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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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SCOTT HUGHES: We're in for a
uncomfortable couple of weeks,

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but we will do our damnedest
to make sure that--

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I'm not going to say
life won't be disrupted.

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Life is going to be
bloody well disrupted.

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No question of that.

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But number one goal is
making sure everyone

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remains healthy, both
physically and mentally.

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When we're forced to
isolate a little bit,

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we lose the social contact
that makes life worth living,

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and so we're going to
be working really hard.

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Look to your social group
as well, and to your peers

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and and mentors and others to
try to find a way to remain--

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if you can't meet in
person quite as much,

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there's the phone, there's
Skype, there's FaceTime.

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Better than nothing,
and that's sort of what

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we're looking at these days.

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And we're, of course,
within the department,

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very committed to figuring
out a way to make sure

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that the education that
we're sort of here to do,

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we can deliver it to you
in some form or another.

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There may be a few bumps in the
road while we work this out,

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but we're getting there.

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Hopefully, this will-- if
there is a disruption coming,

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it will be short-lived.

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If not, let's just
let's focus on what

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the important things are.

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And today, the important things
are the geodesic equation.

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At least that's what we're
going to start things,

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and then we're going to take
it into the next major concept

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that describes manifolds
with curvature,

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I think we were going
to take advantage of.

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So just a quick recap, where I
ended things last time was we

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described geodesic trajectories.

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These are trajectories
which parallel transport

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the tangent--

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as they move along
their world line,

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they parallel transport
the tangent vector

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to that world line.

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And I forgot to put
the physics up here.

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The key reason why
this is interesting

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is that this corresponds
to a free fall trajectory.

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Free fall basically
means you are

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moving under the influence
of nothing but gravity.

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And if you want to
understand gravity

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in a relativistic theory, well,
that's all you care about.

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So these are very
important trajectories.

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And you know, it's
not an exaggeration

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to say that solving
this equation, OK-- so

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in this thing, I've kind of left
agnostic what the spacetime is

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that you use to compute
those covariant derivatives

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and to write your
Christoffel symbols gamma,

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but it could be
any spacetime that

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solves the relativistic
field equations,

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which we haven't derived
yet but we shall soon.

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This is what describes
sort of small bodies moving

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through that kind
of a spacetime.

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That is the starting point to
a tremendous amount of analyses

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in general relativity.

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I made a crack last
time that I think

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65% of my published
research work

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is essentially based on
solving this equation.

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I actually went and checked.

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It's probably more like 75%, OK.

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That wasn't actually
an exaggeration.

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It shows up a lot.

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All right, so this,
as we went through,

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and it's basically
just saying that I'm

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going to take the
covariant derivative

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of the tangent vector u
and contract it with u.

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OK, and there's a
couple of other ways

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of writing this which
I've written out here

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in just sort of notation.

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But if you expand it
out, it looks like this.

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So what you're saying
is that the vector

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u, the four-vector components
u, are parameterized

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by some quantity which if
it's a time-like trajectory,

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you can think of
it as essentially

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the proper time as you
move along that trajectory.

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This is just describing
how this thing

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behaves as a function
of that parameter, OK.

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So we're going to do
some more with that.

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To begin with, there's
a couple of cool results

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that we can derive.

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So there's a nice side note.

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We can rewrite this
in terms of momentum.

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So if you imagine, let's
focus on the version

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where I'm doing it
per units proper time.

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And I'm going to take advantage
of the fact that for a body

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moving on a
time-like trajectory,

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so a body with a rest mass
m, I just take this equation,

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and basically, you
multiply by m twice,

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and it very clearly turns into
something that is pretty much

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exactly the same, but I just
replace my u's with p's.

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Two comments I want
to make about this.

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So first, and you know what,
let me actually expand this out.

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I mean, it's quite obvious.

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Why don't we write it in
terms of the components

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in the Christoffel symbols.

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OK, so I'm going to
write it like this.

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Let's do the following, OK.

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Recall that if I have-- so
this trajectory, assuming, when

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you write like this,
your parameter lambda

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is called an affine parameter.

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And that is a parameter
such that the right hand

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side of [INAUDIBLE]
equation is 0 on a free fall

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trajectory, OK.

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If it's something that's
proportional to you,

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it is a valid
parameter, but it's

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one that's sort of been
defined in a bad way.

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One thing we showed
last time is that I

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can shift lambda
by any constant,

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OK, with the right units.

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That essentially
amounts to just changing

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the origin of my clock.

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And I can multiply it by any
scalar, which essentially

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amounts to changing the units
in which I am measuring time.

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And it's still a good
affine parameter.

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So here's an example
of an affine parameter

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that I could use.

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Suppose that I defined
this such that an interval

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of affine parameter delta
lambda is an interval

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of delta tau divided by m.

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If I do this, well,
then, p alpha is just d--

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so remember this is
now going to be--

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let's write it like this.

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So this is my original
definition of this thing.

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This then becomes
dx alpha d lambda.

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So this is a way of choosing
an affine parameter such

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that I'm essentially writing
my tangent along the world line

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as the momentum rather than
something like the four

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velocity, OK?

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But there's something
cool about this,

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so let's now go
and just write what

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my geodesy equation turns into.

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It's basically exactly the same.

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It's just that I'm going to
absorb the m on the first term.

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Basically, it's exactly the
same geodesy equation as I had

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before, but with
u's promoted to p's.

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What's kind of
cool about this is

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you can take a limit
in which m goes

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to zero as long as your
interval of proper time

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goes to zero at a rate such
that delta t over m is constant.

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So what this allows us to
do is just conceptually

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reformulate the
geodesy equation,

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so that it's perfectly
well behaved.

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Not just for time
like trajectories,

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but for null or light
like trajectories.

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OK, so that's very
important for us.

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A lot of the most important
tests of general relativity

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actually come down to looking
at the behavior of light

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as it moves in some kind
of a curved space time.

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And the geodesy
equation, if you sort of

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interpret the way we thought
about it before, you're

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kind of like, well, let's
go back, and suppose I'm

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running it sort of in this one.

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And I'm thinking of my
lambda as proper time.

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An interval with
proper time is not

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defined along a light
like trajectory, OK?

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So that just kind of makes
it clear that that's fine.

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What we're going
to do when we're

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talking about a
light like trajectory

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is we're just going to find
the parameter along the world

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line, such that the tangent
vector is the momentum

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along that world line.

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Mass doesn't even make sense.

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So the p that goes into
this-- so when I do this,

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this is going to be a p,
such that p alpha p alpha

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equals minus m squared.

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We still have that
rule that it's always

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going to be minus m squared,
which is zero in this case.

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So this gives us
a tool that we can

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use to study the
motion of light as it

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reacts to gravity, for example.

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OK, we'll switch
gears, so I want

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to do one other trick based on
this momentum form of things.

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So I can rewrite the
geodesy equation as follows,

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and it's going to
start out where it just

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looks like I'm essentially doing
what we call index gymnastics.

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I'm just sort of moving
a few indices around.

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So let's write this as p
alpha, and what I'm going to do

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is contract it on this.

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I'm putting the
index and my momentum

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in the downstairs position now.

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First of all, you should
stop and ask yourself,

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am I allowed to do that?

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If I do that, do I not
generate some additional term

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that should then be moved
to the right hand side?

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Well, think about
what I'm doing.

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If I am lowering an index, that
essentially means that I am--

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00:10:53,432 --> 00:10:54,390
let's do the following.

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Let's change this to
a gamma for a second.

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What I have done here
is I have essentially

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00:11:02,590 --> 00:11:09,440
taken the equation
I wrote over there,

197
00:11:09,440 --> 00:11:15,710
and I had hit it with
g beta gamma, OK?

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00:11:15,710 --> 00:11:17,900
I can always multiply
by those things.

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00:11:17,900 --> 00:11:21,380
The covariant derivative
of the metric is zero.

200
00:11:21,380 --> 00:11:24,200
Because the covariant derivative
of the metric is zero,

201
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it commutes with
that derivative,

202
00:11:25,670 --> 00:11:28,266
so I can just walk it inside
the derivative operator.

203
00:11:43,960 --> 00:11:45,793
So I wanted to go through
that just a little

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bit carefully, because
that's actually

205
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a trick that once
you've seen it once,

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00:11:49,090 --> 00:11:50,260
I want you to know it well.

207
00:11:50,260 --> 00:11:52,180
Because I'm just going to
do it many, many times as we

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00:11:52,180 --> 00:11:52,810
move forward.

209
00:11:52,810 --> 00:11:53,620
There's going to be
a bunch of times,

210
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where I'm taking the partial--

211
00:11:54,640 --> 00:11:56,400
it's going to be the covariant
derivative is something.

212
00:11:56,400 --> 00:11:58,900
I know it's going be raising
indices willy nilly on whatever

213
00:11:58,900 --> 00:11:59,670
it's operating on.

214
00:11:59,670 --> 00:12:01,087
But I'm taking it
down to the fact

215
00:12:01,087 --> 00:12:04,090
that I'm effectively moving a
metric inside and outside here.

216
00:12:08,470 --> 00:12:14,160
All right, so let's take
that for the geodesy equation

217
00:12:14,160 --> 00:12:16,320
and expand it out.

218
00:12:16,320 --> 00:12:28,435
So I end up with mdp beta d
tau, so I'm here using the fact

219
00:12:28,435 --> 00:12:30,060
that the p alpha
that's on the outside,

220
00:12:30,060 --> 00:12:32,190
I'm writing that as mu alpha.

221
00:12:32,190 --> 00:12:34,920
And I'm using that to
convert the derivative I get,

222
00:12:34,920 --> 00:12:37,500
expanding that
into a d by d tau.

223
00:12:37,500 --> 00:12:39,180
And then I get a
term that basically

224
00:12:39,180 --> 00:12:42,195
corrects the downstairs index.

225
00:12:49,598 --> 00:12:52,140
Because it's a downstairs index,
it enters with a minus sign.

226
00:12:54,292 --> 00:12:56,750
Let's move this to the other
side, and what I'm going to do

227
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is make all the indices be
in the downstairs position.

228
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Let's see.

229
00:13:02,482 --> 00:13:03,190
Hang on a second.

230
00:13:03,190 --> 00:13:04,130
Did I do this right?

231
00:13:04,130 --> 00:13:04,390
Sorry.

232
00:13:04,390 --> 00:13:05,890
Yeah, I'm going
make all the indices

233
00:13:05,890 --> 00:13:09,377
in the downstairs position
on this capital gamma,

234
00:13:09,377 --> 00:13:10,960
so I'm going to write
this as follows.

235
00:13:24,850 --> 00:13:27,390
OK, so I chose to
write it this way

236
00:13:27,390 --> 00:13:28,800
as you'll see in just a moment.

237
00:13:28,800 --> 00:13:34,440
Because this is now symmetric
on exchange of alpha and gamma.

238
00:13:34,440 --> 00:13:40,320
Let's expand that
Christoffel symbol,

239
00:13:40,320 --> 00:13:46,170
so I'm going to have one term
beta derivative alpha gamma.

240
00:14:04,980 --> 00:14:05,980
We'll put this up above.

241
00:14:10,000 --> 00:14:13,203
OK, so take a look at that
last line of that expression.

242
00:14:23,070 --> 00:14:27,020
So as written here,
I've got a term

243
00:14:27,020 --> 00:14:30,770
that is symmetric on
exchange of alpha and gamma.

244
00:14:30,770 --> 00:14:33,110
But inside my parentheses,
bearing in mind

245
00:14:33,110 --> 00:14:35,450
that my metric is itself
a symmetric object,

246
00:14:35,450 --> 00:14:38,210
I've got two terms,
this one and this one,

247
00:14:38,210 --> 00:14:41,000
where if I exchange alpha and
gamma, I get a minus sign.

248
00:15:06,300 --> 00:15:10,550
So I got a symmetric
contracted with anti-symmetric.

249
00:15:10,550 --> 00:15:18,050
Therefore, I can
simplify this whole thing

250
00:15:18,050 --> 00:15:20,990
to something that
only involves--

251
00:15:20,990 --> 00:15:23,540
the only derivative
I need to compute

252
00:15:23,540 --> 00:15:29,525
is one partial
derivative of the metric.

253
00:15:37,920 --> 00:15:44,393
Now that's nice, but
if you think about it,

254
00:15:44,393 --> 00:15:46,060
it might even be nicer
than you realize.

255
00:15:53,890 --> 00:15:58,150
Suppose you're working in
some coordinate system, such

256
00:15:58,150 --> 00:16:01,130
that for a particular
derivative, for a particular,

257
00:16:01,130 --> 00:16:03,070
let's say, it's the
derivative with respect

258
00:16:03,070 --> 00:16:06,063
to your time coordinate.

259
00:16:06,063 --> 00:16:07,230
Suppose the metric vanishes.

260
00:16:29,490 --> 00:16:32,190
Suppose that equals zero
for some coordinate.

261
00:16:32,190 --> 00:16:35,745
Then you've just learned
that a particular component

262
00:16:35,745 --> 00:16:38,370
of the four momentum, component
of the downstairs four momentum

263
00:16:38,370 --> 00:16:43,591
mind you, that is a constant of
the motion along the worldwide.

264
00:17:11,109 --> 00:17:12,790
I sort of said in
words a few things

265
00:17:12,790 --> 00:17:14,290
about this a couple
lectures ago when we were

266
00:17:14,290 --> 00:17:15,540
talking about Killing vectors.

267
00:17:15,540 --> 00:17:17,260
I'm going to actually tie this
to that discussion in just

268
00:17:17,260 --> 00:17:18,170
a moment.

269
00:17:18,170 --> 00:17:20,140
This is often operationally
the simplest way

270
00:17:20,140 --> 00:17:22,790
to deduce that you, in fact,
have a constant motion,

271
00:17:22,790 --> 00:17:26,260
so there is some space
times, very complicated ones

272
00:17:26,260 --> 00:17:28,840
that play huge roles in many
of the kind of analysis we do.

273
00:17:28,840 --> 00:17:29,560
But you sort of look
at them, and you

274
00:17:29,560 --> 00:17:30,430
kind of go, oh, thank god.

275
00:17:30,430 --> 00:17:31,347
It's time independent.

276
00:17:31,347 --> 00:17:35,260
All right, that means I know
p downstairs t is constant.

277
00:17:35,260 --> 00:17:38,350
It's independent of
the actual angle.

278
00:17:38,350 --> 00:17:43,330
P downstairs phi is a
constant, and that ends up

279
00:17:43,330 --> 00:17:47,110
giving us some quantities
that we can exploit.

280
00:17:47,110 --> 00:17:49,480
And later when we start
talking about certain solutions

281
00:17:49,480 --> 00:17:50,897
of the field
equations and looking

282
00:17:50,897 --> 00:17:52,930
at the behavior of
these things, we're

283
00:17:52,930 --> 00:17:54,790
going to see how
we can exploit them

284
00:17:54,790 --> 00:17:59,740
to understand the motion of
bodies in very strong gravity.

285
00:17:59,740 --> 00:18:03,190
Before I do this, let me
connect what you're saying here

286
00:18:03,190 --> 00:18:06,210
to-- what I'm saying right here
to stuff that we did a lecture

287
00:18:06,210 --> 00:18:08,170
or two ago with
the Killing vector.

288
00:18:13,380 --> 00:18:16,330
So we know that, if a metric--

289
00:18:16,330 --> 00:18:18,830
so this is something that we
wrote down a little bit before.

290
00:18:18,830 --> 00:18:26,810
We also know that,
if the metric is

291
00:18:26,810 --> 00:18:29,210
independent of some
particular coordinate,

292
00:18:29,210 --> 00:18:39,523
there exists a Killing
field, or Killing vector,

293
00:18:39,523 --> 00:18:44,340
which I will call c beta.

294
00:18:44,340 --> 00:18:47,600
What I want to do now
is say, OK, how does--

295
00:18:47,600 --> 00:18:48,810
let's look at how.

296
00:18:53,230 --> 00:19:00,690
I'm going to define a
particular scalar, so what

297
00:19:00,690 --> 00:19:01,930
do I get when I take--

298
00:19:01,930 --> 00:19:02,430
oh, bugger.

299
00:19:06,380 --> 00:19:08,574
That made no sense.

300
00:19:08,574 --> 00:19:10,040
There we go.

301
00:19:10,040 --> 00:19:12,830
What would I get if I
take that Killing vector?

302
00:19:12,830 --> 00:19:15,860
I contract it with
my for momentum.

303
00:19:15,860 --> 00:19:19,928
How does this guy behave as
I evolve along a trajectory?

304
00:19:29,915 --> 00:19:31,790
So the way we're going
to solve this is we'll

305
00:19:31,790 --> 00:19:33,620
just look at the
time of evolution

306
00:19:33,620 --> 00:19:36,243
as we move along the
trajectory as we--

307
00:19:36,243 --> 00:19:37,910
and the way we'll do
that, I'll show you

308
00:19:37,910 --> 00:19:40,580
how to construct that time
evolution in just a moment.

309
00:19:40,580 --> 00:19:44,230
We're going to assume p
solves the geodesy equation,

310
00:19:44,230 --> 00:19:45,502
c solves Killing's equation.

311
00:19:45,502 --> 00:19:46,460
Let's see what happens.

312
00:19:51,880 --> 00:19:56,810
So what we're going to
do is look at d by d tau,

313
00:19:56,810 --> 00:19:59,880
so that the proper
covariant derivative

314
00:19:59,880 --> 00:20:02,450
along the trajectory
of this guy.

315
00:20:06,510 --> 00:20:10,740
Well, this, if I take advantage
of Leibniz's rule, so first

316
00:20:10,740 --> 00:20:12,465
of all, I can just
write this as--

317
00:20:14,887 --> 00:20:15,470
you know what?

318
00:20:15,470 --> 00:20:17,600
Let's throw an m into
here just to make

319
00:20:17,600 --> 00:20:19,085
things nice and symmetric.

320
00:20:19,085 --> 00:20:20,960
The reason I did
that is so that I

321
00:20:20,960 --> 00:20:28,462
can write that derivative
in the following form,

322
00:20:28,462 --> 00:20:29,920
so this is what I
want to evaluate.

323
00:20:29,920 --> 00:20:31,990
So one thing I do is
expand out that derivative

324
00:20:31,990 --> 00:21:01,230
using Leibniz's
rule, so this is--

325
00:21:01,230 --> 00:21:02,980
let's first plot
my Killing vector.

326
00:21:08,150 --> 00:21:11,210
That term is like this.

327
00:21:11,210 --> 00:21:17,506
Then I got a term, and
it looks like this.

328
00:21:20,820 --> 00:21:22,610
Well, the first term,
it's going to die.

329
00:21:22,610 --> 00:21:24,360
Because like I said,
I'm going to assume p

330
00:21:24,360 --> 00:21:28,830
solves the geodesy
equation, so p is a geodesy.

331
00:21:28,830 --> 00:21:31,350
I kill that.

332
00:21:31,350 --> 00:21:33,820
What about the second term?

333
00:21:33,820 --> 00:21:36,270
Well, for the second
term, what I'm going to do

334
00:21:36,270 --> 00:21:42,420
is note that whenever you have
some general two index object,

335
00:21:42,420 --> 00:21:46,080
so suppose I have some two
index tensor, m alpha beta.

336
00:21:46,080 --> 00:21:53,790
I can always write this in
the following way, right?

337
00:21:53,790 --> 00:22:05,340
Where remember, the parentheses
denote the symmetric part,

338
00:22:05,340 --> 00:22:08,970
and the braces denote
the anti-symmetric part.

339
00:22:15,795 --> 00:22:17,170
So this is just
a theorem, right?

340
00:22:17,170 --> 00:22:18,003
You add it together.

341
00:22:18,003 --> 00:22:20,810
The 1/2s combine, and
you get this thing back

342
00:22:20,810 --> 00:22:21,607
for the first term.

343
00:22:21,607 --> 00:22:23,690
And they combine the minus
with the other one, OK?

344
00:22:23,690 --> 00:22:25,400
Very simple identity.

345
00:22:25,400 --> 00:22:47,020
So if I do that,
applying it up here,

346
00:22:47,020 --> 00:22:49,990
this is symmetric under
exchange of indices.

347
00:22:49,990 --> 00:22:51,190
This is anti-symmetric.

348
00:22:51,190 --> 00:22:52,270
It dies.

349
00:22:52,270 --> 00:22:54,400
The only thing that
is left is this term.

350
00:23:18,380 --> 00:23:23,930
But if this is a Killing
vector by Killing's equation,

351
00:23:23,930 --> 00:23:26,540
this equals zero by
Killing's equation.

352
00:23:37,720 --> 00:23:39,760
So the importance
of this, you've

353
00:23:39,760 --> 00:23:49,810
just shown that what you
get when you contract

354
00:23:49,810 --> 00:23:53,160
for momentum with
the Killing vector

355
00:23:53,160 --> 00:23:54,870
gives you a constant motion.

356
00:23:54,870 --> 00:23:59,160
You've also shown that the
component of the four momentum,

357
00:23:59,160 --> 00:24:02,250
the downstairs component of
the four momentum associated

358
00:24:02,250 --> 00:24:05,130
with whatever
coordinate the metric

359
00:24:05,130 --> 00:24:09,120
happens to be independent of, is
also a constant of the motion.

360
00:24:09,120 --> 00:24:12,630
The key thing to note is both
are actually very powerful

361
00:24:12,630 --> 00:24:14,370
and important statements.

362
00:24:14,370 --> 00:24:17,370
One depends on the coordinate
system and the representation

363
00:24:17,370 --> 00:24:18,180
you've chosen.

364
00:24:18,180 --> 00:24:20,880
The other does not, OK?

365
00:24:20,880 --> 00:24:24,900
So this is really
true and useful,

366
00:24:24,900 --> 00:24:27,660
if you happen to have chosen
the coordinate system such

367
00:24:27,660 --> 00:24:33,120
that this derivative
is equal to zero.

368
00:24:33,120 --> 00:24:35,040
This is true,
though, independent

369
00:24:35,040 --> 00:24:37,620
of your representation,
so these are just

370
00:24:37,620 --> 00:24:42,330
two different ways of calling
out constants of motion.

371
00:24:42,330 --> 00:24:44,942
And we actually find both
of them to be very useful,

372
00:24:44,942 --> 00:24:46,650
so we're going to take
advantage of them.

373
00:24:46,650 --> 00:24:48,483
There's a variation on
this calculation that

374
00:24:48,483 --> 00:24:55,890
is on the next p set, and
you will come back to this,

375
00:24:55,890 --> 00:24:59,070
again, when we start talking
about motion in certain space

376
00:24:59,070 --> 00:25:01,710
times in the second
half of this course.

377
00:25:05,188 --> 00:25:07,230
So let me just do a couple
really quick examples.

378
00:25:07,230 --> 00:25:09,330
I've already kind
of mentioned these,

379
00:25:09,330 --> 00:25:11,190
but let me give names
to what these are.

380
00:25:14,280 --> 00:25:18,170
So if your space time has
a time coordinate, such

381
00:25:18,170 --> 00:25:22,670
that the time derivative of
any metric element is zero,

382
00:25:22,670 --> 00:25:26,660
then you know that a time
like Killing vector, which

383
00:25:26,660 --> 00:25:30,940
I will call ct, and I'm
leaving the vector sign on it.

384
00:25:30,940 --> 00:25:46,630
This thing exists, and you
also know that p downstairs t

385
00:25:46,630 --> 00:25:49,000
is constant.

386
00:25:49,000 --> 00:25:56,500
Now the name that is given
to this is negative energy.

387
00:25:56,500 --> 00:25:59,090
Why negative?

388
00:25:59,090 --> 00:26:00,830
Well, the main reason
why it's negative

389
00:26:00,830 --> 00:26:02,210
is that we will often--

390
00:26:02,210 --> 00:26:04,700
so let me just caution that
this is not always an identity

391
00:26:04,700 --> 00:26:05,470
that we're going to use.

392
00:26:05,470 --> 00:26:07,553
We're going to use it in
a huge number of problems

393
00:26:07,553 --> 00:26:09,500
that we care about.

394
00:26:09,500 --> 00:26:17,420
Many of the space times that
we are going to work with

395
00:26:17,420 --> 00:26:20,870
are those that when you get
really far away from whatever

396
00:26:20,870 --> 00:26:22,820
source is generating
your gravity,

397
00:26:22,820 --> 00:26:25,100
it looks just like
special relativity.

398
00:26:25,100 --> 00:26:29,480
We call such space times
asymptotically flat.

399
00:26:29,480 --> 00:26:36,590
In other words, as you get
asymptotically far away,

400
00:26:36,590 --> 00:26:38,993
it reduces to the
space time that we

401
00:26:38,993 --> 00:26:40,910
studied in the first
couple weeks of the class

402
00:26:40,910 --> 00:26:43,320
when we were doing geometric
spatial relativity.

403
00:26:43,320 --> 00:26:47,975
And in that case, we knew the
timelike component was energy.

404
00:26:47,975 --> 00:26:49,850
And in a flat space
time, you lower that time

405
00:26:49,850 --> 00:26:51,200
like component's index.

406
00:26:51,200 --> 00:26:53,030
You get minus energy.

407
00:26:53,030 --> 00:26:55,160
It just so happens when
you go through the math

408
00:26:55,160 --> 00:26:58,220
carefully that
negative of the energy

409
00:26:58,220 --> 00:27:00,640
ends up defined in this way.

410
00:27:00,640 --> 00:27:02,570
It's going to be the
quantity that is actually

411
00:27:02,570 --> 00:27:03,910
conserved everywhere.

412
00:27:03,910 --> 00:27:05,628
What's kind of
cool is that we use

413
00:27:05,628 --> 00:27:07,295
this associated with
asymptotic flatness

414
00:27:07,295 --> 00:27:08,930
to give you some intuition.

415
00:27:08,930 --> 00:27:10,730
But this is actually
true, even if you're

416
00:27:10,730 --> 00:27:13,610
right outside of the vicinity of
a rapidly rotating black hole.

417
00:27:13,610 --> 00:27:15,290
It is still the case
that p downstairs

418
00:27:15,290 --> 00:27:17,660
t for the right choice
of t is a constant.

419
00:27:23,690 --> 00:27:25,390
In my notes, I also
show you that there

420
00:27:25,390 --> 00:27:30,470
is an example of an
actual Killing vector that

421
00:27:30,470 --> 00:27:31,850
corresponds to angle momentum.

422
00:27:31,850 --> 00:27:34,950
Again, we'll come back to
that a little bit later,

423
00:27:34,950 --> 00:27:37,633
so let me do one
example geodesy before I

424
00:27:37,633 --> 00:27:39,050
sort of change
topic a little bit.

425
00:27:44,710 --> 00:27:47,140
I'm going to write
down a space time

426
00:27:47,140 --> 00:27:53,170
that we either in
person or on video

427
00:27:53,170 --> 00:27:56,290
are going to derive basically
right after Spring break.

428
00:28:04,390 --> 00:28:07,497
So suppose I hand you
the following space time.

429
00:28:35,060 --> 00:28:38,190
This function phi, I'm not going
to say too much about it quite

430
00:28:38,190 --> 00:28:39,330
yet.

431
00:28:39,330 --> 00:28:43,673
What I will say is that
it is small in the sense

432
00:28:43,673 --> 00:28:45,840
that when you're doing
various calculations with it,

433
00:28:45,840 --> 00:28:49,770
feel free to discard terms of
order five squared or higher.

434
00:28:49,770 --> 00:28:57,330
And it only depends on the
coordinates x, y, and z.

435
00:28:57,330 --> 00:28:58,719
No time dependence.

436
00:29:03,610 --> 00:29:07,050
I want to examine slow
motion in the space time.

437
00:29:12,610 --> 00:29:14,110
So what I'm going
to do is I'm going

438
00:29:14,110 --> 00:29:20,392
to imagine that my four
momentum has the usual form.

439
00:29:20,392 --> 00:29:22,850
And I can think of it as having
an energy, and sort of time

440
00:29:22,850 --> 00:29:28,420
like component, and
momentum in the space like.

441
00:29:28,420 --> 00:29:31,090
The magnitude of
the energy is always

442
00:29:31,090 --> 00:29:35,950
going to be much greater than
the magnitude of the momentum,

443
00:29:35,950 --> 00:29:40,060
and in fact, will
be approximately

444
00:29:40,060 --> 00:29:47,060
equal to the mass, where that
is the mass of whatever body is

445
00:29:47,060 --> 00:29:49,010
actually undergoing this
motion in this space

446
00:29:49,010 --> 00:29:49,843
that I've given you.

447
00:29:53,250 --> 00:29:58,017
OK, so the reason why I'm doing
this is what we're going to do

448
00:29:58,017 --> 00:29:59,850
is you want to say,
what does free fall look

449
00:29:59,850 --> 00:30:01,050
like in this space?

450
00:30:05,160 --> 00:30:29,380
Well, I look at geodesics, so
there is my geodesy equation.

451
00:30:29,380 --> 00:30:32,350
This slow motion
condition that I've

452
00:30:32,350 --> 00:30:35,590
applied over here, that tells
me that when I expand all

453
00:30:35,590 --> 00:30:43,670
these terms out here, this
is going to be dominated

454
00:30:43,670 --> 00:30:47,090
by the time time term, OK?

455
00:30:47,090 --> 00:30:49,280
So it'll be dominated
because of the fact

456
00:30:49,280 --> 00:30:53,240
that when you just look at the
numerical magnitude of the size

457
00:30:53,240 --> 00:30:59,000
of the components
of the momentum,

458
00:30:59,000 --> 00:31:01,940
those are going to be the ones
that dominate this calculation.

459
00:31:01,940 --> 00:31:04,610
Everything else, if you put
your factors of c back in,

460
00:31:04,610 --> 00:31:06,530
they're going to
be down by factors

461
00:31:06,530 --> 00:31:08,960
that look like v over c.

462
00:31:12,110 --> 00:31:18,420
So my geodesic
equation turns into--

463
00:31:31,190 --> 00:31:33,410
so all I did was say,
it's dominated by this.

464
00:31:33,410 --> 00:31:35,702
I'm going to move it to the
other side of the equation.

465
00:31:55,680 --> 00:31:59,960
So without giving away
the plot, the beta

466
00:31:59,960 --> 00:32:01,710
equals zero component
is going to turn out

467
00:32:01,710 --> 00:32:04,030
to be very uninteresting.

468
00:32:04,030 --> 00:32:05,270
Can you see why?

469
00:32:05,270 --> 00:32:07,790
Beta equals zero is energy.

470
00:32:07,790 --> 00:32:09,280
When I evaluate
that Christoffel,

471
00:32:09,280 --> 00:32:10,930
I'm going to end up
taking a bunch of-- you

472
00:32:10,930 --> 00:32:12,555
know, I'm going to
have all these zero,

473
00:32:12,555 --> 00:32:14,222
zero, zero, time,
time, time components.

474
00:32:14,222 --> 00:32:16,597
It's time independent, though,
so all my time derivatives

475
00:32:16,597 --> 00:32:17,600
are going to be zero.

476
00:32:17,600 --> 00:32:19,490
It's going to vanish,
but we expect that.

477
00:32:19,490 --> 00:32:22,572
Because it's a time
independent metric,

478
00:32:22,572 --> 00:32:24,530
all the crap I went
through a couple months ago

479
00:32:24,530 --> 00:32:27,800
guarantees that the time like
components, energy's conserved,

480
00:32:27,800 --> 00:32:28,340
right?

481
00:32:28,340 --> 00:32:30,180
So that's kind of
what we expect,

482
00:32:30,180 --> 00:32:32,750
so let's just focus
on-- and if we

483
00:32:32,750 --> 00:32:35,870
had infinite time, which
we clearly don't, it

484
00:32:35,870 --> 00:32:37,220
would be fun to talk about.

485
00:32:37,220 --> 00:32:40,370
Let's just move on and look at
the spatial component of this.

486
00:32:40,370 --> 00:32:46,954
So as I look at the spatial
component of this guy,

487
00:32:46,954 --> 00:32:48,920
let's focus on beta equals i.

488
00:32:51,510 --> 00:32:55,050
What you find when you
actually evaluate this guy is

489
00:32:55,050 --> 00:32:56,040
this turns into--

490
00:33:13,960 --> 00:33:16,313
OK, so go ahead and look up
your Christoffel formulas.

491
00:33:16,313 --> 00:33:18,730
Again, these are one of those
things that, eventually, you

492
00:33:18,730 --> 00:33:20,480
get this memorized by
the end of the term,

493
00:33:20,480 --> 00:33:23,890
but don't feel bad if
you keep forgetting it.

494
00:33:23,890 --> 00:33:24,820
No time derivative.

495
00:33:24,820 --> 00:33:27,130
No time derivative.

496
00:33:27,130 --> 00:33:29,410
The only thing that's
left at the end of the day

497
00:33:29,410 --> 00:33:32,680
is essentially a gradient of the
time time piece of the metric.

498
00:33:35,990 --> 00:33:40,720
So taking advantage
of the fact gi

499
00:33:40,720 --> 00:33:45,470
alpha staying in the
upstairs position,

500
00:33:45,470 --> 00:33:46,880
you can write this as--

501
00:33:50,500 --> 00:33:52,580
it looks like this.

502
00:33:52,580 --> 00:33:55,310
Now, if you like-- remember,
phi is a small value.

503
00:33:55,310 --> 00:33:57,080
You can do binomial
expansion on that.

504
00:33:57,080 --> 00:33:57,920
Knock yourself out.

505
00:33:57,920 --> 00:33:59,920
It's not going to be
important in just a second.

506
00:34:06,780 --> 00:34:10,909
Excuse me, minus one
half one minus two,

507
00:34:10,909 --> 00:34:12,440
five to the minus one power.

508
00:34:22,587 --> 00:34:24,170
So let me talk a
little bit about what

509
00:34:24,170 --> 00:34:27,230
I did here in this last line.

510
00:34:27,230 --> 00:34:30,989
My delta i alpha is coupling
to a partial derivative,

511
00:34:30,989 --> 00:34:33,830
so that partial derivative,
the zero component of that

512
00:34:33,830 --> 00:34:35,360
is the time derivative.

513
00:34:35,360 --> 00:34:37,610
Everything's time
independent, so that's done.

514
00:34:37,610 --> 00:34:39,409
Let's just skip
it, so what I did

515
00:34:39,409 --> 00:34:41,270
was I changed my alpha to a j.

516
00:34:41,270 --> 00:34:43,757
Because I only want
spatial derivatives,

517
00:34:43,757 --> 00:34:44,840
so I'm allowed to do that.

518
00:34:44,840 --> 00:34:46,840
Because I'm just acknowledging
the fact that all

519
00:34:46,840 --> 00:34:48,830
the time derivatives
are uninteresting.

520
00:34:48,830 --> 00:34:52,940
When I do
differentiate g00, I am

521
00:34:52,940 --> 00:34:57,300
differentiating negative
quantity one plus two phi.

522
00:34:57,300 --> 00:34:58,850
The one doesn't contribute.

523
00:34:58,850 --> 00:35:01,162
All that is left is I took
the minus on the inside,

524
00:35:01,162 --> 00:35:02,120
and there's my two phi.

525
00:35:19,380 --> 00:35:21,810
So putting all of these
ingredients together,

526
00:35:21,810 --> 00:35:30,140
I at last get my
Christoffel is delta ij.

527
00:35:30,140 --> 00:35:35,305
I'm saying it looks like
spatial gradient of phi.

528
00:35:35,305 --> 00:35:38,138
And for keeping score, there
are higher order terms,

529
00:35:38,138 --> 00:35:40,180
which we're going to
collect under the assumption

530
00:35:40,180 --> 00:35:42,280
that this phi is small.

531
00:35:42,280 --> 00:35:44,155
Plug it back into my
equation of motion.

532
00:36:00,100 --> 00:36:02,740
I end up with this.

533
00:36:02,740 --> 00:36:05,800
Now let's cancel out the
m's that appeared in here.

534
00:36:10,040 --> 00:36:14,270
If we were not doing relativity,
we would write this as--

535
00:36:14,270 --> 00:36:16,880
ignore the fact that this
is per unit proper time.

536
00:36:16,880 --> 00:36:22,670
This is dpdt is minus gradients
of something that sure as hell

537
00:36:22,670 --> 00:36:23,974
looks like a potential.

538
00:36:47,220 --> 00:36:49,740
What we are going to
do in the lecture rate

539
00:36:49,740 --> 00:36:54,860
after Spring break, so
going into Spring break,

540
00:36:54,860 --> 00:36:56,940
depending on which-- as
I said at the beginning,

541
00:36:56,940 --> 00:36:58,140
it's a little unclear
how these lectures

542
00:36:58,140 --> 00:36:58,710
are going to be delivered.

543
00:36:58,710 --> 00:36:59,418
But bear with me.

544
00:36:59,418 --> 00:37:01,377
We're going to essentially
put together-- we're

545
00:37:01,377 --> 00:37:03,000
going to take all
the last ingredients

546
00:37:03,000 --> 00:37:04,500
and develop the
field equations that

547
00:37:04,500 --> 00:37:06,455
describe relativistic gravity.

548
00:37:06,455 --> 00:37:07,830
The first thing
we're going to do

549
00:37:07,830 --> 00:37:10,530
is solve this in a
particular limit that

550
00:37:10,530 --> 00:37:13,590
describes a body that
is weakly gravitating.

551
00:37:13,590 --> 00:37:17,400
This will emerge
from this with phi

552
00:37:17,400 --> 00:37:21,600
being equal to Newtonian
gravitational potential.

553
00:37:21,600 --> 00:37:24,070
What this is showing is
that the geodesic equation,

554
00:37:24,070 --> 00:37:26,220
this equation that
describes a trajectory that

555
00:37:26,220 --> 00:37:30,060
is as straight as possible in
space time when it is given

556
00:37:30,060 --> 00:37:32,040
that particular
space time, it does

557
00:37:32,040 --> 00:37:35,960
give you the Newtonian
equation of motion, OK?

558
00:37:35,960 --> 00:37:36,551
Yeah?

559
00:37:36,551 --> 00:37:37,964
AUDIENCE: Is there
still supposed

560
00:37:37,964 --> 00:37:40,710
to be a factor of m on
the right-hand side?

561
00:37:40,710 --> 00:37:42,860
SCOTT HUGHES: No, so
it's possible I dropped

562
00:37:42,860 --> 00:37:43,860
an m somewhere in there.

563
00:37:43,860 --> 00:37:46,950
Go through that just a
little bit carefully here,

564
00:37:46,950 --> 00:37:49,590
but you know, it's meant to be--

565
00:37:53,180 --> 00:37:54,180
actually, you know what?

566
00:37:54,180 --> 00:37:54,870
I take it back.

567
00:37:54,870 --> 00:37:56,722
Sorry, so I think I
may have messed up.

568
00:37:56,722 --> 00:37:58,680
I remember what it was,
I remember what it was.

569
00:37:58,680 --> 00:38:00,210
There's an m squared here.

570
00:38:00,210 --> 00:38:01,620
Thank you, Alex.

571
00:38:01,620 --> 00:38:03,450
There was an m squared here.

572
00:38:03,450 --> 00:38:06,858
There was an m and
an m squared here.

573
00:38:06,858 --> 00:38:08,400
That's what I screwed
up, and I think

574
00:38:08,400 --> 00:38:10,110
I have that wrong in my
handwritten notes, which is

575
00:38:10,110 --> 00:38:11,480
probably why I messed that up.

576
00:38:11,480 --> 00:38:13,410
Yeah, so this
equation was correct.

577
00:38:13,410 --> 00:38:15,270
And this should have
been here like so.

578
00:38:15,270 --> 00:38:17,690
We'll clear this
out and get this.

579
00:38:17,690 --> 00:38:20,978
Yeah, so module of
that little bobble.

580
00:38:20,978 --> 00:38:22,770
I just want to show
you this is essentially

581
00:38:22,770 --> 00:38:24,390
the Newtonian limit.

582
00:38:24,390 --> 00:38:27,390
Just give you a little
look at where we go ahead,

583
00:38:27,390 --> 00:38:28,890
there are two ways
that we are going

584
00:38:28,890 --> 00:38:31,920
to derive the field equations
of general relativity.

585
00:38:31,920 --> 00:38:34,440
The first one
essentially boils down

586
00:38:34,440 --> 00:38:38,610
to looking for certain tensors
that have the right symmetries

587
00:38:38,610 --> 00:38:42,270
and allow us to have
sort of a quantity that

588
00:38:42,270 --> 00:38:45,390
looks like derivatives on
the field equaling the stress

589
00:38:45,390 --> 00:38:47,040
energy tensor as the source.

590
00:38:47,040 --> 00:38:51,310
That only works up to
an overall constant,

591
00:38:51,310 --> 00:38:54,030
and this is actually the
way that Einstein originally

592
00:38:54,030 --> 00:38:57,450
developed the field equations,
was worked out all this stuff,

593
00:38:57,450 --> 00:39:00,390
and then by insisting that the
solution that emerge from this

594
00:39:00,390 --> 00:39:02,037
reproduced the
Newtonian equation

595
00:39:02,037 --> 00:39:03,870
and the Newtonian motion,
he was able to fix

596
00:39:03,870 --> 00:39:06,675
what that constant actually is.

597
00:39:06,675 --> 00:39:08,550
There's a more sophisticated
way of doing it,

598
00:39:08,550 --> 00:39:10,820
which I'm going to
also go through.

599
00:39:10,820 --> 00:39:13,260
But it's worth noting that is
the way Einstein originally

600
00:39:13,260 --> 00:39:14,550
did it.

601
00:39:14,550 --> 00:39:18,040
I had the privilege a
couple of years ago--

602
00:39:18,040 --> 00:39:20,680
I was at a conference
in Jerusalem,

603
00:39:20,680 --> 00:39:24,120
where the Einstein Papers
archive is located.

604
00:39:24,120 --> 00:39:27,080
And the guy who is the
main curator of this

605
00:39:27,080 --> 00:39:28,830
was allowing those of
us at the conference

606
00:39:28,830 --> 00:39:29,910
to look through them.

607
00:39:29,910 --> 00:39:31,753
And I actually found--

608
00:39:31,753 --> 00:39:33,420
they had not yet quite
categorized that,

609
00:39:33,420 --> 00:39:36,200
but it was the papers very
much related to working

610
00:39:36,200 --> 00:39:37,450
with these peculiar equations.

611
00:39:37,450 --> 00:39:39,575
I don't think he was trying
to fix the coefficient,

612
00:39:39,575 --> 00:39:43,860
but this spacetime can also be
used to compute the perihelion

613
00:39:43,860 --> 00:39:46,387
precession of mercury.

614
00:39:46,387 --> 00:39:48,720
And so it actually showed
Einstein working through that.

615
00:39:48,720 --> 00:39:53,230
And the thing which is really
cool was he screwed up a lot.

616
00:39:53,230 --> 00:39:55,570
The page that I was looking
at was full of errors.

617
00:39:55,570 --> 00:39:56,880
It would say things like--

618
00:39:56,880 --> 00:39:58,680
big things crossed out
and then "Nein nein nein!"

619
00:39:58,680 --> 00:39:59,520
written on the side.

620
00:39:59,520 --> 00:40:03,670
And so it made me feel
better about myself.

621
00:40:03,670 --> 00:40:05,383
All right.

622
00:40:05,383 --> 00:40:07,050
So everything that
we have done so far--

623
00:40:07,050 --> 00:40:10,950
we've been dancing
around this notion

624
00:40:10,950 --> 00:40:12,708
of what is called "curvature."

625
00:40:16,700 --> 00:40:19,280
So I have used this
word several times,

626
00:40:19,280 --> 00:40:21,530
but I haven't made
this precise yet.

627
00:40:21,530 --> 00:40:24,170
Curvature is going to
be the precise idea

628
00:40:24,170 --> 00:40:28,540
of how two initially
parallel trajectories

629
00:40:28,540 --> 00:40:29,450
cease to be parallel.

630
00:40:53,950 --> 00:40:59,650
So there's a couple of ways
that we can quantify this.

631
00:40:59,650 --> 00:41:01,990
The one which I
am going to use is

632
00:41:01,990 --> 00:41:06,280
one that's amendable
to, with relative ease,

633
00:41:06,280 --> 00:41:10,960
developing a particularly
important tensor, which

634
00:41:10,960 --> 00:41:12,910
characterizes curvature.

635
00:41:12,910 --> 00:41:14,650
And so what we're
going to do is look

636
00:41:14,650 --> 00:41:17,950
at the behavior of a
vector that is parallel

637
00:41:17,950 --> 00:41:26,310
transported in a
non-infinitesimal region

638
00:41:26,310 --> 00:41:27,185
of a curved manifold.

639
00:41:46,330 --> 00:41:47,890
So for the purpose
of this sketch,

640
00:41:47,890 --> 00:41:50,710
I'm going to make this
closed figure be a triangle.

641
00:41:50,710 --> 00:41:52,120
When I actually do the
calculation in just a moment,

642
00:41:52,120 --> 00:41:53,830
I'm going to use a
little parallelogram.

643
00:41:53,830 --> 00:41:59,940
So around a closed figure,
I want a curved manifold.

644
00:42:07,320 --> 00:42:09,300
So suppose my
curvature's actually 0,

645
00:42:09,300 --> 00:42:20,990
and I do this for a triangle
that is on the blackboard.

646
00:42:20,990 --> 00:42:26,258
So let's say I start out with a
vector that points from A to B.

647
00:42:26,258 --> 00:42:28,550
And what I'm going to do is
just parallel transport it.

648
00:42:28,550 --> 00:42:32,190
And in the case, goes, doo, doo,
doo, doo, doo, doo, doo, doo,

649
00:42:32,190 --> 00:42:35,452
doo, doo, doo, doo.

650
00:42:35,452 --> 00:42:37,160
This is an experiment
you can do at home.

651
00:42:37,160 --> 00:42:39,620
When it comes back,
it's pointing exactly

652
00:42:39,620 --> 00:42:41,098
the way it was initially.

653
00:42:57,710 --> 00:43:02,510
Let me also just note
that this triangle--

654
00:43:02,510 --> 00:43:07,820
the sum of its internal
angles is 180 degrees.

655
00:43:14,033 --> 00:43:15,200
Hopefully you all know that.

656
00:43:18,810 --> 00:43:21,750
Now, the next one--

657
00:43:21,750 --> 00:43:24,520
if I'd had a little
bit more time,

658
00:43:24,520 --> 00:43:28,710
I would have grabbed one
of my daughter's balls

659
00:43:28,710 --> 00:43:30,000
to demonstrate this.

660
00:43:30,000 --> 00:43:31,708
But hopefully, if you
guys have something

661
00:43:31,708 --> 00:43:34,890
like a soccer ball
or a basketball,

662
00:43:34,890 --> 00:43:38,310
this is a little experiment
you can do by yourself.

663
00:43:38,310 --> 00:43:44,505
Now imagine a triangle
that is embedded

664
00:43:44,505 --> 00:43:45,630
on the surface of a sphere.

665
00:43:51,810 --> 00:43:56,940
So let's say this is the
North Pole of my sphere.

666
00:43:59,750 --> 00:44:00,605
Here's the equator.

667
00:44:03,730 --> 00:44:06,540
So what I'm going
to imagine is--

668
00:44:06,540 --> 00:44:08,100
let's say I start up here.

669
00:44:08,100 --> 00:44:11,035
Let's make the North
Pole be point A.

670
00:44:11,035 --> 00:44:14,610
I move on a trajectory
that is as straight as I

671
00:44:14,610 --> 00:44:15,510
am allowed to be.

672
00:44:15,510 --> 00:44:18,030
And remember, if I'm a
one-dimensional being living

673
00:44:18,030 --> 00:44:20,490
on the surface of this thing,
that's a straight line.

674
00:44:20,490 --> 00:44:22,885
It only looks straight
looks curved to us

675
00:44:22,885 --> 00:44:25,260
because we see a third dimension
that this whole thing is

676
00:44:25,260 --> 00:44:26,873
embedded in.

677
00:44:26,873 --> 00:44:28,290
And this thing's
going to come in,

678
00:44:28,290 --> 00:44:31,680
and it actually hits the
equator at a right angle, OK?

679
00:44:31,680 --> 00:44:33,360
No ifs, ands, or buts about it.

680
00:44:33,360 --> 00:44:35,790
It's a bloody right angle.

681
00:44:35,790 --> 00:44:36,930
And then I'm going to--

682
00:44:36,930 --> 00:44:38,970
let's call this point B--

683
00:44:38,970 --> 00:44:41,640
walk back along the
equator here till I

684
00:44:41,640 --> 00:44:45,200
reach a point which
I will call C.

685
00:44:45,200 --> 00:44:48,680
And then I'm going to go
straight north until I

686
00:44:48,680 --> 00:44:52,450
come back up to the North Pole.

687
00:44:55,780 --> 00:44:59,590
This is a triangle in which all
three angles are 90 degrees.

688
00:45:11,080 --> 00:45:12,690
So here is a great
little experiment

689
00:45:12,690 --> 00:45:15,127
that's very easy for
you to do at home.

690
00:45:15,127 --> 00:45:16,960
Does anyone happen to
have a ball with them?

691
00:45:16,960 --> 00:45:17,640
OK, never mind.

692
00:45:22,340 --> 00:45:24,090
Let me look at my notes
for just a second.

693
00:45:30,080 --> 00:45:33,205
So let's say I start
out here at point A,

694
00:45:33,205 --> 00:45:37,610
and I have my vector pointing
in the south direction.

695
00:45:37,610 --> 00:45:43,305
So this guy goes down here.

696
00:45:43,305 --> 00:45:45,430
And what you'll see is it
goes down to the equator,

697
00:45:45,430 --> 00:45:47,390
and it keeps pointing south.

698
00:45:47,390 --> 00:45:55,260
Then I bring it along over here,
bring it back up to the north.

699
00:45:55,260 --> 00:45:58,000
The vector has been
rotated by 90 degrees

700
00:45:58,000 --> 00:46:00,680
as it goes around that pass.

701
00:46:00,680 --> 00:46:03,112
It's a really,
fun, exciting demo.

702
00:46:03,112 --> 00:46:05,570
If you've got a ball at home,
you can do this over and over

703
00:46:05,570 --> 00:46:06,070
again.

704
00:46:06,070 --> 00:46:07,888
It's endless fun.

705
00:46:07,888 --> 00:46:10,430
I'm being slightly silly here,
but there's an important point

706
00:46:10,430 --> 00:46:11,300
to be made.

707
00:46:11,300 --> 00:46:21,110
When you do this operation,
parallel transport

708
00:46:21,110 --> 00:46:22,160
rotates the vector.

709
00:46:25,410 --> 00:46:27,960
It turns out that, if you are
working on a two-dimensional

710
00:46:27,960 --> 00:46:30,720
manifold-- particularly, I think
it's a two-dimensional manifold

711
00:46:30,720 --> 00:46:31,763
that is--

712
00:46:31,763 --> 00:46:33,180
it may have to be
of what's called

713
00:46:33,180 --> 00:46:34,680
constant curvature--
in other words,

714
00:46:34,680 --> 00:46:38,450
either a surface, a
plane, or a hyperbola.

715
00:46:42,190 --> 00:46:52,480
It actually rotates by
an angle of whatever

716
00:46:52,480 --> 00:46:58,360
is internal angle of the
triangle minus 180 degrees.

717
00:46:58,360 --> 00:47:01,108
So in this case, it would
rotate it by 90 degrees.

718
00:47:01,108 --> 00:47:03,150
If you took this thing
and you actually opened it

719
00:47:03,150 --> 00:47:05,392
up all the way, you
could basically, just

720
00:47:05,392 --> 00:47:07,600
by taking this leg and making
it as long as you want,

721
00:47:07,600 --> 00:47:08,580
you can make it to 0.

722
00:47:08,580 --> 00:47:11,590
You can make it huge.

723
00:47:11,590 --> 00:47:14,430
And when you do so, you'll just
rotate that vector all the more

724
00:47:14,430 --> 00:47:15,180
as it goes around.

725
00:47:18,120 --> 00:47:24,170
This operation, by the
way, is called a holonomy.

726
00:47:24,170 --> 00:47:27,120
I throw that out there
because, last time I looked,

727
00:47:27,120 --> 00:47:28,710
there was a decent
Wikipedia page

728
00:47:28,710 --> 00:47:32,900
on this that has some cool
animated graphics on it.

729
00:47:42,550 --> 00:47:45,340
Also, MathWorld.Wolfram.com
had some good stuff.

730
00:47:52,920 --> 00:47:54,900
So this has good
descriptions that you

731
00:47:54,900 --> 00:47:57,900
can find it all on Google.

732
00:47:57,900 --> 00:47:59,350
All right.

733
00:47:59,350 --> 00:48:02,340
What I want to do is take
some of these somewhat vague

734
00:48:02,340 --> 00:48:02,980
notions--

735
00:48:02,980 --> 00:48:04,980
so hopefully, I made
it intuitively clear

736
00:48:04,980 --> 00:48:06,180
that there's something
very interesting that

737
00:48:06,180 --> 00:48:07,590
happens when I
parallel transport

738
00:48:07,590 --> 00:48:11,580
a vector around these figures,
depending upon the underlying

739
00:48:11,580 --> 00:48:14,350
geometry of the manifold
that they're embedded in.

740
00:48:14,350 --> 00:48:16,137
Let's try to make
it more precise now.

741
00:48:16,137 --> 00:48:17,970
And I'm going to start
all the way over here

742
00:48:17,970 --> 00:48:20,800
because I'm going to want big,
clean boards to illustrate

743
00:48:20,800 --> 00:48:21,300
this.

744
00:48:33,130 --> 00:48:33,630
OK.

745
00:48:33,630 --> 00:48:38,790
So suppose I'm in some
coordinate system,

746
00:48:38,790 --> 00:48:44,160
and this line I've written here
represents a line of constant.

747
00:48:44,160 --> 00:48:48,708
So lambda is one
particular member

748
00:48:48,708 --> 00:48:50,250
of your set of
spacetime coordinates,

749
00:48:50,250 --> 00:48:52,885
so it might be time
or radius or maybe

750
00:48:52,885 --> 00:48:54,510
you work in some
crazy querying system.

751
00:48:54,510 --> 00:48:57,270
But lambda is meant to
represent some particular member

752
00:48:57,270 --> 00:49:00,180
of your coordinate system.

753
00:49:00,180 --> 00:49:04,580
And then there's another track
over here, which is displaced

754
00:49:04,580 --> 00:49:10,160
from it by delta x lambda, OK?

755
00:49:10,160 --> 00:49:12,380
So everywhere along here,
one of your coordinates

756
00:49:12,380 --> 00:49:14,323
is equal to the value, x lambda.

757
00:49:14,323 --> 00:49:16,490
Everywhere along here wanted,
that same coordinate's

758
00:49:16,490 --> 00:49:18,350
equal to x lambda
plus dx lambda.

759
00:49:21,720 --> 00:49:26,350
Along this trajectory,
there's a different coordinate

760
00:49:26,350 --> 00:49:27,820
that is kept constant.

761
00:49:27,820 --> 00:49:29,330
Lambda and sigma
are not the same.

762
00:49:29,330 --> 00:49:31,480
So there is some
coordinate whose value

763
00:49:31,480 --> 00:49:35,190
I will label as sigma that
is constant along there.

764
00:49:35,190 --> 00:49:40,050
And along this one,
it is also constant.

765
00:49:43,080 --> 00:49:57,290
Let me label the four
vertices, A, B, C, and D.

766
00:49:57,290 --> 00:50:00,590
And let me number
these four edges--

767
00:50:00,590 --> 00:50:08,190
one, two, three, and four.

768
00:50:08,190 --> 00:50:11,460
What I am going to
imagine doing is parallel

769
00:50:11,460 --> 00:50:15,180
transporting some vector,
v alpha, around this loop.

770
00:50:27,772 --> 00:50:29,230
So what I'm going
to do is generate

771
00:50:29,230 --> 00:50:33,170
the equations that describe
how it changes as a transport.

772
00:50:33,170 --> 00:50:35,770
I'm going to start at A, so
v is pointing along here.

773
00:50:35,770 --> 00:50:51,680
Transport it to B to C
to D and then back to A.

774
00:50:51,680 --> 00:50:54,250
So let me very carefully
do the first leg.

775
00:50:54,250 --> 00:50:56,547
Once you get the
pattern, the others

776
00:50:56,547 --> 00:50:58,130
can be done a little
bit more quickly.

777
00:51:04,500 --> 00:51:07,640
So the coordinate-- let's see.

778
00:51:07,640 --> 00:51:09,400
Hang on just one moment.

779
00:51:09,400 --> 00:51:12,065
Yeah, so I am going
from A to B first.

780
00:51:16,090 --> 00:51:20,920
So as I move from A to B,
x lambda remains constant,

781
00:51:20,920 --> 00:51:23,810
and the coordinate x
sigma is increasing.

782
00:51:23,810 --> 00:51:26,470
So I am moving in a
direction that points

783
00:51:26,470 --> 00:51:30,265
along the unit vector associated
with the sigma coordinate.

784
00:51:39,190 --> 00:51:42,373
So I'm going to say that
there's a basis vector.

785
00:51:42,373 --> 00:51:43,790
I shouldn't have
said unit vector.

786
00:51:43,790 --> 00:51:46,090
I don't know its magnitude.

787
00:51:46,090 --> 00:51:48,080
I'm pointing along
the direction in which

788
00:51:48,080 --> 00:51:50,490
coordinate sigma is increasing.

789
00:51:50,490 --> 00:51:53,120
And so parallel
transporting this vector

790
00:51:53,120 --> 00:52:00,070
amounts to requiring that
my covariant derivative

791
00:52:00,070 --> 00:52:05,890
along the sigma
basis vector is 0.

792
00:52:09,690 --> 00:52:10,890
This can be written out.

793
00:52:10,890 --> 00:52:15,100
Turn this into index form.

794
00:52:23,828 --> 00:52:24,620
It looks like this.

795
00:52:41,500 --> 00:52:43,560
OK, no surprises.

796
00:52:43,560 --> 00:52:45,560
So now what I'm going
to do is, essentially,

797
00:52:45,560 --> 00:52:48,650
I'm going to write down an
integral that would describe

798
00:52:48,650 --> 00:53:20,510
how v alpha changes as I move
from A to B. When I do this,

799
00:53:20,510 --> 00:53:24,040
I will then get the value
of the vector at point B.

800
00:53:24,040 --> 00:53:28,990
So the way I'm going to
write this is v alpha at B

801
00:53:28,990 --> 00:53:33,670
is equal to v alpha, the
initial value of this thing,

802
00:53:33,670 --> 00:53:38,410
minus what I get when I
integrate along leg one--

803
00:53:38,410 --> 00:53:47,390
gamma alpha sigma
mu phi mu dx sigma.

804
00:53:47,390 --> 00:53:49,140
Everyone happy with that?

805
00:53:49,140 --> 00:53:52,480
So everything I've done
over here, so far, I think,

806
00:53:52,480 --> 00:53:53,740
is probably just fine.

807
00:53:53,740 --> 00:53:56,000
When you've got a differential
equation, integrate it.

808
00:53:56,000 --> 00:53:56,500
Boom.

809
00:53:56,500 --> 00:53:57,208
You integrate it.

810
00:53:57,208 --> 00:53:58,166
You got your new thing.

811
00:53:58,166 --> 00:54:00,041
We're going to actually
solve these integrals

812
00:54:00,041 --> 00:54:02,890
in a few moments, but we'll
just leave it like this for now.

813
00:54:02,890 --> 00:54:04,420
So that's the first step.

814
00:54:04,420 --> 00:54:07,050
I got a couple more to
do, but hopefully you

815
00:54:07,050 --> 00:54:08,050
can now see the pattern.

816
00:54:13,680 --> 00:54:18,180
If I go from B to
C, I am now moving

817
00:54:18,180 --> 00:54:24,350
in the direction
of lambda, and I'm

818
00:54:24,350 --> 00:54:26,970
holding the value of that
coordinate constant at x

819
00:54:26,970 --> 00:54:28,230
sigma plus dx sigma.

820
00:54:33,200 --> 00:54:36,670
So the vector at
C is going to be

821
00:54:36,670 --> 00:54:40,620
equal to this thing
at B minus what

822
00:54:40,620 --> 00:54:45,250
I get when I integrate along
path two, gamma alpha gamma mu.

823
00:54:54,630 --> 00:54:55,950
We got two more to go.

824
00:55:08,820 --> 00:55:10,580
So this one I am,
again, integrating

825
00:55:10,580 --> 00:55:14,091
along the sigma direction.

826
00:55:18,910 --> 00:55:21,382
But notice, I switched the sign.

827
00:55:21,382 --> 00:55:23,340
I switched the sign
because now my coordinate's

828
00:55:23,340 --> 00:55:25,173
going in the direction
where it's decreasing

829
00:55:25,173 --> 00:55:26,195
rather than increasing.

830
00:55:46,350 --> 00:55:47,360
Get some fresh chalk.

831
00:56:00,810 --> 00:56:05,190
So we've taken it
from A to B, B to C, C

832
00:56:05,190 --> 00:56:11,090
to D. Let's take it
all the way around.

833
00:56:23,940 --> 00:56:27,350
So take it all the way
around my second value

834
00:56:27,350 --> 00:56:36,900
at point A. This is
going to be v alpha at D.

835
00:56:36,900 --> 00:56:40,233
And again, this guy is coming
in the other direction.

836
00:56:40,233 --> 00:56:41,900
So I'll enter this
one with a plus sign.

837
00:56:49,630 --> 00:56:52,040
And I get this.

838
00:56:52,040 --> 00:56:55,630
OK, so the way I'm going
to quantify curvature

839
00:56:55,630 --> 00:56:57,310
is buried in all this stuff.

840
00:56:57,310 --> 00:56:58,338
Let's dig it out.

841
00:56:58,338 --> 00:57:00,130
So the first thing
which you're going to do

842
00:57:00,130 --> 00:57:06,240
is I'm going to say, if I take
v alpha final, basically what

843
00:57:06,240 --> 00:57:08,150
I want to do is
write this guy out,

844
00:57:08,150 --> 00:57:11,025
substitute in for v alpha d,
which requires me to substitute

845
00:57:11,025 --> 00:57:13,440
in for v alpha C, [INAUDIBLE].

846
00:57:13,440 --> 00:57:16,200
So I'm going to get
a big, old mess here.

847
00:57:16,200 --> 00:57:22,090
But in the end, the first
term will be v alpha initial.

848
00:57:22,090 --> 00:57:24,120
So let's subtract that off.

849
00:57:24,120 --> 00:57:25,890
That is the change.

850
00:57:30,858 --> 00:57:32,400
When you actually
work this out, it's

851
00:57:32,400 --> 00:57:35,792
going to involve four integrals.

852
00:57:44,470 --> 00:57:49,678
I have chosen to write
this in a way that

853
00:57:49,678 --> 00:57:51,720
highlights a property I'm
going to take advantage

854
00:57:51,720 --> 00:57:52,553
of in just a moment.

855
00:58:21,330 --> 00:58:24,070
OK, so the reason I
wrote it in this way--

856
00:58:24,070 --> 00:58:26,440
so I have the
integral along four

857
00:58:26,440 --> 00:58:29,020
minus that along two
plus integral along three

858
00:58:29,020 --> 00:58:30,742
minus that along one--

859
00:58:30,742 --> 00:58:32,450
is that each one that
I've written here--

860
00:58:32,450 --> 00:58:34,330
they represent
parts that are sort

861
00:58:34,330 --> 00:58:36,240
of parallel to each
other on the figure, just

862
00:58:36,240 --> 00:58:39,610
offset from each other
by a little bit, parallel

863
00:58:39,610 --> 00:58:40,630
but offset paths.

864
00:58:52,720 --> 00:58:54,210
Yeah, let's put this one high.

865
00:59:16,630 --> 00:59:17,630
Hang on just one moment.

866
00:59:17,630 --> 00:59:20,080
I have a thing in my notes that
said I needed to fix something.

867
00:59:20,080 --> 00:59:21,000
Did I actually fix it?

868
00:59:21,000 --> 00:59:21,320
Yeah, I did.

869
00:59:21,320 --> 00:59:21,820
OK.

870
00:59:32,090 --> 00:59:35,000
So schematically, let's
look at that first line.

871
00:59:35,000 --> 00:59:37,490
The integral along four of--

872
00:59:37,490 --> 00:59:43,910
I have a something, dx
lambda, minus the integral

873
00:59:43,910 --> 00:59:45,350
along path two--

874
00:59:48,690 --> 00:59:51,030
of a something, dx lambda.

875
00:59:55,650 --> 00:59:59,160
So it's the same basic
function inside each of these,

876
00:59:59,160 --> 01:00:06,940
but this one is being
evaluated at x sigma.

877
01:00:06,940 --> 01:00:14,260
This one is being evaluated
at x sigma plus dx sigma.

878
01:00:14,260 --> 01:00:16,670
I can combine them.

879
01:00:16,670 --> 01:00:22,590
So this becomes the integral--
let's say it's along two.

880
01:00:25,540 --> 01:00:27,050
Pardon me for a second.

881
01:00:27,050 --> 01:00:28,300
Make that a little bit bigger.

882
01:00:41,290 --> 01:00:45,660
So what I'm doing
is I'm saying that I

883
01:00:45,660 --> 01:00:47,540
have a function
evaluated at x sigma

884
01:00:47,540 --> 01:00:50,948
minus a function evaluated
at x sigma plus dx sigma.

885
01:00:50,948 --> 01:00:52,490
Let's do a little
binomial expansion.

886
01:00:52,490 --> 01:00:54,950
It's equivalent to an
integral along a single path

887
01:00:54,950 --> 01:00:57,567
of essentially what I
get, the first order

888
01:00:57,567 --> 01:00:58,400
Taylor term of that.

889
01:01:11,390 --> 01:01:12,950
Do the same thing
for the other guy.

890
01:01:20,290 --> 01:01:25,370
Integral along three,
I have a something,

891
01:01:25,370 --> 01:01:33,010
dx sigma minus integral along
1, same something, dx sigma.

892
01:01:33,010 --> 01:01:42,390
This guy is being evaled at
x lambda x delta x lambda.

893
01:01:42,390 --> 01:01:47,150
This guy is being
evaluated at x lambda.

894
01:01:47,150 --> 01:01:49,050
And so this whole
thing is approximately

895
01:01:49,050 --> 01:01:53,590
equal to integral along one.

896
01:02:05,270 --> 01:02:07,930
So it looks like this, OK?

897
01:02:07,930 --> 01:02:10,180
So if you want to do this a
little bit more carefully,

898
01:02:10,180 --> 01:02:10,990
knock yourself out.

899
01:02:14,025 --> 01:02:16,400
Part of that-- I probably
should've said this explicitly,

900
01:02:16,400 --> 01:02:18,190
but hopefully the
notation made it clear--

901
01:02:18,190 --> 01:02:21,012
I'm treating these little
deltas as small quantities, OK?

902
01:02:21,012 --> 01:02:23,470
So it makes sense that I can
introduce a little first order

903
01:02:23,470 --> 01:02:24,130
expansion here.

904
01:02:28,000 --> 01:02:30,895
Let's leave the picture up, but
I'm going to clear this board.

905
01:02:30,895 --> 01:02:35,006
With this way of doing things,
let's rewrite my integrals.

906
01:02:40,970 --> 01:02:47,740
So what this gives me is
delta vx alpha equals--

907
01:02:47,740 --> 01:02:49,660
should really be an
approximately equal

908
01:02:49,660 --> 01:02:52,550
because we're truncating
this expansion.

909
01:02:52,550 --> 01:02:53,860
So the integral from x sigma--

910
01:02:56,710 --> 01:02:59,850
x sigma equals dx sigma.

911
01:02:59,850 --> 01:03:11,670
Alpha x lambda [INAUDIBLE] x
lambda of gamma alpha sigma mu

912
01:03:11,670 --> 01:03:17,195
phi mu dx sigma minus--

913
01:03:42,770 --> 01:03:44,540
So this is just taking
what I wrote there.

914
01:03:44,540 --> 01:03:46,748
Schematically, this is what
you get when you actually

915
01:03:46,748 --> 01:03:47,930
expand all those guys out.

916
01:03:47,930 --> 01:03:48,430
All right.

917
01:03:51,380 --> 01:03:53,770
So I've got a couple
derivatives here.

918
01:03:53,770 --> 01:03:56,090
And I'm doing an
infinite test of a couple

919
01:03:56,090 --> 01:03:58,145
of infinitesimal integrals.

920
01:03:58,145 --> 01:03:59,770
When I'm doing
infinitesimal integrals,

921
01:03:59,770 --> 01:04:01,520
they're very simple to evaluate.

922
01:04:01,520 --> 01:04:04,810
So let's just go ahead, evaluate
them, and also expand out

923
01:04:04,810 --> 01:04:06,057
those derivatives.

924
01:04:36,380 --> 01:04:38,150
Doing so, this
cleans up a fair bit.

925
01:04:38,150 --> 01:04:39,860
First one, I'm going to
be able to finally get rid

926
01:04:39,860 --> 01:04:41,068
of those damn integral signs.

927
01:04:46,793 --> 01:04:49,210
So I'm going to wind up with
something that is essentially

928
01:04:49,210 --> 01:04:50,535
quadratic in these things.

929
01:04:50,535 --> 01:04:51,910
It's going to look
as the product

930
01:04:51,910 --> 01:04:54,560
of my little infinitesimal
displacements.

931
01:04:54,560 --> 01:04:57,910
And I'm going to wind
up with a term that

932
01:04:57,910 --> 01:05:00,490
involves a partial derivative
of my connection here.

933
01:05:12,030 --> 01:05:14,340
So we've got one
term looks like this.

934
01:05:14,340 --> 01:05:17,570
We've got another term
that looks like this.

935
01:05:27,588 --> 01:05:29,880
So don't worry about the
index gymnastics a little bit.

936
01:05:29,880 --> 01:05:32,290
If you go through it
carefully, you'll see it.

937
01:05:32,290 --> 01:05:33,820
Pause here for a second.

938
01:05:33,820 --> 01:05:36,640
This expression sucks.

939
01:05:36,640 --> 01:05:37,682
The reason why it sucks--

940
01:05:37,682 --> 01:05:39,515
it's not just because
there's lots of terms.

941
01:05:39,515 --> 01:05:40,960
There's a bajillion
indices on it.

942
01:05:40,960 --> 01:05:43,120
But it's because I've
got one term that

943
01:05:43,120 --> 01:05:45,460
is linear in the vector
and one that's linear

944
01:05:45,460 --> 01:05:47,120
in the derivative of vector.

945
01:05:47,120 --> 01:05:51,430
However, don't forget-- we get
the derivatives of the vector

946
01:05:51,430 --> 01:05:53,620
by parallel transport.

947
01:05:53,620 --> 01:05:56,710
So we parallel
transported this guy,

948
01:05:56,710 --> 01:06:11,190
which tells us that these
derivatives are simply related

949
01:06:11,190 --> 01:06:12,553
to the vectors themselves.

950
01:06:12,553 --> 01:06:14,220
If I move that to the
other side, that's

951
01:06:14,220 --> 01:06:18,180
equivalent to covariant
derivative of v equals 0.

952
01:06:18,180 --> 01:06:20,200
So if I want to get
rid of my derivative

953
01:06:20,200 --> 01:06:23,620
with respect to x lambda,
here's I you'd write that.

954
01:06:23,620 --> 01:06:26,160
Likewise, if I want to
get rid of-- and I do--

955
01:06:26,160 --> 01:06:32,920
my derivative with
respect to x sigma,

956
01:06:32,920 --> 01:06:34,660
just replace lambda with sigma.

957
01:06:45,950 --> 01:06:47,950
So now let's sub these in.

958
01:06:59,880 --> 01:07:09,530
Now, in 1980, the person
who became my PhD supervisor

959
01:07:09,530 --> 01:07:13,840
wrote this giant review article
on gravitational radiation.

960
01:07:13,840 --> 01:07:16,872
And either the last or second
to last section of the paper--

961
01:07:16,872 --> 01:07:18,080
I'm reminded of it right now.

962
01:07:18,080 --> 01:07:21,830
It begins with the
sentences, "The end is near.

963
01:07:21,830 --> 01:07:24,470
Redemption is at hand.

964
01:07:24,470 --> 01:07:25,670
The end is near.

965
01:07:25,670 --> 01:07:27,860
We shall soon be redeemed."

966
01:07:27,860 --> 01:07:28,430
All right.

967
01:07:28,430 --> 01:07:29,330
Let's plug these in.

968
01:07:32,620 --> 01:07:35,310
So we plug these guys in here.

969
01:07:35,310 --> 01:07:36,040
What do we get?

970
01:07:36,040 --> 01:07:41,190
Delta v alpha
equals these things.

971
01:07:44,560 --> 01:07:46,706
We're supposed to put
this in a parenthesis.

972
01:08:00,443 --> 01:08:02,360
And now, what's going
to happen when I get rid

973
01:08:02,360 --> 01:08:03,560
of all those
derivatives is I'm going

974
01:08:03,560 --> 01:08:06,185
to have a bunch of terms that
look like Christoffel squared.

975
01:08:32,210 --> 01:08:34,410
As Scooby Doo
would say, ruh-roh,

976
01:08:34,410 --> 01:08:37,808
but that is just
what we have to have.

977
01:08:37,808 --> 01:08:40,350
Incidentally, what you see when
you do something like this is

978
01:08:40,350 --> 01:08:42,479
I now have terms entering
into this whole thing that

979
01:08:42,479 --> 01:08:44,146
involved derivatives
of the metric times

980
01:08:44,146 --> 01:08:45,765
derivative of the metric.

981
01:08:45,765 --> 01:08:47,640
Many of you may have
heard sort of the slogan

982
01:08:47,640 --> 01:08:50,890
that general relativity is a
nonlinear theory of gravity.

983
01:08:50,890 --> 01:08:52,770
There's where your
nonlinearity is actually

984
01:08:52,770 --> 01:08:54,145
going to turn out
to be entering,

985
01:08:54,145 --> 01:08:57,000
is the fact you add these
squared terms in here that

986
01:08:57,000 --> 01:08:59,130
involve metric times
itself entering

987
01:08:59,130 --> 01:09:02,827
in such a non-trivial
and important way.

988
01:09:02,827 --> 01:09:04,410
What I'm going to
do, finally, on this

989
01:09:04,410 --> 01:09:06,029
is-- so this is
slightly annoying

990
01:09:06,029 --> 01:09:08,859
because I have one term
in v mu, one term in v nu.

991
01:09:08,859 --> 01:09:09,359
But look.

992
01:09:09,359 --> 01:09:11,883
Both mu and nu
are dummy indices.

993
01:09:11,883 --> 01:09:13,300
They're dummy
indices, so what I'm

994
01:09:13,300 --> 01:09:22,069
going to do is-- on the last
term or the last two terms,

995
01:09:22,069 --> 01:09:26,590
I'm just going to
exchange mu for nu.

996
01:09:26,590 --> 01:09:33,960
And what I finally get is that
the change in the vector v

997
01:09:33,960 --> 01:09:42,260
transported along a loop
whose sides are delta x lambda

998
01:09:42,260 --> 01:09:44,990
and delta sigma--

999
01:09:44,990 --> 01:09:48,220
it's a quantity that is linear
in those two displacements.

1000
01:09:48,220 --> 01:09:58,590
It's linear in the vector and
involves this four index tensor

1001
01:09:58,590 --> 01:10:15,930
whose value depends on
derivatives of the connection

1002
01:10:15,930 --> 01:10:21,873
and two nonlinear terms
in the connection.

1003
01:10:40,620 --> 01:10:48,810
This quantity is a mathematical
entity known as the Riemann

1004
01:10:48,810 --> 01:10:49,680
curvature tensor.

1005
01:11:04,200 --> 01:11:07,170
Even though it involves
connection coefficients,

1006
01:11:07,170 --> 01:11:09,030
Christoffel symbols,
and we argued before--

1007
01:11:09,030 --> 01:11:10,890
and you guys did a homework
exercise where you show this--

1008
01:11:10,890 --> 01:11:12,540
that the connection,
the Christoffel,

1009
01:11:12,540 --> 01:11:16,500
is not tensorial, this
combination of them,

1010
01:11:16,500 --> 01:11:20,550
basically that the terms come
together in such a way that,

1011
01:11:20,550 --> 01:11:22,290
when you change
your representation,

1012
01:11:22,290 --> 01:11:24,750
the nontensorial bits
cancel each other out

1013
01:11:24,750 --> 01:11:27,520
from the terms that are being
subtracted against one another.

1014
01:11:27,520 --> 01:11:29,250
So this is, indeed,
a true tensor.

1015
01:11:37,525 --> 01:11:38,900
There's an equivalent
definition,

1016
01:11:38,900 --> 01:11:40,067
if you are reading Carroll--

1017
01:11:42,390 --> 01:11:43,980
so essentially,
what I just walked

1018
01:11:43,980 --> 01:11:47,340
through here is an
integral equivalent

1019
01:11:47,340 --> 01:11:50,490
of the following
commutator being applied

1020
01:11:50,490 --> 01:11:55,890
to the vector v.
Some textbooks simply

1021
01:11:55,890 --> 01:12:00,430
state the Riemann
tensor is related

1022
01:12:00,430 --> 01:12:03,610
to the commutator of
partial derivatives acting

1023
01:12:03,610 --> 01:12:09,812
upon a four-vector like so.

1024
01:12:09,812 --> 01:12:12,270
With a little bit of effort,
you can show that what this is

1025
01:12:12,270 --> 01:12:15,172
is, essentially, a way of--

1026
01:12:15,172 --> 01:12:17,130
what I worked out over
there is a geometric way

1027
01:12:17,130 --> 01:12:20,310
of understanding what
that commutator means.

1028
01:12:20,310 --> 01:12:23,640
Incidentally, one thing, which
I think is worth calling out--

1029
01:12:28,160 --> 01:12:32,630
when you apply
this to a one-form

1030
01:12:32,630 --> 01:12:42,040
or a downstairs component, you
get this with a minus sign.

1031
01:12:42,040 --> 01:12:44,860
If you are reading the
textbook by Schutz,

1032
01:12:44,860 --> 01:12:47,530
Schutz has this sign wrong
in its first edition.

1033
01:12:54,580 --> 01:12:56,950
Hopefully all copies
of the first edition

1034
01:12:56,950 --> 01:13:00,040
are rare enough now that, if
you are looking at Schutz--

1035
01:13:00,040 --> 01:13:01,870
Schutz it's actually
a wonderful textbook

1036
01:13:01,870 --> 01:13:04,492
for an early introduction
to this field,

1037
01:13:04,492 --> 01:13:06,700
but if you happen to get a
hold of the first edition,

1038
01:13:06,700 --> 01:13:07,950
just be aware that there is--

1039
01:13:07,950 --> 01:13:10,370
I think it's on page
171 of the textbook,

1040
01:13:10,370 --> 01:13:12,940
you'll see this written.

1041
01:13:12,940 --> 01:13:14,650
So I've actually
written a couple papers

1042
01:13:14,650 --> 01:13:17,102
with Bernard Schutz, and so
I'm allowed to tease him.

1043
01:13:17,102 --> 01:13:18,310
Not only did he get it wrong.

1044
01:13:18,310 --> 01:13:19,990
He actually came up with
an intuitive argument

1045
01:13:19,990 --> 01:13:20,575
that is wrong.

1046
01:13:22,818 --> 01:13:25,360
Sometimes you just need to sit
down and bloody well calculate

1047
01:13:25,360 --> 01:13:27,400
something because you
can almost always come up

1048
01:13:27,400 --> 01:13:30,113
with an argument to convince you
of something that's not true.

1049
01:13:30,113 --> 01:13:32,530
And I'm afraid that's what he
did in this particular case.

1050
01:13:35,045 --> 01:13:37,420
So I have a couple notes in
there about what is sometimes

1051
01:13:37,420 --> 01:13:40,750
called curvature coupling,
which essentially tells us--

1052
01:13:40,750 --> 01:13:42,650
I pointed out in
the last lecture

1053
01:13:42,650 --> 01:13:44,410
that when we're
dealing with geodesics,

1054
01:13:44,410 --> 01:13:48,700
strictly speaking they describe,
completely point-like, almost

1055
01:13:48,700 --> 01:13:52,090
just a monopole and no structure
and no shape whatsoever moving

1056
01:13:52,090 --> 01:13:53,710
through spacetime.

1057
01:13:53,710 --> 01:13:56,080
If you have a larger
body or a body that

1058
01:13:56,080 --> 01:13:58,550
has any kind of multipolar
structure associated with it,

1059
01:13:58,550 --> 01:13:59,300
those multipoles--

1060
01:13:59,300 --> 01:14:01,675
you can think of that additional
structure is essentially

1061
01:14:01,675 --> 01:14:03,250
filling up part of
the local Lorentz

1062
01:14:03,250 --> 01:14:06,820
frame around the center
of mass of that point,

1063
01:14:06,820 --> 01:14:10,840
and they couple the
spacetime and push it away

1064
01:14:10,840 --> 01:14:12,220
from the geodesic.

1065
01:14:12,220 --> 01:14:14,380
This Riemann tensor
actually describes the way

1066
01:14:14,380 --> 01:14:17,200
in which that body couples
to the background spacetime,

1067
01:14:17,200 --> 01:14:19,360
that it might be falling in.

1068
01:14:19,360 --> 01:14:21,590
So this ends up playing
a really important role.

1069
01:14:21,590 --> 01:14:25,330
For instance, when you study
the precession of equinoxes,

1070
01:14:25,330 --> 01:14:27,992
we learn how to do this
Newtonian theory using

1071
01:14:27,992 --> 01:14:29,950
the action of tides from
the Earth and the moon

1072
01:14:29,950 --> 01:14:31,480
on a planet like the Earth.

1073
01:14:31,480 --> 01:14:32,980
This ends up being
the quantity that

1074
01:14:32,980 --> 01:14:35,750
mathematically encapsulates
tides and general relativity.

1075
01:14:35,750 --> 01:14:37,790
So it enters into there.

1076
01:14:37,790 --> 01:14:41,720
So I'm going to sketch
through this very,

1077
01:14:41,720 --> 01:14:45,700
very quickly, simply because
we don't have a lot of time,

1078
01:14:45,700 --> 01:14:48,910
and there's good discussion of
this in various other places.

1079
01:14:48,910 --> 01:14:51,462
But let me just point out that--

1080
01:14:51,462 --> 01:14:52,420
you look at this thing.

1081
01:14:52,420 --> 01:14:56,410
It's a four-index
tensor, and each index

1082
01:14:56,410 --> 01:14:57,355
can take four values.

1083
01:15:04,550 --> 01:15:09,410
That makes it look like
it has 256 components.

1084
01:15:12,188 --> 01:15:14,230
Now, I'm not going to step
through this in detail

1085
01:15:14,230 --> 01:15:15,070
right now.

1086
01:15:15,070 --> 01:15:18,700
This will either be in the
next lecture that I do this,

1087
01:15:18,700 --> 01:15:20,603
or you'll watch me
on a video once this

1088
01:15:20,603 --> 01:15:22,270
gets recorded, depending
upon how things

1089
01:15:22,270 --> 01:15:24,930
unroll in the next 24 hours.

1090
01:15:24,930 --> 01:15:27,040
Riemann has a lot of symmetries.

1091
01:15:27,040 --> 01:15:29,067
I will go through those
symmetries carefully,

1092
01:15:29,067 --> 01:15:30,400
either in lecture or on a video.

1093
01:15:40,400 --> 01:15:42,650
So symmetries-- Riemann--

1094
01:15:42,650 --> 01:15:43,400
and you know what?

1095
01:15:43,400 --> 01:15:53,620
Let me write it
out in n dimensions

1096
01:15:53,620 --> 01:15:59,350
from n to the 4, which is what
you'd expect for a four index

1097
01:15:59,350 --> 01:16:06,100
object in n
dimensions, down to n

1098
01:16:06,100 --> 01:16:12,330
squared times m squared
minus 1 over 12th.

1099
01:16:12,330 --> 01:16:13,705
So where I want
to conclude today

1100
01:16:13,705 --> 01:16:15,360
is let's just take
a look at what that

1101
01:16:15,360 --> 01:16:17,730
turns into for a couple
of different numbers

1102
01:16:17,730 --> 01:16:18,794
and dimensions.

1103
01:16:30,170 --> 01:16:33,650
So if you do n
equals 1, you get 0.

1104
01:16:33,650 --> 01:16:36,030
So the Riemann tensor
has no components

1105
01:16:36,030 --> 01:16:37,655
on a one-dimensional manifold.

1106
01:16:37,655 --> 01:16:39,030
There's a simple
reason for that.

1107
01:16:39,030 --> 01:16:40,488
Remember the way
we defined it, OK?

1108
01:16:40,488 --> 01:16:42,480
We did this by
parallel transporting

1109
01:16:42,480 --> 01:16:44,160
around a particular figure.

1110
01:16:44,160 --> 01:16:47,860
If you're in one dimension,
this is all you can do.

1111
01:16:47,860 --> 01:16:50,280
There's no holonomy
operation in one dimension.

1112
01:16:50,280 --> 01:16:52,050
You can't do that.

1113
01:16:52,050 --> 01:16:55,830
So no curvature.

1114
01:16:55,830 --> 01:16:58,228
If you want to be a real
pedant and someone says,

1115
01:16:58,228 --> 01:17:00,020
well, look at a curved
line, you'll go, ah,

1116
01:17:00,020 --> 01:17:01,103
but lines can't be curved.

1117
01:17:04,060 --> 01:17:06,120
n equals 2.

1118
01:17:06,120 --> 01:17:09,950
So you get 2 squared, 2
squared minus 1 over 12--

1119
01:17:09,950 --> 01:17:12,410
you get 1.

1120
01:17:12,410 --> 01:17:15,380
So if you're working
in two dimensions,

1121
01:17:15,380 --> 01:17:17,600
there is a single number
that characterizes

1122
01:17:17,600 --> 01:17:22,370
the curvature at every
point, and this is often

1123
01:17:22,370 --> 01:17:28,430
thought of as just a
radius of curvature.

1124
01:17:28,430 --> 01:17:30,620
Simplest example is
if you have a sphere.

1125
01:17:30,620 --> 01:17:34,040
Sphere is completely
characterized by its radius.

1126
01:17:34,040 --> 01:17:39,290
But if you imagine that it's
like a sphere that you squash,

1127
01:17:39,290 --> 01:17:41,570
well, you can imagine
at every point

1128
01:17:41,570 --> 01:17:43,340
that there is a
particular sphere that

1129
01:17:43,340 --> 01:17:45,590
is tangent to that
point, and the radius

1130
01:17:45,590 --> 01:17:48,620
of curvature of
the tangent sphere

1131
01:17:48,620 --> 01:17:52,057
is the one that defines the
curvature at that point.

1132
01:17:52,057 --> 01:17:54,640
I'm going to skip three because
it's not all that interesting.

1133
01:17:54,640 --> 01:17:58,330
The one that is more important
is, if you do n equals 4,

1134
01:17:58,330 --> 01:18:06,220
you'll wind up with 16 times
15 over 12, which is 20.

1135
01:18:06,220 --> 01:18:11,410
This is exactly the
number of derivatives

1136
01:18:11,410 --> 01:18:14,230
that we could not
cancel out when

1137
01:18:14,230 --> 01:18:18,010
we did the exercise a couple
of lectures ago of assessing

1138
01:18:18,010 --> 01:18:23,530
how well we can make spacetime
have a flat representation

1139
01:18:23,530 --> 01:18:24,910
in the vicinity of some point.

1140
01:18:34,070 --> 01:18:43,150
It's the number of leftover
constraints at second order

1141
01:18:43,150 --> 01:18:44,410
in a freely falling frame.

1142
01:18:53,320 --> 01:18:55,690
All right so I'm going
to stop there for today.

1143
01:18:55,690 --> 01:18:57,465
That's a nice place
for us to stop.

1144
01:19:00,720 --> 01:19:02,390
Keep watching your emails.

1145
01:19:02,390 --> 01:19:04,470
We're in an
interesting situation.

1146
01:19:04,470 --> 01:19:07,950
Life at MIT is evolving.

1147
01:19:07,950 --> 01:19:10,380
But when we pick it up
in one form or another,

1148
01:19:10,380 --> 01:19:13,230
what I'm going to do first
is talk a little bit more

1149
01:19:13,230 --> 01:19:14,640
about the symmetry
of this object

1150
01:19:14,640 --> 01:19:17,460
because there's a couple
of explicit symmetries

1151
01:19:17,460 --> 01:19:20,330
that lead to that reduction
from n to the 4th to n squared

1152
01:19:20,330 --> 01:19:22,800
n squared minus 1 over
12, and it's useful for us

1153
01:19:22,800 --> 01:19:25,350
to go through them and
see what they look like.

1154
01:19:25,350 --> 01:19:27,900
And then I also want to talk
about a couple of variants

1155
01:19:27,900 --> 01:19:29,520
on this curvature tensor, OK?

1156
01:19:29,520 --> 01:19:31,920
So just to give you a
little bit of a preview--

1157
01:19:31,920 --> 01:19:36,270
the curvature tensor--
it's a four index object.

1158
01:19:36,270 --> 01:19:44,165
We have argued
already that we're

1159
01:19:44,165 --> 01:19:45,540
going to end up
doing things that

1160
01:19:45,540 --> 01:19:47,880
look like looking at
derivatives of the metric

1161
01:19:47,880 --> 01:19:49,440
being equal to our source.

1162
01:19:49,440 --> 01:19:52,560
Our source is a two-index
object, the stress energy

1163
01:19:52,560 --> 01:19:53,770
tensor.

1164
01:19:53,770 --> 01:19:55,687
We've got gotta get
rid of two indices.

1165
01:19:55,687 --> 01:19:57,270
And so what we're
going to do is we're

1166
01:19:57,270 --> 01:20:00,570
going to essentially contract
this guy with a couple

1167
01:20:00,570 --> 01:20:04,080
of powers of the metric
in order to trace over

1168
01:20:04,080 --> 01:20:08,070
certain combinations of indices
and make two index variants

1169
01:20:08,070 --> 01:20:09,683
of the curvature tensor.

1170
01:20:09,683 --> 01:20:11,850
And we're also going to
look at derivatives of this,

1171
01:20:11,850 --> 01:20:14,520
because it turns out that there
is a particular combination

1172
01:20:14,520 --> 01:20:16,020
of derivatives at
the Riemann tensor

1173
01:20:16,020 --> 01:20:18,182
that has an important
geometrical meaning.

1174
01:20:18,182 --> 01:20:20,640
What we're going to find is
that, when we combine these two

1175
01:20:20,640 --> 01:20:24,660
notions, there is a
particular divergence

1176
01:20:24,660 --> 01:20:28,503
of a particular variant of the
curvature tensor that is 0.

1177
01:20:28,503 --> 01:20:30,420
In other words, we can
make a curvature tensor

1178
01:20:30,420 --> 01:20:32,490
that is divergence-free.

1179
01:20:32,490 --> 01:20:36,300
Our stress energy tensor
is divergence-free.

1180
01:20:36,300 --> 01:20:38,910
I wonder if one is
related to the other.

1181
01:20:38,910 --> 01:20:40,935
That, in a nutshell,
is how Einstein came up

1182
01:20:40,935 --> 01:20:43,080
with general relativity,
by asking that question

1183
01:20:43,080 --> 01:20:45,840
and then just seeing
what happened.

1184
01:20:45,840 --> 01:20:48,350
So that's what we're
going to go through next.