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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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SCOTT HUGHES: So
we're just picking up

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where we stopped last time.

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So we are beginning
to discuss how

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we are going to sort of
do a geometrical approach

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to physics, using a more
general set of coordinates now.

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So we began talking about how
things change when I discuss

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special relativity,
so for the moment

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keeping ourselves just
at special relativity.

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We, by the way, are going to
begin lifting our assumptions

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that it is simply special
relativity fairly soon.

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But to set that
up, I need to start

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thinking about how to work
in more general coordinate

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systems.

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So we're going to do it in the
simplest possible curvilinear

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coordinates.

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So it's basically just going
from Cartesian coordinates

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in the spatial sector to
plane polar coordinates.

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One of the things which I
have emphasized a few times,

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and I'm going to
continue to hammer on,

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is that these are a little bit
different from the curvilinear

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coordinates that you are
used to in your past life.

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In particular, if I write
out the displacement,

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the little vector of
the displacement element

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in the usual way,
I am using what's

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called a "coordinate basis,"
which means that the vector

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dx is related to the
displacement, the differential

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of the coordinates, by
just that thing contracted

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with all the basis vectors.

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And so what that means is I have
a little displacement in time,

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which looks normal.

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Displacement in radius,
which looks normal.

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Displacement in the z
direction, which looks normal,

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and a displacement in an
angle, which does not.

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In order for this whole thing
to be dimensionally consistent,

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that's telling me e phi has to
have the dimensions of length.

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And that is a
feature, not a bug.

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Last time, we
introduced the matrix

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that allows me to convert
between one coordinate system

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and another, so just
basically the matrix--

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it's sort of a Jacobi matrix.

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It's a matrix of
partials between the two

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coordinate systems.

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And this idea that things are--

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they look a little weird.

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So the way I did that was I
didn't actually write it out,

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but I did the usual mapping
between x, y and r and phi,

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worked out all of
my derivatives.

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And sure enough,
you've got something

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that looks very standard,
with the possible exception

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of these r's that are
appearing in here.

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So notice the elements
of this matrix.

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These do not have
consistent units--

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again, feature, not bug.

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This guy is basically
just the inverse of that.

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This is the matrix that
affects the coordinates

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in the opposite direction.

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And notice in this case,
you have some elements where

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their units are 1 over length.

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So let's just continue
to sort of catalog

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what some of the things we are
going to be working with look

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like in this new
coordinate representation.

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And this will lead
us to introduce

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one of the mathematical
objects that we

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are going to use extensively
as we move forward

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in studying this subject.

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So what I want to do
next is look at what

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my basis vectors look like.

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So what I want to do
is characterize what

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my e r and my e phi look like.

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And these are going to look very
familiar from your intuition,

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from having studied
things like E&M

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in non-Cartesian coordinates.

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So your e r is just related to
the original Cartesian basis

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vectors, like so.

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And if you like, you
can easily read that out

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by performing the
following matrix

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multiplication on the original
Cartesian basis vectors.

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Your e phi perhaps
looks a little wacky.

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So you can see the length
coming into play there.

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A good way to think about
this is if your intuition

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about basis factors--

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I have to be careful with
this language myself--

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your intuition
about basis vectors

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is typically that
they are unit vectors.

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These are not unit vectors.

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They do form a nice basis,
but they are not unit vectors.

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In particular, the basic idea
we're going to go with here

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is that e phi, it's always
going to sort of point

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in the tangential direction.

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But no matter where
I put it in radius,

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I want that vector
to always sort

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of subtend the same
amount of angle.

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In order to do that, its
length needs to grow with r.

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So that's where
that's a little bit

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different from your intuition.

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And there's a very
good reason for this,

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which we will get to, hopefully,
well before the end of today's

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class.

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So last time, when
we first began

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to talk about tensors a
couple of lectures ago,

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the first tensor I gave you--

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so confining ourselves just
to Cartesian coordinates--

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was the metric,
which was originally

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introduced as this
mathematical object that

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came out of looking at dot
products between basis vectors.

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It's essentially a
tensor that allows

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me to feed in two
displacements and get

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the invariant interval between
those displacements that

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comes out of that.

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I am going to continue to call
the dot product of two basis

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vectors the "metric."

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But I'm going to use a slightly
different symbol for this.

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I'm going to call
this g alpha beta.

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In the coordinate representation
that we are using right now,

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so in plane polar
coordinates, this becomes--

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you can work it out from what
I've written down right here.

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This is just the diagonal
of minus 1, 1 r squared 1.

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So this equals dot here.

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This is-- I'll put PPC for plane
polar coordinates under that.

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And then using that, you
know that you can always

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work out the invariant
displacement between two

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events.

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It's always going to be
the metric contracted

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with the differential
displacement element.

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And this is going to
be minus dt squared

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plus dr squared plus r squared
d phi squared plus dz squared.

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That, I hope, makes
a lot of sense.

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This is exactly
what you'd expect

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if I have two events
that are separated

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in plane polar coordinates
by dt, dr, d phi, dz.

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This is what the distance
between them should be.

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So the fact that
my basis vectors

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have this slightly annoying
form associated with them,

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it all sort of comes
out in the wash here.

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Remember at the end of the day,
if we think about quantities

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that are representation
independent--

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and that's going to
be the key thing.

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When you assemble scalars
out of these things,

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the individual
tensor components,

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they can be a little
bit confusing sometimes.

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They are not things
that we measure.

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They are not things that
really characterize what we

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are going to be working with.

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And we really want to
get into the physics,

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unless we're very
careful about it.

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This is something
you can measure.

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And so sure enough,
it comes out,

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and it's got a
good meaning to it.

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Let me just wrap
up one last thing

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before I talk about sort of
where we're going with this.

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So just for completeness, let me
write down the basis one forms.

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Just as the basis vectors had a
bit of a funny form associated

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with them, you're going to find
the basis one forms likewise

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have a bit of a funny
form associated with them.

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And the way I'm
going to get these--

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and so these are going to be
the Cartesian basis one forms--

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basically, I'm not carefully
proving all these relations

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at this point, because you
all know how to do that.

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I'm just using line
up the indices rule.

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And when you do
that, you get this.

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And likewise, your basis one
form for the axial direction,

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I'll just write down the result.
It's going to look like this.

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So the key place where all
of this-- so right now, these

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are all just sort
of definitions.

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Nothing I've done here
should be anything

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that even approaches a
surprise, I hope, just

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given the you guys have done--

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the key thing
that's probably new

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is all this garbage associated
with coordinate bases,

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this extra factors of r and
1 over r that are popping up.

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But provided you're willing to
sort of swallow your discomfort

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and go through
the motions, these

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are not difficult calculations.

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The key place where all
of this really matters

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is going to be when we
calculate derivatives of things.

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It'll turn out there is
an important rule when

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we talk about integrals as
well a little bit later,

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but let's just say that.

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So for now, we'll
focus on derivatives.

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So all the derivatives that
we've been looking at so far,

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we have, indeed, done a couple
of calculations where we've

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computed the derivatives
of various vector valued

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and tensor valued quantities.

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And it was helped by the
fact that all the bases, when

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I work in Cartesian
coordinates, are constant.

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Well, that's not the case now.

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So now, we need to
account for the fact

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that the bases all vary
with our coordinates.

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So let me just
quickly make a catalog

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00:10:26,390 --> 00:10:29,540
of all the non-trivial--
there's basically four.

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In this one, where I'm just
doing plane polar coordinates,

195
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there are four
non-trivial derivatives

196
00:10:34,040 --> 00:10:36,140
we need to worry about.

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One of them actually
turns out to be 0.

198
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So the radial derivative of
the radial unit vector is 0.

199
00:10:44,930 --> 00:10:50,310
But the phi derivative of
the phi unit vector is not.

200
00:10:50,310 --> 00:10:54,300
you go and take the phi
derivative of this guy,

201
00:10:54,300 --> 00:10:56,340
and you basically get--

202
00:10:56,340 --> 00:10:58,080
take the phi derivative
of this, you're

203
00:10:58,080 --> 00:11:01,650
going to get this back,
modulo factor of radius.

204
00:11:01,650 --> 00:11:07,920
So I can write d e r
d phi as e phi over r.

205
00:11:13,990 --> 00:11:17,290
If I take the derivative
of e phi with respect to r,

206
00:11:17,290 --> 00:11:20,020
I get e phi back, divided by r.

207
00:11:20,020 --> 00:11:25,420
So the simplest way to
write this is like so.

208
00:11:31,015 --> 00:11:34,180
And finally, if I take the
phi derivative of the phi unit

209
00:11:34,180 --> 00:11:38,706
vector, I get e r back, with
an extra factor of r thrown in.

210
00:11:44,420 --> 00:11:45,320
And a minus sign.

211
00:11:52,320 --> 00:11:54,900
So we're going to see a way of
doing this that's a little bit

212
00:11:54,900 --> 00:11:56,400
more systematic
later, but I want

213
00:11:56,400 --> 00:11:59,070
to just keep the simple example,
where you can just basically

214
00:11:59,070 --> 00:12:03,150
by hand calculate all the
non-trivial derivatives easily.

215
00:12:03,150 --> 00:12:05,775
Of course, there's also a t
unit vector and a z unit vector.

216
00:12:05,775 --> 00:12:07,400
But they're constants,
so I'm not going

217
00:12:07,400 --> 00:12:08,525
to bother writing them out.

218
00:12:08,525 --> 00:12:11,370
All the derivatives associated
with them are equal to 0.

219
00:12:11,370 --> 00:12:17,600
So let's imagine now that I
have assembled some vector.

220
00:12:17,600 --> 00:12:21,250
So I have some vector field
that lives in this spacetime.

221
00:12:21,250 --> 00:12:22,580
And I'm using this basis.

222
00:12:22,580 --> 00:12:26,830
And so I would write this
vector with components v alpha.

223
00:12:26,830 --> 00:12:27,628
And let's let the--

224
00:12:27,628 --> 00:12:29,920
so this is going to be a
curvilinear coordinate system,

225
00:12:29,920 --> 00:12:33,490
so this will be plane polar
coordinates being used here,

226
00:12:33,490 --> 00:12:37,060
plane polar coordinate
basis vectors.

227
00:12:40,030 --> 00:12:43,845
And what I would like to
do is assemble the tensor

228
00:12:43,845 --> 00:12:45,220
that you can think
of essentially

229
00:12:45,220 --> 00:12:47,790
as the gradient of this vector.

230
00:12:47,790 --> 00:12:51,130
So let's begin by doing this
in a sort of abstract notation.

231
00:12:51,130 --> 00:12:55,030
So the gradient of this guy--
this is sort of ugly notation,

232
00:12:55,030 --> 00:12:57,860
but live with it.

233
00:12:57,860 --> 00:13:01,420
Following what we have
been doing all along, what

234
00:13:01,420 --> 00:13:04,150
you would want to
do is just take

235
00:13:04,150 --> 00:13:06,100
the root of this whole thing.

236
00:13:06,100 --> 00:13:08,560
It's going to have a
downstairs component on it.

237
00:13:08,560 --> 00:13:12,490
So attach to it
the basis one form.

238
00:13:12,490 --> 00:13:15,160
If you prefer, you can write
it using the d notation

239
00:13:15,160 --> 00:13:16,848
like I have there,
but I just want

240
00:13:16,848 --> 00:13:18,640
to stick with the form
I wrote in my notes.

241
00:13:23,810 --> 00:13:27,290
Looking at this the way I've
sort of got this right now,

242
00:13:27,290 --> 00:13:32,600
I can think of, if I
don't include the basis

243
00:13:32,600 --> 00:13:41,290
one forms here, this should be
the components of a one form.

244
00:13:41,290 --> 00:13:43,750
So this should be
a kind of object.

245
00:13:50,490 --> 00:13:52,884
So let's just expand
out that derivative.

246
00:13:55,610 --> 00:13:56,950
Let's write it like this.

247
00:13:59,540 --> 00:14:03,535
So you just-- we haven't
changed calculus.

248
00:14:03,535 --> 00:14:05,410
So when I do this, I'm
going to basically use

249
00:14:05,410 --> 00:14:08,170
the old-fashioned Leibniz
rule for expanding

250
00:14:08,170 --> 00:14:09,840
the derivative
product of two things.

251
00:14:26,290 --> 00:14:29,020
Here's the key thing which
I want to emphasize--

252
00:14:29,020 --> 00:14:30,790
in order for this
whole thing to be--

253
00:14:30,790 --> 00:14:34,780
for this to be a tensorial
object, something that I couple

254
00:14:34,780 --> 00:14:41,310
to this basis one form, the
sum of these two objects

255
00:14:41,310 --> 00:14:44,400
must obey the rules for
transforming tensors.

256
00:14:44,400 --> 00:14:46,680
But the two objects
individually will not.

257
00:15:01,822 --> 00:15:03,280
So this is an
important point which

258
00:15:03,280 --> 00:15:05,650
I'm going to emphasize in
slightly different words

259
00:15:05,650 --> 00:15:07,005
in just a few moments again.

260
00:15:07,005 --> 00:15:09,130
This is one of the key
things I want you to get out

261
00:15:09,130 --> 00:15:11,297
of this lecture, is that
when I'm taking derivatives

262
00:15:11,297 --> 00:15:13,278
of things like this,
you've got to be

263
00:15:13,278 --> 00:15:15,820
a little bit careful about what
you consider to be components

264
00:15:15,820 --> 00:15:17,110
of tensors and what is not.

265
00:15:19,870 --> 00:15:23,050
Now as written like that,
this is kind of annoying.

266
00:15:25,640 --> 00:15:30,100
So my first object has a nice
basis vector attached to it.

267
00:15:30,100 --> 00:15:33,820
My second object involves a
derivative of the basis vector.

268
00:15:33,820 --> 00:15:35,710
However, something
we saw over here

269
00:15:35,710 --> 00:15:37,600
is that derivatives
of basis vectors

270
00:15:37,600 --> 00:15:39,910
are themselves proportional
to basis vectors.

271
00:15:44,858 --> 00:15:47,150
So what I'm going to do is
introduce a bit of notation.

272
00:15:59,440 --> 00:16:01,510
So let me switch
notation slightly here.

273
00:16:01,510 --> 00:16:12,220
So the beta derivative of
e alpha can be written as--

274
00:16:12,220 --> 00:16:15,970
in general, it can be written
as a linear combination

275
00:16:15,970 --> 00:16:16,900
of basis vectors.

276
00:16:33,100 --> 00:16:37,350
So what we're going
to do is define d--

277
00:16:37,350 --> 00:16:39,850
I want to make sure my Greek
letters are legible to everyone

278
00:16:39,850 --> 00:16:40,480
in the room here.

279
00:16:40,480 --> 00:16:42,063
So let me write this
nice and clearly.

280
00:16:45,610 --> 00:16:55,860
d beta of e alpha, I'm going to
write that as capital gamma mu

281
00:16:55,860 --> 00:17:01,200
beta alpha e mu.

282
00:17:01,200 --> 00:17:05,670
This gamma that I've just
introduced here in this context

283
00:17:05,670 --> 00:17:08,310
is known as the
Christoffel symbol.

284
00:17:15,138 --> 00:17:17,680
Fact I'm calling this a symbol,
it's got three indices on it.

285
00:17:17,680 --> 00:17:20,129
You might look at it and go,
ooh, smells like a tensor.

286
00:17:20,129 --> 00:17:21,129
Be a little bit careful.

287
00:17:24,099 --> 00:17:28,420
In much the same way that these
two terms are not individually

288
00:17:28,420 --> 00:17:31,600
components of a tensor,
but their sum is,

289
00:17:31,600 --> 00:17:33,460
this guy individually
is actually

290
00:17:33,460 --> 00:17:35,620
not a component of
a tensor, but when

291
00:17:35,620 --> 00:17:39,340
combined with other things, it
allows us to assemble tensors.

292
00:17:48,220 --> 00:18:10,530
So for our plane
polar coordinates,

293
00:18:10,530 --> 00:18:15,070
there are exactly three
non-zero Christoffel symbols.

294
00:18:15,070 --> 00:18:22,970
So gamma phi r phi is equal
to 1 over r, which is also

295
00:18:22,970 --> 00:18:28,130
equal to gamma phi phi r.

296
00:18:31,290 --> 00:18:35,820
Gamma r phi phi is minus r.

297
00:18:35,820 --> 00:18:38,070
And you can basically just
read that out of that table

298
00:18:38,070 --> 00:18:40,080
that I wrote down over there.

299
00:18:40,080 --> 00:18:41,640
All the others
will be equal to 0.

300
00:18:50,310 --> 00:18:53,226
Now from this
example, this is what

301
00:18:53,226 --> 00:18:55,500
it makes it smell like
every time you introduce

302
00:18:55,500 --> 00:18:59,070
a new coordinate representation.

303
00:18:59,070 --> 00:19:02,072
You're going to need to sit
down for an hour and a half,

304
00:19:02,072 --> 00:19:03,780
or something like
that, and just work out

305
00:19:03,780 --> 00:19:06,405
all the bloody derivatives, and
then go, oh, crap, and read out

306
00:19:06,405 --> 00:19:08,197
all the different
components of this thing,

307
00:19:08,197 --> 00:19:09,557
and assemble them together.

308
00:19:09,557 --> 00:19:11,140
There actually is
an algorithm that we

309
00:19:11,140 --> 00:19:12,848
will get to at the
end of this class that

310
00:19:12,848 --> 00:19:17,310
allows you to easily extract
the Christoffel symbols provided

311
00:19:17,310 --> 00:19:18,420
you know the metric.

312
00:19:18,420 --> 00:19:21,060
But right now, I just want
to illustrate this thing

313
00:19:21,060 --> 00:19:21,660
conceptually.

314
00:19:21,660 --> 00:19:23,610
The key thing which you
should know about it

315
00:19:23,610 --> 00:19:27,280
is that it is essentially the--

316
00:19:27,280 --> 00:19:30,120
I almost said the word "matrix,"
but it's got three indices.

317
00:19:30,120 --> 00:19:34,470
It's a table of
functions that allows

318
00:19:34,470 --> 00:19:38,040
me to relate derivatives
of basis vectors

319
00:19:38,040 --> 00:19:38,970
to the basis vectors.

320
00:19:44,580 --> 00:19:47,690
So before I go on to talk
about some of that stuff,

321
00:19:47,690 --> 00:19:51,440
let's take a look at the
derivative a little bit more

322
00:19:51,440 --> 00:19:52,323
carefully.

323
00:20:20,255 --> 00:20:21,630
So the derivative
of the vector--

324
00:20:21,630 --> 00:20:24,230
so let's basically take what
I've written out up there.

325
00:20:24,230 --> 00:20:28,070
I'm going to write this as
the beta derivative of vector

326
00:20:28,070 --> 00:20:35,510
v. And I can write that as
the beta derivative of e of v

327
00:20:35,510 --> 00:20:36,950
alpha--

328
00:20:36,950 --> 00:20:39,290
so the first term
where the derivative

329
00:20:39,290 --> 00:20:40,901
hits the vector components.

330
00:20:43,550 --> 00:20:45,490
And then I've got
a second term where

331
00:20:45,490 --> 00:20:50,300
the derivative hits the basis.

332
00:20:50,300 --> 00:20:51,760
I'm going to write this like so.

333
00:20:58,840 --> 00:21:00,040
This is sort of annoying.

334
00:21:00,040 --> 00:21:02,200
One term is
proportional to e alpha,

335
00:21:02,200 --> 00:21:03,970
one is proportional to e mu.

336
00:21:03,970 --> 00:21:08,560
But notice, especially in the
second term, both alpha and mu

337
00:21:08,560 --> 00:21:11,740
are dummy indices, so
I'm free to relabel them.

338
00:21:11,740 --> 00:21:13,870
So what I'm going to
do is relabel alpha

339
00:21:13,870 --> 00:21:20,340
and mu by exchanging them.

340
00:21:23,570 --> 00:21:31,660
As long as I do
that consistently,

341
00:21:31,660 --> 00:21:33,490
that is totally kosher.

342
00:21:33,490 --> 00:21:46,140
And when I do that, I can
factor out an overall factor

343
00:21:46,140 --> 00:21:47,040
of the basis object.

344
00:21:49,980 --> 00:21:54,340
This combination
that pops up here--

345
00:21:57,290 --> 00:21:59,480
so we give this a name.

346
00:21:59,480 --> 00:22:01,660
And this is a combination
which, by the time

347
00:22:01,660 --> 00:22:05,140
you have finished this semester,
if you don't have at least

348
00:22:05,140 --> 00:22:07,180
one nightmare in which
this name appears,

349
00:22:07,180 --> 00:22:08,960
I will not have done
my job properly.

350
00:22:08,960 --> 00:22:12,580
This shows up a
lot at this point.

351
00:22:12,580 --> 00:22:17,133
This is called the
"covariant derivative."

352
00:22:24,240 --> 00:22:27,090
And it shows up enough
that we introduce

353
00:22:27,090 --> 00:22:30,780
a whole new notation
for the derivative

354
00:22:30,780 --> 00:22:32,550
to take it into account.

355
00:22:32,550 --> 00:22:34,290
I'm going to call
this combination

356
00:22:34,290 --> 00:22:37,230
of the partial
derivative of v and v

357
00:22:37,230 --> 00:22:40,940
coupled to the
Christoffel symbol--

358
00:22:43,560 --> 00:22:50,830
I'm going to write this using
the, if you're talking LaTeX,

359
00:22:50,830 --> 00:22:54,460
this would be the
nabla operator.

360
00:22:54,460 --> 00:22:57,530
So I made a point
earlier when we

361
00:22:57,530 --> 00:22:59,530
were talking about
derivatives a couple of weeks

362
00:22:59,530 --> 00:23:05,740
ago that we were reserving
the gradient symbol

363
00:23:05,740 --> 00:23:07,570
for a special purpose later.

364
00:23:07,570 --> 00:23:08,980
Here it is.

365
00:23:08,980 --> 00:23:11,320
So whenever I make
a derivative that

366
00:23:11,320 --> 00:23:13,990
involves the gradient
symbol like this,

367
00:23:13,990 --> 00:23:16,000
it is this covariant derivative.

368
00:23:16,000 --> 00:23:20,890
And the covariant derivative
acting on vector components,

369
00:23:20,890 --> 00:23:23,710
it generates tensor components.

370
00:23:23,710 --> 00:23:26,177
Partial derivative does not.

371
00:23:26,177 --> 00:23:28,510
And what I'm going to do,
just in the interest of time--

372
00:23:28,510 --> 00:23:30,718
it's one of those calculations
that's straightforward

373
00:23:30,718 --> 00:23:31,600
but fairly tedious--

374
00:23:31,600 --> 00:23:34,150
I have a set of
notes that I meant

375
00:23:34,150 --> 00:23:36,550
to make live before I headed
over here, but I forgot.

376
00:23:36,550 --> 00:23:39,160
I have a set of notes that I'm
going to show on the website

377
00:23:39,160 --> 00:23:43,810
by this evening which explicitly
works out what happens when you

378
00:23:43,810 --> 00:23:46,360
apply the coordinate
transformation using that--

379
00:23:46,360 --> 00:23:47,410
it's been erased--

380
00:23:47,410 --> 00:23:49,520
when you use that L
matrix to construct

381
00:23:49,520 --> 00:23:52,270
the coordinate transformation
between two representations.

382
00:23:52,270 --> 00:23:55,990
If you try to do it to
partial derivatives of vector

383
00:23:55,990 --> 00:23:57,880
components, basically
what you find

384
00:23:57,880 --> 00:24:00,910
is that there's an extra
term that spoils your ability

385
00:24:00,910 --> 00:24:04,510
to call that-- it spoils the
tensor transformation law,

386
00:24:04,510 --> 00:24:07,360
spoils your ability to call
that a tensor component.

387
00:24:07,360 --> 00:24:09,790
So the partial on its
own doesn't let you.

388
00:24:09,790 --> 00:24:13,840
You get some extra terms that
come along and mess it all up.

389
00:24:13,840 --> 00:24:15,700
On next p set,
you guys are going

390
00:24:15,700 --> 00:24:20,800
to show that if you then try to
apply the tensor transformation

391
00:24:20,800 --> 00:24:23,500
law to the Christoffel
symbols, you

392
00:24:23,500 --> 00:24:25,390
get something that
looks tensorial,

393
00:24:25,390 --> 00:24:27,880
but with an extra term
that spoils your ability

394
00:24:27,880 --> 00:24:29,210
to call it tensorial.

395
00:24:29,210 --> 00:24:31,660
There's a little bit
of extra junk there.

396
00:24:31,660 --> 00:24:33,670
But the two terms
exactly conspire

397
00:24:33,670 --> 00:24:37,990
to cancel each other out so
that the sum is tensorial.

398
00:24:37,990 --> 00:24:39,610
So part one of
this will be notes

399
00:24:39,610 --> 00:24:42,490
that I post to the website
no later than this evening.

400
00:24:42,490 --> 00:24:44,265
Part two, you guys
will do on the p set.

401
00:25:02,880 --> 00:25:06,270
So just saying in
math what I just

402
00:25:06,270 --> 00:25:10,290
said in words, if I
do this, like I said,

403
00:25:10,290 --> 00:25:13,530
you basically will
eventually reach the point

404
00:25:13,530 --> 00:25:17,910
where what I am
writing out right now

405
00:25:17,910 --> 00:25:24,200
will become so automatic
it will haunt your dreams.

406
00:25:24,200 --> 00:25:25,580
Wait a minute, I
screwed that up.

407
00:25:25,580 --> 00:25:27,622
It's so automatic I can't
even write it properly.

408
00:25:35,050 --> 00:25:38,482
Anyhow, something like that
will-- modulo my typo--

409
00:25:38,482 --> 00:25:39,690
that should become automatic.

410
00:25:39,690 --> 00:25:41,273
And the key thing
which I want to note

411
00:25:41,273 --> 00:25:51,840
is that if I take these
guys, and I attach

412
00:25:51,840 --> 00:25:58,590
the appropriate basis
objects to them,

413
00:25:58,590 --> 00:26:00,390
this is an honest-to-god tensor.

414
00:26:00,390 --> 00:26:03,820
And so this derivative is
itself an honest-to-god tensor.

415
00:26:03,820 --> 00:26:05,850
A typical application
of this, so one

416
00:26:05,850 --> 00:26:09,180
that will come up a
fair bit, is how do you

417
00:26:09,180 --> 00:26:12,600
compute a spacetime divergence
in each coordinate system?

418
00:26:19,940 --> 00:26:38,148
So suppose I take the divergence
of some vector field v.

419
00:26:38,148 --> 00:26:40,350
So you're going to
have four terms that

420
00:26:40,350 --> 00:26:42,930
are just the usual, like
when you guys learned

421
00:26:42,930 --> 00:26:46,840
how to do divergence in freshman
E&M in Cartesian coordinates.

422
00:26:46,840 --> 00:26:51,840
You get one term that's just
dv x dx, dv y dy, et cetera.

423
00:26:51,840 --> 00:26:54,000
So you got one term that
looks just like that,

424
00:26:54,000 --> 00:26:55,917
and you're going to have
something that brings

425
00:26:55,917 --> 00:26:57,240
in all of Christoffel symbols.

426
00:26:57,240 --> 00:27:01,090
Notice the symmetry that
we have on this one.

427
00:27:01,090 --> 00:27:03,090
Actually, there is Einstein
summation convention

428
00:27:03,090 --> 00:27:04,620
being imposed here.

429
00:27:04,620 --> 00:27:06,240
But when we look
at this, there's

430
00:27:06,240 --> 00:27:09,210
actually only one
Christoffel symbol

431
00:27:09,210 --> 00:27:12,520
that has repeated indices
in that first position.

432
00:27:12,520 --> 00:27:31,827
So when I put all
this together, you

433
00:27:31,827 --> 00:27:33,660
wind up with something
that looks like this.

434
00:27:38,760 --> 00:27:42,323
So go back and check your
copy of Jackson, or Purcell,

435
00:27:42,323 --> 00:27:44,490
or Griffith, whatever your
favorite E&M textbook is.

436
00:27:44,490 --> 00:27:46,930
And you'll see when you work
in cylindrical coordinates,

437
00:27:46,930 --> 00:27:49,430
you indeed find that there's a
correction to the radial term

438
00:27:49,430 --> 00:27:50,660
that involves 1 over r.

439
00:27:50,660 --> 00:27:53,030
That's popped out exactly
like you think it should.

440
00:27:53,030 --> 00:27:56,090
You have a bit of a wacky
looking thing with your phi

441
00:27:56,090 --> 00:27:59,270
component, of course.

442
00:27:59,270 --> 00:28:03,000
And let me just spend a
second or two making sure.

443
00:28:03,000 --> 00:28:05,300
It's often, especially while
we're developing intuition

444
00:28:05,300 --> 00:28:09,380
about working in a
coordinate basis,

445
00:28:09,380 --> 00:28:12,680
it's not a bad idea to
do a little sanity check.

446
00:28:17,660 --> 00:28:21,740
So here's a sanity check
that I would do with this.

447
00:28:21,740 --> 00:28:23,960
If I take the divergence,
I take a derivative

448
00:28:23,960 --> 00:28:26,660
of a vector field, the
final object that comes out

449
00:28:26,660 --> 00:28:29,420
of that should have the
dimensions of that vector

450
00:28:29,420 --> 00:28:31,640
divided by length.

451
00:28:31,640 --> 00:28:33,930
Remembering c equals 1,
that will clearly have

452
00:28:33,930 --> 00:28:35,180
that vector divided by length.

453
00:28:35,180 --> 00:28:36,597
That will clearly
have that vector

454
00:28:36,597 --> 00:28:39,200
divided by length, vector
divided by length, explicitly

455
00:28:39,200 --> 00:28:40,368
vector divided by length.

456
00:28:40,368 --> 00:28:40,910
That's weird.

457
00:28:44,670 --> 00:28:58,930
But remember, the basis objects
themselves are a little weird.

458
00:29:04,870 --> 00:29:07,600
One of the things we
saw was that e phi

459
00:29:07,600 --> 00:29:08,910
has the dimensions of length.

460
00:29:15,980 --> 00:29:19,010
In order for the vector
to itself be consistent,

461
00:29:19,010 --> 00:29:26,910
v phi must have the dimensions
of v divided by length.

462
00:29:26,910 --> 00:29:28,910
So in fact, when I just
take its phi derivative,

463
00:29:28,910 --> 00:29:30,327
I get something
that looks exactly

464
00:29:30,327 --> 00:29:32,493
like it should if it
is to be a divergence.

465
00:29:45,250 --> 00:29:47,290
Let's move on and
think about how

466
00:29:47,290 --> 00:29:50,620
I take a covariant
derivative of other kinds

467
00:29:50,620 --> 00:29:53,200
of tensorial objects.

468
00:29:53,200 --> 00:29:56,440
This is all you need to know
if you are worried about taking

469
00:29:56,440 --> 00:29:58,252
derivatives of vectors.

470
00:29:58,252 --> 00:30:00,460
But we're going to work with
a lot of different kinds

471
00:30:00,460 --> 00:30:01,330
of tensor objects.

472
00:30:07,753 --> 00:30:09,170
One of the most
important lectures

473
00:30:09,170 --> 00:30:11,360
we're going to do
in about a month

474
00:30:11,360 --> 00:30:14,600
actually involves looking at a
bunch of covariant derivatives

475
00:30:14,600 --> 00:30:19,308
of some four-indexed
objects, so it gets messy.

476
00:30:19,308 --> 00:30:20,350
Let's walk our way there.

477
00:30:23,600 --> 00:30:27,620
So suppose I want to take
the derivative of a scalar.

478
00:30:27,620 --> 00:30:29,990
Scalar have no basis
object attached to them.

479
00:30:38,380 --> 00:30:39,490
There's no basis object.

480
00:30:39,490 --> 00:30:40,540
When I take the
derivative, I don't

481
00:30:40,540 --> 00:30:42,415
have to worry about
anything wiggling around.

482
00:30:42,415 --> 00:30:46,040
No Christoffel symbols come in.

483
00:30:46,040 --> 00:30:50,440
If I want to take the covariant
derivative of some field phi,

484
00:30:50,440 --> 00:30:53,230
it is nothing more than
the partial derivative

485
00:30:53,230 --> 00:30:54,370
of that field phi--

486
00:30:54,370 --> 00:30:55,840
boom.

487
00:30:55,840 --> 00:30:56,410
Happy days.

488
00:30:59,125 --> 00:31:00,000
How about a one form?

489
00:31:06,540 --> 00:31:13,030
The long way to do this would
be to essentially say, well,

490
00:31:13,030 --> 00:31:14,860
the way I started
this was by looking

491
00:31:14,860 --> 00:31:18,570
at how my basis vectors varied
as I took their derivatives.

492
00:31:18,570 --> 00:31:23,650
Let's do the same thing
for the basis one forms,

493
00:31:23,650 --> 00:31:28,285
assemble my table, do a lot
of math, blah, blah, blah.

494
00:31:28,285 --> 00:31:30,410
Knock yourselves out if
that's what you want to do.

495
00:31:30,410 --> 00:31:33,280
There's a shortcut.

496
00:31:33,280 --> 00:31:39,730
Let's use the fact that
when I contract a one form

497
00:31:39,730 --> 00:31:41,150
on a vector, I get a scalar.

498
00:31:55,200 --> 00:32:04,260
So let's say I am looking at the
beta covariant derivative of p

499
00:32:04,260 --> 00:32:06,990
alpha on a alpha.

500
00:32:06,990 --> 00:32:07,890
That's a scalar.

501
00:32:07,890 --> 00:32:18,155
So this is just the
partial derivative.

502
00:32:27,380 --> 00:32:30,160
And a partial derivative of
the product of something I

503
00:32:30,160 --> 00:32:31,500
can expand out really easily.

504
00:32:44,550 --> 00:32:46,900
So using the fact that this
just becomes the partial,

505
00:32:46,900 --> 00:32:54,900
I can write this
as a alpha d beta p

506
00:32:54,900 --> 00:32:58,350
alpha plus p downstairs alpha.

507
00:33:03,580 --> 00:33:06,860
So now what?

508
00:33:06,860 --> 00:33:13,260
Well, let's rewrite this using
the covariant derivative.

509
00:33:18,862 --> 00:33:21,070
Pardon me a second while I
get caught up in my notes.

510
00:33:21,070 --> 00:33:24,230
Here we are.

511
00:33:24,230 --> 00:33:31,210
I can write this as the
covariant derivative

512
00:33:31,210 --> 00:33:34,210
minus the correction that comes
from that Christoffel symbol.

513
00:33:45,595 --> 00:33:46,980
Pardon me just a second.

514
00:33:46,980 --> 00:33:49,480
There's a couple lines here I
want to write out very neatly.

515
00:33:55,740 --> 00:33:57,120
So when I put this in--

516
00:34:22,480 --> 00:34:23,170
oops typo.

517
00:34:29,600 --> 00:34:31,880
That last typo is important,
because I'm now going

518
00:34:31,880 --> 00:34:34,491
to do the relabeling trick.

519
00:34:34,491 --> 00:34:36,199
So what I'm going to
do is take advantage

520
00:34:36,199 --> 00:34:40,280
of the fact that in this
last term, alpha and mu

521
00:34:40,280 --> 00:34:42,050
are both dummy indices.

522
00:34:42,050 --> 00:34:47,030
So on this last term that
I have written down here,

523
00:34:47,030 --> 00:34:48,560
I'm going to swap
out alpha and mu.

524
00:35:23,770 --> 00:35:29,570
When I do that, notice that the
first term and the last term

525
00:35:29,570 --> 00:35:32,060
will both be proportional
to the component a alpha.

526
00:35:52,480 --> 00:35:57,130
Now, let's require that
the covariant derivative

527
00:35:57,130 --> 00:36:00,220
when it acts on two things
that are multiplied together,

528
00:36:00,220 --> 00:36:04,430
it's a derivative, so it should
do what derivatives ordinarily

529
00:36:04,430 --> 00:36:04,930
do.

530
00:36:04,930 --> 00:36:09,680
So what we're going
to do is require

531
00:36:09,680 --> 00:36:15,230
that when I take this
covariant derivative,

532
00:36:15,230 --> 00:36:30,040
I should be able to
write the result like so.

533
00:36:30,040 --> 00:36:35,750
It's a healthy thing that
any derivative should do.

534
00:36:35,750 --> 00:36:37,680
So comparing, I look
at that, and go,

535
00:36:37,680 --> 00:36:40,830
oh, I've got the covariant
derivative of my one form

536
00:36:40,830 --> 00:36:41,330
there.

537
00:36:41,330 --> 00:36:42,140
Just compare forms.

538
00:37:06,840 --> 00:37:11,850
Very, very similar, but
notice the minus sign.

539
00:37:11,850 --> 00:37:14,530
There's a minus sign that's
been introduced there,

540
00:37:14,530 --> 00:37:19,470
and that minus sign guarantees,
if you actually expand out

541
00:37:19,470 --> 00:37:22,290
that combination of
covariant derivatives

542
00:37:22,290 --> 00:37:24,660
I have on the previous line,
there's a nice cancellation

543
00:37:24,660 --> 00:37:28,050
so that the scalar that
I get when I contract p

544
00:37:28,050 --> 00:37:31,740
on a, in fact, doesn't have
anything special going on when

545
00:37:31,740 --> 00:37:34,033
I do the covariant derivative.

546
00:37:41,483 --> 00:37:43,150
So I'm going to
generalize this further,

547
00:37:43,150 --> 00:37:45,130
but let me just make
a quick comment here.

548
00:37:45,130 --> 00:37:48,820
I began this little
calculation by saying,

549
00:37:48,820 --> 00:37:53,260
given how we started
our calculation

550
00:37:53,260 --> 00:37:55,240
of the covariant
derivative of a vector,

551
00:37:55,240 --> 00:37:56,920
we could have begun
by just taking

552
00:37:56,920 --> 00:37:59,620
lots of derivatives of the
basis one forms, and assembling

553
00:37:59,620 --> 00:38:01,840
all these various tables,
and things like that.

554
00:38:01,840 --> 00:38:10,960
If you had done this,
it's simple to find,

555
00:38:10,960 --> 00:38:22,050
based on an analysis
like this, that if you

556
00:38:22,050 --> 00:38:34,640
take a partial
derivative of a one form,

557
00:38:34,640 --> 00:38:37,740
that you get sort of a linear
combination of one forms back.

558
00:38:37,740 --> 00:38:39,167
Looks just like
what you got when

559
00:38:39,167 --> 00:38:41,250
you took a partial derivative
of the basis vector,

560
00:38:41,250 --> 00:38:42,560
but with a minus sign.

561
00:38:42,560 --> 00:38:48,670
And what that minus sign does
is it enforces, if you go back

562
00:38:48,670 --> 00:38:51,730
to a lecture from ages ago, when
I first introduced basis one

563
00:38:51,730 --> 00:39:04,480
forms, it enforces the idea that
when I combine basis one forms

564
00:39:04,480 --> 00:39:08,110
with basis vectors, I get an
identity object out of this,

565
00:39:08,110 --> 00:39:09,910
which is itself a constant.

566
00:39:09,910 --> 00:39:12,190
If you are the
kind of person who

567
00:39:12,190 --> 00:39:14,320
likes that sort of
mathematical rigor,

568
00:39:14,320 --> 00:39:16,090
some textbooks will
start with this,

569
00:39:16,090 --> 00:39:18,790
and then derive other
things from that--

570
00:39:18,790 --> 00:39:21,310
sort of six of one, half
a dozen of the other.

571
00:39:21,310 --> 00:39:24,460
So we could go on at this point.

572
00:39:24,460 --> 00:39:27,310
And I could say,
how do I do this

573
00:39:27,310 --> 00:39:30,070
with a tensor that has two
indices in the upstairs

574
00:39:30,070 --> 00:39:30,890
position?

575
00:39:30,890 --> 00:39:32,350
How do I do this
with a tensor that

576
00:39:32,350 --> 00:39:34,373
has two indices in the
downstairs position?

577
00:39:34,373 --> 00:39:35,790
How do I do it for
a tensor that's

578
00:39:35,790 --> 00:39:37,960
got 17 indices in
the upstairs position

579
00:39:37,960 --> 00:39:40,490
and 38 in the
downstairs position?

580
00:39:40,490 --> 00:39:43,480
The answer is easily
deduced from doing

581
00:39:43,480 --> 00:39:44,970
these kinds of
rules, so I'm just

582
00:39:44,970 --> 00:39:46,900
going to write down a
couple of examples and state

583
00:39:46,900 --> 00:39:47,900
what it turns out to be.

584
00:39:56,660 --> 00:40:01,390
So basically, imagine I want to
take the covariant derivative--

585
00:40:01,390 --> 00:40:03,470
let's do the stress
energy tensor--

586
00:40:03,470 --> 00:40:06,747
covariant derivative of T mu nu.

587
00:40:06,747 --> 00:40:09,080
So remember, the way that the
Christoffel got into there

588
00:40:09,080 --> 00:40:11,420
is that when I looked at
the derivative of a vector,

589
00:40:11,420 --> 00:40:15,000
I was looking at derivatives
of basis objects.

590
00:40:15,000 --> 00:40:16,750
Well, now I'm going
to look at derivatives

591
00:40:16,750 --> 00:40:18,870
of two different basis objects.

592
00:40:18,870 --> 00:40:22,140
So I'm going to wind up with
two Christoffel symbols.

593
00:40:29,100 --> 00:40:42,543
You can kind of think of it
as coming along and correcting

594
00:40:42,543 --> 00:40:43,460
each of these indices.

595
00:40:46,440 --> 00:40:50,520
I can do this with the indices
in the downstairs position.

596
00:40:57,290 --> 00:40:58,250
Guess what?

597
00:40:58,250 --> 00:41:07,510
Comes along and corrects
all them with minus signs.

598
00:41:29,860 --> 00:41:31,930
Just for completeness,
let me just

599
00:41:31,930 --> 00:41:35,250
write down the general rule.

600
00:41:35,250 --> 00:41:40,330
If I am looking at the
covariant derivative of a tensor

601
00:41:40,330 --> 00:41:48,330
with a gajillion
upstairs indices

602
00:41:48,330 --> 00:41:51,990
and a gajillion
downstairs indices,

603
00:41:51,990 --> 00:41:59,690
you get one term that's just a
partial derivative of that guy,

604
00:41:59,690 --> 00:42:10,800
and you get a
Christoffel coupling

605
00:42:10,800 --> 00:42:13,270
for every one of these.

606
00:42:13,270 --> 00:42:32,020
Plus sign for all the
upstairs, minus sign

607
00:42:32,020 --> 00:42:35,820
for all the downstairs.

608
00:42:43,380 --> 00:42:44,500
That was a little tedious.

609
00:42:44,500 --> 00:42:46,450
You basically just, when I
give you a tensor like that,

610
00:42:46,450 --> 00:42:47,950
you just kind of
have to go through.

611
00:42:47,950 --> 00:42:49,723
And it becomes sort
of almost monkey work.

612
00:42:49,723 --> 00:42:51,640
You just have to rotely
go through and correct

613
00:42:51,640 --> 00:42:54,500
every one of the indices using
an algorithm that kind of looks

614
00:42:54,500 --> 00:42:55,000
like this.

615
00:42:59,800 --> 00:43:02,300
Oh, jeez, there's absolutely a
minus sign on the second one.

616
00:43:02,300 --> 00:43:05,085
Thank you.

617
00:43:05,085 --> 00:43:06,040
I appreciate that.

618
00:43:14,160 --> 00:43:19,687
So the way that we have
done things so far,

619
00:43:19,687 --> 00:43:22,020
and I kind of emphasized, it
sort of smells like the way

620
00:43:22,020 --> 00:43:26,190
to do this is you pick your
new coordinate representation,

621
00:43:26,190 --> 00:43:33,420
you throw together all of
your various basis objects,

622
00:43:33,420 --> 00:43:35,467
and then you just
start going whee,

623
00:43:35,467 --> 00:43:37,050
let's start taking
derivatives and see

624
00:43:37,050 --> 00:43:39,380
how all these things vary
with respect to each other,

625
00:43:39,380 --> 00:43:42,780
assemble my table of
the gammas, and then do

626
00:43:42,780 --> 00:43:45,550
my covariant derivative.

627
00:43:45,550 --> 00:43:48,220
If that were, in fact,
the way we did it,

628
00:43:48,220 --> 00:43:50,260
I would not have chosen
my research career

629
00:43:50,260 --> 00:43:51,480
to focus on this field.

630
00:43:51,480 --> 00:43:52,990
That would suck.

631
00:43:52,990 --> 00:43:58,030
Certainly prior to Odin
providing us with Mathematica,

632
00:43:58,030 --> 00:44:01,750
it would have been
absolutely undoable.

633
00:44:01,750 --> 00:44:05,290
Even with it, though, it
would be incredibly tedious.

634
00:44:05,290 --> 00:44:28,890
So there is a better
way to do this,

635
00:44:28,890 --> 00:44:33,170
and it comes via the metric.

636
00:44:33,170 --> 00:44:39,800
Before I derive what the
algorithm actually is,

637
00:44:39,800 --> 00:44:43,250
I want to introduce an extremely
important property of tensor

638
00:44:43,250 --> 00:44:46,430
relationships that we are
going to come back to and use

639
00:44:46,430 --> 00:44:47,750
quite a bit in this course.

640
00:44:57,923 --> 00:45:00,340
So this is something that we
have actually kind of alluded

641
00:45:00,340 --> 00:45:03,280
to repeatedly, but I want to
make it a little more formal

642
00:45:03,280 --> 00:45:04,742
and just clearly state it.

643
00:45:12,180 --> 00:45:14,370
So this relationship
that I'm going to use

644
00:45:14,370 --> 00:45:27,350
is some kind of a tensor
equation, a tensorial equation

645
00:45:27,350 --> 00:45:37,620
that holds in one
representation must

646
00:45:37,620 --> 00:45:42,182
hold in all representations.

647
00:45:46,250 --> 00:45:48,350
Come back to the
intuition when I first

648
00:45:48,350 --> 00:45:50,780
began describing physics in
terms of geometric objects

649
00:45:50,780 --> 00:45:51,460
in spacetime.

650
00:45:51,460 --> 00:45:54,560
One of the key points I
tried to really emphasize

651
00:45:54,560 --> 00:45:57,410
the difference of is that
I can have different--

652
00:45:57,410 --> 00:46:00,457
let's say my arm is a
particular vector in spacetime.

653
00:46:00,457 --> 00:46:02,540
Someone running through
the room at three-quarters

654
00:46:02,540 --> 00:46:05,030
the speed of light will use
different representations

655
00:46:05,030 --> 00:46:06,013
to describe my arm.

656
00:46:06,013 --> 00:46:07,430
They will see
length contractions.

657
00:46:07,430 --> 00:46:09,722
They will see things sort of
spanning different things.

658
00:46:09,722 --> 00:46:11,450
But the geometric
object, the thing

659
00:46:11,450 --> 00:46:16,400
which goes between two events
in spacetime, that does not

660
00:46:16,400 --> 00:46:18,920
change, even though the
representation of those events

661
00:46:18,920 --> 00:46:20,138
might.

662
00:46:20,138 --> 00:46:22,430
This remains true not just
for Lorentz transformations,

663
00:46:22,430 --> 00:46:25,070
but for all classes
of transformations

664
00:46:25,070 --> 00:46:28,850
that we might care to
use in our analysis.

665
00:46:28,850 --> 00:46:32,560
Changing the representation
cannot change the equation.

666
00:46:39,480 --> 00:46:43,350
Written that way, it
sounds like, well, duh,

667
00:46:43,350 --> 00:46:47,741
but as we'll see, it's got
important consequences.

668
00:47:01,980 --> 00:47:06,440
So as a warm-up exercise of
how we might want to use this,

669
00:47:06,440 --> 00:47:09,710
let's think about the
double gradient of a scalar.

670
00:47:40,540 --> 00:47:47,710
So let's define-- let's just say
that this is the object that I

671
00:47:47,710 --> 00:47:50,320
want to compute.

672
00:47:50,320 --> 00:47:54,044
Let's first do this in a
Cartesian representation.

673
00:47:57,920 --> 00:47:59,710
In a Cartesian
representation, I just

674
00:47:59,710 --> 00:48:02,875
take two partial derivatives.

675
00:48:05,490 --> 00:48:07,790
I've got a couple basis
one forms for this.

676
00:48:17,520 --> 00:48:20,102
So I've got something like this.

677
00:48:20,102 --> 00:48:21,560
The thing which I
want to emphasize

678
00:48:21,560 --> 00:48:25,220
is that as written, in
Cartesian coordinates, d

679
00:48:25,220 --> 00:48:27,350
alpha d beta of phi--

680
00:48:27,350 --> 00:48:32,564
those are the components of a
tensor in this representation.

681
00:48:40,150 --> 00:48:42,880
And the key thing is
that they are obviously

682
00:48:42,880 --> 00:48:48,690
symmetric on exchange of
the indices alpha and beta.

683
00:48:53,795 --> 00:48:55,420
If I'm just taking
partial derivatives,

684
00:48:55,420 --> 00:48:57,400
doesn't matter what
order I take them in.

685
00:48:57,400 --> 00:48:58,794
That's got to be symmetric.

686
00:49:02,970 --> 00:49:05,780
Let's now look at
the double gradient

687
00:49:05,780 --> 00:49:08,374
of a scalar in a more
general representation.

688
00:49:31,630 --> 00:49:37,430
So in a general
representation, I'm

689
00:49:37,430 --> 00:49:40,220
going to require
these two derivatives

690
00:49:40,220 --> 00:49:42,870
to be covariant derivatives.

691
00:49:42,870 --> 00:49:47,690
Now, we know one of them can
be very trivially replaced with

692
00:49:47,690 --> 00:49:50,090
a partial, but the
other one cannot.

693
00:49:50,090 --> 00:49:51,590
Hold that thought
for just a second.

694
00:49:55,610 --> 00:50:01,220
If this thing is symmetric in
the Cartesian representation,

695
00:50:01,220 --> 00:50:04,822
I claim it must also be true
in a general representation.

696
00:50:17,370 --> 00:50:27,890
In other words, exchanging the
order of covariant derivatives

697
00:50:27,890 --> 00:50:31,853
when they act on a scalar should
give me the same thing back.

698
00:50:31,853 --> 00:50:33,020
Let's see what this implies.

699
00:51:02,540 --> 00:51:11,725
So if I require the following
to be true, that's saying--

700
00:51:18,630 --> 00:51:19,130
oops.

701
00:51:29,230 --> 00:51:31,030
So let's expand this
out one more level.

702
00:51:37,680 --> 00:51:48,700
So now, I'm correcting that
downstairs index and over here.

703
00:52:04,640 --> 00:52:08,740
So the terms involving
nothing but partials,

704
00:52:08,740 --> 00:52:10,940
they obviously cancel.

705
00:52:10,940 --> 00:52:13,247
I have a common
factor of d mu of phi.

706
00:52:13,247 --> 00:52:15,330
So let's move one of these
over to the other side.

707
00:52:18,240 --> 00:52:20,360
What we've learned is
that this requirement,

708
00:52:20,360 --> 00:52:33,097
that this combination of
derivatives be symmetric,

709
00:52:33,097 --> 00:52:35,430
tells me something about the
symmetry of the Christoffel

710
00:52:35,430 --> 00:52:36,140
symbols itself.

711
00:52:43,640 --> 00:52:46,880
If you go back to
that little table

712
00:52:46,880 --> 00:52:49,250
that I wrote down for
plane polar coordinates,

713
00:52:49,250 --> 00:52:50,750
that was one where
I just calculated

714
00:52:50,750 --> 00:52:53,720
only three non-trivial
components,

715
00:52:53,720 --> 00:52:55,202
but there was a
symmetry in there.

716
00:52:55,202 --> 00:52:56,660
And if you go and
you check it, you

717
00:52:56,660 --> 00:53:00,673
will see it's consistent with
what I just found right here.

718
00:53:00,673 --> 00:53:01,840
Pardon me for just a second.

719
00:53:01,840 --> 00:53:06,600
I want to organize
a few of my notes.

720
00:53:06,600 --> 00:53:08,320
These have gotten
all out of order.

721
00:53:15,960 --> 00:53:16,460
Here it is.

722
00:53:49,897 --> 00:53:51,605
So let me just use
this as an opportunity

723
00:53:51,605 --> 00:53:52,897
to introduce a bit of notation.

724
00:53:56,210 --> 00:54:00,770
Whenever I give you a tensor
that's got two indices,

725
00:54:00,770 --> 00:54:10,870
if I write parentheses
around those indices,

726
00:54:10,870 --> 00:54:14,110
this is going to
mean that I do what

727
00:54:14,110 --> 00:54:15,931
is called "symmetrization."

728
00:54:19,020 --> 00:54:21,090
We're going to use
this from time to time.

729
00:54:21,090 --> 00:54:29,310
If I write square braces,
this is what we call

730
00:54:29,310 --> 00:54:31,057
"anti-symmetrization."

731
00:54:33,800 --> 00:54:41,220
And so what we just learned
is that gamma mu alpha beta

732
00:54:41,220 --> 00:54:45,450
is equal to gamma mu alpha
beta with symmetrization

733
00:54:45,450 --> 00:54:48,630
on those last two indices.

734
00:54:48,630 --> 00:54:53,430
We have likewise learned
that if I contract

735
00:54:53,430 --> 00:55:03,780
this against some object, if
these were anti-symmetric,

736
00:55:03,780 --> 00:55:07,027
I must get a 0 out of it.

737
00:55:07,027 --> 00:55:09,360
So that's a brief aside, but
these are important things,

738
00:55:09,360 --> 00:55:11,610
and I want to make sure you
have a chance to see them.

739
00:55:14,640 --> 00:55:19,710
So trying to make a decision
here about where we're going to

740
00:55:19,710 --> 00:55:22,410
want to carry things forward.

741
00:55:22,410 --> 00:55:25,140
We're approaching the
end of one set of notes.

742
00:55:25,140 --> 00:55:27,810
There's still one more
thing I want to do.

743
00:55:27,810 --> 00:55:29,760
So I set this whole
thing up by saying

744
00:55:29,760 --> 00:55:32,760
that I wanted to give you
guys an algorithm for how

745
00:55:32,760 --> 00:55:34,640
to generate the
Christoffel symbols.

746
00:55:51,230 --> 00:56:02,200
The way I'm going to
do this is by examining

747
00:56:02,200 --> 00:56:11,300
the gradient of the metric.

748
00:56:11,300 --> 00:56:16,140
So suppose I want to
compute the following tensor

749
00:56:16,140 --> 00:56:18,390
quantity-- let's say
is g the metric tensor,

750
00:56:18,390 --> 00:56:20,890
written here in the
fairly abstract notation.

751
00:56:20,890 --> 00:56:26,530
And this is my full-on tensor
gradient of this thing.

752
00:56:32,500 --> 00:56:36,380
So if you want to write
this out in its full glory,

753
00:56:36,380 --> 00:56:38,150
I might write this as
something like this.

754
00:56:44,030 --> 00:56:47,320
But if you stop and think
about this for just a second,

755
00:56:47,320 --> 00:56:53,370
let's go back to this principle.

756
00:56:53,370 --> 00:56:55,830
An equation that is tensorial
in one representation

757
00:56:55,830 --> 00:56:57,990
must be tensorial in all.

758
00:56:57,990 --> 00:57:03,600
Suppose I choose the Cartesian
representation of this thing.

759
00:57:11,330 --> 00:57:26,870
Well, then here's what
it looks like there.

760
00:57:26,870 --> 00:57:29,970
But this is a constant.

761
00:57:29,970 --> 00:57:36,030
So if I do this in Cartesian
coordinates, it has to be 0.

762
00:57:36,030 --> 00:57:39,840
The only way that I can
make this sort of comport

763
00:57:39,840 --> 00:57:43,170
with this principle
that an equation that

764
00:57:43,170 --> 00:57:45,480
is tensorial in
one representation

765
00:57:45,480 --> 00:57:48,370
holds in all representations--

766
00:57:48,370 --> 00:57:57,150
this leads me to say,
I need to require

767
00:57:57,150 --> 00:58:02,970
that the covariant derivative
of the metric be equal to 0.

768
00:58:06,410 --> 00:58:07,460
We're going to use this.

769
00:58:07,460 --> 00:58:09,710
And I think this will be the
last detailed calculation

770
00:58:09,710 --> 00:58:11,250
I do in today's lecture.

771
00:58:11,250 --> 00:58:12,830
We're going to use
this to find a way

772
00:58:12,830 --> 00:58:16,027
to get the Christoffel symbol
from partial derivatives

773
00:58:16,027 --> 00:58:16,610
of the metric.

774
00:58:45,860 --> 00:58:48,050
There's a lot of
terms here and there's

775
00:58:48,050 --> 00:58:49,260
a lot of little indices.

776
00:58:49,260 --> 00:58:51,552
So I'm going to do my best
to make my handwriting neat.

777
00:58:54,137 --> 00:58:55,720
I'm going to write
down a relationship

778
00:58:55,720 --> 00:58:57,100
that I call "Roman numeral I."

779
00:58:59,740 --> 00:59:03,750
The covariant derivative in the
gamma direction, G alpha beta--

780
00:59:15,298 --> 00:59:17,590
you know what, let me put
this down a little bit lower,

781
00:59:17,590 --> 00:59:19,507
so I can get these two
terms on the same line.

782
00:59:29,140 --> 00:59:34,020
So I get this thing that
involves two Christoffel

783
00:59:34,020 --> 00:59:38,790
symbols correcting
those two indices.

784
00:59:43,418 --> 00:59:44,460
This is going to equal 0.

785
00:59:49,490 --> 00:59:52,680
I don't really seem to
have gotten very far.

786
00:59:52,680 --> 00:59:57,120
This is true, but I now
have two bloody Christoffel

787
00:59:57,120 --> 00:59:59,760
symbols that I've somehow
managed to work into this.

788
00:59:59,760 --> 01:00:01,410
What I'm trying to
do is find a way

789
01:00:01,410 --> 01:00:04,470
to get one, and equate it to
things involving derivatives

790
01:00:04,470 --> 01:00:05,520
of the metric.

791
01:00:05,520 --> 01:00:09,670
So this is sort of a
ruh-roh kind of moment.

792
01:00:09,670 --> 01:00:11,770
But there's nothing
special about this order

793
01:00:11,770 --> 01:00:14,030
of the indices.

794
01:00:14,030 --> 01:00:18,790
So with the audacity
that only comes

795
01:00:18,790 --> 01:00:21,790
from knowing the answer in
advance, what I'm going to do

796
01:00:21,790 --> 01:00:23,500
is permute the indices.

797
01:00:54,620 --> 01:00:57,800
Then go, oh, let's permute
the indices once more.

798
01:01:25,665 --> 01:01:27,790
So I'll give you guys a
moment to catch up with me.

799
01:01:27,790 --> 01:01:29,520
Don't forget, these notes
will be scanned and added

800
01:01:29,520 --> 01:01:30,187
to the web page.

801
01:01:30,187 --> 01:01:33,240
So if you don't want to
follow along writing down

802
01:01:33,240 --> 01:01:35,760
every little detail,
I understand,

803
01:01:35,760 --> 01:01:38,568
although personally, I
find that these things gel

804
01:01:38,568 --> 01:01:40,110
a little bit better
when you actually

805
01:01:40,110 --> 01:01:41,110
write them out yourself.

806
01:01:45,780 --> 01:01:52,260
So those are three ways that I
can assert that the metric has

807
01:01:52,260 --> 01:01:54,125
no covariant derivative.

808
01:01:54,125 --> 01:01:55,500
They all are
basically expressing

809
01:01:55,500 --> 01:01:56,790
that same physical fact.

810
01:01:56,790 --> 01:01:59,730
I'm just permuting the indices.

811
01:01:59,730 --> 01:02:03,080
Now there's no better
way to describe this

812
01:02:03,080 --> 01:02:06,240
than you sort of just stare
at this for a few moments,

813
01:02:06,240 --> 01:02:10,694
and then go, gee, I wonder
what would happen if--

814
01:02:10,694 --> 01:02:12,760
so stare at this
for a little while.

815
01:02:12,760 --> 01:02:15,850
And then construct-- you
know I have three things that

816
01:02:15,850 --> 01:02:17,320
are equal to 0.

817
01:02:17,320 --> 01:02:20,020
So I can add them together,
I can subtract one

818
01:02:20,020 --> 01:02:20,830
from the other.

819
01:02:20,830 --> 01:02:24,760
I can add two and
subtract one, whatever.

820
01:02:24,760 --> 01:02:26,800
They should all give me 0.

821
01:02:26,800 --> 01:02:33,160
And the particular
combination I want to look at

822
01:02:33,160 --> 01:02:36,530
is what I get when I
take relationship one

823
01:02:36,530 --> 01:02:40,840
and I subtract from
it two and three.

824
01:02:44,760 --> 01:02:48,470
So I'm going to
get one term that

825
01:02:48,470 --> 01:02:54,210
are just these three
combinations of derivatives,

826
01:02:54,210 --> 01:02:54,710
gamma.

827
01:03:03,020 --> 01:03:05,040
And I get something
that looks like--

828
01:03:22,570 --> 01:03:25,490
let me write this out and
then pause and make a comment.

829
01:03:52,470 --> 01:03:54,690
So I sort of made some lame
jokes a few moments ago

830
01:03:54,690 --> 01:03:56,250
that essentially, the only
reason I was able to get

831
01:03:56,250 --> 01:03:57,930
this was by knowing the answer
in the back of the book,

832
01:03:57,930 --> 01:03:58,890
essentially.

833
01:03:58,890 --> 01:04:01,560
And to be perfectly blunt,
for me personally, that's

834
01:04:01,560 --> 01:04:02,220
probably true.

835
01:04:02,220 --> 01:04:04,157
When I first wrote
this down, I probably

836
01:04:04,157 --> 01:04:05,490
did need to follow an algorithm.

837
01:04:05,490 --> 01:04:07,823
But if I was doing this ab
initio, if I was sitting down

838
01:04:07,823 --> 01:04:09,840
to first do this, what's
really going on here

839
01:04:09,840 --> 01:04:12,600
is the reason I wrote out all
these different combinations

840
01:04:12,600 --> 01:04:16,620
of things is that I was trying
to gather terms together

841
01:04:16,620 --> 01:04:20,010
in such a way that I could take
advantage of that symmetry.

842
01:04:20,010 --> 01:04:24,330
A few moments ago, we showed
that the Christoffel symbols

843
01:04:24,330 --> 01:04:28,210
are symmetric on the
lower two indices.

844
01:04:28,210 --> 01:04:30,630
And so by putting out all
these different combinations

845
01:04:30,630 --> 01:04:33,660
of things, I was then able
to combine them in such a way

846
01:04:33,660 --> 01:04:35,510
that certain
terms-- look at this

847
01:04:35,510 --> 01:04:38,610
and go, ah, symmetry on
alpha and gamma means

848
01:04:38,610 --> 01:04:41,490
this whole term dies.

849
01:04:41,490 --> 01:04:45,620
Symmetry on beta and gamma
means this whole term dies.

850
01:04:45,620 --> 01:04:49,020
Symmetry on alpha and beta
means these two guys combine,

851
01:04:49,020 --> 01:04:50,100
and I get a factor of 2.

852
01:05:05,150 --> 01:05:06,400
So let's clean up our algebra.

853
01:05:16,200 --> 01:05:19,060
Move a bunch of our terms
to the other side equation,

854
01:05:19,060 --> 01:05:22,255
since it's a blah,
blah, blah equals 0.

855
01:05:22,255 --> 01:05:27,010
And what we get when we
do this is g mu downstairs

856
01:05:27,010 --> 01:05:32,140
gamma is equal to 1/2.

857
01:05:50,450 --> 01:05:52,770
What we're going to do now
is we will define everything

858
01:05:52,770 --> 01:05:53,770
on the right-hand side--

859
01:05:53,770 --> 01:05:58,380
I've kind of emphasized earlier
that the Christoffels are not

860
01:05:58,380 --> 01:06:01,202
themselves tensors, but we're
going to imagine that we can

861
01:06:01,202 --> 01:06:02,910
nonetheless-- we're
not going to imagine,

862
01:06:02,910 --> 01:06:04,010
we're just going to define--

863
01:06:04,010 --> 01:06:06,240
we're going to say that we're
allowed to raise and lower

864
01:06:06,240 --> 01:06:08,782
their indices using the metric,
in the same way you guys been

865
01:06:08,782 --> 01:06:11,640
doing with vectors and one forms
and other kinds of tensors.

866
01:06:11,640 --> 01:06:14,580
So let's call everything
on the right-hand side

867
01:06:14,580 --> 01:06:21,890
here gamma with all the indices
in the downstairs position,

868
01:06:21,890 --> 01:06:24,240
gamma sub gamma alpha beta.

869
01:06:24,240 --> 01:06:29,500
And then this is
simply what I get

870
01:06:29,500 --> 01:06:37,370
when I click all of these
things together like so.

871
01:06:41,870 --> 01:06:44,390
If you go and you
look up the formulas

872
01:06:44,390 --> 01:06:46,040
for this in various
textbooks that

873
01:06:46,040 --> 01:06:49,070
give these different
kinds of formulas,

874
01:06:49,070 --> 01:06:53,520
you will typically see it
written as 1/2 g upstairs

875
01:06:53,520 --> 01:06:57,590
indices, and then all this
stuff in parentheses after that.

876
01:06:57,590 --> 01:06:59,780
When you look things up,
this is the typical formula

877
01:06:59,780 --> 01:07:01,133
that is given in these books.

878
01:07:01,133 --> 01:07:02,300
This is where it comes from.

879
01:07:08,750 --> 01:07:15,850
So I need to check one thing
because it appears my notes are

880
01:07:15,850 --> 01:07:18,530
a little bit out of order here.

881
01:07:18,530 --> 01:07:22,210
But nonetheless,
this is a good point,

882
01:07:22,210 --> 01:07:29,490
since we've just finished
a pretty long calculation,

883
01:07:29,490 --> 01:07:30,900
this is a good
point to introduce

884
01:07:30,900 --> 01:07:32,175
an important physical point.

885
01:07:32,175 --> 01:07:33,550
We're going to
come back to this.

886
01:07:33,550 --> 01:07:35,835
We're going to start
this on Thursday.

887
01:07:38,940 --> 01:07:41,460
But I want to begin making
some physical points that

888
01:07:41,460 --> 01:07:43,800
are going to take us
from special relativity

889
01:07:43,800 --> 01:07:45,200
to general relativity.

890
01:07:48,433 --> 01:07:50,100
So despite the fact
that I've introduced

891
01:07:50,100 --> 01:07:54,810
this new mathematical
framework, everything

892
01:07:54,810 --> 01:08:04,050
that I have done so
far is in the context

893
01:08:04,050 --> 01:08:06,810
of special relativity.

894
01:08:06,810 --> 01:08:08,880
I'm going to make a
more precise definition

895
01:08:08,880 --> 01:08:10,490
of special relativity right now.

896
01:08:17,390 --> 01:08:18,890
So special relativity--
we are going

897
01:08:18,890 --> 01:08:32,990
to think of this moving forward
as the theory which allows us

898
01:08:32,990 --> 01:08:50,270
to cover the entire
spacetime manifold

899
01:08:50,270 --> 01:08:52,520
using inertial reference frames.

900
01:09:00,865 --> 01:09:02,990
So we use inertial reference
frames or essentially,

901
01:09:02,990 --> 01:09:05,660
Lorenz reference frames, and
saying that Lorentz coordinates

902
01:09:05,660 --> 01:09:07,220
are good everywhere.

903
01:09:07,220 --> 01:09:09,890
We know we can go between
different Lorentz reference

904
01:09:09,890 --> 01:09:13,550
frames using Lorentz
transformations.

905
01:09:13,550 --> 01:09:15,800
But the key thing is that
if special relativity were

906
01:09:15,800 --> 01:09:19,100
correct, the entire universe
would be accurately described

907
01:09:19,100 --> 01:09:23,720
by any inertial reference
frame you care to write down.

908
01:09:23,720 --> 01:09:28,160
And I will probably only be able
to do about half of this right

909
01:09:28,160 --> 01:09:29,330
now.

910
01:09:29,330 --> 01:09:32,109
We'll pick it up next time,
if I cannot finish this.

911
01:09:32,109 --> 01:09:33,800
The key thing which
I want to emphasize

912
01:09:33,800 --> 01:09:38,000
is, gravity breaks this.

913
01:09:45,899 --> 01:09:51,660
As soon as you put gravity
into your theory of relativity,

914
01:09:51,660 --> 01:09:53,220
you cannot have--

915
01:09:53,220 --> 01:09:58,820
so we will call this a
global inertial frame,

916
01:09:58,820 --> 01:10:03,950
an inertial frame that
is good everywhere,

917
01:10:03,950 --> 01:10:06,080
so "global" in the
mathematical sense,

918
01:10:06,080 --> 01:10:08,330
not "global" in the geographic
sense-- not just earth,

919
01:10:08,330 --> 01:10:11,200
the whole universe, essentially.

920
01:10:11,200 --> 01:10:13,580
As soon as we put in
gravity, we no longer

921
01:10:13,580 --> 01:10:16,580
have global reference frames
and global inertial reference

922
01:10:16,580 --> 01:10:18,230
frames.

923
01:10:18,230 --> 01:10:19,977
That word "inertial"
is important.

924
01:10:29,730 --> 01:10:34,650
But we are going to be allowed
to have local inertial frames.

925
01:10:34,650 --> 01:10:36,570
I have not precisely
defined the difference

926
01:10:36,570 --> 01:10:40,035
what "local" means in this case,
and I won't for a few lectures.

927
01:10:48,440 --> 01:10:52,998
But to give you a preview
as to what that means,

928
01:10:52,998 --> 01:10:54,540
it's essentially
going to say that we

929
01:10:54,540 --> 01:10:56,100
can define an
inertial coordinate

930
01:10:56,100 --> 01:11:01,200
system that is good over a
particular region of spacetime.

931
01:11:01,200 --> 01:11:03,120
And we're going to have
to discuss and come up

932
01:11:03,120 --> 01:11:05,738
with ways of understanding what
the boundaries of that region

933
01:11:05,738 --> 01:11:07,155
are, and how to
make this precise.

934
01:11:11,160 --> 01:11:13,020
So the statement
that gravity breaks

935
01:11:13,020 --> 01:11:16,990
the existence of global
Lorentz frames, like I said,

936
01:11:16,990 --> 01:11:18,000
it's a two-part thing.

937
01:11:24,070 --> 01:11:30,730
I'm going to use a very
handwavy argument which

938
01:11:30,730 --> 01:11:32,620
can be made quite
rigorous later,

939
01:11:32,620 --> 01:11:35,380
but I want to keep it to this
handwaving level, because first

940
01:11:35,380 --> 01:11:37,030
of all, it actually
was first done

941
01:11:37,030 --> 01:11:41,205
by a very high-level
mathematical physicist named

942
01:11:41,205 --> 01:11:43,705
Alfred Schild, who worked in
the early theory of relativity.

943
01:11:43,705 --> 01:11:45,413
It's sort of like he
was so mathematical,

944
01:11:45,413 --> 01:11:48,550
if it was good enough for him,
that's good enough for me.

945
01:11:48,550 --> 01:11:51,190
And I think even though it's
a little handwavy, and kind

946
01:11:51,190 --> 01:11:53,890
of goofy in at
least one place, it

947
01:11:53,890 --> 01:11:57,220
gives a good physical sense
as to why it is gravity

948
01:11:57,220 --> 01:11:58,990
begins to mess things up.

949
01:11:58,990 --> 01:12:04,988
So part one is the
fact that there exists

950
01:12:04,988 --> 01:12:06,030
a gravitational redshift.

951
01:12:20,020 --> 01:12:25,760
So here's where I'm going
to be particularly silly,

952
01:12:25,760 --> 01:12:28,753
but I will back up my silliness
by the fact that everything

953
01:12:28,753 --> 01:12:30,170
silly that I say
here has actually

954
01:12:30,170 --> 01:12:33,100
been experimentally
verified, or at least

955
01:12:33,100 --> 01:12:34,800
the key physical output of this.

956
01:12:34,800 --> 01:12:38,840
So imagine you are
on top of a tower

957
01:12:38,840 --> 01:12:49,110
and you drop a rock of rest mass
m off the top of this tower.

958
01:12:59,484 --> 01:13:00,400
So here you are.

959
01:13:04,112 --> 01:13:04,820
Here's your rock.

960
01:13:08,060 --> 01:13:08,720
The rock falls.

961
01:13:15,770 --> 01:13:22,810
There's a wonderful device down
here which I label with a p.

962
01:13:22,810 --> 01:13:24,301
It's called a photonulater.

963
01:13:27,070 --> 01:13:34,820
And what the
photonulater does, it

964
01:13:34,820 --> 01:13:44,370
converts the rock
into a single photon,

965
01:13:44,370 --> 01:13:46,680
and it does so
conserving energy.

966
01:13:53,790 --> 01:13:56,710
So when this rock
falls, the instant

967
01:13:56,710 --> 01:14:05,880
before it goes into
your photonulater,

968
01:14:05,880 --> 01:14:10,590
just use Newtonian physics
plus the notion of rest energy.

969
01:14:10,590 --> 01:14:15,510
So it's got an energy of m--
mC squared, if you prefer,

970
01:14:15,510 --> 01:14:17,100
its rest energy--

971
01:14:17,100 --> 01:14:19,948
plus what it acquired
after falling--

972
01:14:19,948 --> 01:14:21,990
pardon me, I forgot to
give you a distance here--

973
01:14:21,990 --> 01:14:23,450
after falling a distance h.

974
01:14:30,330 --> 01:14:32,432
So that means that the
photon that I shoot up

975
01:14:32,432 --> 01:14:33,140
from this thing--

976
01:14:36,090 --> 01:14:39,300
let me put a few
things on this board.

977
01:14:49,860 --> 01:15:00,350
So the instant that
I create this photon,

978
01:15:00,350 --> 01:15:02,730
this thing goes
out, and it's going

979
01:15:02,730 --> 01:15:05,882
to have a frequency omega
bottom, which is simply

980
01:15:05,882 --> 01:15:06,840
related to that energy.

981
01:15:15,320 --> 01:15:17,880
This photon immediately is
shot back up to the top,

982
01:15:17,880 --> 01:15:20,760
where clever you,
you happen to have

983
01:15:20,760 --> 01:15:23,507
in your hands a rerockulater.

984
01:15:26,310 --> 01:15:29,430
The rerockulater, as the
name obviously implies,

985
01:15:29,430 --> 01:15:31,080
converts the photon
back into a rock.

986
01:15:52,500 --> 01:15:54,930
Now, suppose it does so--
both the photonulater

987
01:15:54,930 --> 01:15:57,660
and the rerockulater are
fine MIT engineering.

988
01:15:57,660 --> 01:16:00,930
There are no losses
anywhere in this thing.

989
01:16:00,930 --> 01:16:02,160
So there's no friction.

990
01:16:02,160 --> 01:16:03,510
There's no extra heat generated.

991
01:16:03,510 --> 01:16:07,830
It does it conserving
energy, 100%.

992
01:16:07,830 --> 01:16:09,620
What is the energy at the top?

993
01:16:18,450 --> 01:16:21,717
Well, you might naively
say, ah, it's just

994
01:16:21,717 --> 01:16:22,800
going to go up to the top.

995
01:16:22,800 --> 01:16:24,300
It's going to have
that same energy.

996
01:16:24,300 --> 01:16:27,780
It might just have it in the
form of a photon and omega b.

997
01:16:30,740 --> 01:16:34,850
There will be some
frequency at the top.

998
01:16:34,850 --> 01:16:37,103
And your initial
guess might be it's

999
01:16:37,103 --> 01:16:39,270
going to be the same as the
frequency at the bottom.

1000
01:16:39,270 --> 01:16:41,478
But if you do that, you're
going to suddenly discover

1001
01:16:41,478 --> 01:16:43,850
that your rock has more energy
than it started out with,

1002
01:16:43,850 --> 01:16:45,890
and you can redirect it
back down, send it back up.

1003
01:16:45,890 --> 01:16:47,480
Next thing you know, you've
got yourself a perpetual motion

1004
01:16:47,480 --> 01:16:48,770
machine.

1005
01:16:48,770 --> 01:16:50,750
So all you need to do
is get your photonulater

1006
01:16:50,750 --> 01:16:52,375
and your rerockulater
going, and you've

1007
01:16:52,375 --> 01:16:54,630
got yourself a perpetual
motion machine here.

1008
01:16:54,630 --> 01:16:56,463
I will grant that's
probably not the weakest

1009
01:16:56,463 --> 01:16:58,550
part of this argument.

1010
01:16:58,550 --> 01:16:59,500
Suppose you had this.

1011
01:16:59,500 --> 01:17:00,542
I mean, you look at this.

1012
01:17:00,542 --> 01:17:05,210
If technology allowed you
to make these goofy devices,

1013
01:17:05,210 --> 01:17:08,000
you would instantly look
at this and say, look,

1014
01:17:08,000 --> 01:17:09,542
if I am not to have--

1015
01:17:09,542 --> 01:17:11,000
let's just say I
live in a universe

1016
01:17:11,000 --> 01:17:12,417
where I'm fine
with photonulaters.

1017
01:17:12,417 --> 01:17:14,780
I'm fine with
rerockulaters, but damn it,

1018
01:17:14,780 --> 01:17:16,100
energy has to be conserved.

1019
01:17:16,100 --> 01:17:19,100
I am not fine with
perpetual motion machines.

1020
01:17:19,100 --> 01:17:26,200
If that's the case, we always
fight perpetual motion.

1021
01:17:31,990 --> 01:17:35,485
We must have that
the energy at the top

1022
01:17:35,485 --> 01:17:37,360
is equal to the energy
this guy started with.

1023
01:17:37,360 --> 01:17:38,980
When it sort of gets back into--

1024
01:17:38,980 --> 01:17:41,140
imagine that your
rerockulater is shaped

1025
01:17:41,140 --> 01:17:42,460
like a baseball catcher's mitt.

1026
01:17:42,460 --> 01:17:45,100
You want that thing to just
land gently in your mitt,

1027
01:17:45,100 --> 01:17:48,170
and just be a perfectly
gentle, little landing there.

1028
01:17:48,170 --> 01:18:07,290
And when you put all this
together, going through this,

1029
01:18:07,290 --> 01:18:10,223
taking advantage of the fact
that if you work in units where

1030
01:18:10,223 --> 01:18:11,640
you've put your
c's back in, there

1031
01:18:11,640 --> 01:18:21,860
will be a factor of g h over c
squared appearing in here, what

1032
01:18:21,860 --> 01:18:25,340
you find is that the
frequency at the top

1033
01:18:25,340 --> 01:18:30,000
is less than the
frequency at the bottom.

1034
01:18:30,000 --> 01:18:33,340
In other words, the light has
gotten a little bit redder.

1035
01:18:33,340 --> 01:18:36,630
now I fully confess, I did
this via the silliest argument

1036
01:18:36,630 --> 01:18:37,650
possible.

1037
01:18:37,650 --> 01:18:39,720
But I want to
emphasize that this

1038
01:18:39,720 --> 01:18:45,060
is one of the most precisely
verified predictions of gravity

1039
01:18:45,060 --> 01:18:47,250
and relativity theory.

1040
01:18:47,250 --> 01:18:51,030
This was first done, actually,
up the street at Harvard,

1041
01:18:51,030 --> 01:18:54,740
by what's called the
Pound-Rebka experiment.

1042
01:18:54,740 --> 01:18:57,240
And the basic principles of
what is going on with this right

1043
01:18:57,240 --> 01:18:57,958
now--

1044
01:18:57,958 --> 01:19:00,000
I just took this out to
make sure my alarm is not

1045
01:19:00,000 --> 01:19:01,667
about to go off, but
I want to emphasize

1046
01:19:01,667 --> 01:19:03,208
it's actually built
into the workings

1047
01:19:03,208 --> 01:19:04,630
of the global
positioning system.

1048
01:19:04,630 --> 01:19:07,170
Because this fact that
light signals may travel out

1049
01:19:07,170 --> 01:19:09,960
of a gravitational
potential, they

1050
01:19:09,960 --> 01:19:12,390
get redshifted, needs to be
taken into account in order

1051
01:19:12,390 --> 01:19:15,210
to do the precise
metrology that GPS allows.

1052
01:19:15,210 --> 01:19:21,630
Now, this is part one,
this idea that light

1053
01:19:21,630 --> 01:19:24,720
gets redder as it climbs out
of a gravitational field.

1054
01:19:24,720 --> 01:19:27,230
Part two, which I
will do on Thursday,

1055
01:19:27,230 --> 01:19:31,500
is to show that if there
is a global inertial frame,

1056
01:19:31,500 --> 01:19:34,830
there is no way for light to
get redder as it climbs out

1057
01:19:34,830 --> 01:19:38,080
of a gravitational potential.

1058
01:19:38,080 --> 01:19:42,280
You cannot have both gravity
and a global inertial reference

1059
01:19:42,280 --> 01:19:42,898
frame.

1060
01:19:42,898 --> 01:19:44,690
That's where I will
pick it up on Thursday.

1061
01:19:44,690 --> 01:19:45,700
So we'll do that.

1062
01:19:45,700 --> 01:19:49,000
And we will then begin
talking about how

1063
01:19:49,000 --> 01:19:52,510
we can try to put the principles
of relativity and gravity

1064
01:19:52,510 --> 01:19:53,695
together.

1065
01:19:53,695 --> 01:19:56,320
And in some sense, this is when
our study of general relativity

1066
01:19:56,320 --> 01:19:58,060
will begin in earnest.

1067
01:19:58,060 --> 01:20:01,110
All right, so let us stop there.