WEBVTT

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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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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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taken the equation
I wrote over there,

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and I had hit it with
g beta gamma, OK?

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I can always multiply
by those things.

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The covariant derivative
of the metric is zero.

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Because the covariant derivative
of the metric is zero,

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it commutes with
that derivative,

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so I can just walk it inside
the derivative operator.

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So I wanted to go through
that just a little

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bit carefully, because
that's actually

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a trick that once
you've seen it once,

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I want you to know it well.

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Because I'm just going to
do it many, many times as we

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move forward.

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There's going to be
a bunch of times,

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where I'm taking the partial--

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it's going to be the covariant
derivative is something.

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I know it's going be raising
indices willy nilly on whatever

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it's operating on.

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But I'm taking it
down to the fact

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that I'm effectively moving a
metric inside and outside here.

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All right, so let's take
that for the geodesy equation

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and expand it out.

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So I end up with mdp beta d
tau, so I'm here using the fact

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that the p alpha
that's on the outside,

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I'm writing that as mu alpha.

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And I'm using that to
convert the derivative I get,

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expanding that
into a d by d tau.

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And then I get a
term that basically

00:12:39.180 --> 00:12:42.195
corrects the downstairs index.

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Because it's a downstairs index,
it enters with a minus sign.

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Let's move this to the other
side, and what I'm going to do

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is make all the indices be
in the downstairs position.

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Let's see.

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Hang on a second.

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Did I do this right?

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Sorry.

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Yeah, I'm going
make all the indices

00:13:05.890 --> 00:13:09.377
in the downstairs position
on this capital gamma,

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so I'm going to write
this as follows.

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OK, so I chose to
write it this way

00:13:27.390 --> 00:13:28.800
as you'll see in just a moment.

00:13:28.800 --> 00:13:34.440
Because this is now symmetric
on exchange of alpha and gamma.

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Let's expand that
Christoffel symbol,

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so I'm going to have one term
beta derivative alpha gamma.

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We'll put this up above.

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OK, so take a look at that
last line of that expression.

00:14:23.070 --> 00:14:27.020
So as written here,
I've got a term

00:14:27.020 --> 00:14:30.770
that is symmetric on
exchange of alpha and gamma.

00:14:30.770 --> 00:14:33.110
But inside my parentheses,
bearing in mind

00:14:33.110 --> 00:14:35.450
that my metric is itself
a symmetric object,

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I've got two terms,
this one and this one,

00:14:38.210 --> 00:14:41.000
where if I exchange alpha and
gamma, I get a minus sign.

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So I got a symmetric
contracted with anti-symmetric.

00:15:10.550 --> 00:15:18.050
Therefore, I can
simplify this whole thing

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to something that
only involves--

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the only derivative
I need to compute

00:15:23.540 --> 00:15:29.525
is one partial
derivative of the metric.

00:15:37.920 --> 00:15:44.393
Now that's nice, but
if you think about it,

00:15:44.393 --> 00:15:46.060
it might even be nicer
than you realize.

00:15:53.890 --> 00:15:58.150
Suppose you're working in
some coordinate system, such

00:15:58.150 --> 00:16:01.130
that for a particular
derivative, for a particular,

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let's say, it's the
derivative with respect

00:16:03.070 --> 00:16:06.063
to your time coordinate.

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Suppose the metric vanishes.

00:16:29.490 --> 00:16:32.190
Suppose that equals zero
for some coordinate.

00:16:32.190 --> 00:16:35.745
Then you've just learned
that a particular component

00:16:35.745 --> 00:16:38.370
of the four momentum, component
of the downstairs four momentum

00:16:38.370 --> 00:16:43.591
mind you, that is a constant of
the motion along the worldwide.

00:17:11.109 --> 00:17:12.790
I sort of said in
words a few things

00:17:12.790 --> 00:17:14.290
about this a couple
lectures ago when we were

00:17:14.290 --> 00:17:15.540
talking about Killing vectors.

00:17:15.540 --> 00:17:17.260
I'm going to actually tie this
to that discussion in just

00:17:17.260 --> 00:17:18.170
a moment.

00:17:18.170 --> 00:17:20.140
This is often operationally
the simplest way

00:17:20.140 --> 00:17:22.790
to deduce that you, in fact,
have a constant motion,

00:17:22.790 --> 00:17:26.260
so there is some space
times, very complicated ones

00:17:26.260 --> 00:17:28.840
that play huge roles in many
of the kind of analysis we do.

00:17:28.840 --> 00:17:29.560
But you sort of look
at them, and you

00:17:29.560 --> 00:17:30.430
kind of go, oh, thank god.

00:17:30.430 --> 00:17:31.347
It's time independent.

00:17:31.347 --> 00:17:35.260
All right, that means I know
p downstairs t is constant.

00:17:35.260 --> 00:17:38.350
It's independent of
the actual angle.

00:17:38.350 --> 00:17:43.330
P downstairs phi is a
constant, and that ends up

00:17:43.330 --> 00:17:47.110
giving us some quantities
that we can exploit.

00:17:47.110 --> 00:17:49.480
And later when we start
talking about certain solutions

00:17:49.480 --> 00:17:50.897
of the field
equations and looking

00:17:50.897 --> 00:17:52.930
at the behavior of
these things, we're

00:17:52.930 --> 00:17:54.790
going to see how
we can exploit them

00:17:54.790 --> 00:17:59.740
to understand the motion of
bodies in very strong gravity.

00:17:59.740 --> 00:18:03.190
Before I do this, let me
connect what you're saying here

00:18:03.190 --> 00:18:06.210
to-- what I'm saying right here
to stuff that we did a lecture

00:18:06.210 --> 00:18:08.170
or two ago with
the Killing vector.

00:18:13.380 --> 00:18:16.330
So we know that, if a metric--

00:18:16.330 --> 00:18:18.830
so this is something that we
wrote down a little bit before.

00:18:18.830 --> 00:18:26.810
We also know that,
if the metric is

00:18:26.810 --> 00:18:29.210
independent of some
particular coordinate,

00:18:29.210 --> 00:18:39.523
there exists a Killing
field, or Killing vector,

00:18:39.523 --> 00:18:44.340
which I will call c beta.

00:18:44.340 --> 00:18:47.600
What I want to do now
is say, OK, how does--

00:18:47.600 --> 00:18:48.810
let's look at how.

00:18:53.230 --> 00:19:00.690
I'm going to define a
particular scalar, so what

00:19:00.690 --> 00:19:01.930
do I get when I take--

00:19:01.930 --> 00:19:02.430
oh, bugger.

00:19:06.380 --> 00:19:08.574
That made no sense.

00:19:08.574 --> 00:19:10.040
There we go.

00:19:10.040 --> 00:19:12.830
What would I get if I
take that Killing vector?

00:19:12.830 --> 00:19:15.860
I contract it with
my for momentum.

00:19:15.860 --> 00:19:19.928
How does this guy behave as
I evolve along a trajectory?

00:19:29.915 --> 00:19:31.790
So the way we're going
to solve this is we'll

00:19:31.790 --> 00:19:33.620
just look at the
time of evolution

00:19:33.620 --> 00:19:36.243
as we move along the
trajectory as we--

00:19:36.243 --> 00:19:37.910
and the way we'll do
that, I'll show you

00:19:37.910 --> 00:19:40.580
how to construct that time
evolution in just a moment.

00:19:40.580 --> 00:19:44.230
We're going to assume p
solves the geodesy equation,

00:19:44.230 --> 00:19:45.502
c solves Killing's equation.

00:19:45.502 --> 00:19:46.460
Let's see what happens.

00:19:51.880 --> 00:19:56.810
So what we're going to
do is look at d by d tau,

00:19:56.810 --> 00:19:59.880
so that the proper
covariant derivative

00:19:59.880 --> 00:20:02.450
along the trajectory
of this guy.

00:20:06.510 --> 00:20:10.740
Well, this, if I take advantage
of Leibniz's rule, so first

00:20:10.740 --> 00:20:12.465
of all, I can just
write this as--

00:20:14.887 --> 00:20:15.470
you know what?

00:20:15.470 --> 00:20:17.600
Let's throw an m into
here just to make

00:20:17.600 --> 00:20:19.085
things nice and symmetric.

00:20:19.085 --> 00:20:20.960
The reason I did
that is so that I

00:20:20.960 --> 00:20:28.462
can write that derivative
in the following form,

00:20:28.462 --> 00:20:29.920
so this is what I
want to evaluate.

00:20:29.920 --> 00:20:31.990
So one thing I do is
expand out that derivative

00:20:31.990 --> 00:21:01.230
using Leibniz's
rule, so this is--

00:21:01.230 --> 00:21:02.980
let's first plot
my Killing vector.

00:21:08.150 --> 00:21:11.210
That term is like this.

00:21:11.210 --> 00:21:17.506
Then I got a term, and
it looks like this.

00:21:20.820 --> 00:21:22.610
Well, the first term,
it's going to die.

00:21:22.610 --> 00:21:24.360
Because like I said,
I'm going to assume p

00:21:24.360 --> 00:21:28.830
solves the geodesy
equation, so p is a geodesy.

00:21:28.830 --> 00:21:31.350
I kill that.

00:21:31.350 --> 00:21:33.820
What about the second term?

00:21:33.820 --> 00:21:36.270
Well, for the second
term, what I'm going to do

00:21:36.270 --> 00:21:42.420
is note that whenever you have
some general two index object,

00:21:42.420 --> 00:21:46.080
so suppose I have some two
index tensor, m alpha beta.

00:21:46.080 --> 00:21:53.790
I can always write this in
the following way, right?

00:21:53.790 --> 00:22:05.340
Where remember, the parentheses
denote the symmetric part,

00:22:05.340 --> 00:22:08.970
and the braces denote
the anti-symmetric part.

00:22:15.795 --> 00:22:17.170
So this is just
a theorem, right?

00:22:17.170 --> 00:22:18.003
You add it together.

00:22:18.003 --> 00:22:20.810
The 1/2s combine, and
you get this thing back

00:22:20.810 --> 00:22:21.607
for the first term.

00:22:21.607 --> 00:22:23.690
And they combine the minus
with the other one, OK?

00:22:23.690 --> 00:22:25.400
Very simple identity.

00:22:25.400 --> 00:22:47.020
So if I do that,
applying it up here,

00:22:47.020 --> 00:22:49.990
this is symmetric under
exchange of indices.

00:22:49.990 --> 00:22:51.190
This is anti-symmetric.

00:22:51.190 --> 00:22:52.270
It dies.

00:22:52.270 --> 00:22:54.400
The only thing that
is left is this term.

00:23:18.380 --> 00:23:23.930
But if this is a Killing
vector by Killing's equation,

00:23:23.930 --> 00:23:26.540
this equals zero by
Killing's equation.

00:23:37.720 --> 00:23:39.760
So the importance
of this, you've

00:23:39.760 --> 00:23:49.810
just shown that what you
get when you contract

00:23:49.810 --> 00:23:53.160
for momentum with
the Killing vector

00:23:53.160 --> 00:23:54.870
gives you a constant motion.

00:23:54.870 --> 00:23:59.160
You've also shown that the
component of the four momentum,

00:23:59.160 --> 00:24:02.250
the downstairs component of
the four momentum associated

00:24:02.250 --> 00:24:05.130
with whatever
coordinate the metric

00:24:05.130 --> 00:24:09.120
happens to be independent of, is
also a constant of the motion.

00:24:09.120 --> 00:24:12.630
The key thing to note is both
are actually very powerful

00:24:12.630 --> 00:24:14.370
and important statements.

00:24:14.370 --> 00:24:17.370
One depends on the coordinate
system and the representation

00:24:17.370 --> 00:24:18.180
you've chosen.

00:24:18.180 --> 00:24:20.880
The other does not, OK?

00:24:20.880 --> 00:24:24.900
So this is really
true and useful,

00:24:24.900 --> 00:24:27.660
if you happen to have chosen
the coordinate system such

00:24:27.660 --> 00:24:33.120
that this derivative
is equal to zero.

00:24:33.120 --> 00:24:35.040
This is true,
though, independent

00:24:35.040 --> 00:24:37.620
of your representation,
so these are just

00:24:37.620 --> 00:24:42.330
two different ways of calling
out constants of motion.

00:24:42.330 --> 00:24:44.942
And we actually find both
of them to be very useful,

00:24:44.942 --> 00:24:46.650
so we're going to take
advantage of them.

00:24:46.650 --> 00:24:48.483
There's a variation on
this calculation that

00:24:48.483 --> 00:24:55.890
is on the next p set, and
you will come back to this,

00:24:55.890 --> 00:24:59.070
again, when we start talking
about motion in certain space

00:24:59.070 --> 00:25:01.710
times in the second
half of this course.

00:25:05.188 --> 00:25:07.230
So let me just do a couple
really quick examples.

00:25:07.230 --> 00:25:09.330
I've already kind
of mentioned these,

00:25:09.330 --> 00:25:11.190
but let me give names
to what these are.

00:25:14.280 --> 00:25:18.170
So if your space time has
a time coordinate, such

00:25:18.170 --> 00:25:22.670
that the time derivative of
any metric element is zero,

00:25:22.670 --> 00:25:26.660
then you know that a time
like Killing vector, which

00:25:26.660 --> 00:25:30.940
I will call ct, and I'm
leaving the vector sign on it.

00:25:30.940 --> 00:25:46.630
This thing exists, and you
also know that p downstairs t

00:25:46.630 --> 00:25:49.000
is constant.

00:25:49.000 --> 00:25:56.500
Now the name that is given
to this is negative energy.

00:25:56.500 --> 00:25:59.090
Why negative?

00:25:59.090 --> 00:26:00.830
Well, the main reason
why it's negative

00:26:00.830 --> 00:26:02.210
is that we will often--

00:26:02.210 --> 00:26:04.700
so let me just caution that
this is not always an identity

00:26:04.700 --> 00:26:05.470
that we're going to use.

00:26:05.470 --> 00:26:07.553
We're going to use it in
a huge number of problems

00:26:07.553 --> 00:26:09.500
that we care about.

00:26:09.500 --> 00:26:17.420
Many of the space times that
we are going to work with

00:26:17.420 --> 00:26:20.870
are those that when you get
really far away from whatever

00:26:20.870 --> 00:26:22.820
source is generating
your gravity,

00:26:22.820 --> 00:26:25.100
it looks just like
special relativity.

00:26:25.100 --> 00:26:29.480
We call such space times
asymptotically flat.

00:26:29.480 --> 00:26:36.590
In other words, as you get
asymptotically far away,

00:26:36.590 --> 00:26:38.993
it reduces to the
space time that we

00:26:38.993 --> 00:26:40.910
studied in the first
couple weeks of the class

00:26:40.910 --> 00:26:43.320
when we were doing geometric
spatial relativity.

00:26:43.320 --> 00:26:47.975
And in that case, we knew the
timelike component was energy.

00:26:47.975 --> 00:26:49.850
And in a flat space
time, you lower that time

00:26:49.850 --> 00:26:51.200
like component's index.

00:26:51.200 --> 00:26:53.030
You get minus energy.

00:26:53.030 --> 00:26:55.160
It just so happens when
you go through the math

00:26:55.160 --> 00:26:58.220
carefully that
negative of the energy

00:26:58.220 --> 00:27:00.640
ends up defined in this way.

00:27:00.640 --> 00:27:02.570
It's going to be the
quantity that is actually

00:27:02.570 --> 00:27:03.910
conserved everywhere.

00:27:03.910 --> 00:27:05.628
What's kind of
cool is that we use

00:27:05.628 --> 00:27:07.295
this associated with
asymptotic flatness

00:27:07.295 --> 00:27:08.930
to give you some intuition.

00:27:08.930 --> 00:27:10.730
But this is actually
true, even if you're

00:27:10.730 --> 00:27:13.610
right outside of the vicinity of
a rapidly rotating black hole.

00:27:13.610 --> 00:27:15.290
It is still the case
that p downstairs

00:27:15.290 --> 00:27:17.660
t for the right choice
of t is a constant.

00:27:23.690 --> 00:27:25.390
In my notes, I also
show you that there

00:27:25.390 --> 00:27:30.470
is an example of an
actual Killing vector that

00:27:30.470 --> 00:27:31.850
corresponds to angle momentum.

00:27:31.850 --> 00:27:34.950
Again, we'll come back to
that a little bit later,

00:27:34.950 --> 00:27:37.633
so let me do one
example geodesy before I

00:27:37.633 --> 00:27:39.050
sort of change
topic a little bit.

00:27:44.710 --> 00:27:47.140
I'm going to write
down a space time

00:27:47.140 --> 00:27:53.170
that we either in
person or on video

00:27:53.170 --> 00:27:56.290
are going to derive basically
right after Spring break.

00:28:04.390 --> 00:28:07.497
So suppose I hand you
the following space time.

00:28:35.060 --> 00:28:38.190
This function phi, I'm not going
to say too much about it quite

00:28:38.190 --> 00:28:39.330
yet.

00:28:39.330 --> 00:28:43.673
What I will say is that
it is small in the sense

00:28:43.673 --> 00:28:45.840
that when you're doing
various calculations with it,

00:28:45.840 --> 00:28:49.770
feel free to discard terms of
order five squared or higher.

00:28:49.770 --> 00:28:57.330
And it only depends on the
coordinates x, y, and z.

00:28:57.330 --> 00:28:58.719
No time dependence.

00:29:03.610 --> 00:29:07.050
I want to examine slow
motion in the space time.

00:29:12.610 --> 00:29:14.110
So what I'm going
to do is I'm going

00:29:14.110 --> 00:29:20.392
to imagine that my four
momentum has the usual form.

00:29:20.392 --> 00:29:22.850
And I can think of it as having
an energy, and sort of time

00:29:22.850 --> 00:29:28.420
like component, and
momentum in the space like.

00:29:28.420 --> 00:29:31.090
The magnitude of
the energy is always

00:29:31.090 --> 00:29:35.950
going to be much greater than
the magnitude of the momentum,

00:29:35.950 --> 00:29:40.060
and in fact, will
be approximately

00:29:40.060 --> 00:29:47.060
equal to the mass, where that
is the mass of whatever body is

00:29:47.060 --> 00:29:49.010
actually undergoing this
motion in this space

00:29:49.010 --> 00:29:49.843
that I've given you.

00:29:53.250 --> 00:29:58.017
OK, so the reason why I'm doing
this is what we're going to do

00:29:58.017 --> 00:29:59.850
is you want to say,
what does free fall look

00:29:59.850 --> 00:30:01.050
like in this space?

00:30:05.160 --> 00:30:29.380
Well, I look at geodesics, so
there is my geodesy equation.

00:30:29.380 --> 00:30:32.350
This slow motion
condition that I've

00:30:32.350 --> 00:30:35.590
applied over here, that tells
me that when I expand all

00:30:35.590 --> 00:30:43.670
these terms out here, this
is going to be dominated

00:30:43.670 --> 00:30:47.090
by the time time term, OK?

00:30:47.090 --> 00:30:49.280
So it'll be dominated
because of the fact

00:30:49.280 --> 00:30:53.240
that when you just look at the
numerical magnitude of the size

00:30:53.240 --> 00:30:59.000
of the components
of the momentum,

00:30:59.000 --> 00:31:01.940
those are going to be the ones
that dominate this calculation.

00:31:01.940 --> 00:31:04.610
Everything else, if you put
your factors of c back in,

00:31:04.610 --> 00:31:06.530
they're going to
be down by factors

00:31:06.530 --> 00:31:08.960
that look like v over c.

00:31:12.110 --> 00:31:18.420
So my geodesic
equation turns into--

00:31:31.190 --> 00:31:33.410
so all I did was say,
it's dominated by this.

00:31:33.410 --> 00:31:35.702
I'm going to move it to the
other side of the equation.

00:31:55.680 --> 00:31:59.960
So without giving away
the plot, the beta

00:31:59.960 --> 00:32:01.710
equals zero component
is going to turn out

00:32:01.710 --> 00:32:04.030
to be very uninteresting.

00:32:04.030 --> 00:32:05.270
Can you see why?

00:32:05.270 --> 00:32:07.790
Beta equals zero is energy.

00:32:07.790 --> 00:32:09.280
When I evaluate
that Christoffel,

00:32:09.280 --> 00:32:10.930
I'm going to end up
taking a bunch of-- you

00:32:10.930 --> 00:32:12.555
know, I'm going to
have all these zero,

00:32:12.555 --> 00:32:14.222
zero, zero, time,
time, time components.

00:32:14.222 --> 00:32:16.597
It's time independent, though,
so all my time derivatives

00:32:16.597 --> 00:32:17.600
are going to be zero.

00:32:17.600 --> 00:32:19.490
It's going to vanish,
but we expect that.

00:32:19.490 --> 00:32:22.572
Because it's a time
independent metric,

00:32:22.572 --> 00:32:24.530
all the crap I went
through a couple months ago

00:32:24.530 --> 00:32:27.800
guarantees that the time like
components, energy's conserved,

00:32:27.800 --> 00:32:28.340
right?

00:32:28.340 --> 00:32:30.180
So that's kind of
what we expect,

00:32:30.180 --> 00:32:32.750
so let's just focus
on-- and if we

00:32:32.750 --> 00:32:35.870
had infinite time, which
we clearly don't, it

00:32:35.870 --> 00:32:37.220
would be fun to talk about.

00:32:37.220 --> 00:32:40.370
Let's just move on and look at
the spatial component of this.

00:32:40.370 --> 00:32:46.954
So as I look at the spatial
component of this guy,

00:32:46.954 --> 00:32:48.920
let's focus on beta equals i.

00:32:51.510 --> 00:32:55.050
What you find when you
actually evaluate this guy is

00:32:55.050 --> 00:32:56.040
this turns into--

00:33:13.960 --> 00:33:16.313
OK, so go ahead and look up
your Christoffel formulas.

00:33:16.313 --> 00:33:18.730
Again, these are one of those
things that, eventually, you

00:33:18.730 --> 00:33:20.480
get this memorized by
the end of the term,

00:33:20.480 --> 00:33:23.890
but don't feel bad if
you keep forgetting it.

00:33:23.890 --> 00:33:24.820
No time derivative.

00:33:24.820 --> 00:33:27.130
No time derivative.

00:33:27.130 --> 00:33:29.410
The only thing that's
left at the end of the day

00:33:29.410 --> 00:33:32.680
is essentially a gradient of the
time time piece of the metric.

00:33:35.990 --> 00:33:40.720
So taking advantage
of the fact gi

00:33:40.720 --> 00:33:45.470
alpha staying in the
upstairs position,

00:33:45.470 --> 00:33:46.880
you can write this as--

00:33:50.500 --> 00:33:52.580
it looks like this.

00:33:52.580 --> 00:33:55.310
Now, if you like-- remember,
phi is a small value.

00:33:55.310 --> 00:33:57.080
You can do binomial
expansion on that.

00:33:57.080 --> 00:33:57.920
Knock yourself out.

00:33:57.920 --> 00:33:59.920
It's not going to be
important in just a second.

00:34:06.780 --> 00:34:10.909
Excuse me, minus one
half one minus two,

00:34:10.909 --> 00:34:12.440
five to the minus one power.

00:34:22.587 --> 00:34:24.170
So let me talk a
little bit about what

00:34:24.170 --> 00:34:27.230
I did here in this last line.

00:34:27.230 --> 00:34:30.989
My delta i alpha is coupling
to a partial derivative,

00:34:30.989 --> 00:34:33.830
so that partial derivative,
the zero component of that

00:34:33.830 --> 00:34:35.360
is the time derivative.

00:34:35.360 --> 00:34:37.610
Everything's time
independent, so that's done.

00:34:37.610 --> 00:34:39.409
Let's just skip
it, so what I did

00:34:39.409 --> 00:34:41.270
was I changed my alpha to a j.

00:34:41.270 --> 00:34:43.757
Because I only want
spatial derivatives,

00:34:43.757 --> 00:34:44.840
so I'm allowed to do that.

00:34:44.840 --> 00:34:46.840
Because I'm just acknowledging
the fact that all

00:34:46.840 --> 00:34:48.830
the time derivatives
are uninteresting.

00:34:48.830 --> 00:34:52.940
When I do
differentiate g00, I am

00:34:52.940 --> 00:34:57.300
differentiating negative
quantity one plus two phi.

00:34:57.300 --> 00:34:58.850
The one doesn't contribute.

00:34:58.850 --> 00:35:01.162
All that is left is I took
the minus on the inside,

00:35:01.162 --> 00:35:02.120
and there's my two phi.

00:35:19.380 --> 00:35:21.810
So putting all of these
ingredients together,

00:35:21.810 --> 00:35:30.140
I at last get my
Christoffel is delta ij.

00:35:30.140 --> 00:35:35.305
I'm saying it looks like
spatial gradient of phi.

00:35:35.305 --> 00:35:38.138
And for keeping score, there
are higher order terms,

00:35:38.138 --> 00:35:40.180
which we're going to
collect under the assumption

00:35:40.180 --> 00:35:42.280
that this phi is small.

00:35:42.280 --> 00:35:44.155
Plug it back into my
equation of motion.

00:36:00.100 --> 00:36:02.740
I end up with this.

00:36:02.740 --> 00:36:05.800
Now let's cancel out the
m's that appeared in here.

00:36:10.040 --> 00:36:14.270
If we were not doing relativity,
we would write this as--

00:36:14.270 --> 00:36:16.880
ignore the fact that this
is per unit proper time.

00:36:16.880 --> 00:36:22.670
This is dpdt is minus gradients
of something that sure as hell

00:36:22.670 --> 00:36:23.974
looks like a potential.

00:36:47.220 --> 00:36:49.740
What we are going to
do in the lecture rate

00:36:49.740 --> 00:36:54.860
after Spring break, so
going into Spring break,

00:36:54.860 --> 00:36:56.940
depending on which-- as
I said at the beginning,

00:36:56.940 --> 00:36:58.140
it's a little unclear
how these lectures

00:36:58.140 --> 00:36:58.710
are going to be delivered.

00:36:58.710 --> 00:36:59.418
But bear with me.

00:36:59.418 --> 00:37:01.377
We're going to essentially
put together-- we're

00:37:01.377 --> 00:37:03.000
going to take all
the last ingredients

00:37:03.000 --> 00:37:04.500
and develop the
field equations that

00:37:04.500 --> 00:37:06.455
describe relativistic gravity.

00:37:06.455 --> 00:37:07.830
The first thing
we're going to do

00:37:07.830 --> 00:37:10.530
is solve this in a
particular limit that

00:37:10.530 --> 00:37:13.590
describes a body that
is weakly gravitating.

00:37:13.590 --> 00:37:17.400
This will emerge
from this with phi

00:37:17.400 --> 00:37:21.600
being equal to Newtonian
gravitational potential.

00:37:21.600 --> 00:37:24.070
What this is showing is
that the geodesic equation,

00:37:24.070 --> 00:37:26.220
this equation that
describes a trajectory that

00:37:26.220 --> 00:37:30.060
is as straight as possible in
space time when it is given

00:37:30.060 --> 00:37:32.040
that particular
space time, it does

00:37:32.040 --> 00:37:35.960
give you the Newtonian
equation of motion, OK?

00:37:35.960 --> 00:37:36.551
Yeah?

00:37:36.551 --> 00:37:37.964
AUDIENCE: Is there
still supposed

00:37:37.964 --> 00:37:40.710
to be a factor of m on
the right-hand side?

00:37:40.710 --> 00:37:42.860
SCOTT HUGHES: No, so
it's possible I dropped

00:37:42.860 --> 00:37:43.860
an m somewhere in there.

00:37:43.860 --> 00:37:46.950
Go through that just a
little bit carefully here,

00:37:46.950 --> 00:37:49.590
but you know, it's meant to be--

00:37:53.180 --> 00:37:54.180
actually, you know what?

00:37:54.180 --> 00:37:54.870
I take it back.

00:37:54.870 --> 00:37:56.722
Sorry, so I think I
may have messed up.

00:37:56.722 --> 00:37:58.680
I remember what it was,
I remember what it was.

00:37:58.680 --> 00:38:00.210
There's an m squared here.

00:38:00.210 --> 00:38:01.620
Thank you, Alex.

00:38:01.620 --> 00:38:03.450
There was an m squared here.

00:38:03.450 --> 00:38:06.858
There was an m and
an m squared here.

00:38:06.858 --> 00:38:08.400
That's what I screwed
up, and I think

00:38:08.400 --> 00:38:10.110
I have that wrong in my
handwritten notes, which is

00:38:10.110 --> 00:38:11.480
probably why I messed that up.

00:38:11.480 --> 00:38:13.410
Yeah, so this
equation was correct.

00:38:13.410 --> 00:38:15.270
And this should have
been here like so.

00:38:15.270 --> 00:38:17.690
We'll clear this
out and get this.

00:38:17.690 --> 00:38:20.978
Yeah, so module of
that little bobble.

00:38:20.978 --> 00:38:22.770
I just want to show
you this is essentially

00:38:22.770 --> 00:38:24.390
the Newtonian limit.

00:38:24.390 --> 00:38:27.390
Just give you a little
look at where we go ahead,

00:38:27.390 --> 00:38:28.890
there are two ways
that we are going

00:38:28.890 --> 00:38:31.920
to derive the field equations
of general relativity.

00:38:31.920 --> 00:38:34.440
The first one
essentially boils down

00:38:34.440 --> 00:38:38.610
to looking for certain tensors
that have the right symmetries

00:38:38.610 --> 00:38:42.270
and allow us to have
sort of a quantity that

00:38:42.270 --> 00:38:45.390
looks like derivatives on
the field equaling the stress

00:38:45.390 --> 00:38:47.040
energy tensor as the source.

00:38:47.040 --> 00:38:51.310
That only works up to
an overall constant,

00:38:51.310 --> 00:38:54.030
and this is actually the
way that Einstein originally

00:38:54.030 --> 00:38:57.450
developed the field equations,
was worked out all this stuff,

00:38:57.450 --> 00:39:00.390
and then by insisting that the
solution that emerge from this

00:39:00.390 --> 00:39:02.037
reproduced the
Newtonian equation

00:39:02.037 --> 00:39:03.870
and the Newtonian motion,
he was able to fix

00:39:03.870 --> 00:39:06.675
what that constant actually is.

00:39:06.675 --> 00:39:08.550
There's a more sophisticated
way of doing it,

00:39:08.550 --> 00:39:10.820
which I'm going to
also go through.

00:39:10.820 --> 00:39:13.260
But it's worth noting that is
the way Einstein originally

00:39:13.260 --> 00:39:14.550
did it.

00:39:14.550 --> 00:39:18.040
I had the privilege a
couple of years ago--

00:39:18.040 --> 00:39:20.680
I was at a conference
in Jerusalem,

00:39:20.680 --> 00:39:24.120
where the Einstein Papers
archive is located.

00:39:24.120 --> 00:39:27.080
And the guy who is the
main curator of this

00:39:27.080 --> 00:39:28.830
was allowing those of
us at the conference

00:39:28.830 --> 00:39:29.910
to look through them.

00:39:29.910 --> 00:39:31.753
And I actually found--

00:39:31.753 --> 00:39:33.420
they had not yet quite
categorized that,

00:39:33.420 --> 00:39:36.200
but it was the papers very
much related to working

00:39:36.200 --> 00:39:37.450
with these peculiar equations.

00:39:37.450 --> 00:39:39.575
I don't think he was trying
to fix the coefficient,

00:39:39.575 --> 00:39:43.860
but this spacetime can also be
used to compute the perihelion

00:39:43.860 --> 00:39:46.387
precession of mercury.

00:39:46.387 --> 00:39:48.720
And so it actually showed
Einstein working through that.

00:39:48.720 --> 00:39:53.230
And the thing which is really
cool was he screwed up a lot.

00:39:53.230 --> 00:39:55.570
The page that I was looking
at was full of errors.

00:39:55.570 --> 00:39:56.880
It would say things like--

00:39:56.880 --> 00:39:58.680
big things crossed out
and then "Nein nein nein!"

00:39:58.680 --> 00:39:59.520
written on the side.

00:39:59.520 --> 00:40:03.670
And so it made me feel
better about myself.

00:40:03.670 --> 00:40:05.383
All right.

00:40:05.383 --> 00:40:07.050
So everything that
we have done so far--

00:40:07.050 --> 00:40:10.950
we've been dancing
around this notion

00:40:10.950 --> 00:40:12.708
of what is called "curvature."

00:40:16.700 --> 00:40:19.280
So I have used this
word several times,

00:40:19.280 --> 00:40:21.530
but I haven't made
this precise yet.

00:40:21.530 --> 00:40:24.170
Curvature is going to
be the precise idea

00:40:24.170 --> 00:40:28.540
of how two initially
parallel trajectories

00:40:28.540 --> 00:40:29.450
cease to be parallel.

00:40:53.950 --> 00:40:59.650
So there's a couple of ways
that we can quantify this.

00:40:59.650 --> 00:41:01.990
The one which I
am going to use is

00:41:01.990 --> 00:41:06.280
one that's amendable
to, with relative ease,

00:41:06.280 --> 00:41:10.960
developing a particularly
important tensor, which

00:41:10.960 --> 00:41:12.910
characterizes curvature.

00:41:12.910 --> 00:41:14.650
And so what we're
going to do is look

00:41:14.650 --> 00:41:17.950
at the behavior of a
vector that is parallel

00:41:17.950 --> 00:41:26.310
transported in a
non-infinitesimal region

00:41:26.310 --> 00:41:27.185
of a curved manifold.

00:41:46.330 --> 00:41:47.890
So for the purpose
of this sketch,

00:41:47.890 --> 00:41:50.710
I'm going to make this
closed figure be a triangle.

00:41:50.710 --> 00:41:52.120
When I actually do the
calculation in just a moment,

00:41:52.120 --> 00:41:53.830
I'm going to use a
little parallelogram.

00:41:53.830 --> 00:41:59.940
So around a closed figure,
I want a curved manifold.

00:42:07.320 --> 00:42:09.300
So suppose my
curvature's actually 0,

00:42:09.300 --> 00:42:20.990
and I do this for a triangle
that is on the blackboard.

00:42:20.990 --> 00:42:26.258
So let's say I start out with a
vector that points from A to B.

00:42:26.258 --> 00:42:28.550
And what I'm going to do is
just parallel transport it.

00:42:28.550 --> 00:42:32.190
And in the case, goes, doo, doo,
doo, doo, doo, doo, doo, doo,

00:42:32.190 --> 00:42:35.452
doo, doo, doo, doo.

00:42:35.452 --> 00:42:37.160
This is an experiment
you can do at home.

00:42:37.160 --> 00:42:39.620
When it comes back,
it's pointing exactly

00:42:39.620 --> 00:42:41.098
the way it was initially.

00:42:57.710 --> 00:43:02.510
Let me also just note
that this triangle--

00:43:02.510 --> 00:43:07.820
the sum of its internal
angles is 180 degrees.

00:43:14.033 --> 00:43:15.200
Hopefully you all know that.

00:43:18.810 --> 00:43:21.750
Now, the next one--

00:43:21.750 --> 00:43:24.520
if I'd had a little
bit more time,

00:43:24.520 --> 00:43:28.710
I would have grabbed one
of my daughter's balls

00:43:28.710 --> 00:43:30.000
to demonstrate this.

00:43:30.000 --> 00:43:31.708
But hopefully, if you
guys have something

00:43:31.708 --> 00:43:34.890
like a soccer ball
or a basketball,

00:43:34.890 --> 00:43:38.310
this is a little experiment
you can do by yourself.

00:43:38.310 --> 00:43:44.505
Now imagine a triangle
that is embedded

00:43:44.505 --> 00:43:45.630
on the surface of a sphere.

00:43:51.810 --> 00:43:56.940
So let's say this is the
North Pole of my sphere.

00:43:59.750 --> 00:44:00.605
Here's the equator.

00:44:03.730 --> 00:44:06.540
So what I'm going
to imagine is--

00:44:06.540 --> 00:44:08.100
let's say I start up here.

00:44:08.100 --> 00:44:11.035
Let's make the North
Pole be point A.

00:44:11.035 --> 00:44:14.610
I move on a trajectory
that is as straight as I

00:44:14.610 --> 00:44:15.510
am allowed to be.

00:44:15.510 --> 00:44:18.030
And remember, if I'm a
one-dimensional being living

00:44:18.030 --> 00:44:20.490
on the surface of this thing,
that's a straight line.

00:44:20.490 --> 00:44:22.885
It only looks straight
looks curved to us

00:44:22.885 --> 00:44:25.260
because we see a third dimension
that this whole thing is

00:44:25.260 --> 00:44:26.873
embedded in.

00:44:26.873 --> 00:44:28.290
And this thing's
going to come in,

00:44:28.290 --> 00:44:31.680
and it actually hits the
equator at a right angle, OK?

00:44:31.680 --> 00:44:33.360
No ifs, ands, or buts about it.

00:44:33.360 --> 00:44:35.790
It's a bloody right angle.

00:44:35.790 --> 00:44:36.930
And then I'm going to--

00:44:36.930 --> 00:44:38.970
let's call this point B--

00:44:38.970 --> 00:44:41.640
walk back along the
equator here till I

00:44:41.640 --> 00:44:45.200
reach a point which
I will call C.

00:44:45.200 --> 00:44:48.680
And then I'm going to go
straight north until I

00:44:48.680 --> 00:44:52.450
come back up to the North Pole.

00:44:55.780 --> 00:44:59.590
This is a triangle in which all
three angles are 90 degrees.

00:45:11.080 --> 00:45:12.690
So here is a great
little experiment

00:45:12.690 --> 00:45:15.127
that's very easy for
you to do at home.

00:45:15.127 --> 00:45:16.960
Does anyone happen to
have a ball with them?

00:45:16.960 --> 00:45:17.640
OK, never mind.

00:45:22.340 --> 00:45:24.090
Let me look at my notes
for just a second.

00:45:30.080 --> 00:45:33.205
So let's say I start
out here at point A,

00:45:33.205 --> 00:45:37.610
and I have my vector pointing
in the south direction.

00:45:37.610 --> 00:45:43.305
So this guy goes down here.

00:45:43.305 --> 00:45:45.430
And what you'll see is it
goes down to the equator,

00:45:45.430 --> 00:45:47.390
and it keeps pointing south.

00:45:47.390 --> 00:45:55.260
Then I bring it along over here,
bring it back up to the north.

00:45:55.260 --> 00:45:58.000
The vector has been
rotated by 90 degrees

00:45:58.000 --> 00:46:00.680
as it goes around that pass.

00:46:00.680 --> 00:46:03.112
It's a really,
fun, exciting demo.

00:46:03.112 --> 00:46:05.570
If you've got a ball at home,
you can do this over and over

00:46:05.570 --> 00:46:06.070
again.

00:46:06.070 --> 00:46:07.888
It's endless fun.

00:46:07.888 --> 00:46:10.430
I'm being slightly silly here,
but there's an important point

00:46:10.430 --> 00:46:11.300
to be made.

00:46:11.300 --> 00:46:21.110
When you do this operation,
parallel transport

00:46:21.110 --> 00:46:22.160
rotates the vector.

00:46:25.410 --> 00:46:27.960
It turns out that, if you are
working on a two-dimensional

00:46:27.960 --> 00:46:30.720
manifold-- particularly, I think
it's a two-dimensional manifold

00:46:30.720 --> 00:46:31.763
that is--

00:46:31.763 --> 00:46:33.180
it may have to be
of what's called

00:46:33.180 --> 00:46:34.680
constant curvature--
in other words,

00:46:34.680 --> 00:46:38.450
either a surface, a
plane, or a hyperbola.

00:46:42.190 --> 00:46:52.480
It actually rotates by
an angle of whatever

00:46:52.480 --> 00:46:58.360
is internal angle of the
triangle minus 180 degrees.

00:46:58.360 --> 00:47:01.108
So in this case, it would
rotate it by 90 degrees.

00:47:01.108 --> 00:47:03.150
If you took this thing
and you actually opened it

00:47:03.150 --> 00:47:05.392
up all the way, you
could basically, just

00:47:05.392 --> 00:47:07.600
by taking this leg and making
it as long as you want,

00:47:07.600 --> 00:47:08.580
you can make it to 0.

00:47:08.580 --> 00:47:11.590
You can make it huge.

00:47:11.590 --> 00:47:14.430
And when you do so, you'll just
rotate that vector all the more

00:47:14.430 --> 00:47:15.180
as it goes around.

00:47:18.120 --> 00:47:24.170
This operation, by the
way, is called a holonomy.

00:47:24.170 --> 00:47:27.120
I throw that out there
because, last time I looked,

00:47:27.120 --> 00:47:28.710
there was a decent
Wikipedia page

00:47:28.710 --> 00:47:32.900
on this that has some cool
animated graphics on it.

00:47:42.550 --> 00:47:45.340
Also, MathWorld.Wolfram.com
had some good stuff.

00:47:52.920 --> 00:47:54.900
So this has good
descriptions that you

00:47:54.900 --> 00:47:57.900
can find it all on Google.

00:47:57.900 --> 00:47:59.350
All right.

00:47:59.350 --> 00:48:02.340
What I want to do is take
some of these somewhat vague

00:48:02.340 --> 00:48:02.980
notions--

00:48:02.980 --> 00:48:04.980
so hopefully, I made
it intuitively clear

00:48:04.980 --> 00:48:06.180
that there's something
very interesting that

00:48:06.180 --> 00:48:07.590
happens when I
parallel transport

00:48:07.590 --> 00:48:11.580
a vector around these figures,
depending upon the underlying

00:48:11.580 --> 00:48:14.350
geometry of the manifold
that they're embedded in.

00:48:14.350 --> 00:48:16.137
Let's try to make
it more precise now.

00:48:16.137 --> 00:48:17.970
And I'm going to start
all the way over here

00:48:17.970 --> 00:48:20.800
because I'm going to want big,
clean boards to illustrate

00:48:20.800 --> 00:48:21.300
this.

00:48:33.130 --> 00:48:33.630
OK.

00:48:33.630 --> 00:48:38.790
So suppose I'm in some
coordinate system,

00:48:38.790 --> 00:48:44.160
and this line I've written here
represents a line of constant.

00:48:44.160 --> 00:48:48.708
So lambda is one
particular member

00:48:48.708 --> 00:48:50.250
of your set of
spacetime coordinates,

00:48:50.250 --> 00:48:52.885
so it might be time
or radius or maybe

00:48:52.885 --> 00:48:54.510
you work in some
crazy querying system.

00:48:54.510 --> 00:48:57.270
But lambda is meant to
represent some particular member

00:48:57.270 --> 00:49:00.180
of your coordinate system.

00:49:00.180 --> 00:49:04.580
And then there's another track
over here, which is displaced

00:49:04.580 --> 00:49:10.160
from it by delta x lambda, OK?

00:49:10.160 --> 00:49:12.380
So everywhere along here,
one of your coordinates

00:49:12.380 --> 00:49:14.323
is equal to the value, x lambda.

00:49:14.323 --> 00:49:16.490
Everywhere along here wanted,
that same coordinate's

00:49:16.490 --> 00:49:18.350
equal to x lambda
plus dx lambda.

00:49:21.720 --> 00:49:26.350
Along this trajectory,
there's a different coordinate

00:49:26.350 --> 00:49:27.820
that is kept constant.

00:49:27.820 --> 00:49:29.330
Lambda and sigma
are not the same.

00:49:29.330 --> 00:49:31.480
So there is some
coordinate whose value

00:49:31.480 --> 00:49:35.190
I will label as sigma that
is constant along there.

00:49:35.190 --> 00:49:40.050
And along this one,
it is also constant.

00:49:43.080 --> 00:49:57.290
Let me label the four
vertices, A, B, C, and D.

00:49:57.290 --> 00:50:00.590
And let me number
these four edges--

00:50:00.590 --> 00:50:08.190
one, two, three, and four.

00:50:08.190 --> 00:50:11.460
What I am going to
imagine doing is parallel

00:50:11.460 --> 00:50:15.180
transporting some vector,
v alpha, around this loop.

00:50:27.772 --> 00:50:29.230
So what I'm going
to do is generate

00:50:29.230 --> 00:50:33.170
the equations that describe
how it changes as a transport.

00:50:33.170 --> 00:50:35.770
I'm going to start at A, so
v is pointing along here.

00:50:35.770 --> 00:50:51.680
Transport it to B to C
to D and then back to A.

00:50:51.680 --> 00:50:54.250
So let me very carefully
do the first leg.

00:50:54.250 --> 00:50:56.547
Once you get the
pattern, the others

00:50:56.547 --> 00:50:58.130
can be done a little
bit more quickly.

00:51:04.500 --> 00:51:07.640
So the coordinate-- let's see.

00:51:07.640 --> 00:51:09.400
Hang on just one moment.

00:51:09.400 --> 00:51:12.065
Yeah, so I am going
from A to B first.

00:51:16.090 --> 00:51:20.920
So as I move from A to B,
x lambda remains constant,

00:51:20.920 --> 00:51:23.810
and the coordinate x
sigma is increasing.

00:51:23.810 --> 00:51:26.470
So I am moving in a
direction that points

00:51:26.470 --> 00:51:30.265
along the unit vector associated
with the sigma coordinate.

00:51:39.190 --> 00:51:42.373
So I'm going to say that
there's a basis vector.

00:51:42.373 --> 00:51:43.790
I shouldn't have
said unit vector.

00:51:43.790 --> 00:51:46.090
I don't know its magnitude.

00:51:46.090 --> 00:51:48.080
I'm pointing along
the direction in which

00:51:48.080 --> 00:51:50.490
coordinate sigma is increasing.

00:51:50.490 --> 00:51:53.120
And so parallel
transporting this vector

00:51:53.120 --> 00:52:00.070
amounts to requiring that
my covariant derivative

00:52:00.070 --> 00:52:05.890
along the sigma
basis vector is 0.

00:52:09.690 --> 00:52:10.890
This can be written out.

00:52:10.890 --> 00:52:15.100
Turn this into index form.

00:52:23.828 --> 00:52:24.620
It looks like this.

00:52:41.500 --> 00:52:43.560
OK, no surprises.

00:52:43.560 --> 00:52:45.560
So now what I'm going
to do is, essentially,

00:52:45.560 --> 00:52:48.650
I'm going to write down an
integral that would describe

00:52:48.650 --> 00:53:20.510
how v alpha changes as I move
from A to B. When I do this,

00:53:20.510 --> 00:53:24.040
I will then get the value
of the vector at point B.

00:53:24.040 --> 00:53:28.990
So the way I'm going to
write this is v alpha at B

00:53:28.990 --> 00:53:33.670
is equal to v alpha, the
initial value of this thing,

00:53:33.670 --> 00:53:38.410
minus what I get when I
integrate along leg one--

00:53:38.410 --> 00:53:47.390
gamma alpha sigma
mu phi mu dx sigma.

00:53:47.390 --> 00:53:49.140
Everyone happy with that?

00:53:49.140 --> 00:53:52.480
So everything I've done
over here, so far, I think,

00:53:52.480 --> 00:53:53.740
is probably just fine.

00:53:53.740 --> 00:53:56.000
When you've got a differential
equation, integrate it.

00:53:56.000 --> 00:53:56.500
Boom.

00:53:56.500 --> 00:53:57.208
You integrate it.

00:53:57.208 --> 00:53:58.166
You got your new thing.

00:53:58.166 --> 00:54:00.041
We're going to actually
solve these integrals

00:54:00.041 --> 00:54:02.890
in a few moments, but we'll
just leave it like this for now.

00:54:02.890 --> 00:54:04.420
So that's the first step.

00:54:04.420 --> 00:54:07.050
I got a couple more to
do, but hopefully you

00:54:07.050 --> 00:54:08.050
can now see the pattern.

00:54:13.680 --> 00:54:18.180
If I go from B to
C, I am now moving

00:54:18.180 --> 00:54:24.350
in the direction
of lambda, and I'm

00:54:24.350 --> 00:54:26.970
holding the value of that
coordinate constant at x

00:54:26.970 --> 00:54:28.230
sigma plus dx sigma.

00:54:33.200 --> 00:54:36.670
So the vector at
C is going to be

00:54:36.670 --> 00:54:40.620
equal to this thing
at B minus what

00:54:40.620 --> 00:54:45.250
I get when I integrate along
path two, gamma alpha gamma mu.

00:54:54.630 --> 00:54:55.950
We got two more to go.

00:55:08.820 --> 00:55:10.580
So this one I am,
again, integrating

00:55:10.580 --> 00:55:14.091
along the sigma direction.

00:55:18.910 --> 00:55:21.382
But notice, I switched the sign.

00:55:21.382 --> 00:55:23.340
I switched the sign
because now my coordinate's

00:55:23.340 --> 00:55:25.173
going in the direction
where it's decreasing

00:55:25.173 --> 00:55:26.195
rather than increasing.

00:55:46.350 --> 00:55:47.360
Get some fresh chalk.

00:56:00.810 --> 00:56:05.190
So we've taken it
from A to B, B to C, C

00:56:05.190 --> 00:56:11.090
to D. Let's take it
all the way around.

00:56:23.940 --> 00:56:27.350
So take it all the way
around my second value

00:56:27.350 --> 00:56:36.900
at point A. This is
going to be v alpha at D.

00:56:36.900 --> 00:56:40.233
And again, this guy is coming
in the other direction.

00:56:40.233 --> 00:56:41.900
So I'll enter this
one with a plus sign.

00:56:49.630 --> 00:56:52.040
And I get this.

00:56:52.040 --> 00:56:55.630
OK, so the way I'm going
to quantify curvature

00:56:55.630 --> 00:56:57.310
is buried in all this stuff.

00:56:57.310 --> 00:56:58.338
Let's dig it out.

00:56:58.338 --> 00:57:00.130
So the first thing
which you're going to do

00:57:00.130 --> 00:57:06.240
is I'm going to say, if I take
v alpha final, basically what

00:57:06.240 --> 00:57:08.150
I want to do is
write this guy out,

00:57:08.150 --> 00:57:11.025
substitute in for v alpha d,
which requires me to substitute

00:57:11.025 --> 00:57:13.440
in for v alpha C, [INAUDIBLE].

00:57:13.440 --> 00:57:16.200
So I'm going to get
a big, old mess here.

00:57:16.200 --> 00:57:22.090
But in the end, the first
term will be v alpha initial.

00:57:22.090 --> 00:57:24.120
So let's subtract that off.

00:57:24.120 --> 00:57:25.890
That is the change.

00:57:30.858 --> 00:57:32.400
When you actually
work this out, it's

00:57:32.400 --> 00:57:35.792
going to involve four integrals.

00:57:44.470 --> 00:57:49.678
I have chosen to write
this in a way that

00:57:49.678 --> 00:57:51.720
highlights a property I'm
going to take advantage

00:57:51.720 --> 00:57:52.553
of in just a moment.

00:58:21.330 --> 00:58:24.070
OK, so the reason I
wrote it in this way--

00:58:24.070 --> 00:58:26.440
so I have the
integral along four

00:58:26.440 --> 00:58:29.020
minus that along two
plus integral along three

00:58:29.020 --> 00:58:30.742
minus that along one--

00:58:30.742 --> 00:58:32.450
is that each one that
I've written here--

00:58:32.450 --> 00:58:34.330
they represent
parts that are sort

00:58:34.330 --> 00:58:36.240
of parallel to each
other on the figure, just

00:58:36.240 --> 00:58:39.610
offset from each other
by a little bit, parallel

00:58:39.610 --> 00:58:40.630
but offset paths.

00:58:52.720 --> 00:58:54.210
Yeah, let's put this one high.

00:59:16.630 --> 00:59:17.630
Hang on just one moment.

00:59:17.630 --> 00:59:20.080
I have a thing in my notes that
said I needed to fix something.

00:59:20.080 --> 00:59:21.000
Did I actually fix it?

00:59:21.000 --> 00:59:21.320
Yeah, I did.

00:59:21.320 --> 00:59:21.820
OK.

00:59:32.090 --> 00:59:35.000
So schematically, let's
look at that first line.

00:59:35.000 --> 00:59:37.490
The integral along four of--

00:59:37.490 --> 00:59:43.910
I have a something, dx
lambda, minus the integral

00:59:43.910 --> 00:59:45.350
along path two--

00:59:48.690 --> 00:59:51.030
of a something, dx lambda.

00:59:55.650 --> 00:59:59.160
So it's the same basic
function inside each of these,

00:59:59.160 --> 01:00:06.940
but this one is being
evaluated at x sigma.

01:00:06.940 --> 01:00:14.260
This one is being evaluated
at x sigma plus dx sigma.

01:00:14.260 --> 01:00:16.670
I can combine them.

01:00:16.670 --> 01:00:22.590
So this becomes the integral--
let's say it's along two.

01:00:25.540 --> 01:00:27.050
Pardon me for a second.

01:00:27.050 --> 01:00:28.300
Make that a little bit bigger.

01:00:41.290 --> 01:00:45.660
So what I'm doing
is I'm saying that I

01:00:45.660 --> 01:00:47.540
have a function
evaluated at x sigma

01:00:47.540 --> 01:00:50.948
minus a function evaluated
at x sigma plus dx sigma.

01:00:50.948 --> 01:00:52.490
Let's do a little
binomial expansion.

01:00:52.490 --> 01:00:54.950
It's equivalent to an
integral along a single path

01:00:54.950 --> 01:00:57.567
of essentially what I
get, the first order

01:00:57.567 --> 01:00:58.400
Taylor term of that.

01:01:11.390 --> 01:01:12.950
Do the same thing
for the other guy.

01:01:20.290 --> 01:01:25.370
Integral along three,
I have a something,

01:01:25.370 --> 01:01:33.010
dx sigma minus integral along
1, same something, dx sigma.

01:01:33.010 --> 01:01:42.390
This guy is being evaled at
x lambda x delta x lambda.

01:01:42.390 --> 01:01:47.150
This guy is being
evaluated at x lambda.

01:01:47.150 --> 01:01:49.050
And so this whole
thing is approximately

01:01:49.050 --> 01:01:53.590
equal to integral along one.

01:02:05.270 --> 01:02:07.930
So it looks like this, OK?

01:02:07.930 --> 01:02:10.180
So if you want to do this a
little bit more carefully,

01:02:10.180 --> 01:02:10.990
knock yourself out.

01:02:14.025 --> 01:02:16.400
Part of that-- I probably
should've said this explicitly,

01:02:16.400 --> 01:02:18.190
but hopefully the
notation made it clear--

01:02:18.190 --> 01:02:21.012
I'm treating these little
deltas as small quantities, OK?

01:02:21.012 --> 01:02:23.470
So it makes sense that I can
introduce a little first order

01:02:23.470 --> 01:02:24.130
expansion here.

01:02:28.000 --> 01:02:30.895
Let's leave the picture up, but
I'm going to clear this board.

01:02:30.895 --> 01:02:35.006
With this way of doing things,
let's rewrite my integrals.

01:02:40.970 --> 01:02:47.740
So what this gives me is
delta vx alpha equals--

01:02:47.740 --> 01:02:49.660
should really be an
approximately equal

01:02:49.660 --> 01:02:52.550
because we're truncating
this expansion.

01:02:52.550 --> 01:02:53.860
So the integral from x sigma--

01:02:56.710 --> 01:02:59.850
x sigma equals dx sigma.

01:02:59.850 --> 01:03:11.670
Alpha x lambda [INAUDIBLE] x
lambda of gamma alpha sigma mu

01:03:11.670 --> 01:03:17.195
phi mu dx sigma minus--

01:03:42.770 --> 01:03:44.540
So this is just taking
what I wrote there.

01:03:44.540 --> 01:03:46.748
Schematically, this is what
you get when you actually

01:03:46.748 --> 01:03:47.930
expand all those guys out.

01:03:47.930 --> 01:03:48.430
All right.

01:03:51.380 --> 01:03:53.770
So I've got a couple
derivatives here.

01:03:53.770 --> 01:03:56.090
And I'm doing an
infinite test of a couple

01:03:56.090 --> 01:03:58.145
of infinitesimal integrals.

01:03:58.145 --> 01:03:59.770
When I'm doing
infinitesimal integrals,

01:03:59.770 --> 01:04:01.520
they're very simple to evaluate.

01:04:01.520 --> 01:04:04.810
So let's just go ahead, evaluate
them, and also expand out

01:04:04.810 --> 01:04:06.057
those derivatives.

01:04:36.380 --> 01:04:38.150
Doing so, this
cleans up a fair bit.

01:04:38.150 --> 01:04:39.860
First one, I'm going to
be able to finally get rid

01:04:39.860 --> 01:04:41.068
of those damn integral signs.

01:04:46.793 --> 01:04:49.210
So I'm going to wind up with
something that is essentially

01:04:49.210 --> 01:04:50.535
quadratic in these things.

01:04:50.535 --> 01:04:51.910
It's going to look
as the product

01:04:51.910 --> 01:04:54.560
of my little infinitesimal
displacements.

01:04:54.560 --> 01:04:57.910
And I'm going to wind
up with a term that

01:04:57.910 --> 01:05:00.490
involves a partial derivative
of my connection here.

01:05:12.030 --> 01:05:14.340
So we've got one
term looks like this.

01:05:14.340 --> 01:05:17.570
We've got another term
that looks like this.

01:05:27.588 --> 01:05:29.880
So don't worry about the
index gymnastics a little bit.

01:05:29.880 --> 01:05:32.290
If you go through it
carefully, you'll see it.

01:05:32.290 --> 01:05:33.820
Pause here for a second.

01:05:33.820 --> 01:05:36.640
This expression sucks.

01:05:36.640 --> 01:05:37.682
The reason why it sucks--

01:05:37.682 --> 01:05:39.515
it's not just because
there's lots of terms.

01:05:39.515 --> 01:05:40.960
There's a bajillion
indices on it.

01:05:40.960 --> 01:05:43.120
But it's because I've
got one term that

01:05:43.120 --> 01:05:45.460
is linear in the vector
and one that's linear

01:05:45.460 --> 01:05:47.120
in the derivative of vector.

01:05:47.120 --> 01:05:51.430
However, don't forget-- we get
the derivatives of the vector

01:05:51.430 --> 01:05:53.620
by parallel transport.

01:05:53.620 --> 01:05:56.710
So we parallel
transported this guy,

01:05:56.710 --> 01:06:11.190
which tells us that these
derivatives are simply related

01:06:11.190 --> 01:06:12.553
to the vectors themselves.

01:06:12.553 --> 01:06:14.220
If I move that to the
other side, that's

01:06:14.220 --> 01:06:18.180
equivalent to covariant
derivative of v equals 0.

01:06:18.180 --> 01:06:20.200
So if I want to get
rid of my derivative

01:06:20.200 --> 01:06:23.620
with respect to x lambda,
here's I you'd write that.

01:06:23.620 --> 01:06:26.160
Likewise, if I want to
get rid of-- and I do--

01:06:26.160 --> 01:06:32.920
my derivative with
respect to x sigma,

01:06:32.920 --> 01:06:34.660
just replace lambda with sigma.

01:06:45.950 --> 01:06:47.950
So now let's sub these in.

01:06:59.880 --> 01:07:09.530
Now, in 1980, the person
who became my PhD supervisor

01:07:09.530 --> 01:07:13.840
wrote this giant review article
on gravitational radiation.

01:07:13.840 --> 01:07:16.872
And either the last or second
to last section of the paper--

01:07:16.872 --> 01:07:18.080
I'm reminded of it right now.

01:07:18.080 --> 01:07:21.830
It begins with the
sentences, "The end is near.

01:07:21.830 --> 01:07:24.470
Redemption is at hand.

01:07:24.470 --> 01:07:25.670
The end is near.

01:07:25.670 --> 01:07:27.860
We shall soon be redeemed."

01:07:27.860 --> 01:07:28.430
All right.

01:07:28.430 --> 01:07:29.330
Let's plug these in.

01:07:32.620 --> 01:07:35.310
So we plug these guys in here.

01:07:35.310 --> 01:07:36.040
What do we get?

01:07:36.040 --> 01:07:41.190
Delta v alpha
equals these things.

01:07:44.560 --> 01:07:46.706
We're supposed to put
this in a parenthesis.

01:08:00.443 --> 01:08:02.360
And now, what's going
to happen when I get rid

01:08:02.360 --> 01:08:03.560
of all those
derivatives is I'm going

01:08:03.560 --> 01:08:06.185
to have a bunch of terms that
look like Christoffel squared.

01:08:32.210 --> 01:08:34.410
As Scooby Doo
would say, ruh-roh,

01:08:34.410 --> 01:08:37.808
but that is just
what we have to have.

01:08:37.808 --> 01:08:40.350
Incidentally, what you see when
you do something like this is

01:08:40.350 --> 01:08:42.479
I now have terms entering
into this whole thing that

01:08:42.479 --> 01:08:44.146
involved derivatives
of the metric times

01:08:44.146 --> 01:08:45.765
derivative of the metric.

01:08:45.765 --> 01:08:47.640
Many of you may have
heard sort of the slogan

01:08:47.640 --> 01:08:50.890
that general relativity is a
nonlinear theory of gravity.

01:08:50.890 --> 01:08:52.770
There's where your
nonlinearity is actually

01:08:52.770 --> 01:08:54.145
going to turn out
to be entering,

01:08:54.145 --> 01:08:57.000
is the fact you add these
squared terms in here that

01:08:57.000 --> 01:08:59.130
involve metric times
itself entering

01:08:59.130 --> 01:09:02.827
in such a non-trivial
and important way.

01:09:02.827 --> 01:09:04.410
What I'm going to
do, finally, on this

01:09:04.410 --> 01:09:06.029
is-- so this is
slightly annoying

01:09:06.029 --> 01:09:08.859
because I have one term
in v mu, one term in v nu.

01:09:08.859 --> 01:09:09.359
But look.

01:09:09.359 --> 01:09:11.883
Both mu and nu
are dummy indices.

01:09:11.883 --> 01:09:13.300
They're dummy
indices, so what I'm

01:09:13.300 --> 01:09:22.069
going to do is-- on the last
term or the last two terms,

01:09:22.069 --> 01:09:26.590
I'm just going to
exchange mu for nu.

01:09:26.590 --> 01:09:33.960
And what I finally get is that
the change in the vector v

01:09:33.960 --> 01:09:42.260
transported along a loop
whose sides are delta x lambda

01:09:42.260 --> 01:09:44.990
and delta sigma--

01:09:44.990 --> 01:09:48.220
it's a quantity that is linear
in those two displacements.

01:09:48.220 --> 01:09:58.590
It's linear in the vector and
involves this four index tensor

01:09:58.590 --> 01:10:15.930
whose value depends on
derivatives of the connection

01:10:15.930 --> 01:10:21.873
and two nonlinear terms
in the connection.

01:10:40.620 --> 01:10:48.810
This quantity is a mathematical
entity known as the Riemann

01:10:48.810 --> 01:10:49.680
curvature tensor.

01:11:04.200 --> 01:11:07.170
Even though it involves
connection coefficients,

01:11:07.170 --> 01:11:09.030
Christoffel symbols,
and we argued before--

01:11:09.030 --> 01:11:10.890
and you guys did a homework
exercise where you show this--

01:11:10.890 --> 01:11:12.540
that the connection,
the Christoffel,

01:11:12.540 --> 01:11:16.500
is not tensorial, this
combination of them,

01:11:16.500 --> 01:11:20.550
basically that the terms come
together in such a way that,

01:11:20.550 --> 01:11:22.290
when you change
your representation,

01:11:22.290 --> 01:11:24.750
the nontensorial bits
cancel each other out

01:11:24.750 --> 01:11:27.520
from the terms that are being
subtracted against one another.

01:11:27.520 --> 01:11:29.250
So this is, indeed,
a true tensor.

01:11:37.525 --> 01:11:38.900
There's an equivalent
definition,

01:11:38.900 --> 01:11:40.067
if you are reading Carroll--

01:11:42.390 --> 01:11:43.980
so essentially,
what I just walked

01:11:43.980 --> 01:11:47.340
through here is an
integral equivalent

01:11:47.340 --> 01:11:50.490
of the following
commutator being applied

01:11:50.490 --> 01:11:55.890
to the vector v.
Some textbooks simply

01:11:55.890 --> 01:12:00.430
state the Riemann
tensor is related

01:12:00.430 --> 01:12:03.610
to the commutator of
partial derivatives acting

01:12:03.610 --> 01:12:09.812
upon a four-vector like so.

01:12:09.812 --> 01:12:12.270
With a little bit of effort,
you can show that what this is

01:12:12.270 --> 01:12:15.172
is, essentially, a way of--

01:12:15.172 --> 01:12:17.130
what I worked out over
there is a geometric way

01:12:17.130 --> 01:12:20.310
of understanding what
that commutator means.

01:12:20.310 --> 01:12:23.640
Incidentally, one thing, which
I think is worth calling out--

01:12:28.160 --> 01:12:32.630
when you apply
this to a one-form

01:12:32.630 --> 01:12:42.040
or a downstairs component, you
get this with a minus sign.

01:12:42.040 --> 01:12:44.860
If you are reading the
textbook by Schutz,

01:12:44.860 --> 01:12:47.530
Schutz has this sign wrong
in its first edition.

01:12:54.580 --> 01:12:56.950
Hopefully all copies
of the first edition

01:12:56.950 --> 01:13:00.040
are rare enough now that, if
you are looking at Schutz--

01:13:00.040 --> 01:13:01.870
Schutz it's actually
a wonderful textbook

01:13:01.870 --> 01:13:04.492
for an early introduction
to this field,

01:13:04.492 --> 01:13:06.700
but if you happen to get a
hold of the first edition,

01:13:06.700 --> 01:13:07.950
just be aware that there is--

01:13:07.950 --> 01:13:10.370
I think it's on page
171 of the textbook,

01:13:10.370 --> 01:13:12.940
you'll see this written.

01:13:12.940 --> 01:13:14.650
So I've actually
written a couple papers

01:13:14.650 --> 01:13:17.102
with Bernard Schutz, and so
I'm allowed to tease him.

01:13:17.102 --> 01:13:18.310
Not only did he get it wrong.

01:13:18.310 --> 01:13:19.990
He actually came up with
an intuitive argument

01:13:19.990 --> 01:13:20.575
that is wrong.

01:13:22.818 --> 01:13:25.360
Sometimes you just need to sit
down and bloody well calculate

01:13:25.360 --> 01:13:27.400
something because you
can almost always come up

01:13:27.400 --> 01:13:30.113
with an argument to convince you
of something that's not true.

01:13:30.113 --> 01:13:32.530
And I'm afraid that's what he
did in this particular case.

01:13:35.045 --> 01:13:37.420
So I have a couple notes in
there about what is sometimes

01:13:37.420 --> 01:13:40.750
called curvature coupling,
which essentially tells us--

01:13:40.750 --> 01:13:42.650
I pointed out in
the last lecture

01:13:42.650 --> 01:13:44.410
that when we're
dealing with geodesics,

01:13:44.410 --> 01:13:48.700
strictly speaking they describe,
completely point-like, almost

01:13:48.700 --> 01:13:52.090
just a monopole and no structure
and no shape whatsoever moving

01:13:52.090 --> 01:13:53.710
through spacetime.

01:13:53.710 --> 01:13:56.080
If you have a larger
body or a body that

01:13:56.080 --> 01:13:58.550
has any kind of multipolar
structure associated with it,

01:13:58.550 --> 01:13:59.300
those multipoles--

01:13:59.300 --> 01:14:01.675
you can think of that additional
structure is essentially

01:14:01.675 --> 01:14:03.250
filling up part of
the local Lorentz

01:14:03.250 --> 01:14:06.820
frame around the center
of mass of that point,

01:14:06.820 --> 01:14:10.840
and they couple the
spacetime and push it away

01:14:10.840 --> 01:14:12.220
from the geodesic.

01:14:12.220 --> 01:14:14.380
This Riemann tensor
actually describes the way

01:14:14.380 --> 01:14:17.200
in which that body couples
to the background spacetime,

01:14:17.200 --> 01:14:19.360
that it might be falling in.

01:14:19.360 --> 01:14:21.590
So this ends up playing
a really important role.

01:14:21.590 --> 01:14:25.330
For instance, when you study
the precession of equinoxes,

01:14:25.330 --> 01:14:27.992
we learn how to do this
Newtonian theory using

01:14:27.992 --> 01:14:29.950
the action of tides from
the Earth and the moon

01:14:29.950 --> 01:14:31.480
on a planet like the Earth.

01:14:31.480 --> 01:14:32.980
This ends up being
the quantity that

01:14:32.980 --> 01:14:35.750
mathematically encapsulates
tides and general relativity.

01:14:35.750 --> 01:14:37.790
So it enters into there.

01:14:37.790 --> 01:14:41.720
So I'm going to sketch
through this very,

01:14:41.720 --> 01:14:45.700
very quickly, simply because
we don't have a lot of time,

01:14:45.700 --> 01:14:48.910
and there's good discussion of
this in various other places.

01:14:48.910 --> 01:14:51.462
But let me just point out that--

01:14:51.462 --> 01:14:52.420
you look at this thing.

01:14:52.420 --> 01:14:56.410
It's a four-index
tensor, and each index

01:14:56.410 --> 01:14:57.355
can take four values.

01:15:04.550 --> 01:15:09.410
That makes it look like
it has 256 components.

01:15:12.188 --> 01:15:14.230
Now, I'm not going to step
through this in detail

01:15:14.230 --> 01:15:15.070
right now.

01:15:15.070 --> 01:15:18.700
This will either be in the
next lecture that I do this,

01:15:18.700 --> 01:15:20.603
or you'll watch me
on a video once this

01:15:20.603 --> 01:15:22.270
gets recorded, depending
upon how things

01:15:22.270 --> 01:15:24.930
unroll in the next 24 hours.

01:15:24.930 --> 01:15:27.040
Riemann has a lot of symmetries.

01:15:27.040 --> 01:15:29.067
I will go through those
symmetries carefully,

01:15:29.067 --> 01:15:30.400
either in lecture or on a video.

01:15:40.400 --> 01:15:42.650
So symmetries-- Riemann--

01:15:42.650 --> 01:15:43.400
and you know what?

01:15:43.400 --> 01:15:53.620
Let me write it
out in n dimensions

01:15:53.620 --> 01:15:59.350
from n to the 4, which is what
you'd expect for a four index

01:15:59.350 --> 01:16:06.100
object in n
dimensions, down to n

01:16:06.100 --> 01:16:12.330
squared times m squared
minus 1 over 12th.

01:16:12.330 --> 01:16:13.705
So where I want
to conclude today

01:16:13.705 --> 01:16:15.360
is let's just take
a look at what that

01:16:15.360 --> 01:16:17.730
turns into for a couple
of different numbers

01:16:17.730 --> 01:16:18.794
and dimensions.

01:16:30.170 --> 01:16:33.650
So if you do n
equals 1, you get 0.

01:16:33.650 --> 01:16:36.030
So the Riemann tensor
has no components

01:16:36.030 --> 01:16:37.655
on a one-dimensional manifold.

01:16:37.655 --> 01:16:39.030
There's a simple
reason for that.

01:16:39.030 --> 01:16:40.488
Remember the way
we defined it, OK?

01:16:40.488 --> 01:16:42.480
We did this by
parallel transporting

01:16:42.480 --> 01:16:44.160
around a particular figure.

01:16:44.160 --> 01:16:47.860
If you're in one dimension,
this is all you can do.

01:16:47.860 --> 01:16:50.280
There's no holonomy
operation in one dimension.

01:16:50.280 --> 01:16:52.050
You can't do that.

01:16:52.050 --> 01:16:55.830
So no curvature.

01:16:55.830 --> 01:16:58.228
If you want to be a real
pedant and someone says,

01:16:58.228 --> 01:17:00.020
well, look at a curved
line, you'll go, ah,

01:17:00.020 --> 01:17:01.103
but lines can't be curved.

01:17:04.060 --> 01:17:06.120
n equals 2.

01:17:06.120 --> 01:17:09.950
So you get 2 squared, 2
squared minus 1 over 12--

01:17:09.950 --> 01:17:12.410
you get 1.

01:17:12.410 --> 01:17:15.380
So if you're working
in two dimensions,

01:17:15.380 --> 01:17:17.600
there is a single number
that characterizes

01:17:17.600 --> 01:17:22.370
the curvature at every
point, and this is often

01:17:22.370 --> 01:17:28.430
thought of as just a
radius of curvature.

01:17:28.430 --> 01:17:30.620
Simplest example is
if you have a sphere.

01:17:30.620 --> 01:17:34.040
Sphere is completely
characterized by its radius.

01:17:34.040 --> 01:17:39.290
But if you imagine that it's
like a sphere that you squash,

01:17:39.290 --> 01:17:41.570
well, you can imagine
at every point

01:17:41.570 --> 01:17:43.340
that there is a
particular sphere that

01:17:43.340 --> 01:17:45.590
is tangent to that
point, and the radius

01:17:45.590 --> 01:17:48.620
of curvature of
the tangent sphere

01:17:48.620 --> 01:17:52.057
is the one that defines the
curvature at that point.

01:17:52.057 --> 01:17:54.640
I'm going to skip three because
it's not all that interesting.

01:17:54.640 --> 01:17:58.330
The one that is more important
is, if you do n equals 4,

01:17:58.330 --> 01:18:06.220
you'll wind up with 16 times
15 over 12, which is 20.

01:18:06.220 --> 01:18:11.410
This is exactly the
number of derivatives

01:18:11.410 --> 01:18:14.230
that we could not
cancel out when

01:18:14.230 --> 01:18:18.010
we did the exercise a couple
of lectures ago of assessing

01:18:18.010 --> 01:18:23.530
how well we can make spacetime
have a flat representation

01:18:23.530 --> 01:18:24.910
in the vicinity of some point.

01:18:34.070 --> 01:18:43.150
It's the number of leftover
constraints at second order

01:18:43.150 --> 01:18:44.410
in a freely falling frame.

01:18:53.320 --> 01:18:55.690
All right so I'm going
to stop there for today.

01:18:55.690 --> 01:18:57.465
That's a nice place
for us to stop.

01:19:00.720 --> 01:19:02.390
Keep watching your emails.

01:19:02.390 --> 01:19:04.470
We're in an
interesting situation.

01:19:04.470 --> 01:19:07.950
Life at MIT is evolving.

01:19:07.950 --> 01:19:10.380
But when we pick it up
in one form or another,

01:19:10.380 --> 01:19:13.230
what I'm going to do first
is talk a little bit more

01:19:13.230 --> 01:19:14.640
about the symmetry
of this object

01:19:14.640 --> 01:19:17.460
because there's a couple
of explicit symmetries

01:19:17.460 --> 01:19:20.330
that lead to that reduction
from n to the 4th to n squared

01:19:20.330 --> 01:19:22.800
n squared minus 1 over
12, and it's useful for us

01:19:22.800 --> 01:19:25.350
to go through them and
see what they look like.

01:19:25.350 --> 01:19:27.900
And then I also want to talk
about a couple of variants

01:19:27.900 --> 01:19:29.520
on this curvature tensor, OK?

01:19:29.520 --> 01:19:31.920
So just to give you a
little bit of a preview--

01:19:31.920 --> 01:19:36.270
the curvature tensor--
it's a four index object.

01:19:36.270 --> 01:19:44.165
We have argued
already that we're

01:19:44.165 --> 01:19:45.540
going to end up
doing things that

01:19:45.540 --> 01:19:47.880
look like looking at
derivatives of the metric

01:19:47.880 --> 01:19:49.440
being equal to our source.

01:19:49.440 --> 01:19:52.560
Our source is a two-index
object, the stress energy

01:19:52.560 --> 01:19:53.770
tensor.

01:19:53.770 --> 01:19:55.687
We've got gotta get
rid of two indices.

01:19:55.687 --> 01:19:57.270
And so what we're
going to do is we're

01:19:57.270 --> 01:20:00.570
going to essentially contract
this guy with a couple

01:20:00.570 --> 01:20:04.080
of powers of the metric
in order to trace over

01:20:04.080 --> 01:20:08.070
certain combinations of indices
and make two index variants

01:20:08.070 --> 01:20:09.683
of the curvature tensor.

01:20:09.683 --> 01:20:11.850
And we're also going to
look at derivatives of this,

01:20:11.850 --> 01:20:14.520
because it turns out that there
is a particular combination

01:20:14.520 --> 01:20:16.020
of derivatives at
the Riemann tensor

01:20:16.020 --> 01:20:18.182
that has an important
geometrical meaning.

01:20:18.182 --> 01:20:20.640
What we're going to find is
that, when we combine these two

01:20:20.640 --> 01:20:24.660
notions, there is a
particular divergence

01:20:24.660 --> 01:20:28.503
of a particular variant of the
curvature tensor that is 0.

01:20:28.503 --> 01:20:30.420
In other words, we can
make a curvature tensor

01:20:30.420 --> 01:20:32.490
that is divergence-free.

01:20:32.490 --> 01:20:36.300
Our stress energy tensor
is divergence-free.

01:20:36.300 --> 01:20:38.910
I wonder if one is
related to the other.

01:20:38.910 --> 01:20:40.935
That, in a nutshell,
is how Einstein came up

01:20:40.935 --> 01:20:43.080
with general relativity,
by asking that question

01:20:43.080 --> 01:20:45.840
and then just seeing
what happened.

01:20:45.840 --> 01:20:48.350
So that's what we're
going to go through next.