed that we were
able to use time dilation

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or length contraction in order
to quantitatively figure out

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the difference in time.

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But we also discussed
that the person

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who is moving away
and then coming back

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needs to describe the journey in
two different reference frames.

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And from the fact that
you don't consistently can

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describe this sequence
of events as two

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reference frames
can see the paradox

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and the extra confusion.

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So here we have
another activity--

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an asymmetric travel.

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So we discussed also the example
where two people move away

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and then they come back
in a symmetric fashion.

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But here we want
to discuss the case

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where there are three trends.

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Carol stays on Earth.

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Bob moves to Star 1.

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And Alice moves to Star 2.

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The distance to Star 1 is longer
than the distance to Star 2.

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So the question is, in this
journey, they both start

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and they both return at
the same time for Carol.

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But which of the
twins is the youngest?

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So again, I invite you
to just work this out.

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You can use some numbers if
you want a quantitative answer.

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Or you can just reason
about [INAUDIBLE]..

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The answer here is that Bob
is the youngest of the three

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once they return to Earth.

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And the reason for this is
the distance he has to travel

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is the longest.

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Hence the velocity he has
to travel in is the largest.

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And hence the effect of
time dilation for him

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is the biggest.

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And therefore he's going to
be the youngest of the three.

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All right.

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We had a rather long discussion
about waves and light, Doppler

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effect, and relativistic
Doppler effect.

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Here, just as a
reminder, the wave

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equation for an electric
field in a vacuum.

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And the solution to
the wave equation

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is use the second derivative
with space and time.

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And the solution
simply can be expressed

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as a cosine, which is a
function of space and time.

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We have talked about
light quite a bit.

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And just as
reminder, [INAUDIBLE]

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the energy of photon is
related via the Planck-Einstein

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relation to the frequency.

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So the higher the frequency,
the higher the energy.

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The higher the frequency,
the higher the energy.

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And here, in this
picture, you can

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see the effect of
the Doppler effect,

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where, when you have
a moving source,

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the observer sees the
waveline modified.

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Objects which move
towards us are

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blueshifted, starting
from white light,

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or green light in this example.

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And objects that move away
from us are redshifted.

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The effect can be
used, for example,

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in speed measurements of cars.

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It can also be used in order
to measure speed or distances

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of stars moving away from us.

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And so that defines,
then, the concept

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of redshift, which is simply
the ratio of the difference

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in wavelength divided
by the wavelength as

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observed by the observer.

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So here there's two
concept questions.

00:13:24.220 --> 00:13:27.490
The first one is, is
the wave equation,

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which you can see
there as an example,

00:13:29.100 --> 00:13:32.250
invariant under
Lorentz transformation?

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And the second question is,
how about the solutions?

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Are the solutions
to the wave equation

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invariant under
Lorentz transformation?

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So I'll have you
work this out again.

00:13:42.100 --> 00:13:48.800
And the answers are yes and no.

00:13:48.800 --> 00:13:50.840
The wave equation is invariant.

00:13:50.840 --> 00:13:54.490
The wave equation
describes the physics.

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It explains how electric
and magnetic fields change.

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And the laws of physics
need to be invariant

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under Lorentz transformation.

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Otherwise they
will not be valid.

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They will violate the postulate
we just made that all reference

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frames are equal to each other.

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However, the solutions of the
wave equation-- light itself--

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is not invariant under
Lorentz transformation.

00:14:16.900 --> 00:14:19.870
We've just discussed
redshift and blueshift,

00:14:19.870 --> 00:14:22.120
which means that the wavelength
and frequency of light

00:14:22.120 --> 00:14:26.290
changes with respect to the
observer, or for each observer.

00:14:26.290 --> 00:14:28.120
So the solutions-- light--

00:14:28.120 --> 00:14:32.613
are not invariant under
Lorentz transformation.

00:14:32.613 --> 00:14:34.780
And then we went a little
bit into particle physics.

00:14:34.780 --> 00:14:37.660
And I have to apologize
for my own preference.

00:14:37.660 --> 00:14:39.630
But elementary
particles, as they

00:14:39.630 --> 00:14:41.080
have been reproduced
or observed,

00:14:41.080 --> 00:14:43.970
are typically moving at
rather large velocities.

00:14:43.970 --> 00:14:45.850
So they are very good
examples to study

00:14:45.850 --> 00:14:48.310
effects of special relativity.

00:14:48.310 --> 00:14:51.520
We looked at energy, the
total energy m0 gamma c

00:14:51.520 --> 00:14:54.010
squared, which also
can be expressed

00:14:54.010 --> 00:14:57.280
as the energy of the rest
energy of the particle,

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m0 c squared, plus
the kinetic energy.

00:15:00.100 --> 00:15:01.930
And we looked at
the total energy

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as being invariant, one
of those invariants,

00:15:05.680 --> 00:15:08.900
as equal to the total
momenta squared.

00:15:08.900 --> 00:15:11.350
The total energy squared is
equal to the total momentum

00:15:11.350 --> 00:15:14.560
squared times c squared
plus the rest mass squared

00:15:14.560 --> 00:15:16.610
times c to the fourth power.

00:15:16.610 --> 00:15:19.980
And then we went through a
larger number of examples,

00:15:19.980 --> 00:15:23.340
from accelerating electrons
to composite particles.

00:15:23.340 --> 00:15:27.490
We talked about deuteron
photon absorption and emission,

00:15:27.490 --> 00:15:31.350
the creation of particle, the
creation of antiparticles,

00:15:31.350 --> 00:15:33.520
and the scattering of particles.

00:15:33.520 --> 00:15:35.580
So here we had another example.

00:15:35.580 --> 00:15:39.270
Oops-- without the solution.

00:15:39.270 --> 00:15:43.710
In 1995, at Fermilab, a
proton-antiproton collider,

00:15:43.710 --> 00:15:47.130
the Tevatron, top
quarks were discovered.

00:15:47.130 --> 00:15:51.960
And we measured the top
quark mass to 175 GeV.

00:15:51.960 --> 00:15:54.120
The center of mass
energy at the Tevatron

00:15:54.120 --> 00:15:58.740
was 1.8, and later almost
2 tera-electronvolt,

00:15:58.740 --> 00:16:02.730
and clearly sufficient for the
production of top and antitop.

00:16:02.730 --> 00:16:05.640
But what is the minimal energy
in order for this process

00:16:05.640 --> 00:16:07.270
to occur?

00:16:07.270 --> 00:16:10.950
And here we went through
a number of examples.

00:16:10.950 --> 00:16:12.420
The minimal energy-- sorry.

00:16:12.420 --> 00:16:14.580
I have to work this out again.

00:16:14.580 --> 00:16:20.070
The minimal energy required
can be derived or extracted

00:16:20.070 --> 00:16:23.340
in the center of mass frame,
where the top and antitop are

00:16:23.340 --> 00:16:25.000
produced at rest.

00:16:25.000 --> 00:16:26.560
And if you do this--
in this example,

00:16:26.560 --> 00:16:29.530
the proton in this
collider experiment--

00:16:29.530 --> 00:16:31.320
the experiment is
already conducted

00:16:31.320 --> 00:16:32.760
in the center of mass frame.

00:16:32.760 --> 00:16:35.760
So the minimal energy
is simply 2 times

00:16:35.760 --> 00:16:39.210
the top mass times c square,
or 2 times gamma times the mass

00:16:39.210 --> 00:16:41.310
of the proton times c
square, which gives you

00:16:41.310 --> 00:16:44.380
a gamma factor of 175.

00:16:44.380 --> 00:16:47.310
But the likelihood
to actually observe

00:16:47.310 --> 00:16:54.070
a top quark and a antitop
quark at that energy, 175 GeV

00:16:54.070 --> 00:16:58.840
proton or antiproton
energy, is rather 0.

00:16:58.840 --> 00:17:01.540
And the reason for this has
to do with the structure

00:17:01.540 --> 00:17:02.620
of the proton.

00:17:02.620 --> 00:17:05.619
The actual interaction between
the proton and the antiproton

00:17:05.619 --> 00:17:10.869
is such that the quarks and
antiquarks inside the proton,

00:17:10.869 --> 00:17:12.730
and also the gluons, interact.

00:17:12.730 --> 00:17:15.040
And they only carry a
fraction of the momentum

00:17:15.040 --> 00:17:16.390
and the energy of the proton.

00:17:16.390 --> 00:17:18.640
And hence this
minimal calculation

00:17:18.640 --> 00:17:21.430
is insufficient to get a
sufficient cross-section

00:17:21.430 --> 00:17:25.359
likelihood for top quarks and
antiquarks to be produced.

00:17:25.359 --> 00:17:28.420
But that is particle physics
and goes beyond the scope

00:17:28.420 --> 00:17:30.640
of this lecture.

00:17:30.640 --> 00:17:33.700
One last point, which leads
sometimes to confusion,

00:17:33.700 --> 00:17:38.660
is the concept of conserved
and invariant properties.

00:17:38.660 --> 00:17:41.260
When we look at the
meaning of the word,

00:17:41.260 --> 00:17:43.630
invariant means never-changing.

00:17:43.630 --> 00:17:46.060
And in the concept of
special relativity,

00:17:46.060 --> 00:17:48.550
properties are invariant
when they do not

00:17:48.550 --> 00:17:52.540
change under Lorentz
transformation or Galilean

00:17:52.540 --> 00:17:56.170
transformation, as we
discussed earlier in the class.