[SQUEAKING]
[RUSTLING]
[CLICKING]
PROFESSOR: OK.
Great.
So at this point,
we basically have
all of the most
important-- well,
we now have the
full understanding
of how to work with tensors.
I haven't really done too
much physics with them at this
point, but we've
very carefully--
one might even argue
excessively carefully--
laid out this
mathematical structure
that is going to
be-- we're going
to use it to contain the
geometric objects that
will describe the
physics that we're
going to study over the
course of the entire semester.
So there was a lot of twiddly
detail in the preceding
lecture, but I would say the
two most important things I want
you to take out of this is the
idea that we now have tensors
as a general class of geometric
objects which map one-forms--
or dual vectors, if you prefer--
a combination of one-forms
and vectors to the Lorentz
invariant real numbers.
And further, the distinction
between one-forms, and vectors,
and what is being mapped to what
is not so important, because I
can always use the metric
to either raise or lower
indices on tensors or
vectors or one forms
in order to convert
from one to another.
So if I raise an index--
if I have a tensor that's
got m indices in the
upstairs position
and n in the downstairs,
I raise an index,
and I've got m plus 1
upstairs, n minus 1 downstairs.
And likewise, I
can lower index--
put one of those ones
in the upstairs, down
to the downstairs.
We wanted to go through all
that stuff with great care,
because we're going to need
the foundations of that
to make a lot of what we
talk about later rigorous.
It is really overkill
for where we're starting,
but it's worthwhile to have
that scaffolding in place.
I'm sick of it,
though, so today I'm
going to try to do
some things that
are more physics, because,
well, not all of you
are necessarily
physics students,
but I'm a physics
professor and I'm
a little tired of doing math.
So let's think about--
when we're studying
physics, generally we
are interested in the
behavior of bodies and fields.
Let's move this to the
relativity framework.
We're looking at the behavior of
bodies and fields in spacetime.
So far, if I think
about the quantities
that we have introduced--
and by the way, I apologize
for the blackboards.
A previous lecturer today used
chalk that cannot be erased
very well, so they're sort
of grayish boards today,
unfortunately.
But we'll manage.
Anyhow, so far, the quantities
that we have introduced
that have solid
physics content are
good for describing particles--
individual particles-- maybe
a handful of them at once.
So the two that I want to
focus on in particular are
the 4-velocity--
so if I have a body
with a 4-velocity u,
a particular Lorentz observer
will describe that as having
a timelike component gamma--
special relativistic gamma-- and
a spacelike component gamma v.
This is defined such that
u dot u equals minus 1.
It is great for
describing-- it is
the tool we use to describe the
trajectory of a material body
that follows a timelike
trajectory through spacetime.
Notice that its norm is minus 1.
The fact that it's 1 means
that it's normalized,
and the minus tells us
that it is timelike.
By the way, because
this is timelike,
the 4-velocity is not going
to be a useful tool for us
when we want to describe
the behavior of light.
If we want to talk
about the motion
of a photon in our spacetime, we
can't use a 4-velocity for it.
The trajectory of
a photon is null.
Whatever quantity that is going
to describe its trajectory--
I take its dot product with
itself, I have to get 0.
And you can
intuitively get a sense
what's going on with that.
If I take the way a
particular observer interprets
these things and
says, what would
this turn into if v
equals c, well, you
have infinity and infinity,
because your gammas
are diverging there.
So it's a singular limit.
It doesn't behave well.
We will overcome
this difficulty.
The other quantity
that we have used--
let me see if I can clean
this a little bit better.
All right.
The other quantity
that we have defined
which is good for particles
is the 4-momentum,
which is just that 4-velocity
multiplied by a rest mass.
So this is defined--
you can clearly,
by trimming together a
bunch of definitions--
you have p dot p
equals minus m squared.
We're actually going to
interpret this, though.
So if a particular
Lorentz observer
makes a measurement
of this, they
will call the timelike component
E and the spatial component p.
And so you put all
these things together,
and this tells me that E
squared minus magnitude of p
squared equals m squared.
Notice this is
perfectly well-behaved
for a massless particle.
This is actually going
to be the trick by which,
when I want to describe
the trajectory of photons
or of light in a spacetime--
I can't use 4-velocity.
I can use 4-momentum.
It works perfectly well.
It's 0 times a 4-velocity, but
if I use this intuition that
this thing is diverging when I
take the speed of light limit,
I sort of--
I guess I cheat,
but I'm basically
getting 0 times infinity.
And it behaves.
We're going to do it a little
bit more rigorously than that,
but if you want a
little bit of intuition,
that's essentially why it works.
So the last thing
which I will say here
is that this is
good for m equals 0.
OK.
And when you do have
m equals 0, it's
often convenient to write
this as h bar omega.
So use the usual formula
for the energy of a photon--
let me move this so
that you can see--
and then I just
put a unit vector
into the spatial direction
that defines the direction
in which this is moving.
OK.
So this is all fine
as far as it goes,
but it's kind of restrictive.
We want to a little bit
more with our physics
than just worry
about the kinetics
of individual particles.
There's a lot we
can do with that,
but one of the
things that we like
to do when we're
studying gravity
is to have a description
of continual matter,
like things that gravitate.
You want to be able to build a
description of something like
a star, or an exotic
star-- a neutron star--
something like that.
And so just having the behavior
of individual particles
is not good enough.
I want to be able to
describe things like fluids.
I want to be able to
have a continuum that
describes things.
So what I'm going
to start doing today
is begin to introduce
the mathematical tools
that we will use.
Pardon me.
I seem to have twisted this
around myself in a crazy way.
We're going to describe
the mathematical tools that
are useful for dealing
with continuous matter,
rather than just particles.
So let's call this more
interesting matter.
So the simplest continuum
form of matter which
we're going to talk about is--
we're going to call it dust.
OK.
When you're here
at the chalkboard,
you can't get away
with it, so you
have an idea of what it means.
Physically, what
I mean by dust is
that it is a collection
of particles that do not
interact with each other.
So they pass through each other.
There's no pressure
that is generated.
So they have energy density
associated with them,
but no interaction.
So it's the most boring kind
of matter that you like.
You can think of it as
essentially just particles,
but it's a ton of
them, and they're
smearing out into a continuum.
In particular, one
thing I'm going to say
is, so I can imagine, if I
take my two erasers here,
I have a nice dust
field in front of me
now, which I helpfully
just inhaled.
And you can think of this
thing as a field of little dust
elements that are
all moving around.
If I go in and I track
an individual element
of that dust--
I go in and I make
a little cube that's
a nanometer on each side--
every little element
has its own rest frame.
In that cloud-- I'm not
going to do that again--
but as I made that
cloud, different elements
within the cloud have
different rest frames,
but for each individual element,
I can define a rest frame.
So we're going to
use the properties
of the dust in that rest
frame as a way of beginning
to normalize and
get a grasp on how
we're going to describe this.
But I just want to emphasize
again-- in a given cloud,
different elements may
have different rest frames.
OK.
But let's suppose that
I clap my erasers,
I make my little cloud,
and I go and I zoom in
on one particular
nanometer by nanometer
by nanometer chunk
of that thing,
and I go into the rest frame
of that particular little bit
of dust.
So how am I going to
characterize that?
So presumably, each
particle has its own mass
associated with it.
Each little bit of dust that's
in that thing has its own mass.
We're going to
worry about how that
comes into the picture
a little bit later.
Let's just begin by saying,
one of the things it's
going to be
interested in knowing
is, how much dust is there in
that little cubic nanometer
of dust in front of me there?
So the first thing
we're going to focus on
is just counting how many bits
of dust are in this element.
In particular, I'm going to
want to know, how many elements
per unit volume?
In other words,
the number density.
So I claim that's a good thing
for us to get started on.
And it looks like
they haven't quite
poisoned this board as badly.
Good.
So let's call n sub
0 the number density
in the rest frame
of that element.
OK.
So this will just be some
number per unit volume.
OK.
So it's a quantity that has
dimensions 1 over volume.
Now, in the rest
frame, like I said,
we're going to talk about how
I treat the mass associated
with each element,
and how much energy is
in that thing, a
little bit later.
There's a few other tools
I want to introduce first
before I get to there.
But in the rest
frame, if I'm not
worrying about the mass
and the energy, this is it.
This is really the only
thing I can say about this.
So what do I'm going to do
now is say, OK, well, if I
have a general cloud here--
as I described, when we made
this little turbulent cloud
of dust in front of us,
it was all swirling around
and doing its own thing.
You're generally not
in the rest frame.
OK.
So you also want to know how
to characterize the dust out
of the rest frame.
So let's move out
of the rest frame.
When I do this,
two things happen.
So first of all, let's
continue to focus on--
so we've got a particular
nanometer cubed of dust
that we are looking at here.
So we're very attached
to that particular thing.
So I now boost
into a frame where
I'm going 3/4 of the speed of
light or something like that,
but I'm going to pay
attention to that--
I'm very attached to the
dust in that little cube,
so I'm going to keep
myself focused on that.
Boosting into frame can't
change the total number,
so the amount of dust
has to be the same.
But of course, there's going
to be a Lorentz contraction,
and so the volume
will get smaller.
So the number in a
particular volume
stays the same while the
volume Lorentz contracts.
And so let's call n the number
density in this new frame.
This is just going to be
the Lorentz gamma times n0.
So again, I'm assuming
you're all perfectly
fluent with special relativity.
So all I did there was say
it's going to contract along
that direction.
The lengths are smaller
by a factor of gamma,
therefore the volume is
larger by a factor of gamma.
But there's a second thing
that happens as well.
When I'm in this
new frame, there
is now a flow associated
with the dust.
So this will now be
flowing through space.
In particular,
what you can do is
define a flux that describes
the number of particles crossing
unit area in unit time.
So let's define n
3-vector to be this.
And you use a little
bit of intuition,
and just think about the flow
of these particles as they
kind of go translating past you.
This can only be
the number density
that you measure in that
frame times the 3-velocity.
OK?
Think about that a little bit.
It's very much
like when you learn
about current density in basic
E&M. Notice it's n and not n0
here.
So when you look at these two
things for a moment or two--
let me just write
down one more step.
I can write this as gamma n0 v.
These are screaming to be put
together to make a 4-vector.
They're looking at
you and saying--
this bit's kind of like, I'm a
timelike piece of a 4-vector.
Spatial piece, I love you.
Let's get together.
They clearly are just
screaming to go together.
And when you do
that, you say, hmm,
let's see what happens here.
Two great tastes that
taste great together.
OK.
That's nice.
Let's take advantage of
the fact that both of these
have a really
simple form in terms
of the density in the
dust's own rest frame.
And you look at this
and go, holy moly.
That is nothing
more than the number
density times a 4-velocity.
So that's pretty cool.
If I have this stuff
in front of me--
you know, again, what the heck.
This time I'm going to
actually do my clap again.
So I do this.
Every little element--
I can think of it as having some
trajectory through spacetime.
I can attach a
4-velocity to that.
And if I know what the density
of dust is in that rest frame,
I put it together-- now I've
got a geometric object that
describes this thing.
And one of the reasons
why this is powerful is,
geometric objects-- as I
have emphasized repeatedly--
have a geometric
meaning that transcends
a particular coordinate
representation.
Another way to think about
this is that all observers--
I don't care whether you're
at rest in this classroom,
or you've had too
much coffee and you're
dashing through at 3/4
the speed of light.
We all agree on N--
capital N vector-- but we choose
different ways of splitting
spacetime into space and time.
And so I will have a
different n from you,
I will have a
different nv from you.
We all have different ways
of splitting this spacetime
into space and
time, and so we have
different ways of splitting
these two things up.
But this is a geometric
object we all agree on.
It's something that we can
hang a lot of our physics
on, and build frame-independent
powerful geometric
representations.
So before going on to do
a few things with this,
let's just explore a couple
of the properties of this.
All right.
So whenever you've got a
4-vector, at a certain point,
you just sort of
say to yourself,
well, I've got this thing.
Let's take the dot
product with itself.
Why not?
So in this case, N dot N
equals minus n0 squared.
Or, turning this
around, this tells you
that if someone comes buy
and goes, psst, here's
a number 4-vector,
you can figure out
density in the rest frame
associated with each element
by the following operations.
So that's nice.
OK.
In other words, the number
density, in a sense,
tells you what the
normalization of this vector is.
A few moments ago,
I defined N as being
kind of a flux of these things.
It's a flux in the
spatial directions.
We want to think about flux
in a more general kind of way.
And so another thing
which we're going to do--
and this is where 1-forms are
going to play an important role
for us--
is a nice systematic way to pick
out a flux across a surface.
So here's the bit where,
when I was talking about it
in Tuesday's lecture, I was
getting the blank stares.
And sorry, I'm coming
right back to this again,
so we're going to harp on this
concept a little bit more.
So recall we talked
about the fact
that if I have a particular
coordinate system--
I have a particular set
of coordinates x alpha
that I use to
represent spacetime.
If I make 1-forms that are the
gradients of those coordinates,
I can represent
them as essentially,
level surfaces at unit ticks
of the coordinate x alpha.
OK.
And basically, the
way to think of it--
I kept emphasizing that
the basis 1-forms are
dual to basis vectors.
And if you have this intuition
that your basis vector is
a little arrow pointing
like so, then the thing
that is dual to it is a surface
that is everywhere but pointing
along like so.
That's kind of the way I
want you to think about it.
And those graphics that are
scans from MTW I put on the web
give better pictures of
that than I am able to draw.
So putting these
concepts together,
it basically tells
me that I can use
these basis 1-forms as a way of
defining in an abstract form.
So if I want to know
the flux of N in the--
let's write it this way--
in the x alpha direction.
So remember, the basis 1-forms--
the alpha in that case is
not like a coordinate index.
It's labeling a particular
member of a set.
And so what you would do is, you
would construct this by saying,
OK, let's take the beta
component of a basis
1-form alpha.
Contract it like so.
And that tells me about the
flux-- how much of this stuff
is flowing in the direction
associated with alpha.
So this is a tool that we're
going to use from time to time.
One little thing which I
want to emphasize-- actually,
two things I want to emphasize.
So if I'm working in an
intelligent coordinate system,
remember, this is basically
just the identity matrix, right?
And so this ends up being a very
simple thing that I actually
work out.
And it says, if I want to
know the timelike component,
then this is 1 in the timelike
direction, 0 everywhere else,
and I just pick out N sub t--
in other words, the density
itself in that frame.
That is the other thing
which I want to emphasize.
So when you do this, if I pick
out the timelike piece of this,
this just gives
me the 0 component
of this thing,
which is the density
that I measure in this frame.
This is very, very simple
in terms of the calculation,
but I want you to stop and think
about what it means physically.
So flux in the timelike
direction-- what
does it mean for a flux to be in
the timelike direction, right?
If i take my water--
actually, I'm kind of thirsty--
so I want a little bit of flux
of water going down my throat.
You have an intuition
about what that means.
The water's actually flowing
in a particular direction.
If it's flowing through time--
there it goes.
That basically means
it's just sitting there.
It's not doing anything,
but just moving in time,
as we all are doing.
One time I remember my wife
saying about our daughter--
she's like, she's
growing up so fast.
I was like, eh, she's growing
up at 1 second per second.
And that's just
the way things go.
It's just sitting there
and it's living its life
at 1 second per
second, as we all do.
That is density.
So we described the
flux of a thing in time
as just being its density.
That's another way-- when we
do a lot of our calculations,
and we talk about
the flow of things
in the timelike
direction-- that tends
to be just the simple density
associated with stuff.
And then if I did
this-- if I pulled out
the x direction of this thing,
I would pull out the x component
of velocity times that.
And that's sort of the flux of
something in the x direction
like you are probably used
to from other classes.
Now, a place where
this turns out--
so doing it when I'm just
using the basis 1-forms
is kind of trivial,
[INAUDIBLE] sucks.
More generally,
what you can do is
define a surface as being
the solution of some scalar
function in spacetime.
So let's say I do something
like psi of t comma x comma y
equals a constant.
OK.
This is very abstract, so let me
just make it a little bit more
concrete.
Suppose my function
psi were square root
of x squared plus y
squared plus z squared,
and my constant was 5.
OK.
Well then, my scalar
field would be picking out
a sphere of radius 5.
OK.
You can make a little bit
more complicated than that,
and people often do.
You can define the unit
1-form that is associated
with the normal to this thing.
Or rather, it's the 1-form
associated with the surface.
If you translate
this to a vector--
you raise the index--
with the vector,
it would be the vector
normal to the surface.
You might need to normalize it.
Let's imagine that we
normalize this thing.
And then we would just need
to contract it along this,
and that tells me
about the flux through
this particular surface.
OK?
This is one of the
things that-- this idea
of 1-forms being
level surfaces--
it tends to be useful
for things like that.
We're going to come back to
a similar sort of picture
in just a moment, because
we'll have to start
talking about integration.
OK.
In prep for that
discussion, now that I'm
talking about
things a little bit
more complicated than just the
kinematics of simple particles,
we're going to want
to have some laws.
And we want to have
geometric ways of describing
those laws that essentially
act as constraints
on what those particles can do.
OK.
So the form in which we're
going to express them-- we're
going to tend to put things into
the form of conservation laws.
So suppose, here's my little
element that's got dust in it,
and it's embedded
in an environment
with a bunch of dust around it.
And over some time interval,
some dust flows in,
some does flows out.
The density can go up,
the density can go down.
The total number--
it may vary depending
on how things are flying.
The spatial flux out of
the sides must come--
and I'm going to say
it's the flux out--
so let's say that that comes
at the expense of the density
of dust already there.
OK.
And so if you were to
just to intuitively
write down what kind
of conservation law
you would expect to see, you
would write it as something
like this, based on simple
Euclidean intuition.
What's kind of nice-- you
look at that for a second.
You go, ooh, if I think
about this as being the time
complement of my 4-vector.
This is the space
component of my 4-vector.
This has a very obvious
form when I write it
in a geometric framework.
This whole thing can be
rewritten as a conservation
law that looks like this.
OK.
So I'll remind you, d sub
alpha equals d by dx alpha.
I'm going to talk a little bit
about some of the derivatives
a little bit later, because
there are a few subtle points
that can get introduced.
But for now, we just
know that d downstairs
t is d by dt. d downstairs x
is d by dx. d by downstairs y,
et cetera.
So this is really nice.
One thing which I want
to emphasize-- again,
coming back to what I said
over there just a moment ago.
When I write down
this conservation law,
I'm assuming that
someone has defined
what time means and someone
has defined what space means.
This is a form that's
covariant, right?
All observers agree
that this goes together.
When they actually make
their own coordinate systems,
they're going to have their own
time coordinate and their own
x-coordinate, but they're all
going to be different flavors--
just different ways of
instantiating what this is.
Now, pardon me for
just one moment.
Depending on the
pace of the course,
we're about to switch over to a
different set of lecture notes,
and I want to make
sure I smoothly
go from one to the other.
OK.
So in many of your physics
classes, you have learned,
when you get a conservation
law, both a differential
form like this--
the rate of change of N in a box
is related to the amount of N
flowing out of that
box and its sides.
And you also learn an integral
form of the conservation law.
So without proof,
let me just say
that it should be
intuitively clear
that what I've written down
over here is equivalent to--
OK, so it looks like this.
You know what?
Let me just fix up my
notation a little bit.
Let me call this lowercase n.
OK.
A few symbols that
I've introduced here--
so V3 is some volume in
three-dimensional space,
and dV like this--
this is a symbol
that means the boundary
of that 3-volume.
And that's basically--
it's a form of Gauss's law.
That's what I've
written down there.
OK?
So you've all seen
things like that.
Again, let me emphasize
that when I write down
a formula like that, I can
only do that having assumed
a particular Lorentz frame.
That t is the t
of some observer.
That volume is the volume
of that particular observer
who is using t.
You jump into a
different Lorentz frame,
their volumes will
not be the same.
Their times will
not be the same.
I'm going to make
some coordinates up.
So an integral form like this,
as I've written it there, only
works in one given
Lorentz frame.
Nonetheless, we are
going to find it useful,
even though in some
sense, when you do it
in an integral form
like this, you're
saying things in the framework
of some particular observer.
Sometimes you want to
know in the framework
of some particular observer.
It could be you, right?
And you might care about
these sorts of things.
And that's good, but the
way I've written it here--
first of all, it's
in language that--
it's in a mathematical
formulation
that's not easy to generalize
as I take things up
to higher dimensions.
And so what I
would like to do is
think about how to step
up a formulation like this
in such a way that things are
put into as frame-independent
a language as is possible, and
that will generalize forward
when we start looking at
more complicated geometries
than just geometry of
special relativity.
So I want to spend
the next roughly 10
or 15 minutes talking about
volumes and volume integrals.
And my goal here is to try to--
I'm going to start by
just doing stuff that
comes from the journal of duh.
It's stuff you have seen
over and over and over again,
but I want to re-express it
using mathematical formulation
that maybe--
you have seen all the
symbols, but perhaps not used
in quite this way.
And then it'll carry forward
in a framework that generalizes
in a very useful way for us.
So let's begin with
just simple 3D space.
So I'm going to
begin in 3D, and I'm
going to consider
a parallelepiped--
parallelepiped.
Ha-ha.
Got it right-- whose sides
are a set of vectors.
So there are three
vectors, A, B, and C. OK.
So here is a vector A. This
one going into the board
is vector B. And this one
going up here is vector C. OK.
So those are my three vectors.
And if I go draw the ghost legs
associated with these things--
OK.
That's a little bit better.
So these three vectors
define a particular volume.
And you guys have
probably all seen--
you know you have three vectors.
You can define a volume
associated with this.
A really easy way
to get that volume
given those three
vectors is to take A
and dot it into the
cross product of B and C.
This is a quantity which is
cyclic, so if you prefer,
you can write it as B
dotted into C cross A,
or C dotted into A cross B.
That can be expressed
as a determinant, or--
you guys can look at my notes to
see the determinant written out
if you like, but it's
not that interesting,
so I'm not going to
use it very much.
An equivalent way
of writing all that
is to use the
Levi-Civita symbol.
So that 3-volume is given by
epsilon ijk Ai Bj Ck, where--
I'll remind you-- epsilon ijk
equals plus 1 if i equals 1,
j equals 2, k equals 3,
and even permutations.
Even permutations means I
swap two pairs of indices.
So 123, 231, 312--
those all give me plus 1.
It gives me minus 1 for any
odd permutations of those.
So 132, 231, et cetera.
Those will all give me minus 1.
And it's 0 if any
index is repeated.
So you probably have all
seen things like this before.
This is fairly basic
vector geometry.
We are going to regard
the Levi-Civita symbol
as the components
of a 0, 3 tensor.
OK.
Bear in mind for just a
moment here I'm working only
in-- sorry, just ran
out of good chalk--
I'm only working in 3-space,
so my tensor definition
is slightly different.
It's not going to be a set of
things that maps to Lorentz
invariants, but it's going
to be invariant with respect
to things like rotations
and translations
in three-dimensional space.
So I'm going to regard these
as the components of a 0, 3
tensor that basically
takes in vectors and spits
out the volume associated with
the element whose edges are
bounded by those vectors.
OK.
So I could say, in this abstract
form I wrote down earlier,
imagine a boldfaced epsilon
which is my volume tensor.
I put these slots
into it, and voila.
I get the volume out of it.
Now, with that in mind,
remember some of the games
that we played with tensors
in the previous lecture.
So when I was talking
about spacetime tensors,
if I filled up all
of their slots,
I got out a Lorentz
invariant number.
In this case, I'm in 3-space,
so I fill up all its slots,
I get an invariant
number in this 3-space.
Suppose I only put
in two vectors.
So suppose I do something
like, I plug in--
let's leave the
first slot blank--
and I put in vectors
B and C. OK, well,
writing this out
in component form,
I know this is
epsilon ijk Bj Ck.
That's just B cross C, right?
That's the area spanned
by the side that
is B cross C. And you guys
have learned in other classes
that you have an
extra index left over,
so it's a vector that has a
direction associated to it.
So it's sort of an
oriented surface.
We put the index in the
downstairs position,
so we're actually going
to think about this
as a 1-form corresponding
to the side whose
edges are B and C. So let's
call this side 1-form sigma.
This is a 1-form whose magnitude
is the area of the side spanned
by the vectors B and C.
And although I can still tell
it hasn't quite gelled yet,
it's useful to think
of 1-forms as being
associated with surfaces.
Guess what?
This is the side of
a parallelepiped.
That's a surface.
So it actually holds together.
All right.
So using all of this, if
I wanted to write down
how to do something like
Gauss's theorem in this kind
of geometric language-- and
again, we emphasize this
is very much in the spirit
right now of mosquito
with a sledgehammer.
We don't need all
this sort of stuff,
but we're about to
step up to something
a little bit more complicated.
So what you would do is say, OK,
well, I know Gauss's theorem.
I pick a particular 3-volume.
I say the divergence of
some vector field integrated
over that volume is
given by integrating
the flux of that vector over
the surface of this thing.
So what you might
want to do then
at this point is say, oh, OK,
well, what I'm going to do,
then, is say that my volume--
oops, pardon me a second.
First thing I'll do is
define a differential triple.
I'll define some x1 that
points along one direction
I care about, and x2, and an x3.
And then I will say dV equals
epsilon ijk dx1i dx2j dx3k.
I can likewise define
a 1-form associated
with my area element, as
I have done over here.
I'm not going to
actually write this out.
The key thing
which I want to say
is you have all the pieces--
you put all these
things together,
and you can define this thing.
It's now very easy for you to
prove Gauss's theorem using
this kind of ingredients.
What I want to move onto--
there's a few more
details in my notes.
It's not super difficult or
interesting to go through this.
What I want to now start
doing is generalize
all of these ideas
to the way we're
going to approach
them in spacetime.
Basically, we're going to
do exactly the same kind
of operations that I
just did in space--
three-dimensional space--
but I'm going to put an
extra index on things,
and I'm going to do all of
my quantities in spacetime.
OK.
So imagine a parallelepiped
with sides A, B, C,
and D. Four dimensions, so
it's going to point along four
different-- these can
be mutually orthogonal.
I'm going to define the
invariant 4-volume associated
with these things like so.
Where now my four-index
Levi-Civita is defined such
that epsilon 0123 equals plus 1.
If I do an odd
permutation of those--
I exchange one pair of
them-- epsilon 1023--
this equals minus 1.
If I repeat any index, I get 0.
And likewise, all even
permutations of this
give me plus 1.
All odd permutations of
this give me minus 1.
Or likewise, just do even
permutations of this one.
So that is how I'm
going to generalize
my Levi-Civita symbol.
As they say on
The Simpsons, it's
a perfectly cromulent object.
I'm going to need to talk
about the area associated
with the faces of
each of these things.
So what is the area of
the face of a 4-volume?
A 3-volume.
So you can define a 1-form
that tells me about the--
you can either call it the
3-volume or the 4-area.
Knock yourselves out as to
how you want to call it.
And the obvious
generalization-- let's
say I leave off edge A. I want
to get something like this.
I do my similar exercise of
defining a-- in this case,
it'll be a differential quartet
associated with directions 0,
1, 2, and 3.
And so by going through a
procedure very similar to this,
you get a generalization
of Gauss's theorem.
It says that if I integrate
the spacetime divergence
of some 4-vector over a
four-dimensional volume,
it looks like what
I get when I sum up
the flux of that guy over
all of the little faces.
So I'm not going to step
through the proof of that.
It's fairly elementary,
and basically it's
just like proofs of Gauss's law
that you have seen elsewhere,
but there's an extra
dimension attached to it.
Really nothing new
that's going on here.
The thing which is
new is, this is now
being done in an
additional dimension,
and where this
tends to be useful
is when there is some
kind of a conservation law
that tells you something about
this left-hand side here.
So the whole starting
point of this discussion
was I wrote down, on
intuitive grounds,
that the rate of change of
the total amount of-- so
what I did was I had an
integral of dust density
over a 3-volume, and I
said, d by dt of that
was balanced by the flux
through the surfaces
on the edge of that 3-volume.
As I argued, that's an
observer-dependent statement,
because you have chosen
a particular time.
You have chosen a particular
space to make those volumes.
This, on the other hand--
so let's switch my
general vector field
V to be N that we started
this discussion off with.
So this guy-- oops.
Let me write it the way
I wrote it over there.
d4x over some 4-volume.
This must be 0,
because we are going
to require that this thing be--
so if my number density is
conserved, this must be 0.
So this tells me that when
I do this flux integral,
it's going to have to be 0.
Let me now break this
integral up, and actually
write this into a form that
is a little closer to the way
that you may have seen
something like this before.
So what I want to do is
actually zoom in and think
about the
four-dimensional volume
that I'm doing
this integral over.
So I'm just going to do a
two-dimensional cut of it
here on the blackboard.
Let's let the time
axis go up, and let's
define the edges of my
volumes to be t1 and t2.
x-axis going across here.
Boundaries are x1 and x2.
OK.
So here is my V4.
And every face here is an
example of my boundary of d4.
OK.
Of course there's one over
here as well, and over here.
So if I were to do
the top line here,
I know that's got
to get me 0, so I'm
going to take advantage of
this and say, let's just
look at what happens
when I do the integral
of the flux of this thing
across the many different faces.
So in a four-dimensional
parallelepiped--
an n-dimensional
parallelepiped has 2n phases,
so there will be eight
integrals we need to do.
They're pretty obvious
though, so I'm just
going to write down
a handful of them.
OK.
So over all those faces--
N alpha sigma alpha.
So what I'm going to
do now is say, OK,
let's evaluate this on the
face that is at t equals t2.
So when I do this, I'm
going to get N0 dx dy dz.
Let's do next the
contribution at moment--
actually, let me move
this over, because I
want a little bit more room.
I also do one on the
slice t equals t1.
When I do this,
though, in the same way
that when you guys do fluxes
in three-dimensional space,
you get a sign associated
with the orientation
of these things, because
the Levi-Civita symbol here
has sort of a right-hand
rule built into it.
And so when I do it
for the side that
is on the future
side of the box,
I'm going to get a plus sign.
Do all the analysis
carefully on the side
that's on the past side the
box, you get a minus sign.
Then I'm going to do it--
I'll pick out the
component N1, and I'm
going to do this along the
face that's at x equals x1.
Sorry.
This is going to be x2.
OK.
You can write down
four other integrals
according to when you do
the y side and the z side
that I've not drawn.
OK.
What I'm going to
do now is imagine
that this box gets very small
in the timelike direction.
So let's let t2
go to t1 plus dt.
And I'm going to do
this and then rearrange
things a little bit.
Again, apologies.
I'm eating chalk here.
So I'm going to
rearrange this integral
so that it's of the form--
this side's being
evaluated at t1 plus t2.
So I'm going to rearrange stuff
so that I can then write--
just one moment.
Oh.
Apologies.
This whole thing,
of course, equals 0.
I'm just looking over
my notes and went--
there was a magic sign flip.
I was just looking at
it and going, where
the hell did that come from?
OK.
So I left off my equals 0 there.
I'm basically moving a bunch
of terms to the other side.
So now I'm integrating
this over the face at x2.
And I do an integral
over the face at x1.
And then I'm not drawing
in a bunch of faces
along y2 and y1, z1 and z2.
Divide both sides by dt.
Take the limit of dt going to 0.
What is this?
This is the derivative of
the volume integral of N0.
So let's just write
that out explicitly.
So if I've got integral
t1 plus dt N0 dx dy dz
minus the integral at t1--
all this divided by dt.
Take the limit as dt goes to 0.
That becomes this.
That i is a bad erasure.
My apologies.
What do I get for
the other term?
Well, when I divide
out that dt, I
am just left with the flux
of the spatial component
of the number vector N through
all six sides at moment t,
or t1.
And at last-- so I will wrap
things up in just a moment
here.
What this finally leads us to
is that this fully covariant
form that I have
at the top there--
this becomes-- so
in the language
that you probably learned
about in an earlier class,
you can think of this as the
area element of each side.
These are completely
equivalent to one another
after you have chosen a
particular Lorentz frame
and invoked this
four-dimensional analog
of Gauss's law.
So that was a lot to take,
but I wanted to do it
for this one particularly--
to be blunt, this was a
particularly simple example,
because--
so right now, the
only matter that I've
introduced beyond particle
kinematics is dust.
And dust is actually
surprisingly important.
When we actually
get to cosmology,
there are essentially
two forms of matter
that we consider when we study
cosmology, and one of them
is dust.
In cosmological
situations, a dust particle
is basically a galaxy, or
even a cluster of galaxies.
We're thinking big.
So it's not trivial to
do this, but we are soon
going to need to introduce
mathematical tools that
are a little better for
describing things like fluids,
and stuff like that.
And when we do
that, we are at last
going to have things like
conservation of energy
and conservation of momentum.
It will be really
easy to write down
a differential conservation
law that describes conservation
of both energy and momentum
using that mathematical object.
Via a process very similar
to what I just did here,
we can then turn
this into integrals
that describe how
energy is conserved
in a particular volume
where things may be flowing
into or flowing out of it,
and how momentum is conserved
per unit volume as things flow
into it and flow out of it.
I'm not going to go
through that in detail
on the board, but
having done this--
I have some notes that I'm going
to post to the website that
describe this, and that will
be the way we communicate this.
And we're going to take
advantage-- to be blunt,
we will mostly just use
this differential form.
We'll use the fact there is a
particular mathematical object
whose divergence-like
derivative there is equal to 0.
OK.
We have a few minutes left and
we've ended early a few times,
so I'd like to take
advantage of this
to switch gears a
little bit and talk
about another
important 4-vector that
plays some role in physics,
and allows me to introduce
a few other very useful
tricks that we will often
take advantage of at various
points in this class.
I forgot there's a straw.
When you tip it, the
straw doesn't work.
Anyway, so the next
example of stuff
that we will occasionally talk
about is an electric current.
So switching gears
very, very much now.
So we will describe
this as a 4-vector whose
timelike component is the
charge density as seen
by some observer, and
whose spatial component
is the current density
as seen by that observer.
Bear in mind, you
might look at this
and twitch a little bit
because the units look wrong.
Don't forget c equals 1.
So if you use this in
other systems of units,
sometimes we call this component
charge density times c.
So you guys have all--
you know what, let me just
go ahead and write it out.
So a couple
properties about this
are important and interesting.
One is that the current
and the charge density
obey a continuity equation.
We can think of it as a
conservation of charge.
It's expressed-- so
in elementary E&M,
you guys all presumably
learned that the rate of change
of charge density is
related to the divergence
of the current density.
Well, this is exactly
the same as saying
that the spacetime divergence
of the current 4-vector
is equal to 0.
This is really useful for us.
And indeed, bearing
this in mind,
we find that if we want to
express Maxwell's equations
in a covariant way, we
can do so such that this
is built in automatically.
So skipping over a few
sets of things in my notes,
we're going to find--
we're not going to find.
We have found that electric
fields and magnetic fields
are inconvenient
objects to describe
using geometric objects that
are appropriate for spacetime.
If I want a
geometric object that
is appropriate for spacetime,
the first thing you think of
is a 4-vector.
A 4-vector has four components.
E-fields and
B-fields have a total
of six components among them.
So what you going to do, have
two 4-vectors and just ignore
two of the components?
That seems sketchy.
So you think, eh, you know what?
Why don't we make a tensor.
Ah crap, a tensor
has 16 components.
That doesn't seem right.
Then you go, ooh, I
can make it symmetric.
If you have a symmetric
4-by-4 tensor,
well, that basically
means that the number
of independent numbers that
go into this thing-- you
have four down the diagonal.
And then you count the
number that are off
the diagonal-- you have six.
There's 16 in total, but the
ones that are off the diagonal
are equal to one another, so you
have four on diagonal, six off.
Too many.
So you say, well, what
about antisymmetric?
If I have an
antisymmetric object,
that means that
component F alpha
beta is the negative of
component F beta alpha.
When you do that, that
forces you the conclusion
that there are,
in fact, actually
only six independent
numbers in that thing.
The diagonal has to be
0, because set beta equal
to alpha here--
F alpha alpha equals
minus F alpha alpha.
That only works if that
component is equal to 0.
So the diagonal becomes zero,
and only the six off-diagonals
survive.
And then you go, holy crap.
Six.
That's exactly what I need to
have a geometric object that
cleanly holds the three
independent electric field
components and the three
independent magnetic field
components.
So many of you
have seen all this.
If this isn't familiar
to you, take a look
at a book like Griffiths
or something like that.
It goes through this.
The punchline is
that what you find
is that the electric and
magnetic field is very nicely
represented by this
antisymmetric 2-index object,
whose components, in the
units that we are using,
are filled with the
E and the B like so.
OK.
Last semester I had
these memorized,
but I have totally
forgotten them.
OK.
So this is a geometric
object that-- whoops.
Anybody know why I just
said whoops and had
to put a dot on that there?
The point is that electric
and magnetic fields
look different to
different observers.
This is its representation
according to one
particular Lorentz observer.
So it's important
to get that right.
So in terms of this--
you're all familiar with the
four Maxwell's equations.
They turn out to
be equivalent to--
so if you take a divergence
of this F, it is--
depending on your
units, so you might
want to put u0's in there,
and things like that.
Basically, I'm
setting everything
that I can never remember to 1.
The divergence of that
thing-- by the way,
actually divergence
on the second index--
becomes the current density.
This will actually
only give you half
of the Maxwell's equations.
The other half-- what you do
is you lower these two indices,
and there's this cyclic
permutation of derivatives
that give you these.
So you put these
two things together,
you apply it to this form
I've written out here,
and you will reproduce
Maxwell's equations
as they are presented
in textbooks
like Purcell and Griffiths.
The thing which I
want to emphasize here
is, this form that we've
got is written in such a way
that the conservation of
source is built into it.
This geometric language requires
that J mu have no divergence.
And let me just show
you can do this.
All you need to know is that
F is an antisymmetric tensor.
You don't need to know
anything about the properties
of the E- and the B-field.
So let's just try it.
Let's look at the
divergence of the current.
So I'm going to
do 4pi d mu J mu.
So I'm going to take a
mu derivative of this.
Now bear in mind, mu and nu--
I'm using Einstein's
summation convention--
they are dummy indices,
so I can relabel them.
I can change them.
I can change mu to an
alpha, nu to a beta,
or I can just change
mu to nu, nu to mu.
So this-- as long as I do it
consistently on all objects
that have those things--
this is the exact same thing.
But I also know that this
tensor is antisymmetric.
So if I switch these guys
back, I get a minus sign.
OK?
Antisymmetry.
What happens if I switch the
order of the derivatives?
Does it matter whether I take
the x derivative first and then
the y derivative, or the
y derivative and then
the x derivative?
Partial derivatives commute
with each other, right?
They are perfectly symmetric.
So this-- so what I've got
is d mu d nu of F mu nu
is minus d mu d nu of
F mu nu, and that only
works if the whole thing is 0.
What I just did is, I
actually just invoked a trick
that we are going
to use many times.
And the one reason why--
this is a bit of a tangent.
Or more than a tangent--
150-degree turn from
what I've been discussing
before-- but I wanted to make
sure you saw this little trick.
Whenever I have an object that
is antisymmetric in its indices
and I contract it
with an object that
is symmetric in its
indices, you just get 0.
You can go through
the little exercise
if you don't feel
fluent in this yet.
You can do the exercise I
just did up there again over.
Take advantage of dummy
indices, swap one.
Work in the antisymmetry,
swap the other.
Work in the symmetry-- boom.
You will necessarily
prove that you've got 0.
So I lay this out
here because there's
going to be several times
later in the course, where
I'm going to get to a
particular calculation
and there's going to
be some godawful mess.
We're going to look at it
and go, oh, this is horrible,
and then go, wait,
symmetry-antisymmetry, boom.
We just killed 13 terms.
Tricks are fun.
So I am going to
stop there for today.
There is another
symmetry-antisymmetry thing
which allows you to--
if you apply it to
the equation of motion
of a charge in an
electromagnetic field,
it just shows you that
that equation of motion
builds in the fact that in
spacetime, the acceleration is
always orthogonal
to the 4-velocity.
I won't do it in class.
It is on page 6 of the notes
that I'm about to post up.
We will pick up next time.
We'll begin by talking about
the stress energy tensor.
