WEBVTT

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Professor Ramamurti
Shankar: So,

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what did we do the last time?

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Here is what we were talking
about.

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We were trying to describe an
event as seen by two observers.

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One will be called S and
here is the x axis for

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S;
the other is called S

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prime.
S prime is sliding to

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the right at velocity u.

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Therefore, at a certain
time--Let me say S is me

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and S prime is you.

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When you pass me,
I'm sitting at the origin of my

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coordinates, x = 0.

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You're at the origin of your
moving coordinates,

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x prime = 0,
and when we crossed each other,

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that time we set as 0 in our
own clocks.

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We synchronized them so that
the event, x = 0,

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t = 0,
also had coordinate x

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prime = 0,
t prime = 0.

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When you and I crossed,
we pushed our stopwatches,

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we synchronized our clocks,
our origins coincided,

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and that's when it began.

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So, after some time,
you are here.

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This is your frame and this is
my frame.

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If something happens here,
something could be anything.

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I hope you understand what an
event is.

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That's not a relativistic
notion.

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That's the very old notion.

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Something happens at some place;
firecracker goes off.

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I ask you, "When did it happen?"

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and "Where did it happen?"

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and you are given the
coordinates.

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So, everything that happens is
an event.

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You saw Elvis,
I say, "Where did you see

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Elvis?"
At the supermarket.

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Okay.
When did you see Elvis?

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I would also ask you what you
were smoking because this third

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question, that's not an extra
coordinate but in this case I

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would like,
well, I should tell you beyond

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all the laughter that's a
serious issue.

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Why is that not a coordinate?

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Why not ask more and more
questions and call them all

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coordinates?
The reason the other things are

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not coordinates is very
important.

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Why is time now suddenly a
coordinate?

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Well, as you could have always
asked 100 years ago when did

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something happen,
and the reason is that you will

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see in the new relativistic
physics, x and t,

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according to one person,
get mixed up into x

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prime and t prime of the
other person.

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The fact that space and time
coordinates can be combined to

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give the new space and time
coordinates is why it‘s called

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a coordinate.
I'll say more about it later.

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Now, by the way,
I've posted some notes on this

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topic for those of you who want
to have a second pass at it.

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I also don't know how much the
textbook covers this topic

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because I spend a lot of time on
this topic, not in proportion to

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what's in the book.

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So, you can look at those notes
which are posted today.

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Before all the Einstein stuff,
if something occurred here,

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you would say it's at a
distance x prime from

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your origin.
I will say it's at a distance

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x from my origin.

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This difference between our
origins would be ut.

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Therefore, before Einstein,
I would say x prime is

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x - ut and you would say
x should be what you

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think is a coordinate +
ut.

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This is just going back and
forth from you to me and you can

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already notice that to go back
and forth from you to me,

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we just change the sign of the
velocity, because I'm going to

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the right and you're going to
the left [Note:

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Left and right mixed up here.].

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That's why the formulas have
sign of u reversed.

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What is the big change after
Einstein?

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First is, I admit,
the possibility that maybe you

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will think the time elapsed
since we synchronized our clocks

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is not necessarily the same.

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I leave that option open that
t and t prime are

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not the same.
Second thing I do is--I have

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already explained to you that
the velocity of light coming out

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the same for different people is
very counterintuitive,

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because if you are going to the
right, I expect you to get a

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smaller speed and yet to get the
same answer.

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So, we know that it's because
we don't agree anymore on clocks

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and meter sticks being the same.

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So in particular,
you will say,

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"I don't buy your prediction
that I expect to get the answer

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to be x - ut.

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I'm going to put a fudge
factor, gamma,

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which depends on the velocity
between you and me,

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because your lengths are not my
lengths."

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Likewise, I will tell you,
"I don't agree with your

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expectation that I should get
x prime + ut

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prime.
I don't believe your lengths

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are really my lengths,
but I'm going to put the same

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fudge factor,
gamma."

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This is a very interesting
result.

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If I think your meter sticks
are short so that I want to blow

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up the answer with some amount;
you are allowed to say the same

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about me.
It's one of the big paradoxes

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in relativity that we can accuse
each other of using meter

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sticks,
which are short,

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and we'll go into a little bit
more how that's even possible.

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But the postulates tell us that
if my fudge factor is gamma,

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yours should also be the same
gamma.

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So, gamma is what we don't know.

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I showed you a trick to find
gamma.

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I said, let's imagine that this
event here, which could've been

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a firecracker,
is triggered by a light pulse

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that was emitted when you and I
met here.

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The light pulse goes racing and
sets off an explosion here.

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That's the event we are talking
about.

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That's not a generic event.

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It's a particular event in
which x has to be

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c times t because
that's the equation for a light

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pulse, according to me.

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Light goes at the velocity of
c, and the event has got

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the same velocity,
according to you.

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This event here should be
connected by a light pulse that

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for x/t is c,
and x prime/t

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prime is the same c.

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Then, I said,
take these three equations,

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multiply the left-hand side by
the right-hand side,

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put in these numbers and
extract gamma,

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and I will not repeat that
part, and gamma was this.

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Once you've got this,
you can take this gamma and put

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it back here.
Let's see what we get.

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So I'm going to put what we get
here into this board.

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You guys got all of that?

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Okay.
So, I'm going to put it back

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here, because this is just
background from last lecture.

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What we have then is x
prime = x - ut divided by

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this famous square root,
and if you go and solve for

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t prime in the other
equation, it's a simple

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algebraic manipulation,
so I don't want to waste my

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time doing that.
You will find t prime

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= t - ux/c^(2) divided by
the same square root.

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These are priceless.

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This is really all of
relativity, all the funny stuff

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you hear about E =
Mc^(2), funny clocks,

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the Twin Paradox.

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Everything comes from this
equation.

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And you people should be very
pleased that within,

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what, six weeks of starting
Physics 200,

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you have all the things you
need to understand these

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equations, to understand where
they came from.

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That's also the remarkable
thing about relativity.

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A lot of the modern things in
physics require a lot of

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mathematics.
In fact, if you want to take a

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beam and you want to load it
with weight and you want to see

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the stresses and strains,
the math in all that is 1,000

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times more difficult than this
one, and yet the mathematics of

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relativity is very,
very simple,

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and accessible to all of you,
and there is nothing that I

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know of that you don't know as
far as how to get these

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equations.
This is what you do.

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Everything follows from taking
these equations and analyzing

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them, extracting the
consequences and that's really

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more an issue of courage than
just intelligence because once

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you've got these equations,
you will have to take them,

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you'll have to follow them
where they take you.

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That's what we're going to do.

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The rest of the whole remaining
lecture, it's all about getting

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information from these
equations.

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Now, some of you are not used
to writing equations.

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I've noticed,
when you see something like

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this, it's not clear what is
being stated.

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So, let me remind you one more
time.

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u is a fixed number.

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That's your speed relative to
mine.

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I see something happening,
I give it a pair of numbers,

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xt, you give the same
event,

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another pair of numbers,
x prime,

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t prime,
and this is how they are

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related.
So, I gave you an analogy,

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but let me repeat the analogy.

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If you're now talking about the
xy plane rather than the

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xt plane,
that guard is not an event.

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That guard is a coordinate of a
point, maybe where something is

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sitting.
I give to it a pair of numbers,

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x and y.

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Now, you have a different
coordinate system,

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rotated relative to mine,
by some angle θ.

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To that same point,
same location,

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you measure x prime up
to here and t prime from

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there.
I'm sorry, x prime and

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y prime are measured
along your axis,

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and the formula there is
x prime = x cos θ + y

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sin θ and y prime is
-x sin θ + y cos θ.

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θ is the analog of the
velocity now.

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If θ is 0,
you and I agree completely.

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If θ is not 0,
your axis is rotated relative

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to mine, and the same point has
two different numbers,

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xy for me,
x prime/y prime

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for you, and the relation
between them is this.

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So, you plug in my x and
y, you can get your

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x prime and y
prime.

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Let me give a concrete example.

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Let's take θ to
π/4 or 45°.

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Then, as you know,
at 45, sine and cosine,

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they're all the same.

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One over root 2.

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So, in that case,
x prime is x over

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root 2 + y over root 2,
and y prime is -x

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over root 2 + y over root
2.

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That's a special case when
θ is 45°.

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So, for every angle,
cosine and sine will reduce to

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usual simple numbers that happen
to be the [one over]

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square root of 2 here.

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Then, it tells you my x
and y are related to your

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x prime and y
prime in this manner,

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and you can test it.

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For example,
take a point with coordinates

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(1,1).
(x,y) is 1,1.

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If (x,y) is (1,1),
and my axis is related to yours

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by 45°, (1,1) lies right there.

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So, what I expect is your
y prime, there's

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no y prime,
and the coordinate should be

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entirely x prime.

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If I put in (1,1) and indeed
you find if I put x = y =

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1, y prime becomes 0.

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And how about x
prime? It becomes 1+1

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over root 2;
that means 2 over root 2 is the

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square root of 2.

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The square root of 2,
of course, would be the lengthy

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measure this way because the
length of the vector is square

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root of 2,
and the vector is entirely

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along the x direction.

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So, the length you will get for
this coordinate will be root 2.

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So, come back to these
equations.

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They're the same thing.

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If you want,
I can write x prime as

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x divided by this number
- u divided by this

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number times t.
So, you can think of 1 over

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this number here as the analog
of cos θ and

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u over this number as the
analog of sine.

12:49.299 --> 12:52.819
So, it's just that instead of
cosine and sine all being

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upstairs, some of the numbers
are down, some of the numbers

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are up.
But for a given u,

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these are also numbers constant
depending on u,

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and x prime is the
linear combination of x

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and t,
and t prime is some

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other linear combination of
x and t.

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But I should warn you that it
is not an ordinary rotation.

13:11.809 --> 13:14.989
In other words,
you cannot treat this as a

13:14.988 --> 13:17.158
cosine and that as a sine.

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They're not cosine or sine of
anything, because if they were,

13:21.378 --> 13:24.818
this squared plus that squared
should add up to 1,

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and they won't.
So, don't even try that.

13:27.490 --> 13:30.440
This is not an ordinary
rotation but it is still what

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you can call a linear
transformation.

13:32.649 --> 13:36.439
Linear transformation means the
new numbers are related to the

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first powers of the old numbers.

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They are linear.
They don't involve t^(2)

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and x^(2).

13:41.559 --> 13:46.259
Okay, so that is the content of
the Lorentz transformation.

13:46.259 --> 13:50.159
Everybody should understand
what they do by way of going

13:50.163 --> 13:52.983
back and forth.
But now, you can go backwards.

13:52.980 --> 13:55.770
You can say,
"Well, how do I write x

13:55.772 --> 13:59.032
in terms of x prime and
t prime?"

13:59.029 --> 14:02.729
There are two options open for
you.

14:02.730 --> 14:05.200
One is, these are simultaneous
equations.

14:05.200 --> 14:08.270
You've got to find a way to
solve for x and t

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in terms of x prime and
t prime and all these

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funny functions involving
u.

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You treat them all as constants
and juggle them around,

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multiply by this,
divide by that,

14:18.389 --> 14:21.509
but you shouldn't do that
because you know what the answer

14:21.512 --> 14:24.092
should be.
The answer should be the same

14:24.085 --> 14:26.855
as what I got with the velocity
reversed.

14:26.860 --> 14:30.570
So, x should be x
prime + ut prime or the

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square root.
If I write the square root,

14:32.819 --> 14:35.359
it means I don't feel like
putting what's in it.

14:35.360 --> 14:37.300
It's the same old thing,
so you got tired of writing

14:37.303 --> 14:40.103
that.
t prime will be t +

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ux over c^(2) divided
by the square root.

14:44.730 --> 14:47.950
So, one should be able to go
back and forth,

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just like in this formula.

14:50.039 --> 14:52.789
If you like,
I gave it to you as an example

14:52.792 --> 14:55.022
long back as a homework problem.

14:55.019 --> 14:59.179
You can write a new formula
that says x = x prime

14:59.178 --> 15:03.258
cos θ - y sin θ and
y prime is equal to

15:03.261 --> 15:05.741
something.
I mean, y is equal to

15:05.744 --> 15:08.244
something something,
and the way to get that is to

15:08.236 --> 15:11.286
either solve the equations or
realize that if I go from me to

15:11.289 --> 15:14.169
you by a θ,
I go from you to me by -θ,

15:14.168 --> 15:17.608
and all you have to do is
change sin θ to -sin

15:17.607 --> 15:20.307
θ and leave the cos
θ alone.

15:20.309 --> 15:22.409
Similarly here,
you change the sign of the

15:22.406 --> 15:25.216
velocity u to get the
reverse transformations.

15:25.220 --> 15:30.080
Okay, so that is the Lorentz
transformation.

15:30.080 --> 15:33.340
Now, we are going to start
milking the transformation.

15:33.340 --> 15:35.550
Everything is going to be
applying it to understand

15:35.548 --> 15:36.298
various things.

15:36.300 --> 15:39.410


15:39.409 --> 15:43.209
The first step in getting the
mileage out of the Lorentz

15:43.209 --> 15:46.179
transformation is,
take a pair of events.

15:46.179 --> 15:49.589
I urge you, whenever you get a
problem with relativity,

15:49.594 --> 15:53.394
to think in terms of events and
quite often think in terms of

15:53.388 --> 15:55.698
pair of events.
Two events.

15:55.700 --> 15:58.960
So, event one is,
let me give concrete events,

15:58.956 --> 16:03.226
and most of the relativistic
examples involve some degree of

16:03.225 --> 16:06.405
violence and so this one
involves a gun.

16:06.410 --> 16:08.620
So, I take this gun.

16:08.620 --> 16:10.940
I'm not going to point it at
any of you guys.

16:10.940 --> 16:13.910
I fire the gun.
That's event 1.

16:13.910 --> 16:18.060
The bullet hits the wall;
that's event 2.

16:18.059 --> 16:21.749
You can take two events
connected by a bullet leaving me

16:21.750 --> 16:25.170
and hitting the wall,
or you can take two events not

16:25.172 --> 16:27.322
connected to anything,
okay.

16:27.320 --> 16:29.670
You can take two unrelated
events.

16:29.670 --> 16:30.600
It doesn't matter.

16:30.600 --> 16:32.830
We're going to call them event
1 and event 2.

16:32.830 --> 16:37.180
So, event 1 will have
coordinate x_1,

16:37.181 --> 16:39.581
t_1,
right?

16:39.580 --> 16:41.410
And according to you,
x_1 prime,

16:41.406 --> 16:42.506
t_1 prime.

16:42.509 --> 16:45.959
So, write the Lorentz
transformation that first

16:45.958 --> 16:50.078
relates x_1
prime is x_1 -

16:50.081 --> 16:53.381
ut_1 over the
square root.

16:53.379 --> 16:57.289
Then right here,
t_1 prime is

16:57.288 --> 17:01.628
t_1ux
_1 or c^(2)

17:01.631 --> 17:04.151
divided by a square root.

17:04.150 --> 17:09.090
Similarly, take the second
event and write the law for

17:09.087 --> 17:11.087
that.
Well, it's the same thing with

17:11.087 --> 17:12.117
the new numbers in it.

17:12.119 --> 17:17.889
So, t_2 prime
is equal to t_2 -

17:17.894 --> 17:23.194
ux_2 over
c^(2) divided by a square

17:23.188 --> 17:28.618
root. Now,
take the difference of 2 - 1.

17:28.619 --> 17:32.439
You can take 1 - 2 but it's
very common to define the

17:32.443 --> 17:34.943
difference to be 2 - 1,
and call it

17:34.943 --> 17:37.373
Δx_2 prime.

17:37.369 --> 17:41.209
Δx prime is
x_2 prime

17:41.206 --> 17:43.846
- x_1 prime.

17:43.849 --> 17:47.249
All the Δs will be
defined to be 2nd – the 1st.

17:47.250 --> 17:50.390
I'm telling you to take
x_1 prime,

17:50.389 --> 17:53.209
subtract it from
x_2 prime on

17:53.214 --> 17:56.734
the left-hand side and call it
Δ of x prime.

17:56.730 --> 17:59.880
Δ of x prime is the
difference in the spatial

17:59.875 --> 18:02.835
coordinates of the two events
according to you.

18:02.839 --> 18:05.319
If you come to the right-hand
side, now this is something you

18:05.317 --> 18:07.007
guys should be able to do in
your head.

18:07.009 --> 18:09.829
I want to subtract that from
that.

18:09.829 --> 18:12.659
They share the same
denominator, so let me put the

18:12.656 --> 18:13.806
denominator there.

18:13.810 --> 18:17.190


18:17.190 --> 18:19.810
In the numerator you will get
x_2 -

18:19.812 --> 18:22.442
x_1,
which in my convention I will

18:22.435 --> 18:25.835
call it Δx and the other
one will be uΔt.

18:25.840 --> 18:30.160


18:30.160 --> 18:34.280
What this tells you is that
differences in coordinates also

18:34.279 --> 18:37.049
obey the same Lorentz
transformation.

18:37.049 --> 18:39.599
Instead of saying the
coordinate of an event was x,

18:39.600 --> 18:41.430
t and you get x
prime,

18:41.430 --> 18:43.800
t prime from the Lorentz
transformation,

18:43.797 --> 18:47.047
if you take a pair of events
and they are separated in space

18:47.046 --> 18:50.346
by Δx and in time by
Δt according to me,

18:50.349 --> 18:54.029
the separation according to
you, your Δx primes and

18:54.031 --> 18:57.901
Δt primes are given by
similar formulas as the Lorentz

18:57.903 --> 19:00.523
transformation;
put a Δ everywhere.

19:00.520 --> 19:05.990


19:05.990 --> 19:08.910
This just came from taking the
difference of two equations

19:08.906 --> 19:10.796
applied to the two separate
events.

19:10.800 --> 19:14.900


19:14.900 --> 19:19.730
All right.
You can also do this backwards,

19:19.727 --> 19:21.877
if you like.
If you want the differences

19:21.875 --> 19:24.165
that I get in terms of yours and
yours in terms of mine,

19:24.166 --> 19:25.996
you have to reverse the sign of
u.

19:26.000 --> 19:27.790
So, I won't do that again.

19:27.790 --> 19:31.340
Yep?
Student:

19:31.342 --> 19:36.202
Shouldn't that be t?

19:36.200 --> 19:37.130
Professor Ramamurti
Shankar: Here?

19:37.130 --> 19:40.810
Student: Yes.

19:40.809 --> 19:42.629
Professor Ramamurti
Shankar: This was the

19:42.625 --> 19:44.895
formula for x in terms of
x prime and t

19:44.903 --> 19:46.963
prime,
and t prime in terms of

19:46.959 --> 19:48.239
t and x.

19:48.240 --> 19:53.700
Student:
[inaudible]

19:53.703 --> 20:02.743
Professor Ramamurti
Shankar: Here?

20:02.740 --> 20:03.570
Professor Ramamurti
Shankar: Okay,

20:03.571 --> 20:04.301
I've got to give it some
numbers.

20:04.299 --> 20:06.369
This is a,
this is b,

20:06.365 --> 20:07.515
this is c.

20:07.520 --> 20:08.450
Where is the problem?

20:08.450 --> 20:14.230
One of you [inaudible]
Professor Ramamurti

20:14.234 --> 20:18.314
Shankar: Oh,
yes yes yes.

20:18.310 --> 20:19.650
Of course.
Yes.

20:19.650 --> 20:22.540
Thank you.
Okay, now a public apology is

20:22.537 --> 20:24.617
forthcoming.
You wanted me to do this.

20:24.620 --> 20:26.560
Student: Yeah.

20:26.559 --> 20:29.009
Professor Ramamurti
Shankar: Yes?

20:29.010 --> 20:31.800
Thank you very much.

20:31.799 --> 20:35.709
Okay, do not hesitate to do
this, okay?

20:35.710 --> 20:39.510
In fact, when I said I don't
know anymore than you do about

20:39.512 --> 20:43.252
this, it looks like I know less
than you do about this.

20:43.250 --> 20:46.070
So, the fact that I know this
doesn't mean I'm going to get it

20:46.070 --> 20:47.990
right.
So, all of you who screwed up

20:47.991 --> 20:50.561
in the Midterm,
remember there is hope for you.

20:50.559 --> 20:53.439
Maybe you won't do well as
students but you can become a

20:53.439 --> 20:56.559
professor here.
Apparently, it's all right for

20:56.563 --> 20:58.353
us to get things wrong.

20:58.350 --> 21:00.170
That's correct.
Very good.

21:00.170 --> 21:04.620
Okay, so if you're following me
that well, I'm very happy now to

21:04.624 --> 21:08.164
know you must be following what
I'm saying here.

21:08.160 --> 21:11.950
So now, let us--Everything now
I keep telling you I'm building

21:11.948 --> 21:15.118
it up in a big way but
everything really is going to

21:15.116 --> 21:18.526
come from this version of the
Lorentz transformation for

21:18.532 --> 21:21.422
differences.
Look, suppose you were smart

21:21.424 --> 21:25.484
enough like Einstein and you did
use these equations for gamma.

21:25.480 --> 21:27.050
What do you do next?

21:27.050 --> 21:28.290
You write them down.

21:28.289 --> 21:30.899
You can publish them and say
according to me,

21:30.896 --> 21:32.846
this is a rule for
transformation,

21:32.851 --> 21:34.511
but you cannot stop now.

21:34.509 --> 21:38.689
You have to say what are the
implications of my equations

21:38.687 --> 21:43.087
because these equations are
dramatic variations of Newtonian

21:43.088 --> 21:44.848
laws.
For example,

21:44.850 --> 21:46.760
let us do one thing.

21:46.759 --> 21:49.859
Two events are separated by 10
meters according to me.

21:49.859 --> 21:53.559
Δx prime,
if Δx is 10 meters,

21:53.556 --> 21:56.996
then Δx prime is not 10
meters.

21:57.000 --> 21:59.830
It is 10 minus something
divided by something.

21:59.829 --> 22:02.999
So, that distance between two
events is changing.

22:03.000 --> 22:06.470
That's not supposed to happen
just because you get into a

22:06.473 --> 22:08.633
train.
Right?

22:08.630 --> 22:11.230
If I hold my hands up and say I
caught a fish that big,

22:11.233 --> 22:14.033
that distance should be the
same for me and the train or if

22:14.030 --> 22:16.920
you look at me riding in the
train from outside the train you

22:16.924 --> 22:18.664
should find the same distance.

22:18.660 --> 22:19.750
We're saying it's not true.

22:19.750 --> 22:23.690
Likewise, if I say two events
took place when I came to Yale,

22:23.687 --> 22:27.097
I'm certain there I got a
degree four years later,

22:27.099 --> 22:29.359
that's supposed to be true for
anybody.

22:29.360 --> 22:31.300
But that's also not true.

22:31.299 --> 22:34.799
If Δt is four years,
Δx is whatever you

22:34.796 --> 22:36.896
like.
Maybe you never left New Haven,

22:36.897 --> 22:37.957
so Δx is 0.

22:37.960 --> 22:39.930
That's four years,
that's not four years.

22:39.930 --> 22:43.510
These are all drastic
consequences and we've got to

22:43.514 --> 22:45.454
explore the consequences.

22:45.450 --> 22:49.410
The first thing I'm going to do
is to make sure that the

22:49.406 --> 22:53.646
velocity transformations from
one frame to other will be the

22:53.651 --> 22:55.811
requirements we set on it.

22:55.810 --> 22:58.340
So, here's what I'm going to do.

22:58.340 --> 23:00.820
Event 1: I fired the gun.

23:00.820 --> 23:05.250
Okay.
Let me just call it Event 1,

23:05.248 --> 23:09.738
I fire the gun.
Event 2: a bullet hits the wall.

23:09.740 --> 23:17.570


23:17.569 --> 23:20.489
So, the separation between
these two events is Δx

23:20.486 --> 23:23.346
for me and the time it took the
bullet is Δt.

23:23.349 --> 23:26.159
You can also see the bullet
from your train and you think

23:26.162 --> 23:29.232
the distance between me and the
wall is something and the time

23:29.226 --> 23:32.336
between the firing and hitting
the wall is something else.

23:32.339 --> 23:36.219
What's the velocity of the
bullet according to you and me?

23:36.220 --> 23:47.360
v velocity of bullet,
for me, would be

23:47.357 --> 23:52.887
Δx/Δt.
Okay, now I should be a little

23:52.889 --> 23:54.159
more careful with my rotation.

23:54.160 --> 23:57.290
Usually Δ is used for
infinitesimal numbers,

23:57.292 --> 24:00.612
especially when you're going to
define the velocity,

24:00.609 --> 24:03.429
velocity is the limit of these
guys going to 0.

24:03.430 --> 24:06.710
But it is the property of
Lorentz transformation that at

24:06.712 --> 24:09.402
this level, when I say
differences in time and

24:09.397 --> 24:13.237
differences in space,
they are not necessarily small.

24:13.240 --> 24:16.550
Nowhere was I assuming that the
difference in x was a

24:16.552 --> 24:18.632
small or the difference in
[inaudible]

24:18.630 --> 24:20.090
was small [inaudible].

24:20.089 --> 24:22.409
They could be events separated
by five light years,

24:22.408 --> 24:23.658
you could still use them.

24:23.660 --> 24:26.150
But at this stage,
don't think Δx has to

24:26.145 --> 24:27.145
be infinitesimal.

24:27.150 --> 24:29.200
It is simply a shorthand for
difference.

24:29.200 --> 24:31.980
If I had all the time in the
world, I would write everything

24:31.982 --> 24:33.822
as x_2 -
x_1,

24:33.821 --> 24:35.851
but I'm using Δ as a
shorthand.

24:35.849 --> 24:38.199
But when I'm going to find the
velocity of a bullet,

24:38.201 --> 24:40.461
which could even be the
instantaneous velocity,

24:40.460 --> 24:45.410
then, here I do want to take
the limit in which they go to 0

24:45.412 --> 24:47.932
so it becomes dx/dt .

24:47.930 --> 24:50.990
So, from now on,
for this purpose of velocity

24:50.990 --> 24:54.890
calculation, you should take
them to be infinitesimal and

24:54.886 --> 24:57.526
approaching 0,
but not in general.

24:57.529 --> 25:01.569
These equations are valid but
arbitrarily big intervals in

25:01.566 --> 25:05.126
time and space.
Now, w better be the

25:05.134 --> 25:06.714
velocity for you.

25:06.710 --> 25:09.810


25:09.809 --> 25:12.209
That is, I am S and you
are S prime.

25:12.210 --> 25:15.170
w is Δx prime
over Δt prime with

25:15.172 --> 25:16.162
suitable limits.

25:16.160 --> 25:19.500


25:19.500 --> 25:21.770
Well, from these
transformational laws,

25:21.770 --> 25:25.360
you can get them because here
is Δx prime and here is

25:25.355 --> 25:26.605
Δt prime.

25:26.609 --> 25:34.799
Let me divide this by this on
the left-hand side to get

25:34.801 --> 25:38.501
w.
When I write Δx prime

25:38.500 --> 25:41.930
over Δt prime,
you guys take the limits of

25:41.930 --> 25:43.680
everything going to 0.

25:43.680 --> 25:45.030
I don't feel like writing that.

25:45.029 --> 25:49.289
In fact, it's true even without
the limit, but let's apply it in

25:49.291 --> 25:51.931
the end to an instantaneous
velocity.

25:51.930 --> 25:53.960
What happens on the right-hand
side?

25:53.960 --> 25:57.820
Here is what Δx prime
is equal to.

25:57.820 --> 25:59.260
Can you do this in your head?

25:59.259 --> 26:04.729
If this Δx - uΔt,
this denominator cancels

26:04.729 --> 26:10.809
between dividing this by this,
so it looks like--I just

26:10.809 --> 26:17.149
divided this guy by this guy,
because that's when you divide

26:17.150 --> 26:21.880
the left-hand side by the
left-hand side,

26:21.880 --> 26:23.800
you have to divide the
right-hand side by the

26:23.796 --> 26:26.936
right-hand side.
Now, you should have an idea of

26:26.938 --> 26:28.838
what I'm planning to do.

26:28.839 --> 26:32.419
What do we do next to this
expression here?

26:32.420 --> 26:38.100


26:38.100 --> 26:39.430
Yes?
Student:

26:39.429 --> 26:40.239
[inaudible]
Professor Ramamurti

26:40.241 --> 26:41.031
Shankar: A limit of what?

26:41.029 --> 26:43.429
If we just take a limit of
Δt going to 0,

26:43.430 --> 26:44.880
you're just going to get 0.

26:44.880 --> 26:46.880
That's not the limit you want
to take.

26:46.880 --> 26:48.760
You've got to bring velocity
into the picture.

26:48.760 --> 26:49.730
Yes?
Student:

26:49.727 --> 26:50.057
u over t?

26:50.059 --> 26:51.239
Professor Ramamurti
Shankar: Pardon me?

26:51.240 --> 26:52.770
Student:
u over the t?

26:52.769 --> 26:53.269
Professor Ramamurti
Shankar: No.

26:53.269 --> 26:54.559
For any value of u--
Student:

26:54.560 --> 26:55.570
[inaudible]
Professor Ramamurti

26:55.574 --> 26:56.224
Shankar: Yes.

26:56.220 --> 27:00.380
What you want to do now is to
divide the top and bottom of

27:00.379 --> 27:04.759
this by Δt because you
got the velocity of the bullet

27:04.757 --> 27:06.287
according to you.

27:06.289 --> 27:09.219
I want to get the velocity of
the bullet according to me into

27:09.217 --> 27:11.307
the picture.
That is Δx over

27:11.310 --> 27:14.990
Δt, but this is not a
typical derivative in calculus,

27:14.989 --> 27:17.689
okay?
You've got to divide everything

27:17.690 --> 27:20.690
by Δt,
then you will get Δx

27:20.685 --> 27:24.955
over Δt there - u
divided by 1 - u over

27:24.963 --> 27:28.603
c^(2) times Δx
over Δt.

27:28.599 --> 27:30.999
This is true even for finite
differences.

27:31.000 --> 27:32.070
Now, take the limit.

27:32.069 --> 27:36.819
Then, this becomes the velocity
of the bullet - u divided

27:36.820 --> 27:39.610
by 1 - uv over
c^(2).

27:39.610 --> 27:43.460


27:43.460 --> 27:45.680
So, you've got to draw a box
around this guy.

27:45.680 --> 27:47.640
This is another great formula.

27:47.640 --> 27:52.970


27:52.970 --> 27:56.930
So, I hope you know why I
divided by Δt.

27:56.930 --> 27:59.830
I want to find the velocity of
the bullet according to you and

27:59.825 --> 28:02.045
according to me.
For you, I took the distance

28:02.051 --> 28:03.851
over time.
Well, I've got all these

28:03.850 --> 28:06.110
distances and times in the top
and bottom.

28:06.109 --> 28:09.049
I divide by the time so
everything turns into velocity.

28:09.049 --> 28:13.079
So, this is the velocity of the
bullet according to me.

28:13.079 --> 28:16.179
If you're going to the right,
at a speed u,

28:16.178 --> 28:18.958
in the good old days,
what would I expect?

28:18.960 --> 28:20.130
You've got to understand that.

28:20.130 --> 28:22.850
My expectation in the old days
would be v - u,

28:22.851 --> 28:24.981
right?
Whether it's going as speed

28:24.984 --> 28:28.544
v, they're going as speed
u to the right.

28:28.539 --> 28:32.369
So, you will see a diminished
speed by an amount equal to your

28:32.367 --> 28:34.887
speed.
But now is the twist.

28:34.890 --> 28:37.940
There's something in the
bottom, and the stuff in the

28:37.944 --> 28:41.414
bottom is a number less than 1,
because of 1 - something.

28:41.410 --> 28:43.980
We're going to jack up velocity.

28:43.980 --> 28:46.720
Therefore, the velocity you
will actually measure is

28:46.717 --> 28:49.667
somewhat bigger than what I
expected in the old days.

28:49.670 --> 28:53.040
If you ever want to get back to
the good old days in any

28:53.039 --> 28:56.409
relativistic calculation,
you should let the velocity of

28:56.408 --> 28:57.938
light go to infinity.

28:57.940 --> 28:59.390
Of course, it's not infinity.

28:59.390 --> 29:03.090
What you really mean is
u/c and v/c are

29:03.094 --> 29:05.204
negligible.
That means my velocity,

29:05.200 --> 29:07.720
the bullet velocity,
they're all small compared to

29:07.716 --> 29:11.066
the velocity of light;
we get back the answer from the

29:11.071 --> 29:13.041
old days.
This is the answer from the new

29:13.038 --> 29:13.288
days.

29:13.290 --> 29:17.450


29:17.450 --> 29:21.190
Now, let's find the beauty of
this result.

29:21.190 --> 29:23.460
Let's get the reverse result.

29:23.460 --> 29:29.190
Let us solve for the velocity I
get in terms of the velocity you

29:29.193 --> 29:31.573
get.
Again, I think you realize you

29:31.573 --> 29:34.513
can do the algebra,
but you guys should know that

29:34.507 --> 29:37.557
to go from me back to you,
I should simply reverse the

29:37.560 --> 29:39.030
sign of the relative velocity.

29:39.030 --> 29:44.420


29:44.420 --> 29:45.710
This is the backwards result.

29:45.710 --> 29:48.220
In other words,
you're in a train and you fired

29:48.224 --> 29:50.524
a bullet towards the front of
the train.

29:50.519 --> 29:55.079
What speed do I attribute to
the bullet from the ground?

29:55.079 --> 29:57.399
Well, I add the bullet speed to
the train speed.

29:57.400 --> 29:59.120
That's what the numerator is.

29:59.119 --> 30:02.589
The numerator says the answer
is somewhat less than that by

30:02.591 --> 30:04.371
this number.
Yes?

30:04.369 --> 30:07.549
Student:
[inaudible]

30:07.545 --> 30:13.525
Professor Ramamurti
Shankar: Thank you very

30:13.530 --> 30:18.250
much.
Yeah.

30:18.250 --> 30:20.230
That's absolutely correct.

30:20.230 --> 30:23.510
What I really need to do--quite
correct--it's not simply to

30:23.507 --> 30:26.917
change u to -u,
but to change the velocities

30:26.919 --> 30:30.419
that you were seeing and put
them in the place of velocities

30:30.415 --> 30:31.655
that I was seeing.

30:31.660 --> 30:36.400
That's correct.
This is what you want to get.

30:36.400 --> 30:40.950
Now, let's look at the strength
of this result.

30:40.950 --> 30:43.320
If someone tells you according
to relatively,

30:43.322 --> 30:45.752
nothing can go faster than the
speed of light,

30:45.749 --> 30:48.229
you can try to beat the system
as follows.

30:48.230 --> 30:51.260
You can come to me and say,
"Can there be a gun whose

30:51.255 --> 30:54.335
bullets go at three-fourths the
velocity of light?"

30:54.339 --> 30:56.069
and I would say,
"yes," and you'd say,

30:56.066 --> 30:58.866
"How about a train that goes at
three-fourths the velocity of

30:58.866 --> 31:00.456
light?"
and I would say that seems to

31:00.455 --> 31:01.825
be allowed.
Then you can say,

31:01.830 --> 31:04.730
"Well, let me get into this
train at three-fourths the

31:04.732 --> 31:07.742
velocity of light and fire a
bullet at three-fourths the

31:07.743 --> 31:10.173
velocity,
then from the ground it should

31:10.166 --> 31:13.256
appear to be going at 1.5 times
the velocity of light."

31:13.259 --> 31:15.399
Well, that's the naive
expectation.

31:15.400 --> 31:19.850
But if you do it now,
let's put w = ¾c

31:19.848 --> 31:24.638
and u = ¾c,
the old answer is disaster;

31:24.640 --> 31:28.790
it says 1.5c,
but the correct answer is that

31:28.790 --> 31:30.700
plus (¾)^(2).

31:30.700 --> 31:37.720


31:37.720 --> 31:39.820
So, what is this guy?

31:39.819 --> 31:48.179
This is 1.5c divided by
1 + 9/16.

31:48.180 --> 31:52.730


31:52.730 --> 31:54.610
You multiply everything by 16.

31:54.609 --> 31:59.289
On the top you'll get 24,
the bottom will get 25 [times]

31:59.293 --> 32:02.513
c.
You see, you can jack up the

32:02.510 --> 32:06.990
velocity as seen by the ground
but it'll never be 1.5.

32:06.990 --> 32:08.370
It'll still be less than the
velocity of light.

32:08.370 --> 32:13.950


32:13.950 --> 32:17.520
And the last thing you want to
check of this velocity addition

32:17.522 --> 32:20.102
formula is why we started the
whole thing.

32:20.099 --> 32:23.409
In other words,
suppose I see a pulse of light.

32:23.410 --> 32:26.690
Let's go back now to this
formula.

32:26.690 --> 32:32.660
w is v - u over 1
- uv over c^(2).

32:32.660 --> 32:35.630
Rather than applying it to a
bullet or whatnot,

32:35.626 --> 32:38.396
let's apply it to the light
pulse itself.

32:38.400 --> 32:42.780
So, I saw a light pulse so I
say the object I saw,

32:42.784 --> 32:47.444
which is what v stands
for, had a value

32:47.436 --> 32:51.456
c.
And this is the pre-Einstein

32:51.461 --> 32:54.641
expectation.
w should be c - u

32:54.637 --> 32:58.207
because the pulse appears to
travel slower to a person moving

32:58.214 --> 33:00.484
in the same direction as the
pulse.

33:00.480 --> 33:05.790
But our new formula says it is
really that uc over

33:05.788 --> 33:07.208
c^(2).

33:07.210 --> 33:10.460


33:10.460 --> 33:14.120
So, that canceled one part of
c, multiply top and

33:14.115 --> 33:16.905
bottom by c,
and you will find it is

33:16.906 --> 33:19.746
c.
So, the velocity of light will

33:19.749 --> 33:21.859
always come out to be the same.

33:21.859 --> 33:24.799
That is built into the formula
but it is a good thing to test.

33:24.799 --> 33:29.879
So, what you find is,
velocities don't add in the

33:29.875 --> 33:32.835
simple way.
If they did, you are in trouble.

33:32.839 --> 33:35.679
You cannot get an upper limit
if they added in the simple way

33:35.684 --> 33:38.344
because you can put a rocket
inside another rocket inside

33:38.339 --> 33:41.179
another rocket and add up all
the speeds and even though each

33:41.184 --> 33:43.784
one is less than 0,
the total could be whatever you

33:43.777 --> 33:45.927
like.
But they don't add that way.

33:45.930 --> 33:48.940
They add this way so that the
answer is either bigger than

33:48.937 --> 33:51.837
what you think if v and
u are opposite,

33:51.839 --> 33:55.369
or smaller than what you think
if the velocities in the top are

33:55.368 --> 33:56.448
of the same sign.

33:56.450 --> 33:59.420
No matter what you do,
the answer will always be less

33:59.424 --> 34:03.524
than c.
Okay, so that is the first

34:03.519 --> 34:06.209
conclusion from this.

34:06.210 --> 34:13.050
Second conclusion,
which is again very staggering,

34:13.054 --> 34:19.624
is that simultaneity is the
relative concept.

34:19.619 --> 34:22.289
In other words,
if two events occur at the same

34:22.290 --> 34:25.600
time for me, they don't occur at
the same time for you.

34:25.600 --> 34:27.440
That is very surprising.

34:27.440 --> 34:30.190
For example,
if you have twins born,

34:30.187 --> 34:33.797
one in New-- sorry,
twins cannot be born in New

34:33.797 --> 34:35.757
York and Los Angeles.

34:35.760 --> 34:40.070
So, I've got to pick two kids
of different motherhood,

34:40.071 --> 34:43.651
just happened to be born at the
same time.

34:43.650 --> 34:46.290
When I say "time," we're not
talking about trivial three-hour

34:46.287 --> 34:48.527
time difference between Los
Angeles and New York.

34:48.530 --> 34:50.040
That's an artificial thing.

34:50.039 --> 34:51.499
In the Einstein world,
you imagine,

34:51.498 --> 34:53.428
we all have clocks that read
the same time.

34:53.429 --> 34:56.719
So, by that measure,
the two kids are born at the

34:56.718 --> 35:00.968
same time but if you watch that
from a moving train or a moving

35:00.966 --> 35:03.566
rocket, you will disagree on
that.

35:03.570 --> 35:06.630
You will, in fact,
say they are not simultaneous.

35:06.630 --> 35:07.710
How does that come?

35:07.710 --> 35:09.750
Well, it just comes from going
to this formula.

35:09.750 --> 35:15.340
Δt prime is Δt -
uΔa x over c^(2)

35:15.339 --> 35:20.529
divided by this number and all
I'm telling you is,

35:20.530 --> 35:25.860
even though Δt is 0,
Δt prime is not 0.

35:25.860 --> 35:32.160
Another shock.
Simultaneity is not absolute.

35:32.160 --> 35:34.530
We used to think it's absolute.

35:34.530 --> 35:37.450
In other words,
two events are occurring in Los

35:37.449 --> 35:38.909
Angeles and New York.

35:38.909 --> 35:42.119
They can be arranged to be
simultaneous for me,

35:42.118 --> 35:44.488
living on the planet,
let's say,

35:44.489 --> 35:47.279
but if you go in a rocket,
I expect you to agree they were

35:47.284 --> 35:49.254
simultaneous.
I mean, how can they be

35:49.251 --> 35:51.321
different?
Two things are happening right

35:51.323 --> 35:53.893
now in different places,
it's got to be right now for

35:53.893 --> 35:55.053
you.
But it's not.

35:55.050 --> 35:56.950
It just comes from that formula.

35:56.949 --> 36:00.379
Once again, if the velocity of
light is made much bigger than

36:00.379 --> 36:03.859
everything in the problem,
simply to set it to infinity,

36:03.855 --> 36:06.825
you will find Δt prime
is Δt.

36:06.829 --> 36:12.109
That goes back to Galilean
times or the old pre-Einstein

36:12.111 --> 36:14.941
times.
That's why in relativity,

36:14.943 --> 36:19.573
space and time form a new
space-time and time is called a

36:19.565 --> 36:24.265
fourth component because if
c goes to infinity,

36:24.269 --> 36:27.119
Δ prime is always
Δt and t prime is

36:27.119 --> 36:29.289
always t.
If the coordinate never mixes

36:29.292 --> 36:31.492
with anything else,
it doesn't deserve to be called

36:31.487 --> 36:33.497
a coordinate,
whereas, it's the x and

36:33.496 --> 36:35.926
y coordinates that mix
with each other when you do

36:35.934 --> 36:37.474
rotations.
After Einstein,

36:37.466 --> 36:41.396
the space and time coordinates
mix with each other to give you

36:41.398 --> 36:45.458
new space and time coordinates
under Lorentz transformation,

36:45.460 --> 36:48.310
which means,
when seen in a moving train.

36:48.309 --> 36:52.639
That's why time is now elevated
to another dimension because it

36:52.644 --> 36:56.634
transforms very much the way
x and y did.

36:56.630 --> 36:58.330
The details of the
transformation are different.

36:58.329 --> 37:01.059
We have sines and cosines
replaced by u and

37:01.062 --> 37:04.302
u over the square root
but it's still mathematically

37:04.297 --> 37:07.027
the way components of vectors
will transform.

37:07.030 --> 37:09.340
Okay, so simultaneity is
relative.

37:09.340 --> 37:14.680
So, you have to ask yourself,
"How did that happen?"

37:14.679 --> 37:18.059
So, here's the famous example
that is given in all the books.

37:18.059 --> 37:20.359
Maybe it was given by Einstein,
I'm not sure.

37:20.360 --> 37:21.530
So, here's a train.

37:21.530 --> 37:24.550


37:24.550 --> 37:28.190
Let's say the train is at rest
and you are standing in the

37:28.190 --> 37:31.380
middle and you want to arrange
for two events to be

37:31.384 --> 37:33.664
simultaneous.
Oh by the way,

37:33.659 --> 37:35.359
there is one catch.

37:35.360 --> 37:41.600
If the two events occurred at
the same time and the same

37:41.601 --> 37:46.141
place, Δx = 0,
Δt = 0;

37:46.139 --> 37:48.619
then, the transformation will
tell you Δx prime is 0

37:48.622 --> 37:49.802
and Δt prime is 0.

37:49.800 --> 37:53.160
Because two events occurring at
the same time,

37:53.158 --> 37:56.068
at the same place means,
for example,

37:56.070 --> 37:58.910
my two hands came and clapped,
the two hands were at the same

37:58.907 --> 38:00.087
time at the same place.

38:00.090 --> 38:02.360
If someone said,
well, they were not at the same

38:02.356 --> 38:04.716
place at the same time,
it means I didn't clap.

38:04.720 --> 38:06.890
Two cars collide.

38:06.889 --> 38:10.289
So, if two things occur at the
same time at the same place,

38:10.288 --> 38:13.158
something happened that they
met in space-time.

38:13.159 --> 38:16.299
You cannot find another
observer who says they did not

38:16.304 --> 38:19.274
meet in space-time,
or they'll say the two cars did

38:19.270 --> 38:21.910
not collide.
Even after relativity,

38:21.911 --> 38:26.291
it is true that two events at
the same time and same place

38:26.294 --> 38:30.374
occur the same time and same
place for all people.

38:30.369 --> 38:36.259
Same time alone is not enough
and same place alone is not

38:36.262 --> 38:38.342
enough.
You see that?

38:38.340 --> 38:41.370
If two cars came and they
collided, they were at the same

38:41.368 --> 38:42.718
time at the same place.

38:42.719 --> 38:45.839
If the cars were at the same
time, here and there,

38:45.842 --> 38:47.182
that's no accident.

38:47.179 --> 38:49.919
If they were at the same place
because if this car went and two

38:49.923 --> 38:51.913
minutes later that car went at
the same place,

38:51.914 --> 38:54.894
nothing happens.
That's why you say this poor

38:54.892 --> 38:58.332
guy was in the wrong place at
the wrong time.

38:58.329 --> 39:00.539
You don't want to say this guy
was in the wrong place.

39:00.540 --> 39:01.530
What does that mean?

39:01.530 --> 39:04.060
You can go to the Battle of
Gettysburg site and stand now

39:04.064 --> 39:06.644
and nothing happens to you
because it's the same place but

39:06.643 --> 39:07.733
it's the wrong time.

39:07.730 --> 39:09.390
Okay?
We all realize,

39:09.393 --> 39:12.163
same place, same time,
is a congruence in space-time

39:12.160 --> 39:14.980
and even after Einstein,
you cannot say something did

39:14.981 --> 39:16.861
not happen.
Okay.

39:16.860 --> 39:18.780
We leave that to politicians
please.

39:18.780 --> 39:21.780
It did happen.
It still happened and the

39:21.784 --> 39:25.254
equations have the property but
I've got several events

39:25.250 --> 39:28.010
separated in space but at the
same time.

39:28.010 --> 39:28.990
You're in the train.

39:28.989 --> 39:31.939
Your job is to make two things
happen at the same time and

39:31.942 --> 39:33.032
here's what you do.

39:33.030 --> 39:36.350
You send a beam of light that
splits into a,

39:36.353 --> 39:39.293
you know, here's a blown up
picture.

39:39.289 --> 39:42.369
Beam of light comes and splits
like that and goes to the back

39:42.374 --> 39:43.664
and front of the train.

39:43.659 --> 39:46.589
You're in the middle of the
train, and you know that events

39:46.586 --> 39:48.146
will be simultaneous for you.

39:48.150 --> 39:51.990
The pulse travels on either
side and hits the two things at

39:51.987 --> 39:56.017
the two ends and sets off two
explosions and you have done the

39:56.022 --> 39:59.002
best you can to have
simultaneous events.

39:59.000 --> 40:03.620
Okay now, I see you from the
ground.

40:03.619 --> 40:05.849
So, all this happening in the
moving train,

40:05.854 --> 40:08.574
you have every right to say
you're not moving but to

40:08.568 --> 40:10.588
me--relative to me you are
moving.

40:10.590 --> 40:13.000
That's a fact.
And I look at how well you did

40:13.002 --> 40:15.892
with this.
So, you sent off these two

40:15.889 --> 40:18.329
light pulses.
Then what happened?

40:18.329 --> 40:21.729
This wall of the train is
moving away from the light

40:21.727 --> 40:23.317
pulse.
This wall of the train,

40:23.324 --> 40:25.224
the rear end,
is rushing to meet the light

40:25.223 --> 40:27.873
pulse.
Now, the velocity of light is

40:27.869 --> 40:32.789
the same for everybody so I can
think in terms of what happens.

40:32.789 --> 40:35.449
If one wall is rushing to meet
the light pulse and one wall is

40:35.450 --> 40:37.020
running away from the light
pulse,

40:37.019 --> 40:39.689
I know very clearly the
firecracker in the back of the

40:39.690 --> 40:42.710
train will go off first and in
the front of the train will go

40:42.714 --> 40:47.534
off later.
That means that Δt will

40:47.533 --> 40:50.793
not be 0.
In fact, you can see Δt

40:50.794 --> 40:54.484
will be negative because the
event with the greater x

40:54.475 --> 40:56.155
coordinate occurs later.

40:56.159 --> 40:59.159
That's also an agreement of the
formula.

40:59.159 --> 41:01.489
If Δt = 0,
then Δx prime will be

41:01.489 --> 41:02.629
some negative number.

41:02.630 --> 41:05.870


41:05.869 --> 41:07.719
Now, why do we bring in the
light pulse?

41:07.719 --> 41:10.889
We bring in the light pulse
because we can tell that they

41:10.890 --> 41:14.230
were simply not simultaneous
because about the light we know

41:14.230 --> 41:16.150
this.
Its velocity is the same for

41:16.147 --> 41:18.237
everybody.
That's why all arguments in

41:18.240 --> 41:21.150
relativity involve doing things
with light pulses or

41:21.150 --> 41:25.030
communicating with a light pulse
because we know what light does.

41:25.030 --> 41:27.780
It travels at one and the same
velocity for all people.

41:27.780 --> 41:28.720
That's a postulate.

41:28.719 --> 41:33.299
Therefore, we know you couldn't
have done any better in making

41:33.300 --> 41:37.130
them simultaneous and I simply
disagree with you.

41:37.130 --> 41:39.820
I'd say they just did not
happen at the same time,

41:39.816 --> 41:42.116
and there's no question of who
is right.

41:42.119 --> 41:45.009
Operationally,
for me, those two events were

41:45.014 --> 41:49.124
separated in time by some amount
and for you they were not.

41:49.119 --> 41:52.759
That's a great new idea that
things did not happen the same

41:52.759 --> 41:55.539
time for all people,
and relativity tells you that

41:55.537 --> 41:58.377
they didn't and tells you by how
much they were different.

41:58.380 --> 42:03.240
Okay.
Then, I take the next surprise

42:03.239 --> 42:05.349
here.
The next surprise has to do

42:05.353 --> 42:06.133
with clocks.

42:06.130 --> 42:09.860


42:09.860 --> 42:12.520
The claim is that clocks will
not run at the same rate.

42:12.519 --> 42:14.979
Okay, when we bought the
clocks, we compared them,

42:14.981 --> 42:18.331
we got them from the same shop,
they were completely in synch

42:18.333 --> 42:21.843
and you put one in your train
and you got into your train and

42:21.837 --> 42:25.167
the claim is that I will find
that your clocks are running

42:25.166 --> 42:29.296
slow.
Let me show that to you.

42:29.300 --> 42:31.970
Again, if you want to do
anything with relativity,

42:31.973 --> 42:33.723
at least learn this one thing.

42:33.719 --> 42:36.259
Go back to Lorentz
transformation and think in

42:36.255 --> 42:39.745
terms of events that are just
going to come out of the wash.

42:39.750 --> 42:42.920
If you try something original
on your own, you people do all

42:42.917 --> 42:44.907
kinds of stuff.
I've seen you going in a

42:44.908 --> 42:47.648
cyclical circular or arguments
from which you cannot even come

42:47.651 --> 42:49.181
out until somebody rescues you.

42:49.180 --> 42:50.460
So don't do that.

42:50.460 --> 42:52.140
So, I'm looking at a clock.

42:52.139 --> 42:56.099
You've got to ask yourself,
how do I turn this issue of

42:56.095 --> 42:58.215
time into a pair of events?

42:58.219 --> 43:01.919
I have my clock,
so it goes tick-tock,

43:01.920 --> 43:04.410
tick-tock.
I pick two events.

43:04.409 --> 43:09.819
Event 1: a clock says tick,
and event 2:

43:09.822 --> 43:13.862
clock says tock.
I tell you why I have tick and

43:13.863 --> 43:17.003
tock because I want to have two
distant kind of events so that

43:17.002 --> 43:18.342
we can talk about them.

43:18.340 --> 43:22.650
So, this event is a clock that
I'm carrying with me.

43:22.650 --> 43:25.660
The clock is with me.

43:25.659 --> 43:29.129
So, let me put a clock at the
center of my coordinates system

43:29.133 --> 43:31.393
or the origin.
It doesn't matter where it is.

43:31.390 --> 43:32.750
Let me put it at the origin.

43:32.750 --> 43:36.850
The space-time coordinate x
= 0, t = 0 is the

43:36.851 --> 43:38.691
first tick of the clock.

43:38.690 --> 43:41.930


43:41.930 --> 43:44.250
How about the next tick?

43:44.250 --> 43:47.440
The next tick,
this is x and t

43:47.438 --> 43:50.848
coordinates, the tock of the
clock, let me call

43:50.848 --> 43:54.628
τ_0 is the
time of the clock.

43:54.630 --> 43:57.390
It's a time period;
how many seconds elapsed

43:57.392 --> 43:59.722
between the tick and the tock.

43:59.719 --> 44:03.809
The main question is,
"What is the location of the

44:03.805 --> 44:06.135
second tick of the clock?"

44:06.140 --> 44:07.480
Where does that happen?

44:07.480 --> 44:10.300
Yes?
Student:

44:10.298 --> 44:11.458
[inaudible]
Professor Ramamurti

44:11.461 --> 44:13.361
Shankar: It doesn't move
with respect to me and I'm

44:13.364 --> 44:15.414
talking about a clock that I'm
holding in my hand so if the

44:15.408 --> 44:17.028
first event took place at x
= 0,

44:17.030 --> 44:19.730
second one also takes place at
x = 0.

44:19.730 --> 44:21.250
You understand?
I'm holding the clock.

44:21.250 --> 44:23.240
I've traveled forward in time.

44:23.239 --> 44:26.039
It's τ_0
seconds later but the clock has

44:26.041 --> 44:27.911
not gone anywhere so the two
events,

44:27.909 --> 44:31.209
the two tickings of the clock,
are separated in space by 0 and

44:31.207 --> 44:33.097
in time by
τ_0.

44:33.099 --> 44:37.579
That means Δx =
0, Δt =

44:37.580 --> 44:39.970
τ_0.

44:39.970 --> 44:42.690
What do you get?
According to you,

44:42.694 --> 44:46.534
the time difference between the
two ticks is Δt,

44:46.526 --> 44:50.006
which is τ_0
- u etc.,

44:50.010 --> 44:52.800
times 0 divided by this factor.

44:52.800 --> 44:57.610
So, that means the time
difference between the two ticks

44:57.606 --> 45:00.696
would be bigger.
If τ_0 is 1

45:00.703 --> 45:02.543
second, if you divide 1 by [root
of]

45:02.543 --> 45:06.133
1 - u^(2) over c^(2),
you're going to find out

45:06.133 --> 45:08.343
it is less.
There's more.

45:08.340 --> 45:11.240
For example,
if this factor in the

45:11.236 --> 45:15.096
denominator is .5,
then Δt would be 2

45:15.097 --> 45:19.307
times τ_0. In
other words,

45:19.309 --> 45:22.459
when your clock has taken 1
second to go from 1 tick to the

45:22.463 --> 45:25.623
next, I will say according to me
the real time elapsed is 2

45:25.617 --> 45:27.537
seconds.
It can be 10 seconds,

45:27.540 --> 45:28.750
it can 100 seconds.

45:28.750 --> 45:31.490
You can make Δt as big
as you like by letting you

45:31.491 --> 45:32.471
approach c.

45:32.470 --> 45:38.270


45:38.270 --> 45:39.580
Yes?
Student:

45:39.583 --> 45:40.763
[inaudible]
Professor Ramamurti

45:40.755 --> 45:41.675
Shankar: Thank you.

45:41.679 --> 45:45.109
I'm having this problem today
with the prime.

45:45.110 --> 45:48.950
t prime is
τ_0 over this.

45:48.949 --> 45:50.279
If you put
τ_0,

45:50.282 --> 45:53.362
I put 1 second.
Question there, guys?

45:53.360 --> 45:55.440
Student:
[inaudible]

45:55.440 --> 45:58.480
Professor Ramamurti
Shankar: No,

45:58.480 --> 46:00.800
u did not go to 0.

46:00.800 --> 46:02.810
u is not 0.

46:02.809 --> 46:06.959
uΔx over c^(2,)
it is the Δx that went

46:06.956 --> 46:09.806
to 0.
Go the Lorentz transformation.

46:09.809 --> 46:17.119
Δt prime is Δt -
uΔx over c^(2).

46:17.120 --> 46:19.200
Δx = 0.

46:19.199 --> 46:24.549
That's where I'm telling the
equation the clock is at rest

46:24.549 --> 46:26.519
with respect to me.

46:26.519 --> 46:28.099
That means that--
Student:

46:28.098 --> 46:29.728
[inaudible]
Professor Ramamurti

46:29.726 --> 46:30.906
Shankar: No,
no.

46:30.909 --> 46:33.999
In this example,
let the unprimed observer be

46:33.996 --> 46:35.956
the one holding the clock.

46:35.960 --> 46:38.620
Then Δx is,
you know Δt is whatever

46:38.615 --> 46:40.045
the time of the clock is.

46:40.050 --> 46:42.380
1 second.
Δt prime is seen by

46:42.381 --> 46:45.381
anybody else going at a speed
u relative to me.

46:45.380 --> 46:48.650
You can also do it backwards by
taking the clock in the hands of

46:48.651 --> 46:51.821
the S prime but you've
got to do a little more work.

46:51.820 --> 46:54.920
It's a lot easier here because
I get to pump into the equation

46:54.920 --> 46:58.010
two facts at one shot,
that the time between the two

46:58.011 --> 47:01.561
ticks is 1 second and it's at
rest with respect to me,

47:01.559 --> 47:05.129
which is why the Δx
vanished between the two ticks.

47:05.130 --> 47:07.650
If I want to use your
equations, Δx prime

47:07.646 --> 47:10.736
won't be 0 because the clock has
moved and you can do it.

47:10.739 --> 47:13.709
I've, I think,
shown in my notes how to do

47:13.711 --> 47:15.741
that but it's unnecessary.

47:15.740 --> 47:16.790
This is the main result.

47:16.790 --> 47:18.930
Now, here is the paradox.

47:18.929 --> 47:22.349
You can go back to the
backwards equations.

47:22.350 --> 47:26.630
Let's do this informally, okay.

47:26.630 --> 47:27.990
Let's not write it again.

47:27.990 --> 47:29.370
It will look like this.

47:29.370 --> 47:31.620
Right.
You know that.

47:31.619 --> 47:33.659
Let's take a clock you are
carrying.

47:33.659 --> 47:36.939
It ticks off 1 second,
so between the tick and the

47:36.944 --> 47:39.964
next tick, the time difference
is 1 second,

47:39.960 --> 47:43.170
space difference is 0,
I'll find Δt is 1 over

47:43.171 --> 47:45.651
a square root.
So, I will say your clock is

47:45.648 --> 47:47.508
slow, you'll say my clock is
slow.

47:47.510 --> 47:50.860
How is that possible?

47:50.860 --> 47:54.740
How can it be that we accuse
each other of having--yes?

47:54.739 --> 47:57.519
Student:
[inaudible]

47:57.520 --> 48:01.590
Professor Ramamurti
Shankar: Oh,

48:01.585 --> 48:04.575
you're absolutely right.

48:04.579 --> 48:07.779
The predictions are in
agreement with that principle,

48:07.775 --> 48:10.845
but I ask you to think about
the conflict you have,

48:10.848 --> 48:12.778
yes?
Student:

48:12.780 --> 48:15.950
[inaudible]
Professor Ramamurti

48:15.947 --> 48:19.497
Shankar: Oh,
that's very good.

48:19.500 --> 48:21.000
Student:
[inaudible]

48:21.001 --> 48:23.201
Professor Ramamurti
Shankar: No,

48:23.195 --> 48:25.865
that is correct.
Let me repeat what he said.

48:25.869 --> 48:28.479
He said, if there are two
people, they were right next to

48:28.477 --> 48:31.357
each other, they are at the same
height and they move apart and

48:31.364 --> 48:34.024
each person looks at the other
person and finds the person

48:34.018 --> 48:36.298
diminished in size because of
the distance,

48:36.300 --> 48:39.590
and each person can tell the
other person you look small from

48:39.587 --> 48:41.867
where I am.
In fact, even more paradoxical

48:41.873 --> 48:45.133
is how in this world two people
can simultaneously look down on

48:45.125 --> 48:47.235
each other.
You figure that out.

48:47.239 --> 48:50.109
There are two people who
simultaneously have higher

48:50.107 --> 48:53.377
opinion of themselves to the
other person and are the same

48:53.376 --> 48:55.666
thing, they look down on each
other.

48:55.670 --> 48:58.360
I have not found how many space
dimensions I should embed but

48:58.358 --> 49:00.998
these people to make it that
possible, but this is certainly

49:01.002 --> 49:03.372
correct.
A spatial resolution does give

49:03.366 --> 49:05.866
the impression.
So, here's the answer that's

49:05.871 --> 49:07.951
usually given to explain this to
you.

49:07.949 --> 49:12.029
If I take a real clock like
this watch here and I ask you

49:12.025 --> 49:16.605
why does it look slow to you
when I'm moving relative to you,

49:16.610 --> 49:19.920
it's difficult because it's got
electronics and stuff and I

49:19.918 --> 49:22.768
still don't know how to set the
clock on my VCR.

49:22.769 --> 49:24.249
I'm not going to figure that
out.

49:24.250 --> 49:27.490
So, for all of us guys who are
challenged, there is a clock

49:27.494 --> 49:30.854
that's particularly simple,
and the clock works like this.

49:30.849 --> 49:36.789
It has got two mirrors and a
light pulse just goes up and

49:36.793 --> 49:43.053
down between the mirrors and
every time it completes a round

49:43.054 --> 49:45.804
trip,
it sets off some detector and

49:45.800 --> 49:48.700
it goes "click."
Now, this is very important to

49:48.701 --> 49:52.221
mention that this is my x
coordinate and that's the

49:52.219 --> 49:55.679
y coordinate and I did
not show you here there are

49:55.676 --> 49:58.936
logical arguments and why,
if you and I are moving along

49:58.942 --> 50:01.512
x, why our y
coordinates, if they agree in

50:01.507 --> 50:03.887
the beginning,
continue to agree.

50:03.889 --> 50:06.839
That's because if you say
something is two meters tall and

50:06.838 --> 50:09.478
they pass each other,
you cannot have a disagreement

50:09.477 --> 50:11.647
on the fact that heights are the
same.

50:11.650 --> 50:14.580
But as in a linear dimension
there is room for discussion,

50:14.580 --> 50:16.740
in the transverse dimension
there isn't.

50:16.739 --> 50:19.499
So, anyway, I have this clock
that goes up and down.

50:19.500 --> 50:21.340
Light pulse goes up and down.

50:21.340 --> 50:24.720
It goes a certain distance,
L, in a vertical

50:24.717 --> 50:28.767
direction, and 2L/c is
the time period of my clock.

50:28.769 --> 50:30.839
That's the time for a round
trip.

50:30.840 --> 50:35.170
You look at my clock and what
do you think is happening?

50:35.170 --> 50:38.520
Remember, you are moving to the
right, relative to me,

50:38.517 --> 50:41.667
so according to you my clock is
moving to the left,

50:41.674 --> 50:43.384
and it looks like this.

50:43.380 --> 50:47.370


50:47.369 --> 50:51.789
Light beam is going on a zigzag
path, according to you.

50:51.790 --> 50:53.340
You agree that's very clear.

50:53.340 --> 50:56.960
For me, the pulse is going up
and down, for you it's going on

50:56.963 --> 50:59.653
the hypotenuse.
You can imagine how with the

50:59.652 --> 51:03.042
hypotenuse u^(2) and
c^(2) are going to come

51:03.036 --> 51:05.526
from drawing the sides of the
triangle.

51:05.530 --> 51:07.890
That's why the time would come
out longer.

51:07.889 --> 51:11.399
Now, why do they like this
clock?

51:11.400 --> 51:13.790
Because we know everything
about the operation of the

51:13.791 --> 51:16.231
clock.
We know that this path is

51:16.232 --> 51:21.462
longer than the straight up and
down path because the transfers'

51:21.455 --> 51:23.855
coordinate is known to us.

51:23.860 --> 51:25.610
Furthermore,
we know the velocity of light

51:25.605 --> 51:27.345
is the same in all frames of
reference;

51:27.349 --> 51:30.249
so, light going on a longer
path is simply going to take

51:30.254 --> 51:33.864
longer.
So, I know your clock would

51:33.856 --> 51:36.236
slow down.
Now, here is the beauty.

51:36.239 --> 51:38.759
The beauty is that,
if you have a clock that's

51:38.762 --> 51:40.952
going up and down,
straight up and down,

51:40.949 --> 51:43.639
when I see it,
it's going to look like this.

51:43.640 --> 51:48.000


51:48.000 --> 51:50.740
So, it's as simple as saying
that if you had these two

51:50.740 --> 51:53.900
clocks, what I think is going up
and down and you think it's a

51:53.895 --> 51:56.245
zigzag this way,
and I look at your clock,

51:56.252 --> 51:59.272
it's a zigzag the other way,
it's perfectly okay for me to

51:59.269 --> 52:01.549
say to you,
your light pulse is going on a

52:01.550 --> 52:04.630
hypotenuse and mine is going up
and down, and why you can say

52:04.631 --> 52:08.221
that to me.
So, at least this light clock

52:08.217 --> 52:13.257
explains to you why clocks will
appear to be slow.

52:13.260 --> 52:16.540
Now, the question you can ask
is, what if I have some other

52:16.544 --> 52:19.494
clock with gears and wheels and
teeth and whatnot?

52:19.490 --> 52:20.830
How does that slow down?

52:20.829 --> 52:24.179
The answer is,
we don't know exactly how to

52:24.184 --> 52:28.984
explain that clock but I know
that if you carry a light clock

52:28.976 --> 52:32.726
and another clock made of wheels
and gears,

52:32.730 --> 52:34.910
they should run at the same
rate.

52:34.909 --> 52:38.259
They should run at the same
rate because if they ran at

52:38.259 --> 52:41.359
different rates,
one slowed down and one didn't,

52:41.360 --> 52:44.970
in comparing the two clocks you
can determine your velocity,

52:44.967 --> 52:48.757
and that we know is impossible
by the postulate that you cannot

52:48.758 --> 52:50.468
detect uniform velocity.

52:50.469 --> 52:53.669
Therefore, if the light clock
does something,

52:53.666 --> 52:57.436
all clocks must do the same
thing regardless of their

52:57.444 --> 53:01.154
mechanism, and that includes
biological clocks.

53:01.150 --> 53:05.000
So, if you have a clock which
is just yourself,

53:05.001 --> 53:08.771
I look at you,
you know, over 15 or 20 years I

53:08.770 --> 53:10.780
notice some changes.

53:10.780 --> 53:13.120
You become taller,
then your hair turns white,

53:13.117 --> 53:14.257
your teeth fall off.

53:14.260 --> 53:17.890
That's a clock and that clock
should also slow down.

53:17.889 --> 53:21.279
I don't care how your life
systems work but you are a clock

53:21.284 --> 53:23.044
and you've got to slow down.

53:23.039 --> 53:26.029
That's why we can make
predictions about what happened

53:26.029 --> 53:29.469
to living systems even though
that's not our main business.

53:29.469 --> 53:31.559
So, in particular,
if you take a clock,

53:31.561 --> 53:34.701
which is made up of mechanical
parts and a human being,

53:34.699 --> 53:38.609
and they travel at high speeds,
the aging of the human being

53:38.605 --> 53:42.435
should slow down just like the
clock's ticking will slow it

53:42.444 --> 53:44.674
down,
without knowing the reasons for

53:44.669 --> 53:46.529
it.
So, this leads to a very famous

53:46.526 --> 53:48.276
paradox called the Twin Paradox.

53:48.280 --> 53:53.320
The Twin Paradox says that
you've got two guys,

53:53.322 --> 53:56.192
twins.
Now, this is actually a valid

53:56.191 --> 53:58.241
example.
They were born at the same

53:58.243 --> 54:00.893
time, same place,
and then the one goes on a trip

54:00.891 --> 54:04.091
at some speed u,
and goes around and comes back.

54:04.090 --> 54:08.740
So, I think the person's been
gone for 20 years,

54:08.740 --> 54:11.660
my twin.
So, let's say he was 20 when he

54:11.655 --> 54:14.755
left, so I expect him to be 40
when he gets back.

54:14.760 --> 54:19.210
But he'll come back younger
because as a clock,

54:19.208 --> 54:21.238
he has slowed down.

54:21.239 --> 54:24.439
So, what I think is 20 years,
this could be,

54:24.442 --> 54:27.722
my time is 20 years,
but his time could be 10

54:27.720 --> 54:30.700
because his factor downstairs is
.5.

54:30.699 --> 54:34.589
So, he can come back being
younger than me.

54:34.590 --> 54:37.170
Yes?
Student:

54:37.171 --> 54:40.361
[inaudible]
Professor Ramamurti

54:40.364 --> 54:45.014
Shankar: Oh,
but we've got to be careful.

54:45.010 --> 54:47.250
So, his point had something to
do with acceleration,

54:47.252 --> 54:48.662
but the real question is this.

54:48.659 --> 54:52.829
See, if I'm on the ground and I
send my twin on the trip and we

54:52.833 --> 54:55.663
meet, now the clocks are being
compared.

54:55.659 --> 54:58.079
As long as the zigzag is going
this way and that zigzag is

54:58.081 --> 54:59.951
going that way,
you can say what you like.

54:59.949 --> 55:02.149
But what if the clocks are
brought head-to-head and

55:02.146 --> 55:04.596
compared?
Can they both be slower than

55:04.604 --> 55:06.844
the other?
We know that's impossible.

55:06.840 --> 55:09.360
So, instead of clocks,
we use human beings and

55:09.355 --> 55:12.425
dramatize the paradox and say,
"Who will be younger?"

55:12.429 --> 55:14.139
Me or the twin who went on the
rocket?

55:14.140 --> 55:17.460


55:17.460 --> 55:20.250
Well, there can only be one
answer to that question,

55:20.249 --> 55:23.309
and yet no matter what the twin
says, I have no reason to

55:23.312 --> 55:24.682
believe that I moved.

55:24.679 --> 55:27.379
You are the one who went on the
trip the opposite way so you've

55:27.380 --> 55:28.600
got to be younger than me.

55:28.600 --> 55:32.230
Yes?
Student:

55:32.230 --> 55:38.630
[inaudible]
Professor Ramamurti

55:38.634 --> 55:43.684
Shankar: Very good.

55:43.680 --> 55:45.440
Okay.
I couldn't have said it better.

55:45.440 --> 55:46.140
Let me repeat it.

55:46.139 --> 55:47.699
The point is,
as long as you've got

55:47.698 --> 55:50.258
oppositely moving clocks,
they can believe what they want

55:50.264 --> 55:53.114
of each other,
but if you want the clocks to

55:53.105 --> 55:56.785
be subject to a comparison,
then you can never compare

55:56.789 --> 56:01.029
objects that are constantly
moving at opposite velocities.

56:01.030 --> 56:04.540
So, somebody's got to turn
around and come back.

56:04.539 --> 56:07.719
So, in the case of the twin,
he and I were abreast,

56:07.722 --> 56:11.612
the twin got into the rocket to
set up the velocity difference

56:11.605 --> 56:13.065
between him and me.

56:13.070 --> 56:15.570
So, in the early stages,
he is the one undergoing

56:15.573 --> 56:16.933
acceleration and not me.

56:16.929 --> 56:20.339
Likewise, the twin went
somewhere and stopped and turned

56:20.335 --> 56:21.755
around and came back.

56:21.760 --> 56:24.160
The twin is suffering the
deceleration.

56:24.159 --> 56:27.069
So, there are two periods at
least, during the voyage,

56:27.066 --> 56:30.186
when the twin has no right to
claim that he is not moving,

56:30.192 --> 56:33.602
that I am.
That's why we don't have the

56:33.602 --> 56:35.702
same status.
Whereas for me,

56:35.701 --> 56:38.961
the whole time Newton's laws
and the laws of inertia were

56:38.957 --> 56:41.417
operating for me,
whereas for him and the rocket

56:41.416 --> 56:43.316
during takeoff,
things started flying off and

56:43.318 --> 56:45.218
during landing things started
flying off.

56:45.219 --> 56:48.639
He cannot possibly claim he has
the same status as me.

56:48.639 --> 56:51.389
That's why in that problem,
the relationship is not

56:51.388 --> 56:54.848
symmetrical, and one person can
say I did not move because I was

56:54.850 --> 56:57.170
always inertial,
the other will have,

56:57.168 --> 56:59.178
I have to concede that he
moved.

56:59.180 --> 57:01.250
Yes?
Student:

57:01.253 --> 57:04.033
[inaudible]
Professor Ramamurti

57:04.031 --> 57:05.801
Shankar: Yeah.

57:05.800 --> 57:07.770
Then if you do it,
if you give them symmetrical

57:07.770 --> 57:09.570
acceleration,
both leave like this and both

57:09.570 --> 57:11.260
come back.
Then, both will agree on their

57:11.260 --> 57:13.370
age but they will disagree with
the person on the ground.

57:13.370 --> 57:14.740
Suppose there's some triplets.

57:14.739 --> 57:17.459
You have triplets and two kids
are sent out that way,

57:17.462 --> 57:20.082
third one stays back,
the third one that stays back

57:20.079 --> 57:21.859
will be older than the others.

57:21.860 --> 57:23.900
So, this is not science fiction
at all.

57:23.900 --> 57:28.000
If you want to be alive for the
year 3000, you can do it.

57:28.000 --> 57:31.500
You just have to get into a
rocket, at sufficient velocity

57:31.498 --> 57:35.198
close to the speed of light,
so that this is 3000 years,

57:35.199 --> 57:38.919
okay, and you figure out how
long you've got to live.

57:38.920 --> 57:40.140
Maybe another 50 years.

57:40.139 --> 57:41.969
Do the math,
you find the speed,

57:41.970 --> 57:44.510
get into the rocket and go and
come back.

57:44.510 --> 57:47.840
Now, this experiment is done
all the time with microscopic

57:47.843 --> 57:49.133
particles, you know.

57:49.130 --> 57:51.230
They are accelerated in
Fermilab, for example.

57:51.230 --> 57:54.310
They go around in a ring and
just by virtue of their motion,

57:54.314 --> 57:55.834
they live a very long time.

57:55.829 --> 57:59.879
So, particles are supposed to
have a short lifetime which you

57:59.880 --> 58:02.310
calculate in their own rest
frame,

58:02.309 --> 58:05.609
live much longer because they
are moving, and one way to keep

58:05.612 --> 58:09.022
them moving is to put them on an
accelerated ring and they live

58:09.024 --> 58:10.404
for a very long time.

58:10.400 --> 58:12.650
Yes?
Student:

58:12.652 --> 58:15.632
[inaudible]
Professor Ramamurti

58:15.627 --> 58:17.337
Shankar: No.

58:17.340 --> 58:20.920
No, the rocket will think I'm
50 years older.

58:20.920 --> 58:24.400
If you went on the rocket,
it's 50 years for you but for

58:24.399 --> 58:26.929
people on the ground it's 3,000
years.

58:26.929 --> 58:28.219
Student:
[inaudible]

58:28.223 --> 58:30.213
Professor Ramamurti
Shankar: Yeah,

58:30.214 --> 58:33.204
but if you don't want to age at
all, you can arrange it.

58:33.200 --> 58:34.460
You want, what, five years?

58:34.460 --> 58:37.710
Okay.
You just fixed the number.

58:37.710 --> 58:41.650
You want to age only .5 years,
that's another number.

58:41.650 --> 58:43.350
You'll never run out of numbers.

58:43.349 --> 58:46.449
You can let u approach
c as much as you like and

58:46.453 --> 58:49.613
you can live as long as you want
and you can come back whenever

58:49.608 --> 58:51.528
you want.
Other issues like going

58:51.531 --> 58:54.221
backwards in time--I'll come
back to that later.

58:54.220 --> 58:55.860
There are problems with that.

58:55.860 --> 58:59.490
We'll come to that,
but you can stop time as far as

58:59.490 --> 59:02.250
we know, and we see it all the
time.

59:02.250 --> 59:06.490
The muons produced in the upper
atmosphere, their lifetime is

59:06.490 --> 59:09.530
something, 10 to - 6 seconds or
whatever.

59:09.530 --> 59:11.680
Even if they travel at the
speed of light,

59:11.677 --> 59:14.767
we see their time is not long
enough to cross the atmosphere

59:14.768 --> 59:16.128
but they make it here.

59:16.130 --> 59:17.270
How did they make it?

59:17.269 --> 59:19.929
They make it because their
lifetime is computed in their

59:19.925 --> 59:20.935
frame of reference.

59:20.940 --> 59:22.880
They still think they lived
only their lifetime,

59:22.876 --> 59:24.686
but I think they lived a much
longer time.

59:24.690 --> 59:27.120
That's how they made it to here.

59:27.119 --> 59:30.319
But how about from their point
of view?

59:30.320 --> 59:32.540
They only lived a short time
and how did they go 100

59:32.543 --> 59:34.673
kilometers?
That's the next thing I'm going

59:34.667 --> 59:36.257
to tell you.
According to them,

59:36.262 --> 59:38.722
the atmosphere is not 100
kilometers but maybe 8

59:38.724 --> 59:41.184
kilometers long.
Now, that's the last point

59:41.184 --> 59:43.724
which has to do with the length
contraction.

59:43.719 --> 59:48.129
So length contraction says,
by the way, so thing to

59:48.125 --> 59:53.495
remember what time dilatation
is: Every clock runs the fastest

59:53.499 --> 59:55.789
in its own rest frame.

59:55.789 --> 59:59.849
In any other frame of reference
in which a clock has any

59:59.850 --> 1:00:04.350
velocity, it will appear to be
slower than the advertised time

1:00:04.354 --> 1:00:06.204
period of that clock.

1:00:06.199 --> 1:00:09.809
The last thing I want to do is
length contraction.

1:00:09.809 --> 1:00:13.899
Length contraction says,
if you have a meter stick and

1:00:13.897 --> 1:00:17.287
you and I bought them from the
same store,

1:00:17.289 --> 1:00:20.959
once you're in this plane or
this rocket, I will claim your

1:00:20.963 --> 1:00:23.943
meter stick is actually shorter
than a meter.

1:00:23.940 --> 1:00:27.180
In fact, a shortening factor is
this.

1:00:27.180 --> 1:00:33.190


1:00:33.190 --> 1:00:34.260
This is the shortening factor.

1:00:34.260 --> 1:00:36.370
In fact, way back,
in the derivation of the laws

1:00:36.370 --> 1:00:38.660
of transformation,
the fudge factor I calculated,

1:00:38.659 --> 1:00:41.969
taking your lengths into my
lengths, is precisely connected

1:00:41.966 --> 1:00:43.046
with this factor.

1:00:43.050 --> 1:00:46.890
If you say the length to
something, I will tell you that

1:00:46.891 --> 1:00:51.221
the length is actually less than
that because your meter sticks

1:00:51.222 --> 1:00:53.672
are short.
But let me prove that to you.

1:00:53.670 --> 1:00:56.980
So, let's take a rod that's
moving.

1:00:56.980 --> 1:01:02.590
It's moving at a speed u;
you're carrying the rod and I

1:01:02.585 --> 1:01:03.965
want to find its length.

1:01:03.970 --> 1:01:06.070
What should I do?

1:01:06.070 --> 1:01:07.920
That's the question, okay?

1:01:07.920 --> 1:01:10.010
So, the rod is moving at speed
u.

1:01:10.010 --> 1:01:11.890
I want to find its length.

1:01:11.889 --> 1:01:23.549
So, Δx prime is Δx
- uΔt over the square root.

1:01:23.550 --> 1:01:26.590
I'm going to make a pair of
events to find the length of the

1:01:26.590 --> 1:01:28.690
rod.
Event 1, remember this rod is

1:01:28.685 --> 1:01:31.515
zooming past me,
Event 1 is when this front end

1:01:31.518 --> 1:01:35.028
of the rod hits a certain
marking on my graduated x

1:01:35.028 --> 1:01:38.538
axis and event 2 is when the
back end of the train hits my

1:01:38.537 --> 1:01:39.397
origin,

1:01:39.400 --> 1:01:44.260


1:01:44.260 --> 1:01:47.410
and the distance between those
two is the length of the rod,

1:01:47.413 --> 1:01:48.753
provided one condition.

1:01:48.750 --> 1:01:51.080
How do you find the length of a
moving rod?

1:01:51.079 --> 1:01:55.509
You find this end and that end
at the same time.

1:01:55.510 --> 1:01:58.680
Otherwise you will screw up,
right?

1:01:58.679 --> 1:02:01.199
If you find this end now,
go on a lunch break and come

1:02:01.202 --> 1:02:02.822
back and the road is over there.

1:02:02.820 --> 1:02:05.300
And I said that's the rear end
of the rod, and you get back to

1:02:05.299 --> 1:02:07.069
be the length;
that's not how you find the

1:02:07.074 --> 1:02:08.204
length of a moving object.

1:02:08.199 --> 1:02:11.489
You find the length by
measuring the coordinates of the

1:02:11.486 --> 1:02:13.186
two ends at the same time.

1:02:13.190 --> 1:02:16.410
Consequently,
at the same time means

1:02:16.412 --> 1:02:20.662
Δt must be 0;
Δx is the length

1:02:20.661 --> 1:02:22.261
according to you.

1:02:22.260 --> 1:02:25.110
According to the person going
with the rod,

1:02:25.111 --> 1:02:28.031
the two events,
since they take place at the

1:02:28.030 --> 1:02:31.240
two ends of the rod,
are separated by the length of

1:02:31.241 --> 1:02:33.401
the rod because the rod,
according to you,

1:02:33.402 --> 1:02:34.722
is not going anywhere.

1:02:34.720 --> 1:02:36.030
Suppose you have the rod.

1:02:36.030 --> 1:02:37.740
It's not moving relative to you.

1:02:37.739 --> 1:02:39.819
Something happens at one end,
something happens with the

1:02:39.816 --> 1:02:41.676
other end.
What's the spatial distance

1:02:41.679 --> 1:02:43.949
between them?
It's the length of the rod.

1:02:43.949 --> 1:02:45.659
That's the meaning of the
length of the rod.

1:02:45.659 --> 1:02:48.639
Therefore, if you
cross-multiply,

1:02:48.636 --> 1:02:51.516
you get this result,
L is

1:02:51.520 --> 1:02:56.750
L_0 times--;So,
a rod will appear longest in

1:02:56.752 --> 1:03:00.722
its rest frame and to anybody
else it will appear shorter.

1:03:00.720 --> 1:03:04.550


1:03:04.550 --> 1:03:06.570
Clocks appear fastest in the
rest frame;

1:03:06.570 --> 1:03:09.050
rods appear longest in the rest
frame.

1:03:09.050 --> 1:03:15.680
For anybody to whom a rod is
moving, the rod appears short.

1:03:15.680 --> 1:03:16.990
So, here's another paradox then.

1:03:16.990 --> 1:03:20.220


1:03:20.219 --> 1:03:22.569
I make a hole in my x
axis.

1:03:22.570 --> 1:03:25.780


1:03:25.780 --> 1:03:27.190
I make the hole to be half a
meter long.

1:03:27.190 --> 1:03:30.660


1:03:30.660 --> 1:03:32.330
This is half a meter.

1:03:32.329 --> 1:03:37.059
You and I bought a meter stick
but you are moving at a velocity

1:03:37.062 --> 1:03:39.812
where this number has become
half.

1:03:39.809 --> 1:03:42.489
u over c is 1
over root 2.

1:03:42.489 --> 1:03:44.299
u^(2) over c^(2)
is whatever.

1:03:44.300 --> 1:03:47.040
Look, it's whatever it takes to
make this factor half.

1:03:47.039 --> 1:03:51.529
So, I expect your meter stick
to have shrunk to half its

1:03:51.525 --> 1:03:54.165
length.
So, I make a hole on the table,

1:03:54.167 --> 1:03:56.257
half a meter long,
the rod goes by,

1:03:56.260 --> 1:03:58.660
I think it would fall into the
hole.

1:03:58.660 --> 1:04:01.590
That's my expectation.

1:04:01.590 --> 1:04:04.410
You come and say,
my meter stick is a meter

1:04:04.412 --> 1:04:06.602
stick.
The hole that you dug that you

1:04:06.595 --> 1:04:09.495
think is half a meter long
actually is a quarter meter

1:04:09.501 --> 1:04:11.431
long.
So, there's no way my meter

1:04:11.426 --> 1:04:14.836
stick is going through that hole
in the table because it's four

1:04:14.838 --> 1:04:17.148
times as long as the hole in the
table.

1:04:17.150 --> 1:04:18.700
Do you understand that?

1:04:18.699 --> 1:04:21.629
Normally, you make a meter
stick, you put it on top of a

1:04:21.631 --> 1:04:24.031
hole of the same length,
it will fall down.

1:04:24.030 --> 1:04:26.800
But if it's a moving meter
stick, which has contracted

1:04:26.796 --> 1:04:29.196
according to me from one meter
to half a meter,

1:04:29.196 --> 1:04:31.646
half a meter hole is enough for
it to fall.

1:04:31.650 --> 1:04:35.310
But from your vantage point,
my half a meter hole looks like

1:04:35.312 --> 1:04:37.052
a quarter of a meter hole.

1:04:37.050 --> 1:04:39.190
Your meter stick still looks
like a meter.

1:04:39.190 --> 1:04:41.710
The question is,
when this experiment occurs,

1:04:41.710 --> 1:04:42.970
will it fall or not?

1:04:42.970 --> 1:04:45.880
Hey, these are all paradoxes.

1:04:45.880 --> 1:04:48.100
If you invented the theory,
you have got to defend these

1:04:48.104 --> 1:04:50.004
paradoxes.
You cannot get two answers to

1:04:49.999 --> 1:04:52.049
one question.
That's a logical issue.

1:04:52.050 --> 1:04:55.550


1:04:55.550 --> 1:04:57.960
So, the answer to these
problems is, if you know one

1:04:57.962 --> 1:05:00.472
point of view in which it is
correct, that's the right

1:05:00.469 --> 1:05:02.259
answer.
Then, the other person has to

1:05:02.255 --> 1:05:03.385
figure out what happened.

1:05:03.389 --> 1:05:06.269
So, in my point of view,
if I made a half a meter hole

1:05:06.273 --> 1:05:09.813
and your rod has shrunk to half
the length, it's going to fall.

1:05:09.810 --> 1:05:11.840
It will fall.
The question is,

1:05:11.838 --> 1:05:14.768
how does the other person
reconcile himself to the fact

1:05:14.771 --> 1:05:17.141
that here is,
according to that person,

1:05:17.137 --> 1:05:20.487
the hole is a quarter meter
long and the rod is one meter

1:05:20.490 --> 1:05:22.700
long?
Well, I should--for it to fall,

1:05:22.696 --> 1:05:25.816
you've got to give it a little
tilt, okay, so that it can

1:05:25.815 --> 1:05:26.925
actually fall in.

1:05:26.930 --> 1:05:29.250
I give it a little here.

1:05:29.250 --> 1:05:31.920
How can this object fall into
that?

1:05:31.920 --> 1:05:36.220
The way other person explains
it is, he said first,

1:05:36.217 --> 1:05:38.277
this end went in here.

1:05:38.280 --> 1:05:40.640
Other end was sticking way out.

1:05:40.639 --> 1:05:44.369
Sometime later,
the second end went in here.

1:05:44.370 --> 1:05:46.040
That's how we got to the hole.

1:05:46.039 --> 1:05:48.259
Namely, the two ends did not go
at the same time.

1:05:48.260 --> 1:05:52.040
That way, it's certainly
possible for a one-meter stick

1:05:52.035 --> 1:05:55.175
to go through a
quarter-centimeter hole in the

1:05:55.180 --> 1:05:56.960
table.
I mean, give the rod a little

1:05:56.962 --> 1:05:58.272
tilt so it can fall into the
hole.

1:05:58.270 --> 1:06:01.390
The tip enters first;
the tail is way outside but

1:06:01.394 --> 1:06:04.524
after a while the tail goes in
and in that manner;

1:06:04.520 --> 1:06:07.940
the rod goes in.
So, they can reconcile this

1:06:07.940 --> 1:06:12.320
paradox by saying the two ends
did not go at the same time.

1:06:12.320 --> 1:06:16.160
Also, if you ask the other
person who thinks something is

1:06:16.159 --> 1:06:20.269
two meters long and the other
person thinks it's half a meter

1:06:20.272 --> 1:06:22.812
long,
how did they reconcile it?

1:06:22.809 --> 1:06:25.749
I say, I measure the two ends
at the same time but you will

1:06:25.748 --> 1:06:28.688
tell me, "You did not measure
the ends of my meter stick at

1:06:28.687 --> 1:06:30.777
the same time.
You measured one end and you

1:06:30.781 --> 1:06:33.231
goofed off and you came back and
measured the second end.

1:06:33.230 --> 1:06:35.440
By the time the rod had slipped
to the right,

1:06:35.440 --> 1:06:37.600
you measured that end and you
goofed off.

1:06:37.599 --> 1:06:40.389
You waited until it came there
and then you measured the other

1:06:40.385 --> 1:06:43.045
end.
That's where you got half

1:06:43.046 --> 1:06:47.176
instead of one."
And nobody is right or wrong.

1:06:47.179 --> 1:06:49.899
I did measure the two ends
simultaneously according to me

1:06:49.902 --> 1:06:52.482
but you don't have to agree they
were simultaneous.

1:06:52.480 --> 1:06:57.880
So, it's by going soft on what
simultaneity means that you are

1:06:57.877 --> 1:07:00.087
able to reconcile this.

1:07:00.090 --> 1:07:02.200
Because in Newtonian days,
simultaneity is absolute;

1:07:02.200 --> 1:07:03.460
length is absolute.

1:07:03.460 --> 1:07:05.850
In relativity,
simultaneity is relative and

1:07:05.848 --> 1:07:09.368
length is also relative because
the operational way to find the

1:07:09.373 --> 1:07:12.333
length of a body will not
satisfy all observers.

1:07:12.329 --> 1:07:14.609
If I find the length of your
moving meter stick,

1:07:14.608 --> 1:07:17.568
the two ends at the same time,
you will say I didn't measure

1:07:17.572 --> 1:07:20.352
them at the same time and the
formulas relative will allow

1:07:20.350 --> 1:07:22.680
that.
That's how we live with the

1:07:22.679 --> 1:07:27.129
fact that we accuse each other
of having relatively short meter

1:07:27.131 --> 1:07:28.911
sticks.
I blame you on your

1:07:28.910 --> 1:07:32.320
measurement, but that blame is
not a genuine blame because you

1:07:32.315 --> 1:07:35.495
can dig a hole equal to the
reduced length and things will

1:07:35.496 --> 1:07:37.806
be falling.
So, you have every right to say

1:07:37.813 --> 1:07:40.203
things are shorter,
and I will say I didn't shrink;

1:07:40.199 --> 1:07:43.009
I fell in because my nose went
in first and then a little later

1:07:43.013 --> 1:07:46.093
the tail went.
There are various paradoxes.

1:07:46.090 --> 1:07:47.410
I'll just leave you with one.

1:07:47.410 --> 1:07:48.380
You have a garage.

1:07:48.380 --> 1:07:51.200
Your parents built you a
garage, 12 meters long.

1:07:51.200 --> 1:07:53.360
You bought a car 14 meters long.

1:07:53.360 --> 1:07:56.420
Can you park it?
Well, if you go at sufficiently

1:07:56.420 --> 1:07:59.330
high velocities,
so the 14 gets shrunk to 12,

1:07:59.329 --> 1:08:02.239
there will be a brief instant
in which both ends of your car

1:08:02.237 --> 1:08:03.417
are inside the garage.

1:08:03.420 --> 1:08:06.100
Of course, you will have to
smash through the back end of

1:08:06.103 --> 1:08:07.783
the garage but you will
maintain,

1:08:07.780 --> 1:08:11.010
"Yes, I smashed into that but
my car was in the garage for

1:08:11.005 --> 1:08:13.095
some time" and the parent will
say,

1:08:13.099 --> 1:08:15.509
no, the front end went in and
smashed the rear.

1:08:15.510 --> 1:08:16.900
The back hadn't even come in.

1:08:16.900 --> 1:08:18.630
A little later the back came.

1:08:18.630 --> 1:08:20.760
So, everyone will agree you
broke the garage,

1:08:20.762 --> 1:08:23.042
you broke the car,
and the lengths will contract

1:08:23.040 --> 1:08:26.050
relatively to each other and the
phenomena is that according to

1:08:26.045 --> 1:08:27.665
you,
there was a time the whole car

1:08:27.673 --> 1:08:29.473
was in the garage,
that according to the parents

1:08:29.466 --> 1:08:29.996
it was not.