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SCOTT HUGHES: OK.
Let's move on to
the next lecture.
Might have gotten a little bit
of me getting prepped here.
I really have to
give a lot of credit
to the people in the Office
of Digital Learning for making
this entire thing possible.
So I'm really
grateful particularly
to Elaine Mello, who is at home
running the show remotely right
now.
This is-- a lot of people at MIT
are doing a tremendous amount
to keep this place functioning,
and everyone really
deserves a huge pat on the back.
All right.
So in the lecture that I just
completed a few moments ago,
we described--
I described how to characterize
the gauge-invariant radiative
degrees of freedom, which we
call gravitational radiation.
With gravitational radiation,
we found at the end--
there's a formula that I will
write down again a little
bit later in this lecture,
but we essentially
found that it looks like certain
projections of a term that
involves two time derivatives
of the quadrupole moment
of a source.
This has in recent
years led to a--
well, we've now been
able to directly measure
this form of radiation.
I'm wearing my LIGO hat in
honor of the facility that first
directly measured these things.
And we have inaugurated
an entirely new field
of observational
astronomy based on being
able to measure those things.
It's pretty exciting.
I hope, though,
that a few people
noticed how restricted the
analysis that I did is.
I defined gravitational
waves only
in the context of linearized
theory on a flat background.
We haven't talked much
about other solutions
to the Einstein field equations
yet, but you know it's coming.
And I hope you can appreciate
that, in fact, our universe is
not described by
a flat background.
Many of the sources
that we study,
it is only weakly curved
away from a flat background.
So for instance,
our solar system
is accurately described as
something that is weakly curved
away from a flat background.
And in that context,
much of what I described
carries over very cleanly.
And so you essentially have--
in that case, you need to--
if you want to analyze things
like gravitational waves
propagating in our solar system
or through a galaxy which
has a spacetime metric, it's
also accurately described
as flat spacetime plus a
weak deviation away from it.
You can take
advantage of the fact
that your gravitational
wave tends
to be something
that's rapidly varying
and your solar system
and your galaxy
is something that has
very slow time variation,
or it's even static.
So you can take
advantage of the fact
that there's these
different time
derivatives to separate terms.
That actually leads
me naturally to,
what do we do if
I want to describe
the very realistic scenario of
gravitational waves propagating
not on a flat background,
but on some kind
of a curved background?
Perhaps they propagate
near a black hole
where spacetime is very
different from the spacetime
of special relativity.
Perhaps we need to
describe them propagating
across a large sector
of our universe.
We haven't talked
about cosmology yet,
but it's the subject of the
next lecture I intend to record.
And the spacetime that describes
our universe on large scales
does not look like the
metric of special relativity.
So how do I describe
gravitational radiation, GWs,
Gravitational Waves, on
a non-flat background?
Schematically, you can
imagine that in this case,
you're working with
a spacetime that can
be broken into a background.
I'll put a little hat on
it to say that this denotes
my background spacetime.
And then some kind of a wave
field propagating on it.
This background
is no longer going
to be that of
special relativity.
This could be
something that varies
with both space and time.
This will be something that in
general is small in the sense
that in most of the cases,
certainly all the cases
we're going to
study in this class,
any term that involves
an h times an h
will be second order
in a very small number,
and so we can discard it.
But it's still going to be
challenging to figure out
how I can even define what
a wave means in this case.
So to wrap your
heads around that,
think about what a local
measurement might do, OK?
So suppose I am in some
region of spacetime.
So first of all,
we had a parable
that we looked at in
our previous lecture
where we looked at geodesics
in a spacetime that was flat
plus a gravitational wave.
And of course,
geodesics are geodesics,
and so we just found that they
are unaccelerated relative
to freefall frames.
That sort of is-- you
know, more or less
defines what a geodesic is,
so it was not too surprising.
But we can then look at,
for example, two relatively
nearby geodesics,
and we can look
at how their geodesic
separation changes.
We can look at the
behavior of light
as it bounces back and forth.
Something that's
important to bear in mind
is that when I make
a measurement like
that, I measure this
or some quantity that
is derived from this,
the curvature tensor
or an integrated light
propagation time.
We may know that
the spacetime metric
has two vary conceptually
different contributions to it.
It's got a background,
and it's got a wave.
How do we-- conceptually, we
might see how to separate them,
but how do we define some
kind of a toolkit that
allows us to separate?
How do we define the separation
between background and wave?
Put it another way.
Suppose what you are sensitive
to is spacetime curvature.
Suppose you are
sensitive to things that
depend on the Riemann tensor.
How do you know from your
measurement that you are
measuring the Riemann tensor
associated to the gravitational
wave and not a Riemann tensor
associated with a curvature--
excuse me--
associated with your background?
That is an extremely
important issue.
If all I can measure is
this, but I want to get this,
what is the trick that I use to
distinguish between these two?
I call this a trick
in my notes, but I
don't like that term, actually.
This is not a trick.
This is a fundamentally
important point
when you are trying to measure
some kind of a perturbation
to a--
not just to spacetime,
but to any kind of a field
that you are studying,
any kind of quantity
you're studying which is
itself spatially and temporally
varying.
What you need to do is take
advantage of a separation
of length and timescales.
When I'm talking about
general relativity
and I'm talking about
gravitational waves,
I am going to use the fact
that a gravitational wave is
oscillatory.
And as such, it varies on a much
smaller length and timescales--
it varies on much smaller
length and timescales
than the background does.
OK?
It's useful to introduce
an analogy here.
Think about a water wave
propagating on the ocean.
In almost all
cases when you look
at a water wave
propagating on the ocean,
it's obvious what
is the wave and what
is the curvature in the ocean.
It's actually
associated with the fact
that the Earth is round.
So it is clear from the
separation of both length
scales and timescales--
not clean-- clear
how to separate wave
from the curvature
of the Earth bending
due to geological structures
and things like that.
A component of the
curvature of the ocean
that has a curvature scale
of, say, 6,000 kilometers--
well, that's just the
fact that the Earth
is a sphere with a radius
of 6,000 kilometers.
OK?
So you can kind of see that.
But if you see something
that is varying
with a period of
about a second that's
got a wavelength
of a meter or two,
that has nothing to do with
the curvature of the Earth.
That's a wave.
We want to introduce a similar
concept here and apply it
to our general
relativistic calculation.
So let's introduce two sets
of length and timescales--
let's call them scales--
we will use in our analysis.
So capital L and
capital T, these
are my long length
in timescales.
These describe the
variation of my background.
Lambda and tau are
my short scales,
and they describe the
wavelength in the period
of gravitational waves.
So to put this in the context
of the kind of measurements
that LIGO makes, the
LIGO interferometers
make measurements
of behavior of light
moving in the spacetime
near the Earth.
Now, there is a
component of that
that varies on a
scale that has to do
with the curvature of spacetime
near the surface of the Earth
due to the Earth's
mass distribution.
That occurs on--
so first of all,
that is practically static.
It does vary somewhat
because of the motion
of the moon around
the Earth and because
of the behavior of the
fluid in the Earth's core.
There are small variations there
which, interestingly enough,
can be measured.
But those tend to
occur on timescales
on the order of hours
at the shortest.
For the moon's orbit,
the moon goes around
in a time period
of about a month.
The Earth's rotating.
It's turning on its axis.
And so at a particular
point, you're
sort of casing the gravitational
field of the moon as the Earth
rotates under it, and
so your time thing here
is on the order of
hours to about a day.
The length scale
associated with this--
you can calculate things
like the Riemann curvature
or tensor components near
the surface of the Earth.
That has the units of
1 over length squared.
So take the square
root and inverse it,
and you're going to find
that this varies on a length
scale that is thousands or--
I should probably work this out
before I try quoting numbers,
but it's many, many, many,
many, many kilometers,
tens of thousands or
millions of kilometers.
Tens of thousands, probably.
My gravitational
wave, by contrast,
it's got wavelengths that
are on the order of-- they
are also on the order of
thousands of kilometers,
but there is a time
variation on them
that, for the LIGO detectors,
is on the order of seconds
at most, really
tenths of a second
at the current sensitivity
of the detectors,
up to milliseconds.
Way faster variation.
So by looking for pieces
of what I can measure
that vary on these length
scales and these time scales,
I can separate it
from effects that
are varying on these long time
scales and long length scales.
So as I move forward
in this lecture,
let me emphasize that
some of the things I'm
going to introduce here are--
first of all, they're a
little bit more advanced.
You are certainly not
responsible for knowing
the gory details of how
some of these tools I'm
going to introduce are used.
All right.
So the point is,
once I've introduced
these long scales and
these short scales,
what we can then do is we
can always remove the wave,
remove the oscillation by
introducing an averaging
procedure.
So if you imagine that you
average over a timescale that
is several times the
wave-- excuse me--
several times the period of the
wave and over a length scale
that is several times the
wavelength of the wave.
These are taken to be
shorter than the long scales,
but longer than
the short scales.
And so when you do that--
let me put it this way.
If I average-- so
these angle brackets
are what I'm going to define
as my averaging procedure.
I will make this a little bit
more quantitative in a moment.
On length scale l
and timescale t,
the background basically
doesn't change.
So when I average it, I just
get the background back.
On the other hand, my wave
oscillates a couple of times.
So if I average
something that oscillates
over a couple of
cycles, I get 0.
So given what I can
actually measure,
which is the spacetime or
things related to the spacetime,
this averaging procedure lets
me pick out the background.
The radiation is
then what I get when
I subtract that average
bit from the field.
So these are all things that,
as a theorist thinking about how
someone is going to go
in and actually measure
properties of the
gravitational wave,
these are tools I can use
to separate one spatially
and temporally varying
quantity from another.
Now, here I'm going to
describe something that's
a little bit more advanced
than we need to get into,
but it's important to discuss.
How do you actually
average a tensor like this?
That's a little bit tricky.
So the averaging was
first made rigorous.
And there have been various
other mathematical formulations
of this that are
presented over the years,
but I like this particular one
because it's conceptually quite
simple.
It was first made rigorous by
Dieter Brill and James Hartle--
Hartle is the author of a
nice elementary textbook
on general relativity--
in 1964.
And the reference is
given in the notes.
And what you basically
do is to define
the average of a quantity
like the spacetime metric,
you integrate it over
some proper volume
in spacetime, proper 4 volume
with a particular weighting
function.
So this is something
that is defined such
that if you integrate it over
that same region, you get 1.
And it's taken to be
something that is--
you can sort of think of as a
Gaussian in all four spacetime
directions that
is peaked around--
it's peaked at a
particular location,
and it has a width of l in
all directions, l and t.
We spent a lot of time
talking about what
makes a tensor a tensor.
And one of the things
that we discussed
was the fact that when I am
working in curved space time
and my basis objects
are functions,
I can't really add a
tensor here to a tensor
there because they live in
different tangent spaces.
That's a poorly
defined operation.
An integral is nothing more
than a sum on steroids.
So how is it that I'm
allowed to do this?
Well, the idea here
is this is not exact.
OK?
When one does this,
you will find--
what comes out-- your average
tensor modulo corrections
or errors that are of
order short length scale
over long time length scale.
This is the best you're
going to be able to do.
You're always going
to be a little bit
off when you define an
averaging procedure like this
because by definition,
an average can't
capture everything.
That's why you're averaging.
You're trying to actually
throw away some piece of it.
So this allows us to
define our tensor calculus
in this averaged way in a
very useful framework provided
with--
we just need to accept the
fact that there's always
going to be--
it really only makes
sense in this regime where
we can cleanly separate our
length scales and our time
scales.
I emphasize this because
sometimes one does an analysis,
and one finds things are
really weird going on.
It's often useful
to sort of step back
and say, wait a minute.
I'm trying to describe
gravitational waves
in a particular regime.
Are these actually
gravitational waves?
And you sort of look
at this, and you
realize you've put yourself
into a regime where
your perturbation is varying on
the same timescales and length
scales as your background.
And in fact, your amplitude
is no longer really small.
You sort of go, oh, crap.
What I've done here is
I've actually pushed this
beyond the regime in which
the radiative approximation is
valid.
So bear this in mind.
You're always
working-- you're always
going to be working in
this approximative form.
Suppose we have now done this.
We have separated length scales.
So we now have cleanly
and conceptually
separated into background
and perturbation.
Now what I would
like to do is just
run through my exercise of
computing all the quantities
that I need to do to
describe radiation-- really,
all of my quantities that I need
to do to develop and then solve
the Einstein field equations.
And I'm going to do
so linearizing in h
about this curved
background rather than
about the flat background.
So let's develop all of
our spacetime curvature
tools, all the
various quantities,
linearizing in h around g hat.
So I'm still doing
linearized theory,
but I am not linearizing around
a flat background anymore.
So to give you an
idea, I'm going
to do one term associated with
this and a little bit of--
one term associated with this
and a little bit of detail
so you can just see what the
complications end up looking
like.
So first thing you might want to
do is compute your connection.
Go back to your definition.
That should be in your notes
from a couple of lectures ago.
So you insert this.
Here is the definition
of the connection.
Let's now insert this split.
So the inverse metric
is going to have
a form that looks very
reminiscent to how
we did the inverse
metric in linearizing
around a flat background,
the exact same logic.
I am raising indices
using the background,
so h alpha beta in
the upstairs position
is what I get when
I raise using g hat.
And you get a minus sign.
Again, this is sort of
like the tensor equivalent
of binomial expansion of
1 over 1 plus epsilon.
Pardon me just one moment.
So I'm going to get two terms.
So here's the first thing
where I'm raising it
to the inverse metric.
This is going to
split into two terms.
Let me just write them all
out for completeness here.
So let's pause for
just a second here.
So when I multiply
all of this out,
I'm going to get one term
that involves background
inverse metric hitting all the
derivatives of my background
metric.
I'm going to call it--
I'm going to call that
Christoffel with a hat.
OK?
That is nothing more than
the Christoffel symbol
that you would have gotten
if you were just working
with the background metric.
You're going to get
another term that
involves h alpha beta acting
on the Christoffel symbol
with all the indices in
the downstairs position
and constructed
from the background.
Then you're going to
get another term that
involves your background
inverse hitting
various derivatives of your
wave or of your probation
around the background.
At this point-- oh, and
there'll be a term of order h
squared which I'm
going to discard.
At this point, you just
have to kind of stare
at this for a little bit.
And when you do so, you find out
something of a miracle occurs.
You can write this as
background plus a perturbation
to the connection.
I shift to the connection.
And this shift-- when you
stare at and you organize all
the terms that appear here,
this turns out-- whoops--
this turns out to
look like something
that is very reminiscent of
the formula for the Christoffel
symbols, but only acting
on the h's and using
covariant derivatives rather
than partial derivatives.
I don't have a good way to
prove this other than to say,
you know what?
Expand out all of these
covariant derivatives
and then compare these terms.
And this ends up being what
you get when the smoke clears.
I go through this
one in some detail
because simplifications of
this sort happen at every level
as we move on.
What we're basically going to
do is sort of brute force expand
things.
We'll find a term that looks
like just a pure background
piece, and then there'll
be a whole bunch
of little bits of crap that
are linear in the field h.
And you can gather
them together and often
write them in a form that's
prettier, kind of like this,
where it's just sort of a shift
to the connection involving
covariant derivatives.
There's no great algorithm for
actually working all these guys
out.
Basically, you just
have a bit of labor
to do to put them together.
So let me write out a couple
of other examples of what
follows from this.
Let's see.
There's a point I want to
get to in just a moment.
I'll go through that
in a bit more detail.
OK.
So when I take my Christoffel
and I work out my Riemann
tensor, I find a
piece that looks
just like Riemann computed
solely from the background.
And then I get a
correction to it.
And to linear order in h, this
looks like when you work out--
OK.
Hang on just a second.
So at this point,
we can now go ahead
and start making things like the
Ricci tensor and the Einstein
tensor.
I got slightly
ahead of myself here
because there's an important
point to make at this point.
So now, as we move
forward, if we're
assuming Einstein's
general relativity,
our curvature tensors end up
coupling to the Einstein field
equations, a couple things
like the Ricci tensor
and the Einstein tensor to
the stress energy tensor.
Accordingly, I am going to make
the following simplification.
Let's set the stress
energy tensor equal to 0.
So the analysis which I'm
going to do in the remaining
time in this lecture, it
only describes vacuum regions
of spacetime.
One doesn't need to do
this, and I strongly,
strongly emphasize this.
Doing so makes this lecture
about one third as long
as it would be if I did not
assume this was equal to 0.
So this is solely being
introduced to simplify things.
And what this means is that
I am essentially describing
gravitational waves as I--
I'm describing gravitational
waves far from their source.
Unfortunately, it does
mean it complicates
a little bit how they work in
some cosmological spacetimes.
But it is still
sufficient for me
to introduce some of the key
concepts that I want to do.
By assuming that t
mu nu is equal to 0,
what I'm going to do is
I'm essentially, then,
assuming that the
background is equal to 0.
The background spacetime
arises from an Einstein tensor
that is equal to 0.
And it's not hard to show that
this corresponds to Ricci being
equal to 0, and of
course, then, the Ricci
scalar being equal to 0.
So as I move forward here,
my background Riemann tensor
is nonzero, but my
background Ricci tensor
will be equal to 0.
And I strongly emphasize that
this is just a simplification
that I am introducing in
order to make today's analysis
a little bit more tractable.
So my background Ricci
tensor is equal to 0.
Let's compute the perturbation
in my Ricci tensor.
This is what I get when I
trace over indices 1 and 3.
I will skip over much of
the algebra that is in this.
But what one finds going through
this sea of various definitions
is that what results
is a delta r mu nu.
I will define this
symbol in just a moment.
OK.
So a couple of definitions.
This box operator that
I'm introducing here,
this is a covariance
wave operator.
And the trace is what I get
when I contract my perturbation
h with the background metric.
OK?
So I just want to
quickly emphasize
that there is nothing
very profound about what
has gone into this.
This was all done by essentially
just throwing together
all the various
definitions and linearizing
in all the quantities
that I care about.
So what I would like
to do now is take--
I'm going to take my Ricci
tensor and my Ricci scalar.
I will assemble them to
make my Einstein tensor.
And by requiring that
that be equal to 0,
I will get an equation
that describes radiation
as it's propagating in
this curved background.
But before I do this, I want
to just pause for a second
here and note that I'm going
to get a mess when I do this.
You can already sort of see I've
got some interesting structures
here.
So here's a trace.
Here's a trace.
Here is a term that
looks like a divergence.
Here is a term that
looks like a divergence.
When we were doing linearize
theory on a flat background
spacetime, we got rid of
those degrees of freedom.
We sort of deduced that these
were just kind of annoying
for doing our analysis.
And by changing gauge, we were
able to rewrite the equations
in a way that allowed
us to get rid of them.
Before I assemble
this, I want to explore
what changing gauge
means now that I'm
working on a curved background.
So brief aside: let's
generalize our notion
of a gauge transformation.
So what I'm going to
do is just as when
I was linearizing around a
flat background spacetime,
I began by introducing an
infinitesimal coordinate
displacement.
And when I did this,
I then had a matrix--
a matrix that affected this
coordinate transformation,
which looks like so.
Just as when I-- whoops, typo.
Just as when I was doing this
around a flat background,
I'm going to assume that these
elements tend to be small.
So let's apply this generalized
gauge transformation
to my metric.
Something worth
highlighting at this point--
my metric depends
on space and time.
It's not flat.
So I have to build
in the fact that it's
a function of these coordinates.
So this-- pardon
me just one second.
So now I need to find the
inverse metric-- excuse me--
the inverse coordinate
transformation
that will essentially
give me the same thing,
flipping my indices
around a little bit
and introducing a minus sign.
And when I expand this out--
again, bearing in mind I have to
be careful about the coordinate
that this is being
evaluated at, OK?--
So I'm going to get one term
here that involves my Kronecker
deltas hitting this guy.
Again, be careful
with that argument.
OK?
So I'm going to get another
term that involves--
I'm going to get two
terms that involve
my derivative of my
infinitesimal displacement
hitting the background.
I'm going to get terms that
involve the Kronecker hitting
my metric perturbation.
I'm going to discard terms
that are quadratically small.
Now, here is where we need
to be a little bit careful.
Notice I have shifted
this guy here.
I'm going to expand this
with a Taylor expansion.
Move this a little higher so
that it's not blocking my view.
OK.
That looks pretty gross.
We can clean it up, though.
If I use the fact that--
for example, looking
at one of these terms,
this can be written as
the covariant derivative
of my infinitesimal displacement
minus a connection coefficient.
You can gather together
a whole bunch of terms.
And what you find--
something that looks
remarkably similar to the form
that we got.
By the way, I may have forgotten
to define that back here.
My sincere apologies to those
of you following along at home!
My covariant
derivatives with a hat
mean this is a covariant
derivative being
taken with respect to
the background spacetime.
So just do your normal
recipe for evaluating
a covariant
derivative, but compute
all of your Christoffels using
the background spacetime g hat.
Carrying back over
to here, what I
find is that doing this
infinitesimal coordinate
transformation, it looks
just like, why did you
do all these steps properly?
And this little bit of
doing the Taylor expansion
in the metric--
I speak from experience-- it's
easy to overlook that bit.
This looks just like
what we did when
we applied an infinitesimal
coordinate transformation
to a perturbation
around flat spacetime.
And it tells us that there
is a kind of gauge that
can be applied to my
gravitational waves
around a curved background
provided I promote
my derivatives of the
infinitesimal coordinate
generator from
partials to covariance.
So this defines-- if I introduce
a generalized coordinate
transformation, this generalizes
the notion of a gauge
transformation when I'm
examining things like radiation
on a curved background.
This is my generalized
gauge transformation.
It will also prove useful for
us to have a notion of a trace
reversed perturbation.
So I'm going to
introduce an h bar mu nu.
This will be my original
h mu nu, the perturbation
around my curved background.
And I subtract off 1/2
of the trace like so.
If you take the trace of this in
the way that I defined earlier,
this will give you
minus little h back.
So again, at this
point, I'm going
to skip over a couple of
lines in my lecture notes
because they're straightforward
but a little bit tedious.
What we're going to do
is take my Ricci tensor,
my Ricci scalar, my
background spacetime.
I'm going to assemble my
Einstein tensor associated
with this metric
perturbation, but I'm
going to write it in terms of
the trace reverse [INAUDIBLE]..
So let's write out delta g
mu nu in terms of this guy.
So assembling all the
pieces, doing the algebra.
OK.
So pardon while I just
write all that junk out.
So this, again, I
hope, reminds us
of a step when we were
doing perturbations
around flat spacetime.
We have all these
terms that are sort
of annoying divergences of
that trace reverse metric
perturbation.
And so a line or two of
algebra will justify this.
I can change gauge.
I can find a gauge in
which my perturbation has
no divergence if I choose
these generators such
that they satisfy a wave
equation in which the source is
the divergence of my
old metric perturbation.
Let's suppose we've done this.
By the way, we will call this
generalized Lorenz gauge.
When you do this,
what you find is that
your metric perturbation--
oh, shoot-- is governed by
the following Einstein tensor.
And if you imagine that this
guy is propagating in vacuum,
what this tells you is
that the wave satisfies
something that's similar to the
flat spacetime wave equation.
Now, the operator is
a little bit different
because you have assembled it
out of covariant derivatives
and with a correction,
which shows
that your wave is
actually now coupling
to the curvature of spacetime.
So I have a few more
notes about this.
It's not really important
that we go through them
in great detail here.
Those of you who
are interested, they
will certainly be made
available through the website.
The key thing which I want to
emphasize is essentially some
of the key bits of--
an important piece
of the technique, which is that
what we are doing is using--
we are linearizing around
this flat background.
We have generalized our
gauge transformation
to allow us to simplify
the mathematical structure
of this equation.
And then what results is
an interesting correction
to the usual wave
equation that one sees.
So I will also emphasize this is
a somewhat more advanced topic.
So one of the reasons
why I'm eliding
over some of these details
is they are somewhat tedious.
You don't need to
know them in detail.
It's good to be
familiar with them
and to be able to
follow along here.
I am going to post
a couple of papers
to the 8.962
website that lay out
some of the foundations
of this stuff.
The key thing which
I want to use this
for is an important aspect
of gravitational waves
is that they carry energy.
Electromagnetic
waves carry energy.
It shouldn't be a surprise if
I have some kind of a source--
we're in a room right now.
There's light
shining down on me.
The energy that is going on
me, it heats me slightly.
It is being drained from some
power source somewhere else.
We're all familiar
with the story
of electromagnetic radiation.
For gravitational
waves, ascertaining
what the energy content
of the wave actually is
is a little bit more subtle.
And indeed, understanding that
there is an unambiguous energy
carried by these
waves is something
that occupied a lot
of the foundations
of gravitational wave theory
for a couple of decades.
In part, this is
driven by the fact
that these things are
so hard to measure
that it was difficult to--
this is one of those things
where if you could just
go out and measure it, even if
your theory was a little bit
uncertain, you would
say, well, goddammit it,
it left an imprint
on my detector here.
It must have carried energy.
And so you would know
where you were going.
But because there
were no measurements
of these things for years
and years and years,
no one quite knew which
direction to step in.
Part of what makes
this complicated
is the fact that when we
are in general relativity,
no matter what
your spacetime is,
you can always go into
a local Lorenz frame.
When you do that, spacetime
is flat at a point.
Furthermore, all the
derivatives associated with it--
the first derivatives
associated with it are 0.
So it's sort of--
how the hell can I have a
field h, a gravitational wave
h that carries energy if I
am free to change coordinates
and make it equal to 0?
Well, there's a couple of
comments I'll make about that.
So in sort of the same way--
well, let's back
up for a moment.
We made an analogy earlier to
the electromagnetic potential
when I was describing
gauge transformations.
So the metric is
a quantity that is
subject to gauge
transformations,
but the Riemann
curvature tensor--
when I linearize around
a flat background,
I found that the
Riemann curvature
tensor was invariant to
those transformations.
It's a little bit
more complicated
when I linearize around
a curved background,
but it still remains the
case that I can never
make my curvature go away.
It's reminiscent of the fact
that I can choose a gauge that
lets my electrostatic--
my electromagnetic potentials do
any number of crazy-ass things.
I must be getting tired because
I'm swearing a little bit more.
So I can let my
electromagnetic potential
do any sorts of crazy
things, but my fields
are the quantities that
ultimately carry energy.
They carry energy and momentum.
So in the same way,
the fact that I
can get rid of the metric
by going into a local Lorenz
frame, that shouldn't
bother us too much.
That's kind of like going
into a frame where--
or it's going into a gauge where
I just make my potentials go
away or make my
potentials become
static or something like that.
This is basically telling
me that the energy
must be something that is
bound up in the curvature.
Furthermore, the fact that
I can always make things
look flat at a
particular point--
I must use non-locality
to actually pull
out and understand
the energy content.
This sort of means that in
a very fundamental sense,
I am never going to be able
to define, in a completely
gauge-invariant way,
the notion of energy
in a gravitational wave
at an event in spacetime.
I may actually come up
with gauges in which there
is some notion of an
energy like quantity
that's defined at
a particular point.
It will not be
gauge-invariant, though.
I am going to need to-- if
I want to really rigorously
define what the energy
content gravitation wave is,
it's going to have to
be based on something
where I'm averaging over a
region that is large compared
to a wavelength,
but small compared
to the scales associated
with my background.
So let me sketch for you
how we can understand this.
And this will conclude
with the derivation
of a second quantity
that is often
called the quadrupole formula.
Again turning to my
electromagnetic analogy,
the energy content of--
well, [INAUDIBLE] the content
of the energy momentum
carried by an
electromagnetic wave
is described by a
pointing vector.
And the pointing vector
looks like the e field
times the b field.
It is quadratic in the field.
In a similar way, we
are going to expect
the gravitational
wave energy content
to be something that is going
to be quadratic in a field.
So to do this
properly, we're going
to need to think about how
to go in the second order
in our theory, so second
order in perturbations
around the background.
So what I'm going
to do is imagine
that my spacetime looks
like some background
plus some epsilon
times h alpha beta.
This h alpha beta is
what we just spent
the past hour or so computing.
And imagine that there is some
additional term j alpha beta.
This epsilon that
I've introduced here
is just an order
counting parameter.
Its value is actually 1.
OK?
But what it does is
it allows us to keep
track of the order in
perturbation theory
to which I go.
So if the gravitational
waves are typically
on the order of, say,
10 to the minus 22,
here's my background curvature.
These terms are all in
the order of 1 or so.
These are all in the order
of 10 to the minus 22 or so.
These are all in the order
of 10 to the minus 44 or so.
What I want to do is
run this through--
and let's just focus on
vacuum spacetime for now.
Let's run this through the
vacuum Einstein equation.
So let's expand g alpha beta.
Actually, let's not have too
much crosstalk between indices.
I'll call this g mu nu.
And I'm going to require
this to be equal to vacuum,
so I'll set that equal to 0.
What I want to do
is expand this guy.
That's a tremendous
amount of work,
but I'm going to
sketch for you what
the highlights of
this look like.
So when I expand
my Einstein tensor,
I'm going to get one term.
That is basically just saying
that my background spacetime
is a vacuum solution.
I'm going to get a
term that looks like--
I'll call it g1 alpha beta.
And this is going to depend
on my first-order perturbation
in the background.
g1 is basically the
delta g that I worked out
in the first hour or
so of this lecture.
I will get another term
that looks like the same g1,
but in which my second-order
term is coming along.
But I'm going to get
an additional term
that results from non-linear
coupling of h to h.
I'll call that g2.
So I emphasize again, this
is just my ordinary Einstein
equation.
This is basically telling me
my background satisfies the--
my background is
a vacuum solution.
This is the linear wave
operator on a curved background
that we just worked out
in the first hour or so
of this lecture.
This is that same
linear operator,
but now applied to the
second-order perturbation.
And this is something new.
It is very messy.
It involves lots of
terms that involve
h hitting covariant derivatives
of h, covariant derivatives
of h hitting each other.
I will post a paper by
Richard Isaacson that steps
through this in some detail.
What we're going to now do is
we require the Einstein equation
to hold order by order.
In other words, at
every order in epsilon,
this equation must work.
So at order epsilon to
the 0 or at order 1--
this is what I just said--
the background is
a vacuum solution.
Groovy.
At order epsilon, this
is my wave equation
on my curved background.
At order epsilon squared,
now something new happens.
Now, that is interesting.
What we are seeing is
that the terms which
involve quadratic things with
the linear perturbation, things
that are quadratic
in h are acting
as the source to the wave
equation that governs j.
Let's try to make
some headway on this.
Let's go back to our separation
of length in timescales.
So let's define
delta j mu nu to be
j mu nu minus what
I get when I average
j mu nu on an
intermediate length scale.
So lambda-- I'll remind you
that lambda and tau are sort of
coming along for the ride here.
This is my gw short scale.
l is my background long scale.
And so what I'm
going to do is say,
I'm going to take my j mu nu--
I don't really know too
much about it quite yet.
And I do know that by
definition, my h only
varies on the short scale.
OK?
So I'm going to take--
I'm going to do this
averaging procedure here.
Pardon me.
That's an error.
And so my l is my
intermediate averaging scale.
So I'm going to average things
on the intermediate scale l
here.
And so this will be
something that only varies
on the short scale lambda.
This, I will take out everything
that varies in the short scale.
This only varies
on the long scale.
So this is interesting.
OK?
By now doing this,
let's regroup my metric.
So this guy--
I'm going to first--
sorry, folks.
Getting a little tired.
Here are all the terms that
vary on the long scale.
And here are all the terms
that vary on my short scale.
With this idea than I am now--
so it's kind of interesting.
What we see is that
there is a piece
of the second-order perturbation
that has kind of become
a correction to the background.
So when I take my perturbation
theory to the next order,
there's no reason--
so at linear order, we define
the linear order perturbation
as being only the bit that
varies on the short scale.
But there is no
reason why it has
to remain the case that it
only varies on the short scale
when I go to the next order.
And any bit that does, in
fact, vary on the long scale,
bearing in mind that the
way we do our measurements--
we separate things and
lengthen time scales--
it's going to look to us like
a correction to the background.
So with that in mind, let's
revisit the second-order term
in the Einstein equation.
So here is my second-order
Einstein equation.
You should just
assume that these also
depend on the background.
I'm not going to write
that out explicitly.
What I would like to do now is--
let's apply this averaging
to this equation.
So the average of this guy
equals the average of this guy.
Does this get us anywhere?
Well, it definitely
gets us somewhere
with this, because this
is a linear operator.
OK?
So when I go and--
I should be a little
bit careful there.
A useful trick which I forgot
to state-- my apologies--
and this trick is worked
out in the paper by Isaacson
that I'm going to post--
is that whenever you take the
average of second derivatives
of various kinds
of fields, it is
equal to second derivatives
of the average of that field
plus corrections that scale as
the square of the short length
scale divided by the
long length scale.
So what that tells me is that at
least up to these corrections,
which the separation of
length scales by default
is assuming is a small number,
I can take my averaging operator
inside here.
I can't do that trick
on the right-hand side.
I've not written out what
this equation actually
looks like here.
It's a nonlinear operator, so
it's going to be a lot messier.
But this is something
that's linear in j.
And so as long as I bear in
mind that I incur a small error
by taking various--
by taking my averaging operation
inside the differential
operator, this is telling
me that the wave equation--
sorry.
My Einstein equation
applied to this operator
here is set by the
quantity that I have there
on the right-hand side.
Now, remember, my background
is itself a vacuum solution.
Since my background
is a vacuum solution,
I can rewrite this whole
thing as the Einstein tensor
on my background metric plus my
average quadratic correction.
This is going to be
equal to the average
of that second-order piece.
This is just the Einstein field
equation with a strange source.
If you look at this-- let's
make the following definition.
I'm going to define
a stress energy
tensor for gravitational
waves as minus 1
over 8 pi g times the
average of this operator.
What this is now telling me is
that as my gravitational wave
propagates through
spacetime, it carries,
it generates a
stress energy that
changes the background
in an amount that
is quadratic in the
wave's amplitude.
Now, to get the details of
what that operator actually is,
I have a formula
or two in my notes
that will be scanned
and made available.
I want to cut to the
punchline of this.
The derivation of this is given
in a paper by Richard Isaacson
that I will be posting
to the course website.
So expand out this operator.
Place in this the
transverse traceless field.
And what you find--
fill it with a transverse
traceless field.
That sort of simplifies
a few things,
and you get this
remarkably simple result.
The angle brackets,
I'll remind you,
they tell you that
this quantity is only
defined under the aegis of a
particular averaging operation.
That aside, what we
now have is a quantity
that tells us about how
gravitational waves carry
energy and momentum
away from their sources.
This is known as the Isaacson
stress energy tensor.
Let me comment,
especially for students
who work on things related
to gravitational waves--
you will encounter
many different notions
of how to compute the energy,
things like energy and angular
momentum and things
like that that
are carried from a radiating
source of gravitational waves.
There is wonderful discussion
of this point in a textbook
by Poisson and Will.
It's just entitled
Gravity, I believe.
The Isaacson stress tensor
has the virtue that it is--
it makes it very clear
that the waves are--
so it's a tensor.
It makes it very clear
that it arises out
of quadratic derivatives
of the wave field.
And I really enjoy going
through this derivation
because you can see the way
in which a term that is--
at least at the
schematic level, you
can see the way in which the
second-order contribution
to the Einstein tensor leads
to a source that modifies
your background spacetime.
That's exactly what
a stress energy
tensor is supposed to do.
For certain practical
applications,
it turns out to be a little
bit hard to work with.
And so you'll see this
discussed in some detail in some
of the more advanced textbooks.
I'm happy to discuss this with
students who are interested.
For our purposes, this
is not of concern.
Let me just conclude
this lecture
by evaluating this for the
solution that we worked out.
So let us imagine that we go
into a nearly flat region where
those covariant
derivatives become simple.
And let's just compute one piece
of the stress energy tensor.
So let's look at the energy
flux in a nearly flat region.
So what I'm going
to compute is the 0,
0 piece of this
which defines how
power flows through spacetime.
So doing this in
this region basically
is going to look like the
time derivative of my wave
field contracted onto itself.
Let's get the total
amount of energy
that is flowing
through a large sphere
by evaluating t0 0 integrated
over a large sphere
of radius r.
And for my hij, I will plug
in that quadrupole amplitude
that I derived in
the previous lecture.
Over dots denote d by dt.
So if you're interested in
doing this integral yourself,
it's actually not that
hard, and that's a pretty--
it's a salubrious exercise.
Let's put it that way.
So I'll remind you
that this projection
tensor is built from--
built from the
vectors that describe
the direction of propagation.
And a few useful
tools to know about--
so should you choose
to work this guy out,
it's worth knowing
that if I integrate
ni nj over the sphere, this
can be written in terms
of just sines and cosines.
These are basically
direction sines--
or direction cosines.
I get a simple result
that depends only
on the value of the
indices i and j.
If I integrate this over
this sphere, I get 0.
And any odd power of the
ends multiplied together
when integrated over the
sphere will give me 0.
And one more is necessary if
you want to do this integral.
If you integrate four of these
buggers multiplied together,
you get this object
totally symmetrized
on the indices i, j, k, and l.
So throw it all together.
Take the derivatives.
Note that the only thing
that depends on time
is that quadrupole moment.
And what you wind up with is
this remarkably simple result.
Whoops.
This is a result
that also is known
as the quadrupole formula.
It's worth comparing this
to the dipole formula
you get in electrodynamics.
So the dipole formula
that describes
the leading electromagnetic--
the leading power
in electromagnetic radiation
that comes off of a source
looks like what you
get when you multiply
two derivatives of the dipole
moment with each other.
The leading radiation in
gravity is quadripolar.
And so rather than being
second derivatives,
you have to take one more
derivative out of this thing.
There are three time derivatives
of a quadrupole moment leading
to this formula.
So this result was
actually originally derived
by Albert Einstein.
It's one of the first
calculations that he did.
He actually did it twice.
He first derived something kind
of like this in, I believe,
1916.
And it was totally wrong.
He just basically
made a huge mistake.
But then he did it more
or less correctly in 1918.
So he got this right.
And it was a small--
a small error.
He was off by a
factor of 2, which
is a simple algebra mistake.
It's actually really
enjoyable to count up
the number of minor
errors that Einstein
made in some of these works.
It just makes you feel a little
bit better about yourself.
So anyway, Einstein got
this essentially right,
and then it was
hugely controversial
for quite a few
decades after that.
Essentially, going back to these
conceptual issues with which I
began this lecture, people
just grappled with the idea of,
what does it mean for
gravitational radiation
to carry energy in the first
place given that I can always
go into a freely falling frame?
It took a little while for those
concepts to really solidify.
Once they did, though,
we were off to the races.
And it's worth noting that
if I take this formula
and I apply it to
a binary system--
I imagine I have two stars
that are orbiting each other.
I can compute the quadrupole
moment and the time derivative
of the quadrupole moment.
I can compute the rate at which
power is leaving these systems.
Now, a somewhat more complicated
variant of this calculation--
well, let me back up.
So you go ahead,
and you do that.
You compute this for two stars
that are orbiting each other.
You will do this on
a future problem set.
What you find is that
gravitational radiation carries
energy and angular momentum
away from the system,
and it causes them to fall
towards one another, which
causes their orbital
frequency to evolve
in a very predictable fashion.
That law that arises to
very good approximation
directly from this formula--
there are corrections
that people have worked
out over the years,
but what you get just doing
this quadrupole formula applied
to two stars orbiting each other
describes this chirp signal
that I have on my hat
to very good accuracy.
And this is pretty much
exactly what the LIDO detectors
measure these days.
They measure the
evolution of systems
that are evolving under
the aegis of this formula.
All right.
So that is all we
are going to say
about gravitational
radiation in 8.962 this term.
As we figure out
the way things are
going to proceed
moving forward, I
will make myself
available to answer
some questions about this.
I do want to emphasize
as I conclude
this lecture that many of these
topics are fairly advanced.
I don't expect you to
be familiar with them.
But I hope you understood
the punchline of this.
And I expect you
to be able to apply
some of these formulas
that arise at the end.
And I will conclude
this lecture here.
