[SQUEAKING]
[RUSTLING]
[CLICKING]
SCOTT HUGHES: So let me
just do a quick recap
of what we did last time.
So today, we're going
to move into things that
are a little bit more physics.
Last time we were really
doing some things that
allows us to establish some
of the critical mathematical
concepts we need to
study the tensors that
are going to be used for
physics on a curved manifold.
So one of the things
that we saw is
that if I wanted to formulate
differential equations
on a curved manifold, if I just
defined my derivative the most
naive way you
might think of, you
end up with objects
that are not tensorial.
And so mathematically you
might say, well, that's fine.
It's just not a tensor anymore.
But we really want
tensors for our physics
because we want to be
working with quantities
that have frame-independent
geometric meaning to them.
So that notion of a derivative--
if I just do it the naive way--
isn't the best.
And so I argued that
what we need to do
is to find some kind of
a transport operation
in which there is a linear
mapping between things
like my vector
field or my tensor
field and the displacement,
which allows me to cancel out
the bits of the partial
derivative transformation
laws that are non tensorial.
There's a lot of
freedom to do that.
One of the ways I suggest we do
that is by demanding that when
I do this, that derivative--
when applied to the metric--
give me 0.
And if we do that,
we see right away
that the transport
law that emerges
gives me the covariant
derivative as one
of my examples.
Now this shouldn't
be a surprise.
We introduced the
covariant derivative
by thinking about flat
spacetime operations,
but with all my basis
objects being functionals.
And this in some way is sort of
a continuation of that notion.
The other thing which I talked
about is telling you we're not
going to use a tremendous amount
here except to motivate one
very important result. And
that is if I define transport
by basically imagining that
I slide my vectors in order
to make the comparison-- along
some specified vector field--
I get what's known as
the Lie derivative.
And so this is an example of
the Lie derivative of a vector.
And you get this form that
looks like a commutator
between the vector field
you're sliding along
and the vector field
you are differentiating.
Similar forms--
which are not really
the form of a commutator--
but similar forms
can be written down
for general tensors.
The key thing that
you should be aware of
is that it's got a similar form
to the covariant derivative,
in that you have
one term-- let's
focus on the top
line for the second--
you have one term
that looks just
like the ordinary
vector contracted
onto a partial
derivative of your field.
And then you have terms which
correct every free index
of your field--
one free index if it's
a vector, one free index
if it's a one form, and
corrections for an end index
tensor--
with the sign doing something
opposite to the sign
that appears in the
covariant derivative.
What's interesting about
this is that so defined,
the Lie derivative is
only written in terms
of partial derivatives.
But if you just
imagine-- you promote
those partial derivatives
to covariant derivatives--
you find the exact
same result holds
because all of your Christoffel
symbols-- or connections
as we like to think of them
when we're using parallel
transport-- all the connective
objects cancel each other out.
And this is nice
because this tells me
that even though this
object, strictly speaking,
only involves partial
derivatives, what emerges out
of it is in fact tensorial.
And it's an object that I
can use for a lot of things
I want to do in physics.
In particular where we're
going to use it the most--
and I said you're going
to do this on the PSET,
but I was wrong-- you're going
to do something related to one
of the PSETs--
but I'm going to actually--
if all goes well--
derive an important result
involving these symmetries
in today's lecture.
We can use this to
understand things
that are related to conserved
quantities in your space time.
And where this comes
from is that there
is a definition of an object we
call the Killing vector, which
is an object where if your
metric is Lie transported
along some field C, we
call C a Killing vector.
And from the fact that the
covariant derivative the metric
is 0, you can turn
the equation governing
the Lie derivative along C into
what I wrote down just there.
And I should write its name.
There's the result known
as Killing's Equation.
If a vector has a
Killing vector--
if a metric has a
Killing vector--
then you know that
your metric is
independent of some
kind of a parameter that
characterizes that spacetime.
The converse also holds.
If your metric is
independent of something--
like say the time
coordinate-- you
know that there is a
Killing vector corresponding
to that independent thing.
And so you'll often see this
described as a differing
amorphism of the
spacetime-- if you
want to dig into some of
the more advanced textbooks
on the subject.
We'll come back to that in a few
more details hopefully shortly
before the end of
today's lecture.
So where I concluded
last time, was we
started talking about
these quantities known
as tensor densities,
which are given
the less-than-helpful
definition-- quantities
that are like tensors,
but not quite.
The example I gave of this--
where we were starting--
was the Levi-Civita symbol.
So let me just write down
again what resulted from that.
So if I have Levi-Civita--
and the tilde here is going
to reflect the fact that
this is not really a tensor--
this guy in some
prime coordinates
is related to this guy in
the unprime coordinates
via the following--
let's get the primes
in the right place--
the following mess
of quantities.
So I'm not going to
go through this again.
This is basically a
theorem from linear algebra
that relates the
determinant of a matrix--
not metric, but matrix--
to what you get when you
contract a bunch of matrices
onto the Levi-Civita symbol.
And so the key thing to note
is that if this were not here,
this would look just like
a tensor transformation.
But that is there.
So it's not.
And so we call this a
tensor density of weight 1.
So the other one-- which I
hinted at the end of the last
lecture, but did not
have time to get into--
is suppose we look
at the metric.
Now, the metric-- no ifs,
ands, or buts about it--
it's a tensor.
And it's actually
the first tensor
we've started talking
about back in our toddler
years of studying flat
spacetime, which by the way,
was about three weeks ago.
Obviously that's a tensor.
It's a simple
tensor relationship.
Let's take the determinant
of both sides of this.
You might look at this and go,
why do you want to do that?
Well when I do this,
I'm going to call
the determinant of the metric
in the primed representation.
Let's call that G prime.
I get 2 powers--
2 powers of this Jacobian
matrix is determinant.
And I get the determinant in
my original representation.
Now I want to write
this in a way that's
similar to the way I
wrote it over here.
Notice I have all
my primed objects
over here on the left-hand side.
And my factor of this
determinant relates--
it's got primed indices in the
upstairs position, unprimed
in the downstairs.
But the determinant of 1
over a metric is just 1
over the determinant of--
the determinant of the
inverse of a matrix is just 1
over the determinant
of that matrix.
And so I can really simply
just say this looks like so.
So the determinant of the
metric is a tensor density
of weight minus 2.
What this basically tells us is
I now have two of these things.
I've been arguing basically
this entire course
that we want to use
tensors because of the fact
that they give me a
covariant way of encoding
geometric concepts.
I've got these two things
that are not quite tensors.
I can put them together and
get a tensor out of this.
So what this tells me now is I
can convert any tensor density
into a proper tensor.
So suppose I have a
tensor density of weight
W. I can convert this
into a proper tensor
by multiplying by a power
of that G. So multiply it
by G to the W over 2.
One slight subtlety here,
when we work in spacetime--
let's just stop for a second and
think about special relativity.
In special relativity in an
inertial reference frame,
my metric is minus 1
1 1 1 on the diagonal.
So its determinant
is negative 1.
And when I take negative 1
to some power that involves
a square root, I get sad.
We all know how to work
with complex numbers.
You might think that's all OK.
It's not in this case.
But the way I can fix that
is that equation's still true
if I multiply both
sides by minus 1.
I want this to be a
positive number when
I take the square root.
So I'm allowed just to
take the absolute value.
So we take the absolute
value to clear out the fact
that in spacetime, we tend to
have an indeterminate metric,
where the sign depends
on the interval.
So remember the only
reason we're doing this--
this is just a--
I don't want to
say it's a trick.
But it's not that
far off from a trick.
I'm just combining two
tensor densities in order
to get a tensor out of it.
And minus a tensor density
is still a tensor density.
So I'm OK to do that.
And I'm just doing this
so that my square root
doesn't go haywire on me.
So a particular example--
in fact the one that in my
career has come up the most--
is making a proper
volume element
converting my Levi-Civita
symbol into a tensor that
gives me a volume element.
So my Levi-Civita symbol has
a tensor density of weight 1.
If I want to make that
into a proper tensor,
I multiply by the square root of
the determinant of the metric.
So now I will no
longer have that tilde
on there, which was meant
to be a signpost that this
as a quantity is a
little bit goofy.
You wind up with
something like this.
When you go-- and by
the way, sometimes
when you're working
with this, you
need to have this thing
with indices in the upstairs
position.
You have to be a
little bit careful.
But I'll just give
you one example.
If you raise all four
of the indices, what
you find when
everything goes through,
this one is not that hard to
see because you're basically
playing with a similar
relationship to the one
that I wrote down over here--
just a short homework
exercise to demonstrate this.
And then you end up
with the tensor density
of the opposite sign.
Weight minus 1, you wind up
with a 1 over square root there.
So like I said
one of the reasons
why this is an important example
is that we use it to form
covariant volume operators.
So in four-dimensional space--
so imagine here's my basis
direction for spatial direction
1, spatial direction 2,
spatial direction 3--
you guys can figure out how
to write spatial direction
0 on your own time--
I would define a
covariant 4 volume--
4 volume element from this.
It'll look like this.
And if this is an
orthogonal basis,
this simply turns
into something like--
these are meant
to be superscripts
because these are coordinates.
So it just turns into
something like this
if I'm working in
an orthogonal basis.
And again for intuition,
I suggest go down
to 3-dimensional
spherical coordinates.
And I wrote this last time.
But let me just
quickly write it up.
I mean everything I did here,
I tend to-- since this is
a course on spacetime--
by default I write
down all my formulas
for three space dimensions,
one time dimension.
But it's perfectly good
in 3 spatial dimensions,
2 spatial dimensions, 17 spatial
dimensions-- whatever crazy
spacetime your physics want
you to put yourself in--
or space your physics
wants to put you in.
So I'll just remind you--
that when you do this,
you've got yourself
a metric across
the diagonal of 1 r
squared r squared
sine squared theta.
And just to be consistent--
I usually use Latin letters
for only spatial things.
So let's do that.
This would be how I would
then write my volume element.
Did I miss something?
AUDIENCE: Yeah.
[INAUDIBLE]
SCOTT HUGHES: Absolutely.
Yeah.
Thank you.
I'm writing quickly.
Yeah?
AUDIENCE: Is there [INAUDIBLE]?
SCOTT HUGHES: This is dx--
oh, bugger.
Yep.
I'm trying to get
to something new.
And I'm afraid I'm
rushing a little bit.
So thank you for catching this.
And so with this, take the
determiner of this thing.
And sure enough you get r
squared sine theta d r d theta
d phi.
So this is the
main thing that we
are going to use this result
for-- this thing with tensor
densities.
I want to go on a
brief aside, which
is relevant to the
problem that I delayed on
this week's problem 7.
So there are three
parts of problem 7
that I moved from PSET 3 to PSET
4 because they rely on a result
that I want to talk about now.
So the main thing that we use
the determinant of the metric
for in a formal way is
this-- that it's a tensor
density of weight minus 2.
And so it's a really
useful quantity
for converting tensor
densities into proper tensors.
And really the most
common application of this
tends to be to volume elements.
But it turns out
that it's actually
really useful for what a
former professor of mine
used to like to
call party tricks.
There's some really-- it
offers a really nice shortcut
to computing certain
Christoffel symbols.
So in honor of Saul Teukolsky
let's call this a party trick.
So we're using the
determinant of the metric
to compute certain Christoffels.
So this is going to
rely on the following.
So suppose I calculate
the Christoffel symbol,
but I'm going to sum
on the raised index.
And bearing in mind it's
symmetric in the lower one,
I'm going to do a contraction
of the raised index with one
of the lower indices.
So let's just throw in
a couple of definitions.
This is equivalent
to the following.
And so throwing in the
definition of the Christoffel
with all the indices in
the downstairs position--
this formula, by the
way, is something
that I've been writing down
now for about 27 years.
And I have to look
it up every time.
Usually by the end of a
semester of teaching 8.962,
I have it memorized.
But it decays by then.
So if you're wondering how
to go from here to here--
this is the kind of thing--
just look it up.
So let's pause for a second.
Remember that the metric is--
it's itself symmetric.
So in keeping with
that, I'm going
to flip the indices on this last
term, which-- hang on a second.
That was stupid.
Wait.
Pardon me.
This is the term I want
to switch indices on.
My apologies.
So the reason I
did that is I want
to have both of
these guys ending
with the alpha because
notice this and this--
they're the same.
But I have interchanged
the beta and the mu.
So these two terms--
the first term and
the third term--
are anti symmetric upon
exchange of beta and mu.
They are contracted
with the metric,
which is symmetric upon
exchange of beta and mu.
Question?
AUDIENCE: Does the metric
have to be symmetric?
SCOTT HUGHES: The metric
has to be symmetric.
[LAUGHS] I don't want to get
into that right now, but yes
[LAUGHS].
So these guys are
anti symmetric.
This guy is symmetric.
And remember the rule.
Whenever you contract some
kind of a symmetric object
with an anti
symmetric m you get 0.
So that means this
term and this term die.
And what we are left
with is gamma mu
mu alpha is 1/2 g u beta, and
the alpha derivative of g u
beta.
There is the way
it is contracting
the indices in the 2
metric with the other one.
Well here's a theorem that I'm
going to prove in a second--
or at least motivate-- that
it's going to rely on a result
that I will pull
out of thin air,
but can be found in most
linear algebra textbooks.
It's not too hard to show that
this can be further written
as 1 over square root of the
determinant times the partial
derivative of the square
root of the determinant,
which is sometimes-- depending
on your applications--
this can be written very
nicely as the derivative
of the logarithm of the
absolute value of the--
the square root of the absolute
value of the determinant.
So before I go on and
actually demonstrate this,
you can see why this
is actually a pretty--
so this actually comes up.
I'm going to show
a few applications
as to why this particular
combination of Christoffel
symbols shows up more
often than you might guess.
It's really important for
certain important calculations.
And this is telling
me that I can get it
by just taking one partial
derivative of a scalar
function.
And if you know your
metric, that's easy.
So this becomes really
easy thing to calculate.
So let's prove it.
So the proof of this
relies on a few results
from linear algebra.
So let's not think about
tensors for a second.
And let's just think
about matrices.
So imagine I've
got some matrix m.
I'm going to be agnostic about
the dimensions of this thing.
And suppose I look at
the following variation
of this matrix.
So suppose I imagine
doing a little variation.
So suppose every element
of m is a function.
And I look at a little variation
of the log of the determinant
of that matrix.
Well this can be written
as log is basically
a definition of this.
Now, if I exploit
properties of logarithms,
this can be written as the
log of the determinant--
m plus delta m--
divided by the determinant of m.
Now I'm going to
use the fact that 1
over the determinant
of m is the determinant
of the inverse of m.
So taking advantage of that,
I can further write this guy
as something like this.
Now I'm going to invoke
an identity, which
I believe you can find proven in
many linear algebra textbooks.
It just occurred to me as
I'm thinking about this,
I don't know if I've ever seen
it explicitly proven myself.
But it's something that's
very easy to demonstrate
with just a quick calculation.
You can just do--
I'm a physicist.
So for me I'll use Mathematica.
I'll look at six or
seven examples and go,
it seems right.
And so I've
definitely done that.
But I believe this
is something that you
can find proven explicitly--
like I said-- in most books.
So remember these
are all matrices.
So this isn't the number 1.
We want to think of this
as the identity matrix.
Oh and I'm also going
to regard this variation
as a small quantity.
So if I regard epsilon
as a small matrix--
this can be made formal
by defining something
like condition number
associated with the matrix
or something like that.
But generally what
I want to mean
by that is if I
take this epsilon
and I add it to 1, all of this--
so my identity is
1 on the diagonal--
0s everywhere else--
all the things
that are put into the
sum of 1 plus epsilon
are much, much smaller than
that 1 that's on the diagonal.
That will be sufficient.
So if epsilon is a small
matrix, then the determinant
of 1 plus epsilon is
approximately equal to 1
plus the trace of epsilon.
What that approximately
refers to is-- of course
you can take that further.
And you'll get
additional corrections
that involve epsilon times
epsilon, epsilon times epsilon,
times epsilon.
I believe when you do
that, the coefficient is
no longer universal.
But it depends upon the
dimensions of the matrix.
But leading order it's
independent of dimensions
of the matrix.
And that's something
that you can
you can play with a
little bit yourself.
Like I said, this is sufficient
for what we want to do here.
So I'm going to think
of my small matrix
as the matrix of inverse
m times a variation of m.
This is our epsilon.
So we're going to apply it to
the line that I have up here.
And this tells me that my delta
on the log of the derivative
of m is the log of 1 plus
the trace of m to the minus 1
on the matrix m.
Log of 1 plus a small
number is that small number.
Now the application.
So this is-- like I said,
this the theorem that you
can find in books that I don't
know about but truly exist.
This is something I've seen
documented in a lot of places.
Let's treat our m as
the metric of spacetime.
So my m will be g alpha beta.
My m inverse will be g
in the upstairs position.
And I will write this
something like so.
And I'm going to
apply this by looking
at variations in my metric.
So delta log--
I'm going to throw my
absolute values in here.
That's perfectly allowed to go
ahead and put that into there.
Applying this to
what I've got, this
is going to be the trace of g
mu beta times the variation of g
beta gamma.
And I forgot to say, how do
I take the trace of a matrix?
So the trace that
we're going to use-- we
want it to be something
that has geometric meaning
and has a tensorial
meaning to it.
So we're going to call the
trace of this thing g alpha
beta epsilon alpha beta.
If you think about
what this is doing,
you're essentially
going to take your--
let's say I apply this
to the metric itself.
I put one index in the
upstairs position, one
the downstairs position,
and then I am summing along
the diagonal when I do this.
You will sometimes see this
written as something like that.
So in this case, when I'm taking
the trace of this guy here,
that is going to force me to--
let's see.
So this gives me
a quantity where
I'm summing over my betas.
And then I'm just going to
sum over the diagonal indices.
I'm forcing my two remaining
indices to be the same.
So putting this
together, this tells me--
so now what I'm
going to do is say,
I basically have part of
a partial derivative here.
All I need to do is now divide
by a variation in my coordinate
and take the limit.
So it comes out of this as
the partial derivative--
looks like this.
Now let's trace it back
to our Christoffel symbol.
My Christoffel
symbol-- the thing
which I'm trying to
compute-- is one half
of this right-hand side.
So it's one half of
the left-hand side.
And I can take that one half,
march it through my derivative,
and use the fact
that 1/2 the log of x
is the log of the
square root of x.
Check.
So like I said, this is
what an old mentor of mine
used to like to
call a party trick.
It is a really useful party
trick for certain calculations.
So I want to make sure you
saw where that comes from.
This is something you will
now use on the problem
that I just moved
from PSET 3 to PSET 4.
It's useful for you to
know where this comes from.
You're certainly not going
to need to go through this
yourself.
But this is a good
type of calculation
to be comfortable with.
Those of you who are more
rigorous in your math than me,
you might want to
run off and verify
a couple of these
identities that I used.
But this is very nice
for physics level rigor--
at least astrophysicists
level rigor.
So let me talk about one
of the places where this
shows up and is quite useful.
So a place where I've
seen this show up the most
is when you're looking at
the spacetime divergence
of a vector field.
So when you're calculating
the covariant derivative
of alpha contracting
on the indices--
let's just throw in--
expand out the full
definition of things--
all you've gotta do
is correct one index.
And voila, this is exactly
the things that change--
where is it-- change my
alpha to a mu-- that's
exactly what I've got before.
And so-- hang on
just one moment.
I know what I'm doing.
These are all dummy indices.
So in order to keep things
from getting crossed,
I'm going to relabel
these over here.
So I can take advantage of
this identity and write this.
So stare at this for a second.
And you'll see that the
whole thing can be rewritten
in a very simple form.
Ta-da.
You haven't done as much work
with covariant in your lives
as I have.
So let me just emphasize
that ordinarily when
you see an expression
like you've
got up there on the top
line, you look at that,
and you kind of go [GROANS]
because you look at that,
and the first thing
that comes to your mind
is you've got to work out
every one of those Christoffel
symbols and sum it up
to get those things.
And in a general spacetime,
there will be 40 of them.
And before Odin
gave us Mathematica,
that was a fair amount of labor.
Even with Mathematica it's
not necessarily trivial
because it's really
easy to screw things up.
With this you calculate the
determinant of the metric,
you take its square root,
you multiply your guy,
and you take a partial
derivative, and you divide.
That is something
that most of us
learned how to do
quite a long time ago.
It cleans the hell
out of this up.
So the fact that this gives us
something that only involves
partial derivatives is awesome.
This also-- it turns out-- so
when you have things like this,
it gives us a nice way to
express Gauss's theorem
in a curved manifold.
So Gauss's theorem-- if I
just look at the integrals
that-- or rather the integral
for Gauss's theorem-- let's
put it that way.
Let's say a
Gauss's-type integral.
So go back to when
you're talking
about conservation laws.
If I imagine I'm integrating the
divergence of some vector field
over a four-dimensional
volume, look at that,
I get a nice cancellation.
So this turns into an integral
of that nice, clean derivative
over my four coordinates.
And then you can take
advantage of the actual content
of Gauss's Theorem to
turn that into an integral
over the three-dimensional
surface that bounds that four
volume.
It's a good point--
so you're emboldened by this.
You say, yay, look at that.
We can do all this awesome
stuff with this identity.
It gives me a great
way to express some
of these conservation laws.
You might think to yourself--
and I realized as I was
looking over these notes--
I'm about to I think give away
a part of one of the problems
on the PSET-- but c'est la vie.
It's an important point.
Can we do something
similar for tensors?
So this is great that you
have this form for vectors.
The divergence of a vector
is a mathematical notion that
comes up in various contexts.
So this is important.
But we've already
talked about the fact
that things like
energy and momentum
are described by a
stress energy tensor.
So can we do this for tensors?
Well the answer
turns out to be no,
except in a handful of cases.
And I have a comment about
those handful of cases.
So suppose I take this--
and I'm taking the
divergence on say
the first index of this guy--
so there's the bit involves
my partial derivative--
I'm going to have
a bit that involves
correcting the first index.
So the first correction
is it's of a form that
does in fact involve
this guy we just
worked out this identity for.
And in principle we
could take advantage
of that to massage this
and use this identity.
But the second one there's
nothing to do with that.
This you just have
to go and work out
all of your 40 different
Christoffel symbols
and sit down and
slog through it.
This spoils your
ability to do anything
with it, with one exception.
What if a is an
anti-symmetric tensor?
If a is an
anti-symmetric tensor,
you've got symmetry, anti
symmetry, and it dies.
So that is one example of where
you can actually apply it.
And I had you guys play with
that a little bit on the PSET.
It's worth noting though that
the main reason why one often
finds this to be a
useful thing to do
is that when you take the
divergence of something
like a vector, you
get a scalar out.
You get a quantity that is--
really its
transformation properties
between different inertial
frames or freely-falling frames
is simple.
So even when you can do this and
take advantage of this thing,
working with the
divergence of a tensor--
exploiting a trick like this
turns out to generally not
be all that useful.
And I'll use the example of
the stress energy tensor.
So conservation of stress
energy in special relativity--
it was the partial derivative--
the divergence of
the stress energy
tensor expressed with
the partial derivative
was equal to 0.
We're going to take this
over to covariant derivative
of the stress energy
tensor being equal to 0.
That's what the equivalence
principle tells us
that we can do.
Now when I take the divergence
of something like the stress
energy tensor, I get a 4 vector.
Every 4 vector
always has implicitly
a set of basis objects
attached to it.
When I've got basis
objects attached to it,
those are defined with
respect to the tangent space
at a particular
point in the manifold
where you are currently working.
And so if I want to try to do
something like an integral like
this-- where I add up the
four vector I get by taking
the divergence of stress
energy and integrate it over
a volume--
I'm going to get nonsense
because what's going on
is I'm combining vector
fields that are defined
in different tangent spaces
that can't be properly compared
to one another.
In order to do that
kind of comparison,
you have to introduce
a transport law.
And when you start
doing transports
over macroscopic regions,
you run into trouble.
They turn out to
be path dependent.
And this is where we
run into ambiguities
that have to do with
the curvature content
of your manifold.
We'll discuss where that
comes into our calculations
a little bit later.
But what it basically
boils down to is
if I use a stress energy
tensor as an example,
this equation tells me
about local conservation
of energy and momentum.
In general relativity I cannot
take the local conservation
of energy and momentum
and promote it to a global
conservation of
energy and momentum.
It's ambiguous.
We'll deal with that and
the conceptual difficulties
that that presents a little
bit later in the course.
But it's a good see the
plant at this point.
So let's switch gears.
We have a new set of
mathematical tools.
I want to take a detour away
from thinking about some more
abstract mathematical notions
and start thinking about how
we actually do some physics.
So what I want to do
is talk today about how
do we formulate the
kinematics of a body
moving in curved spacetime?
So I've already
hinted at this in some
of my previous lectures.
And what I want
to do now is just
basically fill in
some of the gaps.
The way that we do
this really just
builds on Einstein's
insight about what
the weak equivalence
principle means.
So go into a freely
falling frame.
Go in that freely-falling frame.
Put things into locally
Lorentz coordinates.
In other words perform
that little calculation
that make spacetime
look like the spacetime
of special relativity of
the curvature corrections.
And to start with, let's
consider what we always
do in physics, is we'll look
at the simplest body first.
We're going to look at
what we call a test body.
So this is the body that has
no charge, no spatial extent,
it's of zero dimensional
point, no spin--
nothing interesting,
except a mass.
So if you want to
think about this--
I use a way that I find
to think about this
is all these various aspects to
it, you're adding additional--
either charges to it or
additional multipolar structure
to this body.
I'm thinking of this-- this is
sort of like a pure monopole.
It's nothing but
mass concentrated
in a single zero size point.
Obviously it's an idealization.
But you've got to
start somewhere.
So since it's got no
charge, no spatial extent,
it's got nothing
but mass, nothing's
going to couple to it.
It's not going to basically
do anything but freefall.
In this frame the body moves on
a purely inertial trajectory.
And what does a purely
inertial trajectory look like?
Well you take whatever your
initial conditions are.
And you move in a
straight line with respect
to time as measured
on your own clock.
Simplest, stupidest possible
motion that you can.
So we would obviously call that
a straight line with respect
to the parameterization
that's being
used in this representation.
So what does that mean
in a more general sense
of the representation?
So if we think about this a
little bit more geometrically,
when a body is moving in a
straight line, that basically
means that whatever the tangent
vector to its world line is,
it's essentially moving such
that the tangent vector at time
T1 is parallel to the tangent
vector at T1 plus delta T1,
provided that's
actually small enough
that they're sort of within
the same local Lorentz frame.
So a more geometric way of
thinking about this motion
is that it's parallel
transporting its tangent
vector.
Let's make this a little
bit more rigorous.
So let's imagine
this body's moving
on a particular trajectory
through spacetime.
So it's a trajectory
parameterized.
I will define its
parameterization a little bit
more carefully very soon.
So for now, just think of lambda
as some kind of a quantity.
It's a scale that just
accumulates uniformly as it
moves along the world line.
So I'm going to say
the small body has
a path through spacetime,
given by u x of lambda.
Its tangent is given by this.
And if it is parallel
transporting its own tangent
vector, that is--
I'll remind you
that the condition
for parallel
transport was that you
take the covariant
derivative your field.
And as you are moving
along, you contract it
along the tangent vector of the
trajectory you're moving on.
And you get 0.
So in my notes, there's a
couple of equivalent ways
of writing this.
So you will sometimes
see this written
as the gradient along u of u.
And you'll sometimes see
this written as capital u
u lambda equals 0.
So these are just--
I just throw that
out because these
are different forms that
are common in the notation
that you will see.
So let's expand this guy out.
It's something like this.
So what we're going to do-- so
remember this is dx d lambda.
This is d by dx.
That's a total derivative
with respect to the parameter
lambda.
So this becomes--
I'm going to write
it in two forms.
This is often written expanding
out the u into a second order
form.
This is obvious but
sufficiently important.
It's worth calling it out.
And this has earned
itself a box.
This result is known as
the geodesic equation.
The trajectories which
solve these equations
are known as geodesics.
One of the reasons why I
highlight this is it's--
I'm trying to keep a straight
face with the comment I
want to make.
A tremendous amount of
research in general relativity
is based around doing
solutions of this equation
for various spacetimes that
go in to make the Christoffel
symbols.
My career-- [LAUGHS] it's
probably not false to say that
about 65% of my papers have
this equation at its centerpiece
at some point with the
thing that goes into making
my gammas--
things related to
black hole spacetimes.
This is really important
because this gives me the motion
of a freely-falling frame.
What does a freely-falling
frame describe?
Somebody who's
moving under gravity.
So when you're doing things like
describing orbits, for example,
this is your tool.
A tremendous number
of applications
where if what you care
about is the motion
of a body due to
relativistic gravity,
this gives you a
leading solution.
Now bear in mind
when I did this,
this is the motion
of a test body.
This is an object
with no charge,
no spatial extent, no spin--
that describes no object.
So it should be borne
in mind that this is
the leading solution to things.
Suppose the body is charged.
And there is an
electromagnetic field
that this body is
interacting with.
Then what you do is
you are no longer going
to be parallel transporting
this tangent factor.
It will be pushed
away-- we like to say--
from the parallel transport.
And you'll replace the
0 on the right hand side
here with a
properly-constructed force
that describes the
interactions of those charges
with the fields.
Suppose the body has some size.
Well then what ends up happening
is that the body actually
doesn't just couple to a
single-- remember what's
going on here is that in
the freely falling frame,
I'm imagining that spacetime
is flat at some point.
And in a decent enough
vicinity of that point,
the first order
corrections are 0.
But there might be
second order corrections.
Well imagine a body is
so big that it fills
that freely-falling frame.
And it actually tastes those
second order corrections.
Then what's going
to happen is you're
going to get additional
terms on this equation, which
have to do with the coupling of
the spatial extent of that body
to the curvature
of the spacetime.
That is where-- so for people
who study astrophysical systems
involving binaries, when
you have spinning bodies,
that ends up actually--
you cannot describe a body
that's spinning without it
having some spatial extent.
And you find terms here that
involve coupling of those
spins to the curvature
of the spacetime.
So this is the leading piece
of the motion of a body moving
in the current spacetime.
And it's enough to do
a tremendous amount.
Basically because gravity is
just so bloody strong that all
of these various things--
it's the weakest
fundamental force.
But it adds up because
it's only got one sine.
And when you're dealing
with some of these things,
it really ends up being the
coupling to the monopole--
the most important thing.
So all these other terms
that come in and correct this
are small enough that
we can add them in.
And that, to be blunt,
is modern research.
So let me make a couple
of comments about this.
A more general form--
this will help to clarify what
the meaning of that lambda
actually is.
Suppose that as my vector is
transported along itself--
so one way is recall how we
derive parallel transport.
We imagine going into
a freely-falling frame
and a Lorentz representation.
And we said, in that
frame, I'm going
to imagine moving
this thing along,
holding all the components
constants-- that
defined parallel transport.
Imagine that I don't keep
the components constant,
but I hold them all in a
constant ratio with respect
to each other, but I allow
the overall magnitude
to expand or contract.
So suppose we allow the
vector's normalization
to change as it slides along.
Well the way I
would mathematically
formulate this is I'm going
to use a notation that
looks like this.
So recall this capital
D-- it's a shorthand
for this combination
of the tangent
and the covariant derivative.
I'm going to call
the parameterization
I use when I set up
like this lambda star,
for reasons that I hope
will be clear in just about
two minutes.
So what I'm basically saying
is that as I move along,
I don't keep the
components constant.
But I keep them
proportional to where
they were on the previous step.
But I allow their magnitude to
change by some function, which
I'll call a kappa.
So you might look at
that and think, you know,
that's a more general
kind of transport law.
It seems to describe physically
a very similar situation here.
It's kind of annoying that
this normalization is changing.
Is there anything
going on with this?
Well what you guys are going
to do as a homework exercise,
you're going to
prove that if this
is the situation
you're in, you've
chosen a dumb parameterization.
And you can actually
convert this
to the normal geodesic
parameterization
by just relabeling your lambda.
So we can always
reparameterize this, such
that the right-hand side is 0.
And right-hand side
being 0 corresponds
to the transport vector
remaining constant
as it moves along.
So I'll just quickly sketch--
so imagine there exists some
different parameterization,
which I will call lambda.
So imagine something that gives
me my normal parallel transport
exists.
And I have a different one that
involves the star parameter.
You can actually show that these
two things describe exactly
the same motion, but with lambda
and the dumb parameterization,
lambda star, related to each
other by a particular integral.
So what this shows
us is we can always--
as long as I'm talking about
motion where I'm in this
regime-- where there's
no forces acting--
it's not an extended body--
it's just a test body--
I can always put
it into a regime
where it'll [INAUDIBLE]
geodesic and the right-hand side
is equal to 0.
If I'm finding
that's not the case,
I need to adjust my
parameterization.
When you are, in fact,
in a prioritization
such as the
right-hand side is 0,
you are using what is called
an affine parameterization.
That's a name that's
worth knowing about.
So your intuition is that
the affine parameterization--
I described this
in words last time.
And this just helps to make it
a little bit more mathematically
precise what those words mean.
Affine parameters
correspond to the tick marks
on the world line,
being uniformly spaced
in the local Lorentz frame.
If you are working with
time-like trajectories--
which if you're a physicist,
you will be much of the time--
a really good choice
of the affine parameter
is the proper time of a body
moving through the spacetime.
That is something that
is uniformly spaced,
assuming that's--
you don't have to
assume anything.
Just by definition
it's the thing
that uniformly measures
the time as experienced
by that observer.
So this is-- you guys are
going to do on PSET 4--
this is the exercise you
need to do to convert
a nonaffine
parameterized geodesic
to an affine parameterized one.
That kind of
parameterization is not
too hard to show that if we
adjust the parameterization
in a linear fashion--
so in other words, let's say I
go from lambda to some lambda
prime, which is equal
to a lambda plus b,
where and b are both constants--
we get a new affine
parameterization.
But that's the only class
of reparamterizations
that allows me to do that.
And hopefully that makes sense.
If you imagine that
you're using proper time
as your reparameterization,
this is basically
saying that you just chose
a different origin for when
you started your clock.
And this means you changed
the units in which you
are measuring time.
That's all.
So I'm going to skip a
bunch of the details.
But I'm going to scan
and put up the notes
corresponding to one
other route to getting
to the geodesic equation, which
I think it's definitely worth
knowing about.
It connects very nicely to other
work in classical mechanics.
So it's a bit of a shame we're
going to need to skip over it.
But we're a little
bit behind pace.
And this is
straightforward enough
that I feel OK posting the
notes that you can read it.
So there is a second
path to geodesics.
So recall the way that we
argued how to get the geodesic
equation, which we
said we're going to go
into-- it's actually in
the board right above where
I'm writing right now--
go into the
freely-falling frame.
I have a body that isn't
coupling to anything
but gravity.
Therefore in the
freely-falling frame,
it just maintains its momentum.
It's going to go
in a straight line.
Straight means parallel
transporting tangent vector--
math, math, math-- and
that's how we get all that.
So what this boiled
down to is I was
trying to make rigorous in a
geometric sense what straight
meant.
There's another
notion of straight
that one can imagine
applying when you're
working in a curved space.
So your intuition for--
if you're talking about how
do I make a straight line
between two points on a globe--
your intuition is you say,
oh, well the straightest line
that I can make is the
path that is shortest.
We're going to
formulate-- and I'll
leave the details
and the calculation
to the notes-- we're
going to formulate
how one can apply
a similar thing
to the notion of geodesics.
So imagine I've got an event
p here and event q up here.
And I ask myself, what is
the accumulated proper time
experienced by all
possible paths that take me
from event p to event q?
I'm going to need
to restrict myself.
I want it to be something that
an observer can physically
ride-- so all the
time-like trajectories that
connect event p to event q.
So I've got one a path
that goes like this,
got a path that goes like this,
path that goes like this, path
goes like this, path
goes like-- some of them
might have just become
somewhat space like,
so I should rule them out.
But you get the idea.
Imagine I take all the
possible time-like paths
that connect p and q.
Some of those paths will
involve strong accelerations.
So they will not be
the freefall path.
Among them there will be
one that corresponds exactly
to freefall.
So if I were talking about--
imagine I was trying to--
and this is something that
Muslim astronomers worked out
long, long ago-- they wanted
to know the shortest path
from some point on
earth towards Mecca.
And so you need to find what
the shortest distance was
for something like that.
And when you're doing this
on the surface of a sphere,
that's complicated.
And that's where the
qibla arose from,
was working out the mathematics
to know how to do this.
This is a similar
kind of concept.
I'm trying to define--
in this case, it's going to turn
out it's not the shortest path,
but it's the path on which an
observer ages the most because
as soon as you
accelerate someone--
it's not hard.
Go back to some of
those problem sets
you guys did where you look
at accelerated observers.
Acceleration tends to decrease
the amount of aging you have
as you move through some
interval of spacetime.
So the path that has
no acceleration on it,
this is going to be the one on
which an observer is maximally
aged.
Why maximum instead
of a minimum?
Well it comes down to
the bloody minus sign
that enters into the
timepiece of an interval
that we have in relativity.
And that's all I'll
say about that,
is just boils down to that.
So what we want to do
is say, well along all
of these trajectories,
the amount
of proper time that's
accumulated-- so let's
just say that every one
of these is parameterized
by some lambda that describes
the motion along these things.
This is the amount
of proper time
that someone accumulates
as they move from point p--
which is at-- let's say this
is defined as lambda equals 0--
and it's indeterminate what
that top lambda is actually
going to be.
It's whatever it takes when
you get up to lambda of q.
So what the notes
I'm going to post do,
is they define an
action principle that
can be applied to understand
what the trajectory is
that allows you to do this.
So I'll just hit the highlights.
So in notes to be posted,
I show that this delta t--
this delta tau rather--
this can be used to
define an action.
It looks like this.
And then if you
vary the action--
or rather you do a variation
of your trajectory--
where you require that the
action remain stationary under
that variation-- in other words
I require delta i equals 0 as x
goes over to such--
so-- what you wind up with--
is delta i equals--
Notice what I've got in here.
This is just a
Christoffel symbol.
So when I do this
variation, what I find--
and by the way going from
essentially that board
to that board, it's about
2/3 a page of algebra.
Going down to this
one, there's a bunch
of straightforward but
fairly tedious stuff.
It's one reasons why I'm
skipping over the details.
We've got enough G
mu nus on the board.
So the key point is I am going
to require that my action be
stationary, independent of
the nature of the variation
that I make.
For that to be true,
the quantity in braces
here must be equal to 0.
Let me just write
that down over here.
This is a good place to
conclude today's lecture.
So we require this to
be 0 for any variation.
Yet the bracketed term
being equal to 0, pull that
out, and clear out that factor
of the metric with an inverse,
you've got your
geodesic equation back.
So we just quickly wrap this up.
So it's worth looking
over these notes.
It's not worth going
through them in gory detail
on the board, which
is why I'm skipping
a few pages of these things.
But what this demonstrates
is that geodesics--
our original
definition is that they
carry the notion of a straight
line in a straightforward way
from where they are obvious
in a locally Lorentz frame
to a more covariant
formulation of that--
so a generalized straight
line to a curved spacetime.
And they give the trajectory of
extremal aging in other words
a trajectory along which
between two points in spacetime,
an observer moving from
one to the other will
accumulate the most proper time.
So I'm going to stop here.
There's a bit more,
which I would like to do,
but I just don't have the time.
But I'll tell you the key
things that I want to say next.
Everything that I've
done here so far
is I've really fixated on
time-like trajectories.
I've imagined there's a
body with some finite rest
mass where I can make a
sensible notion of proper time.
We are also going
to want to talk
about the behavior of light.
Light moves on
null trajectories.
I cannot sensibly define proper
time on long such a trajectory.
They are massless.
There's all sorts of
properties associated with them
that just make this analysis.
The way I've done
it so far, I'll
need to tweak things a little
bit in order for it to work.
We will do that tweaking.
It's actually quite
straightforward
and allows us to also bring
in a bit more intuition
about what affine parameters
mean when we do that.
So that'll be the
one thing we do.
The other-- it's unfortunate
I wasn't able to get to it
today-- but it's a
straightforward saying,
which I think I may include
in the notes that I post--
is including what happens,
if your spacetime--
so if the metric you use to
generate these Christoffels has
a Killing factor
associated with it,
you can combine
Killing's equation
with the geodesic
equation to prove
the existence of conserved
quantities associated
with that motion.
And that's where
we start to begin
to see that if I
have a spacetime that
is independent of time, there's
a notion of conserved energy
associated with it.
So we will do that on Tuesday.
