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SCOTT HUGHES: Last
Thursday, we began

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the work of moving
from special relativity

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to general relativity,
and we spent

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a lot of time unpacking two
formulations of the principle

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of equivalence.

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So one, which goes
under the name

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"weak equivalence principle"--

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a simpler way of
saying that is that,

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at least over a
sufficiently small region,

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if there is nothing
but gravity acting,

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I cannot distinguish between
freefall under the influence

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of gravity or a
uniform acceleration.

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The two things are
equivalent to one another.

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Basically, this is a
reflection of the fact

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that the gravitational
charge and the inertial mass

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are the same thing.

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That is the main thing
that really underlies

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the weak equivalence principle.

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When I gave my tenure talk
a number of years ago here,

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I pointed out that there was
this wonderful program called

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the Apollo program that was
put together to test this.

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And the way it was done was
that they put astronauts

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on the moon, and
you actually show

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that, if you drop a hammer
and a feather on the moon,

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they fall at the same rate.

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Of course, the Apollo
program probably

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did a few other things as well.

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But I'm a general
relativity theorist,

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so for me, that was the
outcome of the Apollo program,

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was test of the
equivalence principle.

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We also have a
different variation

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of this we called the Einstein
equivalence principle, which

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leads us to a
calculation that we went

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through last time, which
states that we can find

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a representation
over a sufficiently

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small region of spacetime
such that the laws of physics

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are reduced to those
of special relativity.

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And we did a
calculation to examine

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this, where we showed, given
an arbitrary spacetime metric,

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I can find a coordinate system
such that this can be written

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in the form metric
of flat spacetime

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plus terms that are of order
so we have coordinate distance

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squared.

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Let's put it this way-- so this,
in the vicinity of a point pl--

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I'll make that clear
in just a moment.

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It ends up looking like of
order of coordinate displacement

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squared with corrections
that scale as 1

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over second derivative
of the metric.

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That sets the scale for what
these end up looking like,

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or maybe it's
actually times that.

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Sorry.

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It's times that.

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Why did I divide?

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I don't know.

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Oh, I know why, because
I wanted to point out

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that that is what the scale
of 1 over this thing--

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for God's sake, Scott!

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Stop putting your
square roots in there!

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So it looks like that.

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And so this is
what I was saying.

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You have a curvature
scale that is

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on the order of square root of
1 over the second derivative

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of the metric.

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Apologies for botching
that as I was writing it

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up there quickly.

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OK, so we did a calculation
that shows that.

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And indeed, what we
did is we went through

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and we showed that a general
coordinate transformation has

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more than enough
degrees of freedom

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to make the metric flat
to get the flat spacetime

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metric at a particular point.

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And in fact, there are six
degrees of freedom left over,

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corresponding to six
rotations and six boosts that

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are allowable at that
particular point or event.

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Bear in mind we're
working in spacetime.

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We have exactly enough degrees
of freedom to cancel out

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the first order term, but we
cannot cancel out the second

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order term.

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And in fact, we find
there are 20 degrees

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of freedom left over.

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And in a future lecture, we
will derive a geometric object

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that characterizes
the curvature that

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has indeed 20 degrees
of freedom in it

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or 20 independent components
that come out of it.

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So this is the foundation of
where we're moving forward.

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And so what this
basically tells us

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is that, in a
spacetime like this,

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we have what we call curvature.

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Trajectories that start
out parallel to one another

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are not going to remain
parallel as they move forward.

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And where we
concluded last time,

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we were dealing with
the problem that,

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if I want to take
derivatives-- and we're going

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to start with vector field.

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If I want to
differentiate a vector

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field on a curved
manifold, it doesn't work.

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If I do the naive thing of just
taking a partial derivative,

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it does not work.

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So I'll remind you
where we left off.

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So we found partial
derivatives of vectors.

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Let me put that in there.

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Partial derivatives
of vectors, and it

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won't take much
work to show it's

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true for one forms
or any tensor,

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so let's say, partial
derivatives of tensors

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do not yield tensors.

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We're familiar with
this to some extent

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because we already
encountered this

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when we began thinking
about the behavior of, even

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in flat spacetime, flat
spacetime and curvilinear

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coordinates.

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So that's the fact that the
basis objects themselves

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have some functional dependence
associated with them.

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We're interpreting it a
little bit differently now.

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And so what we're
doing is we're going

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to say that what's
going on here is that,

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on my curved
manifold-- and I want

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you to visualize
something that's

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like a bumpy surface or, if
you like, maybe a sphere.

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Think about if you are a
two-dimensional being confined

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to the surface of a sphere.

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There's only two directions
at any given point.

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You can go up, or you can
go on the left-right axis

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or the north-south axis, right?

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And you would define unit
vectors pointing along there.

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But as you move around that
sphere, those of us who

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have a three-dimensional life,
and can step back and see this,

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we see that these basis objects
point in different directions

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at different locations
on the sphere.

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The two-dimensional beings
aren't aware of that.

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They just know that they're
on a surface that's curved.

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And so they would
say that the tangent

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space is different
for all these objects.

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They've imagined that every
one of these basis objects

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lives in a plane that is tangent
to the sphere at any given

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point, and that plane is
different at every point

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along the sphere.

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So we interpret this by saying
that all of our basis objects

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live in this tangent space,
and the tangent space

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is different at every
point on the surface.

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So let me just write
out one equation here.

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So when we looked at
the transformation

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of a partial
derivative of a vector,

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if we looked at just
the components--

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so this calculation
is in the notes,

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so I would just write down
the result. What we found

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was that, if I'm taking,
say, the beta derivative

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of component A alpha,
what I found when

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I want to go into a
coordinate transformation

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is that there's
an extra term that

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ruins the tensoriality of this.

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So this goes over to
something that looks like--

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So the first term is what we'd
expect if this were a tensor

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relationship.

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That's exactly the matrix
of the Jacobian between two

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different coordinate
representations

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that we expect to
describe how components

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change if they indeed obey
a tensorial relationship.

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This extra thing here I
wrote on the second line--

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that is spoiling it for us.

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So I began to give you the
physical notion of what we

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were going to do to fix this.

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So let me just
reset that up again.

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So let's imagine I have
a particular curve that

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goes along my manifold.

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I have a point P
here on the curve,

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and I have a point Q over here.

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And let's say that P
is at event x alpha.

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Q is at x alpha plus dx alpha.

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Here's my vector
A at the event Q.

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And here's my vector
A at the event P.

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So what we discussed last time
is that, to get this thing,

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I'm just doing the normal
partial derivative.

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I'm basically imagining that
these are close to one another,

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that I can just subtract A at
Q from A at P, divide by dx

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and take the limit.

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That's the definition
of a derivative.

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And what this is telling us
is, mathematically, yeah,

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it's a derivative, but
it's not a derivative

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that yields a tensor quantity.

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And so we are beginning
to discuss the fact that,

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to compare things that have
different tangent spaces, that

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live in different points
in my curve manifold,

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I need a notion of transport
to take one from the other

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in order to compare them.

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So there are two
notions of transport

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that we're going
to talk about here.

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The first one is called
parallel transport.

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So transport notion one--

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we call this parallel transport.

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I'm going to actually focus a
little bit on the math first

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and then come back to what is
parallel about this afterwards.

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So what I essentially need to
do is say, what I want to do

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is find some kind of
a way of imagining

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that I take the vector at
P and transport it over

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to the point Q, and I will
compare the transported object

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rather than the object
originally at point P.

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Abstractly, what I'm
essentially going

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to do is I'm going to do
what we always do in this.

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I'm going to imagine
that there is

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some kind of an
operation that is

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linear in the separation
between the two of them

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that allows me to
define this transport.

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So let's do the following.

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So I'm going to assume that
we can define an object, which

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I will call Pi, capital
Pi, alpha, beta, mu, which

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is going to do the following.

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00:12:06,410 --> 00:12:12,207
So what I'm going to do is
say that A alpha transported--

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and to make it even clearer,
how it's being transported.

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Let's say it's being
transported from P to Q.

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00:12:21,480 --> 00:12:25,470
This is given by
alpha at P, and what

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I'm going to do is say that,
whatever this object is,

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it is linear in both the
coordinate separation

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of those two events
and the vector field.

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So far, I've said nothing
about physics, by the way.

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00:12:47,460 --> 00:12:49,543
I'm just laying out some
mathematical definitions.

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I'm going to bring it all
together in a few moments.

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00:12:51,840 --> 00:12:53,460
I'm then going to
say, OK, I know

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that I had trouble with my
standard partial derivative.

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Let's define a derivative
operator in the following way.

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Define a derivative by
comparing the transported vector

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00:13:29,980 --> 00:13:37,860
to the field at Q.
So what I'm going

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to do for the moment is
just denote this notion

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of a new kind of derivative
with a capital D. Right now,

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00:13:43,980 --> 00:13:46,260
it's just another
symbol that we would

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pronounce with a "duh" sound so
that it sounds like derivative.

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So don't read too much
into that for a moment.

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So I'm going to define
this as A at Q minus A

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00:14:01,350 --> 00:14:13,340
transported from P to Q and
then divide by the separation.

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00:14:13,340 --> 00:14:16,660
Take the limit-- usual thing.

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And when you do
this, you're going

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to get something that looks
like the partial derivative

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00:14:26,440 --> 00:14:29,010
plus an additional term on here.

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00:14:34,730 --> 00:14:37,370
It looks like this.

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So I've said nothing
about what properties

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00:14:39,440 --> 00:14:41,240
I'm going to demand
of this thing.

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00:14:41,240 --> 00:14:43,130
And in fact, there
are many ways that one

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00:14:43,130 --> 00:14:47,300
could define a transport
of an object like this.

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00:14:47,300 --> 00:14:52,280
In general, when you do this,
this thing I'm calling Pi here

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00:14:52,280 --> 00:14:54,111
is known as the connection.

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00:14:58,440 --> 00:15:10,080
It is the object that
connects point P to point Q.

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00:15:10,080 --> 00:15:13,327
So let's make a couple
demands on its properties.

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00:15:13,327 --> 00:15:15,410
So now I'll start to put
a bit of physics in this.

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00:15:28,990 --> 00:15:40,010
So demand one is I'm going to
require that, when I change

248
00:15:40,010 --> 00:15:51,500
my coordinate representation,
when I evaluate this

249
00:15:51,500 --> 00:15:56,160
in my new coordinate
system, that I get something

250
00:15:56,160 --> 00:15:57,190
that looks like this.

251
00:16:34,470 --> 00:16:39,530
If I do this, I'm going
to find that, when

252
00:16:39,530 --> 00:16:43,910
I change coordinates and apply
it to the entire derivative

253
00:16:43,910 --> 00:16:47,750
I've defined over here, that
little extra bit of schmutz

254
00:16:47,750 --> 00:16:51,170
that's on the second line
there is exactly what you need

255
00:16:51,170 --> 00:16:55,940
to cancel out this annoying
bugger so that you have

256
00:16:55,940 --> 00:16:58,280
a nice tensor relationship left.

257
00:16:58,280 --> 00:17:01,070
So I'm going to demand that a
key part of whatever this guy

258
00:17:01,070 --> 00:17:05,510
turns out to be is something
that cancels out the irritating

259
00:17:05,510 --> 00:17:07,866
garbage that came along
with the partial derivative.

260
00:17:45,490 --> 00:17:48,490
Combine that with what I'm
about to say in just a moment--

261
00:17:48,490 --> 00:17:51,430
that pins things
down significantly.

262
00:17:51,430 --> 00:17:54,040
I'm going to make
one more demand,

263
00:17:54,040 --> 00:17:56,205
and this demand is going
to then be connected

264
00:17:56,205 --> 00:17:58,330
to the physical picture
that I'm going to introduce

265
00:17:58,330 --> 00:17:59,920
in about five minutes.

266
00:17:59,920 --> 00:18:04,270
My final demand is I'm
going to require that

267
00:18:04,270 --> 00:18:07,040
whatever this derivative is--

268
00:18:07,040 --> 00:18:11,960
when I apply it to
the metric, I get 0.

269
00:18:11,960 --> 00:18:23,630
If I do that, then it turns out
that the connection is exactly

270
00:18:23,630 --> 00:18:28,370
the Christoffel symbol
that we worked out earlier.

271
00:18:28,370 --> 00:18:50,920
If I put in this demand, the
connection is the Christoffel,

272
00:18:50,920 --> 00:18:53,260
and this derivative
is nothing more

273
00:18:53,260 --> 00:18:55,000
than the covariant derivative.

274
00:18:55,000 --> 00:18:56,050
Der-iv-a-tive.

275
00:19:00,920 --> 00:19:02,600
This is just the
covariant derivative

276
00:19:02,600 --> 00:19:03,517
we worked out earlier.

277
00:19:11,560 --> 00:19:13,020
So much for mathematics.

278
00:19:13,020 --> 00:19:15,192
The key thing which
I want to emphasize

279
00:19:15,192 --> 00:19:16,650
at this point in
the conversation--

280
00:19:16,650 --> 00:19:18,870
I hope we can see the logic
behind the first demand.

281
00:19:18,870 --> 00:19:21,328
That's just something which
I'm going to introduce in order

282
00:19:21,328 --> 00:19:23,580
to clean this guy up.

283
00:19:23,580 --> 00:19:25,020
This is my choice.

284
00:19:25,020 --> 00:19:26,520
Not just my choice--
it's the choice

285
00:19:26,520 --> 00:19:29,160
of a lot of people who have
helped to develop this subject.

286
00:19:29,160 --> 00:19:33,240
But it's a good one,
because, when I do this,

287
00:19:33,240 --> 00:19:38,730
there is a particularly good
physical interpretation to what

288
00:19:38,730 --> 00:19:40,850
this notion of transport means.

289
00:19:40,850 --> 00:19:42,570
And bear with me
for a second while I

290
00:19:42,570 --> 00:19:43,830
gather my notes together.

291
00:19:49,020 --> 00:19:56,280
So let's do the top one first.

292
00:20:38,990 --> 00:20:42,140
OK, so let's make
that curve again.

293
00:20:45,210 --> 00:20:49,260
And actually, let's just go
ahead and redraw my vector

294
00:20:49,260 --> 00:20:49,980
field.

295
00:20:49,980 --> 00:20:51,772
I do want to have a
different copy of this.

296
00:20:59,210 --> 00:21:02,360
Let me introduce
one other object.

297
00:21:02,360 --> 00:21:04,022
So I'm going to give
this curve a name.

298
00:21:04,022 --> 00:21:05,480
I'm going to call
this curve gamma.

299
00:21:11,062 --> 00:21:12,520
And this is a notion
that I'm going

300
00:21:12,520 --> 00:21:14,820
to make much more precise
in a future lecture,

301
00:21:14,820 --> 00:21:18,710
but imagine that there is
some kind of a tape measure

302
00:21:18,710 --> 00:21:25,340
that reads out along the
curve gamma that is uniformly

303
00:21:25,340 --> 00:21:27,800
ticked in a way that
we will make precise

304
00:21:27,800 --> 00:21:28,740
in a future lecture.

305
00:21:36,940 --> 00:21:38,950
I will call the
parameter that is uniform

306
00:21:38,950 --> 00:21:40,900
that denotes these
uniform tick marks--

307
00:21:45,990 --> 00:21:47,577
I will call that lambda.

308
00:21:47,577 --> 00:21:49,410
If you want to read a
little bit about this,

309
00:21:49,410 --> 00:21:52,140
this is what's known
as an affine parameter.

310
00:21:52,140 --> 00:21:54,870
We will make this a little
bit more precise very soon.

311
00:21:59,590 --> 00:22:02,800
With this in mind, you should
be able to convince yourself

312
00:22:02,800 --> 00:22:06,760
that this field U defines the
tangent vector to this curve.

313
00:22:21,170 --> 00:22:26,140
So let's say that my original
point here, that this

314
00:22:26,140 --> 00:22:29,830
is at lambda equals 2--

315
00:22:29,830 --> 00:22:33,790
and let's say this is
at lambda equals 7.

316
00:22:33,790 --> 00:22:36,610
And what I want to do
is transport the vector

317
00:22:36,610 --> 00:22:39,510
from 2 to 7.

318
00:22:39,510 --> 00:22:41,570
Well, a way that I can
do this is by saying,

319
00:22:41,570 --> 00:22:44,500
OK, I now know
that the derivative

320
00:22:44,500 --> 00:22:47,890
that goes over here-- this is
just the covariant derivative.

321
00:22:47,890 --> 00:22:50,590
So I forgot to write this
down but from now on,

322
00:22:50,590 --> 00:22:52,000
this derivative I wrote down--

323
00:22:52,000 --> 00:22:54,430
I'm going to go back to
the gradient symbol we used

324
00:22:54,430 --> 00:22:55,923
for the covariant derivative.

325
00:22:59,240 --> 00:23:04,590
What I can do is take
the covariant derivative

326
00:23:04,590 --> 00:23:10,560
of my vector field, contract
it with this tangent vector.

327
00:23:10,560 --> 00:23:11,405
And what this does--

328
00:23:17,050 --> 00:23:20,050
I'm going to define this as
capital D alpha D lambda.

329
00:23:20,050 --> 00:23:24,370
This is a covariant derivative
with respect to the parameter

330
00:23:24,370 --> 00:23:27,860
lambda as I move along this
constrained trajectory.

331
00:23:27,860 --> 00:23:42,240
What this does is this
tells me how A changes as it

332
00:23:42,240 --> 00:23:47,250
is transported along the curve.

333
00:24:19,590 --> 00:24:20,820
I'm now going to argue--

334
00:24:20,820 --> 00:24:23,220
what parallel
transport comes down to

335
00:24:23,220 --> 00:24:42,840
is when you require that, as
you slide this thing along here,

336
00:24:42,840 --> 00:24:45,810
the covariant total derivative
of this thing as you

337
00:24:45,810 --> 00:24:49,280
move along the curve,
that it's equal to 0.

338
00:24:49,280 --> 00:24:51,340
OK, now let me motivate
where that's coming from.

339
00:24:51,340 --> 00:24:52,110
Why is that?

340
00:25:00,970 --> 00:25:04,930
So let's put all of our
definitions back in.

341
00:25:04,930 --> 00:25:07,865
Let's apply the definition
of covariant derivative

342
00:25:07,865 --> 00:25:09,490
that we discussed in
a previous lecture

343
00:25:09,490 --> 00:25:10,990
and we've all come
to know and love.

344
00:25:28,610 --> 00:25:30,610
Now, this is the bit where
we begin to introduce

345
00:25:30,610 --> 00:25:32,210
a little bit of physics.

346
00:25:32,210 --> 00:25:35,320
Let's imagine that
points P and Q

347
00:25:35,320 --> 00:25:37,420
are sufficiently close
to each other that they

348
00:25:37,420 --> 00:25:40,210
fit within a single,
freely falling frame.

349
00:25:40,210 --> 00:25:42,340
They can go into the
same local Lorentz frame.

350
00:26:05,820 --> 00:26:08,730
So remember, when I go into
my local Lorentz frame,

351
00:26:08,730 --> 00:26:11,790
the metric becomes the
metric of flat spacetime.

352
00:26:11,790 --> 00:26:15,230
G goes over to eta.

353
00:26:15,230 --> 00:26:17,730
There is a second order
correction to that.

354
00:26:17,730 --> 00:26:19,660
So the second derivatives--

355
00:26:19,660 --> 00:26:22,340
I cannot find a [INAUDIBLE]
that gets rid of them.

356
00:26:22,340 --> 00:26:25,070
So there'll be a little bit
of that there, which tells me

357
00:26:25,070 --> 00:26:28,200
how large this
Lorentz frame can be.

358
00:26:28,200 --> 00:26:31,390
But I can get rid of all
the first derivatives.

359
00:26:31,390 --> 00:26:34,160
And if you get rid of all
the first derivatives,

360
00:26:34,160 --> 00:26:37,740
you zero out the Christoffel
symbols in that frame.

361
00:26:37,740 --> 00:26:40,550
So in that frame, there
are no Christoffel symbols.

362
00:26:50,110 --> 00:27:16,060
So what this means is
that, in this frame,

363
00:27:16,060 --> 00:27:21,645
this just becomes the idea
that a simple total derivative

364
00:27:21,645 --> 00:27:24,270
of this object-- you don't need
to include all the garbage that

365
00:27:24,270 --> 00:27:27,150
comes along with the
covariant derivative--

366
00:27:27,150 --> 00:27:28,230
this is equal to 0.

367
00:27:28,230 --> 00:27:37,810
And that's equivalent to saying
that, as I take this vector

368
00:27:37,810 --> 00:27:42,250
and transport it along, I
hold all of its components

369
00:27:42,250 --> 00:27:46,150
constant as I slide it
from one step to the other.

370
00:28:09,410 --> 00:28:14,757
So I start out with
my A at point P here.

371
00:28:14,757 --> 00:28:16,340
And then I slide it
over a little bit,

372
00:28:16,340 --> 00:28:19,790
holding all the components
constant just like this--

373
00:28:19,790 --> 00:28:23,750
slide over again,
slide it over again.

374
00:28:23,750 --> 00:28:25,788
Da, da, da, da da.

375
00:28:25,788 --> 00:28:27,216
Da, da, da, da, da.

376
00:28:31,498 --> 00:28:33,040
Till finally, I get
it over to there.

377
00:28:33,040 --> 00:28:34,240
What this is doing is that--

378
00:28:37,360 --> 00:28:42,730
any two vectors in the middle
of this transport process--

379
00:28:42,730 --> 00:28:45,100
I am holding them
as parallel as it

380
00:28:45,100 --> 00:28:48,850
is possible to hold them,
given that, to be blunt,

381
00:28:48,850 --> 00:28:51,130
you can't even really
define a notion

382
00:28:51,130 --> 00:28:54,250
of two objects being
parallel on a curved surface

383
00:28:54,250 --> 00:28:57,140
if there's a macroscopic
separation between them.

384
00:28:57,140 --> 00:28:59,640
But if you think about
just a little region that's

385
00:28:59,640 --> 00:29:02,740
sufficiently small,
that it's flat

386
00:29:02,740 --> 00:29:05,620
up to quadratic
corrections, then a notion

387
00:29:05,620 --> 00:29:08,710
of these things being parallel
to each other makes sense.

388
00:29:08,710 --> 00:29:13,060
And this idea, that I'm going
to demand that, upon transport,

389
00:29:13,060 --> 00:29:15,370
the derivative of
the metric equals 0,

390
00:29:15,370 --> 00:29:18,820
thereby yielding my connection
being the Christoffel

391
00:29:18,820 --> 00:29:21,700
and this derivative being
the covariant derivative--

392
00:29:21,700 --> 00:29:24,040
it tells us that this
notion of transport

393
00:29:24,040 --> 00:29:26,080
is one in which
objects are just kept

394
00:29:26,080 --> 00:29:29,230
as parallel as possible
as they slide along here.

395
00:29:29,230 --> 00:29:33,020
And that is why this is
called "parallel transport."

396
00:29:33,020 --> 00:29:35,530
It's as parallel as
it's possible to be,

397
00:29:35,530 --> 00:29:36,806
given the curvature.

398
00:29:59,140 --> 00:30:00,690
So as I switch
gears, I just want

399
00:30:00,690 --> 00:30:05,730
to emphasize I went through
the mathematics of that

400
00:30:05,730 --> 00:30:09,000
with a fair amount
of care because it

401
00:30:09,000 --> 00:30:14,190
is important to keep this stuff
as rigorous as possible here.

402
00:30:14,190 --> 00:30:15,930
This notion of
parallel transport

403
00:30:15,930 --> 00:30:19,680
gets used a lot when we start
talking about things moving

404
00:30:19,680 --> 00:30:24,090
around in a curved spacetime.

405
00:30:24,090 --> 00:30:26,820
In particular, if you
think about an object

406
00:30:26,820 --> 00:30:29,280
that is freely falling,
and it is experiencing

407
00:30:29,280 --> 00:30:32,100
no forces other
than gravity, which

408
00:30:32,100 --> 00:30:35,160
we are going to no longer regard
as a force before too long--

409
00:30:35,160 --> 00:30:38,170
if you just go back to Newtonian
intuition, what does it do?

410
00:30:38,170 --> 00:30:41,190
Well, you give it an initial
velocity or initial momentum,

411
00:30:41,190 --> 00:30:42,060
and it maintains it.

412
00:30:42,060 --> 00:30:44,880
It just continues going
in a straight line.

413
00:30:44,880 --> 00:30:47,280
In spacetime, going
in a straight line

414
00:30:47,280 --> 00:30:51,120
basically means, at
every step, I move

415
00:30:51,120 --> 00:30:55,620
and I take the tangent to my
world line, my four-velocity,

416
00:30:55,620 --> 00:30:58,060
and I move it
parallel to itself.

417
00:30:58,060 --> 00:30:59,730
So this notion of
parallel transport

418
00:30:59,730 --> 00:31:01,350
is going to be the
key thing that we

419
00:31:01,350 --> 00:31:03,990
use to actually define
the kinetics of bodies

420
00:31:03,990 --> 00:31:06,470
in curved spacetime.

421
00:31:06,470 --> 00:31:09,380
There's a tremendous
amount of work

422
00:31:09,380 --> 00:31:13,040
being done by all sorts
of things these days

423
00:31:13,040 --> 00:31:16,190
that's based on studies of
orbits in curved spacetime,

424
00:31:16,190 --> 00:31:19,500
and they all come back to this
notion of parallel transport.

425
00:31:19,500 --> 00:31:20,000
All right.

426
00:31:20,000 --> 00:31:20,620
Bear with me a second.

427
00:31:20,620 --> 00:31:22,160
I just want to take
a sip of water.

428
00:31:22,160 --> 00:31:28,270
And then I'm going to talk about
the other notion of transport,

429
00:31:28,270 --> 00:31:30,173
which we are going to discuss--

430
00:31:30,173 --> 00:31:32,590
I always say, briefly, and
then I spend three pages on it,

431
00:31:32,590 --> 00:31:34,970
so we're going to discuss.

432
00:31:34,970 --> 00:31:36,090
All right.

433
00:31:36,090 --> 00:31:39,970
So parallel transport
is extremely important,

434
00:31:39,970 --> 00:31:41,620
and there's a huge
amount of physics

435
00:31:41,620 --> 00:31:45,330
that is tied up in
this, but one thing

436
00:31:45,330 --> 00:31:49,330
which I really want to emphasize
is that it is not unique.

437
00:31:49,330 --> 00:31:51,025
And there is one
other one which we

438
00:31:51,025 --> 00:31:54,490
are going to really use
to define one particularly

439
00:31:54,490 --> 00:31:57,970
important notion, instead of
quantities, for our class.

440
00:32:10,210 --> 00:32:14,020
So suppose I've
got my curve gamma,

441
00:32:14,020 --> 00:32:16,300
and I'm going to, again,
take advantage of the fact

442
00:32:16,300 --> 00:32:23,680
that I can define a set
of tick marks along it

443
00:32:23,680 --> 00:32:28,190
and make that vector U be
the tangent to this curve.

444
00:32:28,190 --> 00:32:32,770
And I'm going to, again,
have my favorite points,

445
00:32:32,770 --> 00:32:43,520
x alpha at point P plus
dx alpha at point Q.

446
00:32:43,520 --> 00:32:46,760
There's another notion of
transport that is-- basically

447
00:32:46,760 --> 00:32:50,030
what you do is
you cheat, and you

448
00:32:50,030 --> 00:32:56,840
imagine that moving from point
x alpha to x alpha plus dx alpha

449
00:32:56,840 --> 00:32:59,910
is a kind of coordinate
transformation.

450
00:32:59,910 --> 00:33:01,790
So let's do the following.

451
00:33:01,790 --> 00:33:09,772
Let's say that x
alpha plus dx alpha--

452
00:33:09,772 --> 00:33:11,480
we're going to take
advantage of the fact

453
00:33:11,480 --> 00:33:16,040
that, since we have this tangent
notion built into the symbols

454
00:33:16,040 --> 00:33:21,380
we've defined, we'll
just say that it's

455
00:33:21,380 --> 00:33:25,580
going to be the tangent
times the interval of lambda.

456
00:33:25,580 --> 00:33:29,180
And what I'm going
to do is define

457
00:33:29,180 --> 00:33:33,750
this as a new coordinate
system, x prime.

458
00:33:33,750 --> 00:33:38,750
So that's the alpha component
of coordinate system x prime.

459
00:33:38,750 --> 00:33:41,540
It's a little bit weird
because your x prime has

460
00:33:41,540 --> 00:33:43,250
a differential built into it.

461
00:33:43,250 --> 00:33:44,000
Just bear with me.

462
00:33:46,700 --> 00:33:52,390
So what we're going to
do is regard the shift,

463
00:33:52,390 --> 00:33:58,150
or the transport, if
you prefer, from P to Q

464
00:33:58,150 --> 00:33:59,727
as a coordinate transformation.

465
00:34:17,780 --> 00:34:20,000
It's the best eraser, so
I'll just keep using it.

466
00:34:26,889 --> 00:34:36,040
So what I mean by that is
I'm going to regard x alpha,

467
00:34:36,040 --> 00:34:38,830
and I'm going to use a
slightly different symbol.

468
00:34:38,830 --> 00:34:40,364
I will define what the L is.

469
00:34:40,364 --> 00:34:42,739
So this is transported, but
I'm going to put an L in here

470
00:34:42,739 --> 00:34:44,739
for reasons that I will
define in just a moment.

471
00:34:48,210 --> 00:35:02,830
This, from P to Q, is what
I get if I regard the change

472
00:35:02,830 --> 00:35:05,620
from point P to point Q
as a simple coordinate

473
00:35:05,620 --> 00:35:09,030
transformation and do my
usual rule for changing

474
00:35:09,030 --> 00:35:11,670
coordinate representation.

475
00:35:11,670 --> 00:35:15,950
So expand what the definition
of x prime is there,

476
00:35:15,950 --> 00:35:19,770
and what you'll see is that
you get a term that's just

477
00:35:19,770 --> 00:35:22,080
basically dx alpha dx beta.

478
00:35:25,570 --> 00:35:27,000
Then you're going
to get something

479
00:35:27,000 --> 00:35:29,790
that looks like the
partial derivative

480
00:35:29,790 --> 00:35:30,840
of that tangent vector.

481
00:35:36,990 --> 00:35:41,030
And remember, this
is being acted on.

482
00:35:41,030 --> 00:35:47,980
I should've said this is
this thing evaluated at P.

483
00:35:47,980 --> 00:35:48,480
Great.

484
00:35:48,480 --> 00:35:49,530
So we fill this out.

485
00:36:07,960 --> 00:36:08,460
OK.

486
00:36:14,920 --> 00:36:19,760
So that's what I get when I
use this notion to transport

487
00:36:19,760 --> 00:36:22,100
the field from P to Q.

488
00:36:22,100 --> 00:36:24,110
Let's think about
it in another way.

489
00:36:24,110 --> 00:36:28,080
Now, these fields are
all just functions.

490
00:36:28,080 --> 00:36:33,260
So I can also express the field
at Q in terms of the field

491
00:36:33,260 --> 00:36:37,078
at P using a Taylor expansion.

492
00:36:39,710 --> 00:36:42,950
I'm assuming that these are
close enough that everything

493
00:36:42,950 --> 00:37:37,840
is accurate to first
order in small quantities,

494
00:37:37,840 --> 00:37:40,000
so nothing controversial
about this.

495
00:37:40,000 --> 00:37:47,945
I'm assuming dx is small
enough that I can do this.

496
00:38:01,990 --> 00:38:05,260
But now I'm going to
get rid of my dx beta

497
00:38:05,260 --> 00:38:27,290
using the tangent field U.

498
00:38:27,290 --> 00:38:32,810
Now, before I move on, I
just want to emphasize--

499
00:38:32,810 --> 00:38:35,750
these two boards here,
over the way the left--

500
00:38:35,750 --> 00:38:38,610
we're talking about two
rather different quantities.

501
00:38:38,610 --> 00:38:40,250
The one I just
moved to the top--

502
00:38:40,250 --> 00:38:49,670
that actually is the field-- if
you were some kind of a gadget

503
00:38:49,670 --> 00:38:51,620
that managed your field A--

504
00:38:51,620 --> 00:38:53,540
that would tell you
what the value is

505
00:38:53,540 --> 00:38:58,520
that you measure at
point Q. This would tell

506
00:38:58,520 --> 00:39:00,710
you-- what do we
get if you picked P

507
00:39:00,710 --> 00:39:05,180
up and, via this transport
mechanism, moved it over to Q?

508
00:39:05,180 --> 00:39:09,560
They are two potentially
different things.

509
00:39:09,560 --> 00:39:13,088
So this motivates another
kind of derivative.

510
00:39:35,040 --> 00:39:41,400
So suppose I look at A--

511
00:39:41,400 --> 00:39:44,160
value it at a Q--

512
00:39:44,160 --> 00:39:49,530
minus A transported--
whoops, that's supposed to be

513
00:39:49,530 --> 00:39:50,730
transported from P to Q--

514
00:39:59,490 --> 00:40:02,210
defined by D lambda.

515
00:40:02,210 --> 00:40:04,440
I will expand this
out in just a moment.

516
00:40:04,440 --> 00:40:06,970
Now I will, at last,
give this a name.

517
00:40:06,970 --> 00:40:14,750
This is written with
a script L. This

518
00:40:14,750 --> 00:40:24,580
is known as the Lie
derivative of the vector A

519
00:40:24,580 --> 00:40:30,120
along U. Anyone heard of
the Lie derivative before?

520
00:40:30,120 --> 00:40:31,110
Yeah.

521
00:40:31,110 --> 00:40:34,375
So at least in the context where
we're going to be using it,

522
00:40:34,375 --> 00:40:36,750
this is a good way to understand
what's going on with it.

523
00:40:36,750 --> 00:40:39,420
We'll see how it is
used, at least in 8.962

524
00:40:39,420 --> 00:40:41,730
in just a few moments.

525
00:40:41,730 --> 00:40:44,670
Filling in the details-- so
plug in these definitions,

526
00:40:44,670 --> 00:40:47,040
subtract, take limits,
blah, blah, blah.

527
00:40:47,040 --> 00:40:54,540
What you find is
that this turns out

528
00:40:54,540 --> 00:41:02,530
to be U contracted on the
partial derivative of A

529
00:41:02,530 --> 00:41:13,320
minus A contracted on the
partial derivative of U.

530
00:41:13,320 --> 00:41:15,530
Exercise for the reader--

531
00:41:15,530 --> 00:41:18,540
it is actually
really easy to show

532
00:41:18,540 --> 00:41:22,400
that you can promote
these partial derivatives

533
00:41:22,400 --> 00:41:23,977
to covariant derivatives.

534
00:41:40,050 --> 00:41:42,755
And what this means is that,
when you evaluate the Lie

535
00:41:42,755 --> 00:41:44,630
derivative-- so
notice, nowhere in here

536
00:41:44,630 --> 00:41:46,838
did I introduce anything
with a covariant derivative.

537
00:41:46,838 --> 00:41:48,890
There was no connection,
nothing going on there.

538
00:41:48,890 --> 00:41:50,432
If you just go ahead
and work it out,

539
00:41:50,432 --> 00:41:53,600
basically, when you
expand this guy out,

540
00:41:53,600 --> 00:41:56,312
you'll find you have connection
coefficients or Christoffel

541
00:41:56,312 --> 00:41:57,770
symbols that are
equal and opposite

542
00:41:57,770 --> 00:41:59,270
and so they cancel each other.

543
00:41:59,270 --> 00:42:01,930
So you can just go from
partials to covariants.

544
00:42:01,930 --> 00:42:04,070
Give me just a second, Trey.

545
00:42:04,070 --> 00:42:05,990
And this is telling us
that the Lie derivative

546
00:42:05,990 --> 00:42:08,800
is perfectly tensorial.

547
00:42:08,800 --> 00:42:11,330
So the Lie derivative
of the vector field

548
00:42:11,330 --> 00:42:13,610
is also a tensor quantity.

549
00:42:13,610 --> 00:42:16,681
You were asking
a question, Trey.

550
00:42:16,681 --> 00:42:20,078
AUDIENCE: In the second term,
did you miss the D lambda?

551
00:42:24,297 --> 00:42:25,130
SCOTT HUGHES: I did.

552
00:42:25,130 --> 00:42:25,630
Yes, I did.

553
00:42:25,630 --> 00:42:26,300
Thank you.

554
00:42:26,300 --> 00:42:28,970
There should be a D
lambda right here.

555
00:42:28,970 --> 00:42:29,850
Thank you.

556
00:42:29,850 --> 00:42:30,540
Yes.

557
00:42:30,540 --> 00:42:31,040
Yeah.

558
00:42:31,040 --> 00:42:34,010
If you don't have
that, then you get what

559
00:42:34,010 --> 00:42:35,670
is technically called "crap."

560
00:42:35,670 --> 00:42:38,360
So thank you for
pointing that out.

561
00:42:38,360 --> 00:42:48,230
For reasons that I hope you
have probably seen before,

562
00:42:48,230 --> 00:42:49,850
you always compute
the Lie derivative

563
00:42:49,850 --> 00:42:53,630
of some kind of an object
along a vector field.

564
00:42:57,698 --> 00:42:59,990
So when you're computing the
Lie derivative of a vector

565
00:42:59,990 --> 00:43:06,690
field along a vector,
sometimes this

566
00:43:06,690 --> 00:43:09,762
is written using a commutator.

567
00:43:09,762 --> 00:43:11,970
I just throw that out there
because you may encounter

568
00:43:11,970 --> 00:43:13,220
this in some of your readings.

569
00:43:22,550 --> 00:43:25,820
It looks like this.

570
00:43:25,820 --> 00:43:28,245
So let me just do
a few more things

571
00:43:28,245 --> 00:43:30,620
that are essentially fleshing
out the definition of this.

572
00:43:30,620 --> 00:43:33,120
So I'm not going to go through
and apply this definition

573
00:43:33,120 --> 00:43:35,840
very carefully to
higher order objects.

574
00:43:35,840 --> 00:43:37,550
What I will just say
is that, if I repeat

575
00:43:37,550 --> 00:43:43,687
this exercise and, instead
of having a vector field

576
00:43:43,687 --> 00:43:45,770
that I'm transporting from
point to point, suppose

577
00:43:45,770 --> 00:43:46,937
I do it for a scalar field--

578
00:44:02,360 --> 00:44:04,750
well, what you actually get--
pardon me for a second--

579
00:44:04,750 --> 00:44:11,468
is this on the partial, but the
partial derivative of a scalar

580
00:44:11,468 --> 00:44:13,260
is the covariant
derivative because there's

581
00:44:13,260 --> 00:44:14,700
no Christoffel that couples in.

582
00:44:21,840 --> 00:44:29,320
If you do this for a one-form,
where it's a 1 indexed

583
00:44:29,320 --> 00:44:48,910
object in the
downstairs position,

584
00:44:48,910 --> 00:44:51,030
you get something
that looks like this.

585
00:44:51,030 --> 00:44:54,980
And again, when you expand out
your covariant derivatives,

586
00:44:54,980 --> 00:44:58,530
you find that your Christoffel
symbols cancel each other out.

587
00:44:58,530 --> 00:45:00,440
And so, if you
like, you can just

588
00:45:00,440 --> 00:45:05,540
go ahead and replace
these with partials.

589
00:45:05,540 --> 00:45:08,381
And likewise, let me just write
one more out for completeness.

590
00:45:11,030 --> 00:45:12,080
Apply this to a tensor.

591
00:45:27,040 --> 00:45:30,280
So it's a very similar
kind of structure

592
00:45:30,280 --> 00:45:36,190
to what you saw when we did the
covariant derivative in which

593
00:45:36,190 --> 00:45:40,900
every index essentially gets
corrected by a factor that

594
00:45:40,900 --> 00:45:43,270
looks like the covariant
derivative of the field

595
00:45:43,270 --> 00:45:44,888
that you are
differentiating along.

596
00:45:44,888 --> 00:45:46,430
The signs are a
little bit different.

597
00:45:46,430 --> 00:45:51,290
So it's a similar tune, but
it's in a different key.

598
00:45:57,500 --> 00:46:04,490
OK, so that's great.

599
00:46:04,490 --> 00:46:08,117
And if you get your
jollies just understanding

600
00:46:08,117 --> 00:46:09,950
different mathematical
transport operations,

601
00:46:09,950 --> 00:46:13,368
maybe this is
already fun enough.

602
00:46:13,368 --> 00:46:15,410
But we're in a physics
class, and so the question

603
00:46:15,410 --> 00:46:18,020
that should be to
your mind is, is there

604
00:46:18,020 --> 00:46:20,910
a point to all this analysis?

605
00:46:20,910 --> 00:46:27,887
So in fact, the most important
application of the Lie

606
00:46:27,887 --> 00:46:30,340
derivative for our purposes--

607
00:46:33,040 --> 00:46:36,130
in probably the
last lecture or two,

608
00:46:36,130 --> 00:46:42,610
I will describe some stuff
related to modern research

609
00:46:42,610 --> 00:46:43,810
that uses it quite heavily.

610
00:46:43,810 --> 00:46:47,980
But to begin with in our class,
the most important application

611
00:46:47,980 --> 00:47:01,610
will be when we
consider cases where,

612
00:47:01,610 --> 00:47:05,880
when I compute the Lie
derivative of some tensor

613
00:47:05,880 --> 00:47:09,200
along a vector U and I get 0.

614
00:47:24,932 --> 00:47:26,890
I'm just going to leave
it schematic like that.

615
00:47:26,890 --> 00:47:30,400
So L U of the tensor
is equal to 0.

616
00:47:30,400 --> 00:47:37,104
If this is the case, we say that
the tensor is Lie transported.

617
00:47:52,600 --> 00:47:54,350
This is, incidentally,
just a brief aside.

618
00:47:54,350 --> 00:47:56,900
It shows up a lot
in fluid dynamics.

619
00:47:56,900 --> 00:48:01,670
In that case, U often
defines the flow lines

620
00:48:01,670 --> 00:48:03,110
associated with
the velocity field

621
00:48:03,110 --> 00:48:04,735
of some kind of a
fluid that is flowing

622
00:48:04,735 --> 00:48:06,808
through your physical situation.

623
00:48:06,808 --> 00:48:08,600
And you would be
interested in the behavior

624
00:48:08,600 --> 00:48:13,530
of all sorts of quantities that
are embedded in that fluid.

625
00:48:13,530 --> 00:48:15,350
And as we're going
to see, when you

626
00:48:15,350 --> 00:48:19,010
find that those quantities are
Lie transported in this way,

627
00:48:19,010 --> 00:48:21,693
there is a powerful
physical outcome

628
00:48:21,693 --> 00:48:23,360
associated with that,
which we are going

629
00:48:23,360 --> 00:48:25,410
to derive in just a moment.

630
00:48:25,410 --> 00:48:34,819
So suppose I, in fact, have a
tensor that is Lie transported.

631
00:49:05,290 --> 00:49:08,170
So suppose I have some tensor
that is Lie transported.

632
00:49:08,170 --> 00:49:16,830
If that's the
case, what I can do

633
00:49:16,830 --> 00:49:19,740
is define a particular
coordinate system

634
00:49:19,740 --> 00:49:22,765
centered on the curve for
which U is the tangent.

635
00:49:42,840 --> 00:49:47,250
So what I'm going to do is I'm
going to define this curve such

636
00:49:47,250 --> 00:49:55,350
that x0 is equal to lambda,
that parameter that defines

637
00:49:55,350 --> 00:49:58,530
my length along the curve in
a way that, I will admit I've

638
00:49:58,530 --> 00:50:00,540
not made very precise
yet but will soon.

639
00:50:07,130 --> 00:50:13,370
And then I'm going to require
that my other three coordinates

640
00:50:13,370 --> 00:50:15,080
are all constant on that curve.

641
00:50:20,820 --> 00:50:47,780
So if I do that, then my tangent
vector is simply delta x0.

642
00:50:47,780 --> 00:50:50,980
In other words, it's only got
one non-trivial component,

643
00:50:50,980 --> 00:50:52,780
and its value of
that component is 1.

644
00:50:58,210 --> 00:50:59,650
And this is the constant.

645
00:50:59,650 --> 00:51:03,580
So the derivatives
of the tangent field

646
00:51:03,580 --> 00:51:06,520
are all equal to 0.

647
00:51:06,520 --> 00:51:26,360
And when you trace this through
all of our various definitions,

648
00:51:26,360 --> 00:51:28,790
what you find is that
it boils down to just

649
00:51:28,790 --> 00:51:31,970
looking at how the
tensor field varies

650
00:51:31,970 --> 00:51:36,350
with respect to that parameter
along the curve itself.

651
00:51:36,350 --> 00:51:39,375
If it's Lie transported,
then this is equal to 0.

652
00:51:42,600 --> 00:51:47,870
And so this means that,
whatever x0 represents,

653
00:51:47,870 --> 00:51:53,430
it's going to be a constant
along that curve with respect

654
00:51:53,430 --> 00:51:54,776
to this tensor field.

655
00:51:59,060 --> 00:52:01,003
Oh, excuse me.

656
00:52:01,003 --> 00:52:01,670
Screwed that up.

657
00:52:05,470 --> 00:52:13,980
The tensor does not vary with
this parameter along the curve.

658
00:52:27,070 --> 00:52:30,100
This was a lot, so let's
just step back and think

659
00:52:30,100 --> 00:52:32,380
about what this is saying.

660
00:52:32,380 --> 00:52:37,255
One of the most important things
that we do in physics when

661
00:52:37,255 --> 00:52:39,040
we're trying to
analyze systems is

662
00:52:39,040 --> 00:52:40,960
we try to identify
quantities that

663
00:52:40,960 --> 00:52:44,080
are constants of the motion.

664
00:52:44,080 --> 00:52:47,340
This is really tricky
in a curved spacetime

665
00:52:47,340 --> 00:52:52,440
because much of
our intuition gets

666
00:52:52,440 --> 00:52:56,010
garbled by all of the facts
that different points have

667
00:52:56,010 --> 00:52:58,530
different tangent spaces.

668
00:52:58,530 --> 00:53:00,750
You worry about whether
something being true,

669
00:53:00,750 --> 00:53:03,240
and is it just a function
of the coordinate system

670
00:53:03,240 --> 00:53:04,560
that I wrote this out in?

671
00:53:04,560 --> 00:53:07,980
What the hell is going on here?

672
00:53:07,980 --> 00:53:12,390
The Lie derivative is giving us
a covariant, frame-independent

673
00:53:12,390 --> 00:53:18,250
way of identifying things that
are constants in our spacetime.

674
00:53:18,250 --> 00:53:22,950
So we're going to wrap
up this discussion.

675
00:53:22,950 --> 00:53:25,890
Let's suppose that the tensor
that I'm looking at here

676
00:53:25,890 --> 00:53:27,180
is called the metric.

677
00:53:51,470 --> 00:54:06,750
Suppose there exists a vector
C such that the metric is Lie

678
00:54:06,750 --> 00:54:08,960
transported along this thing.

679
00:54:27,310 --> 00:54:28,420
What does this tell us?

680
00:54:28,420 --> 00:54:43,570
So first, it means there
exists some coordinate such

681
00:54:43,570 --> 00:54:52,270
that the metric does not vary.

682
00:54:52,270 --> 00:54:55,090
The metric is constant with
respect to that coordinate.

683
00:54:55,090 --> 00:54:58,120
Essentially, if you go through
what I sketched a moment ago,

684
00:54:58,120 --> 00:54:59,920
this is telling us
that the existence

685
00:54:59,920 --> 00:55:02,470
of this kind of a vector, which
I'm going to give a name to

686
00:55:02,470 --> 00:55:05,240
in just a moment--

687
00:55:05,240 --> 00:55:07,000
the existence of
this thing demands

688
00:55:07,000 --> 00:55:10,274
that my metric is constant with
respect to some coordinate.

689
00:55:14,360 --> 00:55:16,650
I am not going to prove
the following statement.

690
00:55:16,650 --> 00:55:19,920
I will just state it,
because, in some ways,

691
00:55:19,920 --> 00:55:23,025
the converse of that statement
is even more powerful.

692
00:55:38,690 --> 00:55:54,640
If there is a coordinate,
such that dgd,

693
00:55:54,640 --> 00:56:01,318
whatever that coordinate
is, is equal to 0,

694
00:56:01,318 --> 00:56:02,985
then a vector field
of this type exists.

695
00:56:14,860 --> 00:56:18,423
So the second thing I want to
do is expand the Lie derivative.

696
00:56:25,530 --> 00:56:32,810
So if I require that my metric
be transported along the vector

697
00:56:32,810 --> 00:56:44,047
C, well, insert my definition
of the Lie derivative.

698
00:57:02,570 --> 00:57:05,120
Now, what is the main
defining characteristic

699
00:57:05,120 --> 00:57:06,980
of the covariant derivative?

700
00:57:06,980 --> 00:57:10,142
How did I get my connection
in the first place?

701
00:57:10,142 --> 00:57:11,850
In other words, what
is this going to be?

702
00:57:14,750 --> 00:57:17,300
OK, students who took
undergraduate classes with me,

703
00:57:17,300 --> 00:57:19,300
I'll remind you of one
of the key bits of wisdom

704
00:57:19,300 --> 00:57:20,175
I always tell people.

705
00:57:20,175 --> 00:57:24,040
If the professor asks you a
question, 90% of the time,

706
00:57:24,040 --> 00:57:27,220
if you just shout out, "0,"
you are likely to be right.

707
00:57:27,220 --> 00:57:28,973
[LAUGHING]

708
00:57:28,973 --> 00:57:30,640
Usually, there's some
kind of a symmetry

709
00:57:30,640 --> 00:57:33,190
that we want you to understand,
which allows you to go,

710
00:57:33,190 --> 00:57:34,448
oh, it's equal to 0.

711
00:57:34,448 --> 00:57:36,490
By the way, whenever I
point that out to a class,

712
00:57:36,490 --> 00:57:39,040
I then work really hard
to make a non-zero answer

713
00:57:39,040 --> 00:57:40,310
for the next time I ask it.

714
00:57:40,310 --> 00:57:45,640
So the covariant derivative
of g is 0, so this term dies.

715
00:57:45,640 --> 00:57:48,850
Because the covariant
derivative of g is 0,

716
00:57:48,850 --> 00:57:52,940
I can always commute the metric
with covariant derivatives.

717
00:57:52,940 --> 00:57:55,540
So I can take this, move
it inside the derivative.

718
00:57:55,540 --> 00:57:58,129
I can take this, move it
inside the derivative.

719
00:58:19,180 --> 00:58:23,590
So what this means is this
Lie derivative equation,

720
00:58:23,590 --> 00:58:35,090
after all the smoke clears,
can be written like this.

721
00:58:35,090 --> 00:58:38,210
Or, if I recall,
there's this notation

722
00:58:38,210 --> 00:58:42,633
for symmetry of indices,
which I introduced

723
00:58:42,633 --> 00:58:43,550
in a previous lecture.

724
00:58:43,550 --> 00:58:47,660
The symmetric covariant
derivative of this C

725
00:58:47,660 --> 00:58:48,920
is equal to 0.

726
00:58:48,920 --> 00:58:58,350
This equation is known
as Killing's equation,

727
00:58:58,350 --> 00:59:07,620
and C is a Killing vector.

728
00:59:07,620 --> 00:59:10,710
Now, this was a fair
amount of formalism.

729
00:59:10,710 --> 00:59:14,520
I was really laying out a lot of
the details to get this right.

730
00:59:14,520 --> 00:59:20,807
So to give you some context
as to why this matters,

731
00:59:20,807 --> 00:59:22,890
there's a bit more that
needs to come out of this,

732
00:59:22,890 --> 00:59:26,460
but we're going to
get to it very soon.

733
00:59:26,460 --> 00:59:28,560
Suppose I have a
body that is freely

734
00:59:28,560 --> 00:59:30,900
falling through some spacetime.

735
00:59:30,900 --> 00:59:31,650
And you know what?

736
00:59:31,650 --> 00:59:32,858
I'm going to leave this here.

737
00:59:32,858 --> 00:59:35,427
So this is a slightly
advanced tangent,

738
00:59:35,427 --> 00:59:36,510
so I'll start a new board.

739
01:00:00,370 --> 01:00:03,130
So if I have some body
that is freely falling,

740
01:00:03,130 --> 01:00:05,840
what we are going to
show in, probably,

741
01:00:05,840 --> 01:00:10,720
Thursday's lecture is that the
equation of motion that governs

742
01:00:10,720 --> 01:00:12,830
it is-- you can argue
this on physical grounds,

743
01:00:12,830 --> 01:00:14,510
and that's all I
will do for now--

744
01:00:14,510 --> 01:00:18,710
it's a trajectory that parallel
transports its own tangent

745
01:00:18,710 --> 01:00:19,580
factor.

746
01:00:19,580 --> 01:00:22,010
For intuition, go into
the freely falling frame

747
01:00:22,010 --> 01:00:24,500
where it's just the trajectory
from special relativity.

748
01:00:24,500 --> 01:00:26,900
It's a straight
line in that frame,

749
01:00:26,900 --> 01:00:29,580
and parallel transporting
its own tangent vector

750
01:00:29,580 --> 01:00:34,430
basically means it just moves
on whatever course it is going.

751
01:00:34,430 --> 01:00:49,010
So this is a
trajectory for which

752
01:00:49,010 --> 01:00:52,610
I demand that the four-velocity
governing it parallel

753
01:00:52,610 --> 01:00:55,790
transports along itself.

754
01:00:55,790 --> 01:00:59,150
Now, suppose you are
moving in a spacetime that

755
01:00:59,150 --> 01:01:00,530
has a Killing vector.

756
01:01:15,470 --> 01:01:17,240
So this will be
Thursday's lecture.

757
01:01:22,080 --> 01:01:24,600
Suppose the spacetime
has a Killing vector.

758
01:01:24,600 --> 01:01:27,990
Well, so there will be some
goofy C that you know exists,

759
01:01:27,990 --> 01:01:31,860
and you know C
obeys this equation.

760
01:01:31,860 --> 01:01:33,900
By combining these
things, you can

761
01:01:33,900 --> 01:01:41,220
show that there is
some quantity, C,

762
01:01:41,220 --> 01:01:43,830
which is given by
taking the inner product

763
01:01:43,830 --> 01:01:47,400
of the four-velocity of
this freely falling thing

764
01:01:47,400 --> 01:01:49,260
and the Killing vector.

765
01:01:49,260 --> 01:01:51,960
And you can prove that this
is a constant of the motion.

766
01:02:06,018 --> 01:02:08,560
So let's think about where this
goes with some of the physics

767
01:02:08,560 --> 01:02:11,500
that you presumably all
know and love already.

768
01:02:11,500 --> 01:02:15,100
Suppose you look at a spacetime.

769
01:02:15,100 --> 01:02:18,465
So you climb a
really high mountain.

770
01:02:18,465 --> 01:02:20,590
You discover that there's
a spacetime metric carved

771
01:02:20,590 --> 01:02:22,660
into the stone
into the top of it.

772
01:02:22,660 --> 01:02:24,760
You think, OK, this
probably matters.

773
01:02:24,760 --> 01:02:28,870
You look at it and you notice it
depends on, say, time, radius,

774
01:02:28,870 --> 01:02:29,710
and two angles.

775
01:02:34,150 --> 01:02:39,250
Suppose you have a metric
that is time-independent.

776
01:02:46,160 --> 01:02:49,040
Hey, if it's
time-independent, then I

777
01:02:49,040 --> 01:02:53,450
know that the derivative of
that thing with respect to time

778
01:02:53,450 --> 01:02:54,500
is 0.

779
01:02:54,500 --> 01:02:57,320
There must exist
a Killing factor

780
01:02:57,320 --> 01:03:00,020
that is related to the
fact that there is no time

781
01:03:00,020 --> 01:03:01,150
dependence in this metric.

782
01:03:22,800 --> 01:03:25,675
So you go and you calculate it.

783
01:03:25,675 --> 01:03:27,800
So this thing, that C is
a constant of the motion--

784
01:03:35,838 --> 01:03:36,880
I believe that's P set 4.

785
01:03:41,560 --> 01:03:42,340
It's not hard.

786
01:03:42,340 --> 01:03:44,170
You combine that equation
that we're going to derive,

787
01:03:44,170 --> 01:03:46,000
called the geodesic equation,
with Killing's equation.

788
01:03:46,000 --> 01:03:46,600
Math happens.

789
01:03:46,600 --> 01:03:47,408
You got it.

790
01:03:47,408 --> 01:03:49,575
So suppose you've got a
metric that's time-dependent

791
01:03:49,575 --> 01:03:51,210
and you know you've
got this thing.

792
01:03:51,210 --> 01:03:51,918
So you know what?

793
01:03:51,918 --> 01:03:53,860
Let's work it out
and look at it.

794
01:03:53,860 --> 01:03:56,080
It becomes clear,
after studying this

795
01:03:56,080 --> 01:04:10,340
for a little bit, that the C for
this Killing vector is energy.

796
01:04:10,340 --> 01:04:12,620
In the same way that, if
you have a time-independent

797
01:04:12,620 --> 01:04:15,710
Lagrangian, your system
has a conserved energy,

798
01:04:15,710 --> 01:04:18,110
if you have a
time-independent metric,

799
01:04:18,110 --> 01:04:19,777
there is a Killing
vector, which--

800
01:04:19,777 --> 01:04:21,110
the language we like to use is--

801
01:04:21,110 --> 01:04:23,690
we say the motion
of that spacetime

802
01:04:23,690 --> 01:04:26,660
emits a conserved energy.

803
01:04:26,660 --> 01:04:30,890
Suppose you find that the metric
is independent of some angle.

804
01:04:30,890 --> 01:04:33,710
We'll call it phi.

805
01:04:33,710 --> 01:04:35,310
Three guesses what's
going to happen.

806
01:04:35,310 --> 01:04:37,580
In this case-- just
one guess, actually.

807
01:04:37,580 --> 01:04:38,105
Conserved--

808
01:04:38,105 --> 01:04:39,230
AUDIENCE: Angular momentum.

809
01:04:39,230 --> 01:04:41,490
SCOTT HUGHES: Angular momentum
pops out in that case.

810
01:04:41,490 --> 01:04:45,290
So this ends up being the
way in which we, essentially,

811
01:04:45,290 --> 01:04:47,750
make very rigorous
and geometric the idea

812
01:04:47,750 --> 01:04:53,572
that conservation laws are put
into general relativity, OK?

813
01:04:53,572 --> 01:04:55,530
So I realize there's a
lot of abstraction here.

814
01:04:55,530 --> 01:04:57,238
So I want to go on a
bit of an aside just

815
01:04:57,238 --> 01:04:58,970
to tie down where we
are going with this

816
01:04:58,970 --> 01:05:01,760
and why this actually matters.

817
01:05:01,760 --> 01:05:04,575
OK.

818
01:05:04,575 --> 01:05:05,110
Let's see.

819
01:05:05,110 --> 01:05:07,230
So we got about 10 minutes left.

820
01:05:07,230 --> 01:05:12,000
So what we're going to do
at the very end of today--

821
01:05:12,000 --> 01:05:14,050
and we'll pick this up
beginning of next time.

822
01:05:14,050 --> 01:05:17,870
So for the people who walked
in a few minutes late,

823
01:05:17,870 --> 01:05:21,165
the stuff that I'm actually
about to start talking about we

824
01:05:21,165 --> 01:05:23,540
need to get through before
you can do one of the problems

825
01:05:23,540 --> 01:05:24,370
on the P set.

826
01:05:24,370 --> 01:05:26,120
So I'm probably going
to take that problem

827
01:05:26,120 --> 01:05:28,016
and move it on to
P set 4, but I'm

828
01:05:28,016 --> 01:05:30,560
going to start talking
about it right now.

829
01:05:30,560 --> 01:05:39,980
So we've really focused a
lot, so far, on tensors.

830
01:05:42,650 --> 01:05:48,530
We're going to now start talking
about a related quantity called

831
01:05:48,530 --> 01:05:49,850
tensor densities.

832
01:05:56,080 --> 01:05:58,523
There's really only two that
matter for our purposes,

833
01:05:58,523 --> 01:06:00,190
but I want to go
through them carefully.

834
01:06:00,190 --> 01:06:01,780
So I will set up with
one, and then we'll

835
01:06:01,780 --> 01:06:03,460
conclude the other
one at the beginning

836
01:06:03,460 --> 01:06:04,900
of Thursday's lecture.

837
01:06:04,900 --> 01:06:07,950
So let me define this first.

838
01:06:07,950 --> 01:06:09,700
I'm going to give a
definition that I like

839
01:06:09,700 --> 01:06:11,260
but that's actually
kind of stupid.

840
01:06:11,260 --> 01:06:24,250
So these are quantities that
transform almost like tensors--

841
01:06:28,890 --> 01:06:31,060
a little bit lame, but,
as you'll see in a moment,

842
01:06:31,060 --> 01:06:32,100
it's kind of accurate.

843
01:06:32,100 --> 01:06:35,520
What you'll find is that
the transformation law

844
01:06:35,520 --> 01:06:53,003
is off by a factor
that is the determinant

845
01:06:53,003 --> 01:06:54,670
of the coordinate
transformation matrix.

846
01:07:08,233 --> 01:07:09,150
Take it to some power.

847
01:07:13,493 --> 01:07:15,660
So there's is an infinite
number of tensor densities

848
01:07:15,660 --> 01:07:17,130
that one could define.

849
01:07:17,130 --> 01:07:19,800
Two are important
for this class.

850
01:07:42,642 --> 01:07:44,350
So the two that are
most important for us

851
01:07:44,350 --> 01:07:54,780
are the Levi-Civita symbol and
the determinant of the metric.

852
01:08:00,870 --> 01:08:05,580
So we use Levi-Civita already
to talk about volumes.

853
01:08:05,580 --> 01:08:08,012
And it was a tensor
when we were working

854
01:08:08,012 --> 01:08:10,470
in rectilinear coordinates,
where the underlying coordinate

855
01:08:10,470 --> 01:08:14,100
system was essentially
Cartesian plus time.

856
01:08:14,100 --> 01:08:15,484
It's not in general, OK?

857
01:08:15,484 --> 01:08:16,859
And we'll go
through why that is.

858
01:08:16,859 --> 01:08:19,710
That'll probably be the last
thing we can fit in today.

859
01:08:19,710 --> 01:08:21,450
So let me remind you--

860
01:08:24,677 --> 01:08:29,490
Levi-Civita-- I'm going to
write it with a tilde on it

861
01:08:29,490 --> 01:08:34,680
to emphasize that
it is not tensorial.

862
01:08:37,229 --> 01:08:40,550
So this is equal to
plus 1 if the indices

863
01:08:40,550 --> 01:08:45,529
are 0, 1, 2, 3 and
even permutations

864
01:08:45,529 --> 01:08:53,960
of that equals minus 1 for
odd permutations of that.

865
01:08:56,710 --> 01:08:59,090
And it's 0 for any
index repeated.

866
01:09:11,160 --> 01:09:14,380
Now, this symbol has a
really nice property when

867
01:09:14,380 --> 01:09:16,090
you apply it to any matrix.

868
01:09:16,090 --> 01:09:17,979
In fact, this is a theorem.

869
01:09:35,260 --> 01:09:36,979
So I'm working in
four-dimensional space.

870
01:09:36,979 --> 01:09:44,920
So let's say I've got a 4-by-4
matrix, which I will call m.

871
01:09:44,920 --> 01:09:48,580
Write a new [INAUDIBLE]
notation, m alpha mu.

872
01:09:48,580 --> 01:10:11,500
If I evaluate Levi-Civita,
contract it on these guys,

873
01:10:11,500 --> 01:10:18,050
I get Levi-Civita
back, multiply it

874
01:10:18,050 --> 01:10:21,360
by the determinant
of the matrix m.

875
01:10:31,620 --> 01:10:34,050
Now, suppose what I
choose for my matrix

876
01:10:34,050 --> 01:10:36,480
m is my coordinate
transformation matrix.

877
01:11:12,780 --> 01:11:14,930
So I'm just going to
write down this result,

878
01:11:14,930 --> 01:11:20,480
and I'll leave it since we're
running a little short on time.

879
01:11:20,480 --> 01:11:23,908
You can just double check
that I've moved things

880
01:11:23,908 --> 01:11:25,700
from one side of the
equation to the other,

881
01:11:25,700 --> 01:11:27,825
and you can just double-check
I did that correctly.

882
01:11:31,010 --> 01:11:34,430
What that tells me
is that Levi-Civita

883
01:11:34,430 --> 01:11:39,450
and a new set of
prime coordinates

884
01:11:39,450 --> 01:11:43,070
is equal to this guy
in the old, unprimed

885
01:11:43,070 --> 01:11:53,590
coordinates with all my usual
factors of transformation

886
01:11:53,590 --> 01:12:11,990
matrices and then
an extra bit that

887
01:12:11,990 --> 01:12:14,780
is the determinant of the
coordinate transformation

888
01:12:14,780 --> 01:12:16,340
matrix.

889
01:12:16,340 --> 01:12:18,050
If it were just
the top line, this

890
01:12:18,050 --> 01:12:20,100
is exactly what you would
need for Levi-Civita

891
01:12:20,100 --> 01:12:23,205
to be a tensor in the way
that we have defined tensors.

892
01:12:23,205 --> 01:12:23,705
It's not.

893
01:12:44,390 --> 01:12:48,370
So the extra factor pushes away
from a tensor relationship.

894
01:12:48,370 --> 01:12:51,100
And so what we would say
is, because this is off

895
01:12:51,100 --> 01:13:02,610
by a factor of what's
sometimes called the Jacobian,

896
01:13:02,610 --> 01:13:07,290
we call this a tensor
density of weight 1.

897
01:13:16,590 --> 01:13:19,320
So in order to do
this properly--

898
01:13:19,320 --> 01:13:21,740
I don't want to rush--

899
01:13:21,740 --> 01:13:24,390
at the beginning of
the next lecture,

900
01:13:24,390 --> 01:13:27,890
we're going to look at how
the determinant of the metric

901
01:13:27,890 --> 01:13:29,167
behaves.

902
01:13:29,167 --> 01:13:31,250
And what we'll see is that,
although the metric is

903
01:13:31,250 --> 01:13:34,100
a tensor, its
determinant is a tensor

904
01:13:34,100 --> 01:13:36,950
density of weight negative 2.

905
01:13:36,950 --> 01:13:39,920
And so what that tells
us is that I can actually

906
01:13:39,920 --> 01:13:43,370
put together a combination
of the Levi-Civita

907
01:13:43,370 --> 01:13:46,730
and the determinant of
the metric in such a way

908
01:13:46,730 --> 01:13:49,328
that their product is tensorial.

909
01:13:49,328 --> 01:13:51,245
And that turns out to
be real useful because I

910
01:13:51,245 --> 01:13:54,050
can use this to define,
in a curved spacetime,

911
01:13:54,050 --> 01:13:57,170
covariant volume elements, OK?

912
01:13:57,170 --> 01:13:58,923
With this as written,
my volume elements--

913
01:13:58,923 --> 01:14:00,590
if I just use this
like I did when we're

914
01:14:00,590 --> 01:14:03,020
taught about special relativity,
my volume elements won't

915
01:14:03,020 --> 01:14:06,800
be elements of a tensor,
and a lot of the framework

916
01:14:06,800 --> 01:14:09,170
that we've developed
goes to hell.

917
01:14:09,170 --> 01:14:11,510
So an extra factor of the
determining of the metric

918
01:14:11,510 --> 01:14:13,220
will allow us to correct this.

919
01:14:13,220 --> 01:14:16,770
And this seems kind of abstract.

920
01:14:16,770 --> 01:14:19,490
So let me just, as a really
brief aside, before we conclude

921
01:14:19,490 --> 01:14:22,200
today's class--

922
01:14:22,200 --> 01:14:26,980
suppose I'm just in
Euclidean three-space

923
01:14:26,980 --> 01:14:30,070
and I'm working in
spherical coordinates.

924
01:14:38,490 --> 01:14:41,390
So here's my line element.

925
01:14:41,390 --> 01:14:51,240
My metric is the
diagonal of 1r squared

926
01:14:51,240 --> 01:14:55,080
r squared sine squared theta.

927
01:14:55,080 --> 01:14:59,284
The determinant of the
metric, which I will write g--

928
01:15:02,450 --> 01:15:06,560
it's r to the fourth
sine squared theta.

929
01:15:06,560 --> 01:15:08,060
What we're going
to learn when we do

930
01:15:08,060 --> 01:15:14,380
this is that the metric is a
tensor density of weight 2.

931
01:15:14,380 --> 01:15:16,710
And so to correct it to
get something of weight 1,

932
01:15:16,710 --> 01:15:17,670
we take a square root.

933
01:15:23,153 --> 01:15:24,820
If you're working in
circle coordinates,

934
01:15:24,820 --> 01:15:26,720
does that look familiar?

935
01:15:26,720 --> 01:15:28,990
This is, in fact,
exactly what allows

936
01:15:28,990 --> 01:15:31,853
us to convert differentials
of our coordinates.

937
01:15:31,853 --> 01:15:33,770
Remember, we're working
in a coordinate basis.

938
01:15:33,770 --> 01:15:36,220
And so we think of
our little element

939
01:15:36,220 --> 01:15:38,200
of just the coordinates.

940
01:15:38,200 --> 01:15:41,290
It's just dr, d theta, d phi.

941
01:15:41,290 --> 01:15:42,880
This ends up being
the quantity that

942
01:15:42,880 --> 01:15:46,390
allows us to convert the little
triple of our coordinates

943
01:15:46,390 --> 01:15:48,730
into something that has
the proper dimensions

944
01:15:48,730 --> 01:15:51,880
and form to actually be
a real volume element.

945
01:15:51,880 --> 01:15:54,078
And so dr, de theta, d phi--

946
01:15:54,078 --> 01:15:55,120
that ain't enough volume.

947
01:15:55,120 --> 01:15:58,720
But r squared sine theta,
dr, d theta, d phi--

948
01:15:58,720 --> 01:16:00,273
that's a volume element, OK?

949
01:16:00,273 --> 01:16:02,440
So basically, that's all
that we're doing right now,

950
01:16:02,440 --> 01:16:04,510
is we're making that
precise and careful.

951
01:16:04,510 --> 01:16:06,135
And that's where I
will pick things up.

952
01:16:06,135 --> 01:16:08,190
We'll finish that
up on Thursday.