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SCOTT HUGHES: All right, so
in today's recorded lecture,
I would like to pick
up where we started--
excuse me.
I'd like to pick up where
we stopped last time.
So I discussed the
Einstein field equations
in the previous two lectures.
I derived them first
from the method
that was used by Einstein in his
original work on the subject.
And then I laid out the way of
coming to the Einstein field
equations using an
action principle,
using what we call the
Einstein Hilbert action.
Both of them lead us to this
remarkably simple equation,
if you think about it in terms
simply of the curvature tensor.
This is saying that a particular
version of the curvature.
You start with the
Riemann tensor.
You trace over two indices.
You reverse the trace
such that this whole thing
has zero divergence.
And you simply equate that
to the stress energy tensor
with a coupling factor
with a complex constant
proportionality that
ensures that this
recovers the Newtonian limit.
The Einstein Hilbert
exercise demonstrated
that this is in a very
quantifiable way, the simplest
possible way of developing
a theory of gravity
in this framework.
The remainder of
this course is going
to be dedicated to
solving this equation,
and exploring the properties
of the solutions that
arise from this.
And so let me continue
the discussion
I began at the end of
the previous lecture.
We are going to
find it very useful
to regard this as a set
of differential equations
for the spacetime
metric given a source.
That, after all,
is how we typically
think of solving for fields
given a particular source.
And just pardon me while
I make sure this is on.
It is.
I give you a
distribution of mass.
You compute the Newtonian
gravitational potential
for that.
I give you a distribution
of currents and fields.
You calculate the electric
and magnetic fields
that arise from that.
So I give you some distribution
of mass and energy.
You compute the spacetime
that arises from that.
But let's stop before
we dig into this,
and look at what
this actually means
given the mathematical
equations that we have.
So G alpha beta is
the Einstein tensor.
I construct it by taking several
derivatives of the metric.
I first make my
Christoffel symbols.
I combine those Christoffel
symbols and derivatives
of the Christoffel symbols
to make the Riemann tensor.
I hit it with another
power of the metric
in order to trace and
get the Ricci tensor.
I combine it with the trace of
the Ricci tensor and the metric
to get the Einstein.
Schematically, I can
think of G alpha beta
as some very
complicated linear--
excuse me, some very complicated
nonlinear differential
operator acting on the metric.
So thinking about this is
just a differential equation
for the metric.
The left hand side of this
equation is a bit of a mess.
Unfortunately, the right
hand side can be a mess too.
Let's think about this
for a particular example.
Suppose I choose as my force--
my source-- a perfect fluid.
Well, then my right
hand side is going
to be something that involves
the density and the pressure
of that fluid, the fluid
velocity, and then the metric.
OK, so if I'm thinking
about this as a differential
equation for the
metric, the metric
is appearing under this
differential operator
on the left hand
side, and explicitly
in the source on
the right hand side.
Oh, and by the way, don't
forget my fluid needs
to be normalized.
My fluid flow velocity
needs to be normalized.
So I have a further constraint,
that the complement to the four
velocity are related to each
other by the spacetime metric.
So if I am going to regard this
as just a differential equation
for the space time metric.
In general, we're in
for a world of pain.
So as I described at the
end of the previous lecture,
we are going to examine
how to solve this equation.
In what's left of
8.962, we're going
to look at three routes
to solving this thing.
The one that we will
begin to talk about today
is we solve this for what I
will define a little bit more
precisely in a moment
as weak gravity.
And what this is going
to mean is that I only
consider space times that are
in a way that can be quantified
close to flat space time.
Method two will be to
consider symmetric solutions.
Part of the reason the general
framework is so complicated
is that there are in
general, 10 of these
coupled non-linear
differential equations.
When we introduce
symmetries, or we
consider things like
static solutions
or stationary solutions that
don't have any time dynamics.
That at least reduces
the number of equations
we need to worry about.
They may still be coupled
non-linear and complicated,
but hopefully,
maybe we can reduce
those from 10 of
these things we need
to worry about to just a small
number them, one, or two,
or three.
Makes it at least
a little easier.
In truth, symmetric solutions--
if you can then add techniques
for perturbing around them,
these turn out to be
tremendously powerful.
My own research career is--
uses this technique
a tremendous amount.
Finally, just basically
say, you know what?
Let's just dive in
and solve this puppy.
Just do a numerical solution
of the whole monster,
no simplifications.
Eminent members of our field
have dedicated entire careers
to item three.
I will give you an introduction
to the main concepts
in the last lecture.
But it's not
something we're going
to be able to explore in
much detail in this class.
We're going to do one in
a fair amount of detail.
We will look at a couple
of the most important
symmetric solutions,
so that you can
see how these techniques work.
And then in my
last two lectures,
I'm going to describe
a little bit about what
happens when you perturb
some of the most interesting
symmetric solutions.
And we'll talk about numerical
solutions for the general case.
All right, so let's begin.
We'll begin with choice one.
Look at weak gravity,
also known as
linearized general relativity.
So linearized general
relativity is a situation
in which we are only going
to consider space times that
are nearly flat.
If I am in this situation,
then I can choose coordinates,
such that my space time metric
is the metric of flat space
time plus a tensor H-alpha
beta, all of whose components--
so this notation that I'm
sort of inventing here,
double bars around H-alpha beta.
This means the magnitude of
H-alpha beta's components.
These all must be much,
much less than one.
When you are in such
a coordinate system
you are in what we call
nearly Lorenz coordinates.
Such a coordinate
system is as close
to a globally inertial
coordinate system
as is possible to make.
There are other coordinate
choices we could make.
So for instance, you're
working in a system like that.
This basically boils
down to coordinates
that are Cartesian like
on their spatial slices.
You could work in other ones.
These are particularly
convenient.
Because for instance, if I work
in a coordinate system whose
spatial sector is
spherical like,
well, then there's going
to be some components that
grow very large as I go to
large radius away from some--
the source of my gravitation.
And this just makes my
analysis quite convenient.
In particular, where I'm going
to take advantage of this.
Whenever I come
across a term that
involves the perturbations
squared or to a higher power,
I'm just going
approximate it as zero.
I will always neglect
terms beyond linear,
hence the term linearized
GR, in my analysis.
Now, there are a
couple of properties,
before I get into how to develop
weak gravity, linearized GR.
I want to discuss a little
bit some of the properties
of spacetimes of this form.
What are the particularly
important properties of these?
So let's consider
coordinate transformations.
My space time metric is
a tensor, like any other.
And so the usual rules pertain
here, that I can change my
coordinates using
some matrix that
relates my original
coordinate system, which
I've denoted without bars, to
some new coordinate system that
is barred.
Now, recall that when we
worked in flat space time,
there was one category of
coordinate transformations
that was special.
Those were the Lorentz
transformations.
So we are not working
in flat space time.
So on the face of it, we don't
expect Lorentz transformations
to play a particularly
special role,
except perhaps in the domain
of a freely falling frame.
But you know what?
This is a nearly flat space
time, so just for giggles,
let's see what happens
if you apply the Lorentz
transformation to your
nearly flat spacetime.
So if I look at G mu
bar nu bar as being
a Lorentz transformation applied
to my nearly flat metric.
Well, what I get
[INAUDIBLE] side is this.
Now, one of the reasons why
the Lorentz transformation was
special in flat
spacetime is that it
leaves the metric of
flat spacetime unchanged.
And so this just
maps to eta mu nu.
I'm going to define what
comes out here as h mu nu.
I'm doing this in a fairly,
I hope, obvious way.
This is interesting.
What this has told me is
that when I apply the Lorentz
transformation to my
nearly flat space time,
the background is unchanged.
And the perturbation
to the background
transforms just like
any tensor field
would transform in flat space.
Now, it should be emphasized, we
are not working in flat space.
We can compute
curvature tensors.
If we parallel transport--
If we consider two geodesics
moving through the space time,
we will see that parallel
transport along those two
geodesics, if they start
out initially parallel,
they do not remain parallel.
So this is not flat space time.
But in many ways,
it's close enough
that we can borrow many of
the mathematical tools that
were used in flat space time.
In particular, we can
introduce the following.
Think of it as a
useful fiction, which
is that in this
framework, we can regard
the perturbation that
pushes us away from flat
spacetime as just an
ordinary tensor field
living in the manifold
of special relativity,
living in the eta
alpha beta metric.
It's worth bearing in mind in
a fundamental sense, it's not.
Space time is curved.
h alpha beta is telling
me about that curvature.
But the mathematics
works in such a way
that you can borrow
a lot of tools
that we used in
special relativity.
And just imagine h
as a tensor field
living in that special
relativity manifold.
OK, so that's useful
fact one that we
want to bear in mind as we work
in this nearly flat space time.
Useful fact two is
we want to think
about what happens when we
raise and lower indices.
So suppose now that
I'm going to regard
h alpha beta as
an ordinary tensor
field living in this thing.
I might want to
know what it looks
like with its indices raised.
So I'm going to
do what I usually
do when I want to raise
the indices on a tensor.
I hit it with the metric.
We're going to talk about
what my upstairs metric looks
like in just a moment.
But clearly, it's
going to be something
that looks like the metric
of flat space time plus terms
that are on the order of h.
Because I always drop terms
of order h squared and higher,
I can immediately say
that this must simply
be the metric of flat
space time with the indices
in the upstairs
position acting on h.
In other words,
at least when I am
acting on tensors that are
built from the space time
metric itself, I'm
going to always want
to raise and lower them using
my flat space time, eta alpha
beta.
Bearing this in
mind, let's carefully
think about what the metric
inverse actually looks like.
I actually used this in
one of my calculations
in the previous lecture.
And I said, I'm going to justify
this in the next lecture.
So here we are.
Now we're going to justify it.
So let's use this definition.
So the upstairs
metric is defined such
that when it contracts
with the downstairs metric,
I get the identity back.
This is the definition
of the metric inverse.
Working in linear theory,
I know that this thing
is going to be something
like eta alpha beta
plus a term of order h.
I don't know what that
term is yet, so let me just
give it a new name.
I'm going to call
it m alpha beta.
Whoops.
Hopefully, the math
will soon show me
what this m actually is.
It will be of order h,
but as of yet, unknown.
OK, so you know what?
Let's rewrite that over here.
So let's now multiply
this guy out.
Eta alpha beta hitting
eta beta gamma.
That gives me delta alpha gamma.
m hitting eta.
Now, remember what I
just said over here.
When I'm working with
spacetime tensors,
I always raise and lower
indices using eta alpha beta.
So when m alpha beta
hits eta beta gamma,
this is going to give
me m alpha gamma.
This eta hits that h.
I get h, alpha upstairs,
gamma downstairs.
And then this guy is of order
h, that guy is of order h.
So additional terms
of order h squared.
So these guys cancel.
And what I am left
with is m alpha gamma
equals negative h alpha gamma.
I can raise my two indices,
raise the gammas on both sides
here.
And we deduce from this
that my inverse metric
is eta alpha beta
in the upstairs
position, minus h alpha beta.
At least two linear order in h.
I'll just comment
that what this is--
essentially, the matrix
or tensor equivalent
of a binomial expansion.
1 over 1 plus epsilon is
approximately 1 minus epsilon.
That's all this is.
But this is important to do.
In fact, I will just sort of
remark somewhat anecdotally
that when I work with
graduate students on projects,
where we have to do things
in linearized theory,
getting a sign wrong here is
one of the most common mistakes
that people make.
All right, one final detail.
So this detail, as
I just labeled it,
is a particularly important one.
We talked about general
coordinate transformations.
And I immediately went
in to discuss a Lorentz
transformation.
There's a different category
of coordinate transformation
that plays a very important
role in understanding
the physics of systems that we
analyze in linearized theory.
So let's consider
a different kind
of coordinate transformation.
Let's consider a
coordinate shift,
which I'm going to
define by x alpha prime.
Really, they should
be x prime alpha,
but you'll see why I need to
name it this way right now.
So this is my original
coordinate, x alpha,
plus some little offset that's
a function of the coordinates x
beta.
So think of this as suppose
I have a coordinate grid that
looks like this.
And I have some
function that says,
I want to consider a different
system of coordinates that
maybe pushes me a little bit
along away from each of these
coordinate lines in
a way that varies
as a function of position here.
This notation is kind of
an abuse of the indices.
I am really not trying to
define a coordinate invariant
relationship here.
I am just trying to
connect two quantities,
and I'm trying to
connect quantities in two
specific coordinate systems.
And as we'll see, even
though this is a bit ugly.
It works well for
what we want to do.
So my coordinate
transformation matrix.
OK, so I just take
the matrix of--
I developed the
Jacobian-- my matrix
of partial derivatives--
of the new coordinate
with respect to the old
coordinate in the usual way.
This is going to
be my first term.
It's just a Kronecker delta.
And then I'm going
to get a term that
looks like matrix of
derivatives of the function that
defines my infinitesimal--
defines my shift.
I just gave away what
I was about to say.
I'm going to require [in?] my
work in these nearly Lorentz
coordinates, all of these
entries need to be small.
These will all be
much, much less than 1.
And so we call this
an infinitesimal
coordinate transformation.
We are going to need to use
the inverse of this guy.
And using the definition
of the inverse of this,
saying essentially that when
I take this, and I contract--
Let me put it this way.
I'll just write it out.
So if I compute this.
I get my Kronecker delta back.
Taking advantage
of the smallness
of the transformation.
It's not terribly hard
to demonstrate that
what comes out of it is this.
The minus sign is the key thing
which I want to emphasize here.
That minus sign is very
similar to the minus sign
that I have here.
What we're doing
is, again, just kind
of the matrix equivalent
of expanding 1 over 1
plus epsilon for small epsilon.
The reason that I
am doing this is
that I would now
like to look at how
the metric changes under this
coordinate transformation.
So what I'm going to
do is define g mu nu
in the new coordinate system.
Usual operation.
Let's now insert the many
different definitions
that we have introduced here.
Notice that what I am using
for my transformation matrix
there is the inverse
that I just wrote down.
So let's fill that in.
So I'm going to get
a term that involves
a Kronecker minus a matrix
of partial derivatives.
My other one gives me a
nether Kronecker matrix
of partial derivatives.
And then finally,
don't forget we
are working in this nearly
flat space time metric.
And so I insert in my
last term, eta alpha
beta plus h alpha beta.
So now, let's go and expand
all of these terms out.
My Kronecker,
first, I get a term
where both of the Kroneckers hit
the metric of flat space time.
So what I get is eta mu nu.
Then I get a term which
both the Kroneckers hit,
the perturbation h alpha beta.
Gives me h mu nu.
Then, I'm going
to get terms that
involve these matrices of
partial derivatives hitting
the metric of flat space time.
And what that's going to do is
in keeping with our principle
that when we're dealing with
spacetime quantities, we raise
and lower indices with eta.
This is going to now give me--
pardon me just one moment--
a term that looks like partial--
everything in the
downstairs position, d
mu xi nu minus d nu xi mu.
And then all the other
terms are on the order
of h times derivatives
of the generators
of my coordinate transmission.
Small times small.
These are infinitesimal squared.
We are going to neglect them.
Suppose that I insist that
I have gone from one nearly
flat spacetime to another.
Bear in mind this picture.
I'm just changing my
representation a little bit.
I've not changed the physics.
So if I write this as
eta mu prime nu prime,
plus h mu prime nu prime.
Well, I've got etas on both.
The thing which
is interesting is
that I have generated a shift to
my perturbation to the metric.
Let's drop the
primes for a second.
And I'll just say
that my nu, h mu
nu is the old h mu nu minus
the symmetrized combination
of derivatives of--
the symmetrized combination of
derivatives of infinitesimal
coordinate transformation.
Does this remind us of anything?
This is starkly
reminiscent of the way
in which when we work with
electromagnetic fields,
I can take a
potential, and shift it
by the gradient of some scalar
to generate a new potential.
In so doing, what
we find is that this
leaves the fields unchanged.
If you compute your Faraday
tensor associated with this,
it is unchanged.
Similarly, we're
going to write out
the details of this
in just a moment.
When I generate the Riemann
curvature from this,
we find that although the metric
has been tweaked a little bit
by this coordinate
transformation,
Riemann is left unchanged.
In acknowledgment of this,
we call an infinitesimal
coordinate transformation
of this kind a gauge
transformation.
What the gauge
transformation does
is it allows us to
change the metric,
or change the way that we
are representing our metric.
And it's going to
turn out to leave
curvature tensors unchanged,
in the same way that
changing the potential
and electrodynamics
with a gauge transformation
leaves our fields unchanged.
And we're going to exploit
this in exactly the same way
that we exploit this
in electrodynamics.
We use this in
electrodynamics in order
to recast the equations
governing our potentials
into a form that is maximally
convenient for whatever
calculation we are
doing right now.
We're going to find-- and then
we're going to derive this
probably in about 20 minutes--
that the equations
that govern h mu nu.
If we leave things as
general as possible,
they're a bit of a mess.
But by choosing
the right gauge, we
can simplify them,
and wind up with a set
of equations that are--
they cover all physical
situations that matter,
and that allow us
to just cast things
into a form that is much
better for us to work with.
All right.
So we have now developed all
of the sort of linguistics
of linearized geometry
that I want to use.
Let's now go from
linearized geometry
to linearized gravity by
running this through, and making
some physics.
What I want to do
is look at the field
equations in this framework.
I am not going to run through
every step of the next couple
calculations.
Doing so is a good illustration
of the kind of calculation
that a physicist likes to call
straightforward but tedious.
So I'm going to just write down
what the results turned out
to be.
So let's run the metric
through the machinery
that we need to make all
of our curvature tensors.
OK, I'll remind you when we
do this, we are linearizing.
So anytime we see a term that
looks like h squared, it dies.
So we're only keeping
things to linear order in h.
So the first thing we find
is the Riemann tensor turns
into the following combination
of partial derivatives
of the metric perturbation h.
In my notes, I have
written out what
happens when you switch
from some original tensor h
to a modified one using
this gauge transformation.
And what I show is that--
just a quick aside--
the gauge
transformation generates
a delta Riemann
that looks like it's
a whole bunch of-- let's see.
Let's count them up.
1, 2, 3, 4, 5, 6, 7, 8.
You have eight terms.
Of course there's eight,
because there's four terms here,
and you get two
more for each one.
So you're going to wind up with
eight additional terms that
involve three partial
derivatives of the gauge
generator.
So they're of the
form d cubed on xi.
And it's not hard to show.
You just sort of look at them.
They cancel in pairs.
And so delta Riemann is zero.
The Riemann tensor is invariant
to the gauge transformation.
All right, we want
to take this Riemann
and use it to build
the Einstein tensor.
Our goal here is to
make the field equation
in linearized coordinates.
So let's start by
making the Ricci tensor.
So we're going to
raise and lower
indices in linearized theory
with the flat spacetime.
So when we make this
guy, what we get is this.
I've introduced a couple
of definitions here.
One of them, you've seen before.
The box operator is just a
flat spacetime wave operator.
And h with no
indices is what I get
when I trace over h using the
flat background spacetime.
And let's do one more.
Evaluating r, I get one
further contraction.
And this turns out to be
d alpha, d mu, h alpha mu
minus box of h.
So we now have all
the pieces we need
to make the Einstein tensor.
So I'm going to
write out the result.
And then we're going to stop and
just look at it for a second.
Einstein is Ricci minus
1/2 metric Ricci scalar.
Keeping things to
leading order in h.
This becomes flat spacetime
metric going into there.
So when you put all these
ingredients together,
there's an overall
prefactor of 1/2.
And then there are 1,
2, 3, 4, 5, 6 terms.
Let me write them out.
OK.
So recall at the
beginning of the lecture,
I pointed out that when
one regards G alpha beta
as just a differential operator
on the spacetime metric,
it's kind of a mess.
Bearing in mind that what
I have here is a simplified
version of that,
I have discarded
all of the terms that are
higher order in h than linear.
This is already pretty much
a bloody mess as it is.
So you can sort of
see my point there.
If this were done in
its full generality,
it would be kind of a disaster.
Now, in linearized theory, there
is a bit of sleight of hand
that lets us clean
this up a little bit.
Let me emphasize that the next
few lines of calculation I'm
going to write down,
there's nothing profound.
All I'm going to
do is show a way
of reorganizing the
terms, which simplifies
this in an important way.
So what we're going to do is
define the following tensor.
h bar is h minus 1/2
eta alpha beta h.
So this is a good point to
go, well, who ordered that?
Let's take the trace of this.
Let's define h bar
with no indices
is what I get when
I trace on this.
That's going to
be the trace of h.
This would be the
trace of h alpha beta,
so I just get h back,
minus 1/2 h times the trace
of eta alpha beta.
And the trace of eta alpha beta.
This is what I get
when I raise one index,
and sum over the diagonal.
That is 4.
So the trace of h bar is
negative the trace of h.
We call h bar alpha
beta the trace reversed
metric perturbation.
It's got exactly
the same information
as my original
metric perturbation,
but I've just redefined
a couple terms in order
to give it a trace that
has the opposite sign
of the original perturbation h.
The reason why this
is useful is recall
the Einstein tensor is itself
the trace reversed Ricci
tensor.
What we're going to
see is that if we--
in acknowledgment that it's
sort of a trace reverse thing,
if I plug in a trace
reverse metric perturbation,
a couple of terms are
going to get cleaned up.
So here's how we do this.
So let's now insert h bar.
This guy is going
to be equal to--
oops, pardon me.
Insert h.
This guy is h bar plus
1/2 eta alpha beta h.
So just move that
to the other side.
All I'm doing is
taking the definition,
and I am moving part of
it to the other side,
so that I can
substitute in for h.
When you plug this
into here, you'll
see that there are
certain cancellations.
In particular, every term
that involves the trace of h,
h without any indices,
is canceled out.
And so what you find doing this
algebra is that your Einstein
tensor turns into--
that can't be right.
OK.
So now, my Einstein tensor
has no trace of h in it.
Every h that appears
on its right hand side
is the tensor with
both of the indices.
But now, it's the trace
reverse version of that.
This is still a bit of a mess.
Now, we're going to do something
that's got a little bit more--
it's not just sleight of hand.
This is something that's
got a little bit more
of sort of the meaning of
some of these manipulations
that we've worked out.
It's going to play a role in
helping us to understand this.
Notice this term involves
delta mu on h mu.
Excuse me, partial mu on
h mu, partial mu on h mu.
Partial mu and
partial nu on h mu.
This is the only term that does
not look like a divergence.
Three of the terms
in my Einstein tensor
look like divergences of
the trace reverse metric.
Wouldn't it be nice if we
could eliminate them somehow?
Well, if you studied
gauge transformations
and electrodynamics,
you'll note that there's
something similar that is done.
You can choose a gauge, such
the divergence of the vector
for potential vanishes.
Can we set the divergence
of this guy equal to zero?
So if you look at this,
mu is a dummy index.
This is four conditions
that we are trying to set.
This has to happen for mu--
well, we're going
to sum over mu.
Pardon me.
It's going to happen
for nu equal time,
and for my three spaces.
These are four conditions.
My gauge generators, my xi
nu are four free functions.
That suggests that the
gauge generators give me
enough freedom that I
can adjust my gauge such
that if I start out
with some original,
I have an h old that
is not divergence free.
Perhaps I can make
an h new that is.
Well, let's try it.
So remember, I just erased it.
But in fact, I'll just
write it down right now.
The shift to the metric
perturbation arising
from the gauge
transformation, it's on h.
We need to look
at how it affects
the trace reverse stage.
So if I start with
my new perturbation
is related to my old
perturbation as follows.
It's not too hard to show
that your trace reversed
metric perturbation.
Pardon, pardon, pardon.
My trace reverse
perturbation transforms
in almost the exact same way.
I get one extra term.
So now, what I
want to do is look
at how the divergence
of this transforms.
So I'm going to get one
term here, d mu of this.
It gives me a wave operator
acting on my gauge generator.
And then I get another
term here that looks like--
remember, partial
derivatives commute.
So you can think of this as d nu
of the divergence of xi, eta mu
nu acting on this
changes this into d
nu on the convergence of xi.
And I messed up the
sign, my apologies.
That plus sign should
have come down here.
These are equal but opposite.
They cancel.
So let me just
highlight the result.
So what this tells me is if I
choose my gauge generators just
right, I can adjust my
trace reverse metric,
so that it is divergence free.
If I do that, then the first
three terms in my Einstein
tensor here vanish.
And if I do that, then
here is my Einstein tensor.
So just as in e and m, all
that you need to do is say,
I'm going to change
my gauge such
that the following
condition holds.
The condition that
describes going
into this gauge such
that the divergence
of your trace reverse
perturbation vanishes.
This is a simple wave equation.
So solutions to this
are guaranteed to exist.
If you sit down
and you ask, can I
come up with some kind of
a pathological spacetime,
or a pathological--
no.
Imagine I'm in some
original spacetime
sufficiently pathological that
doesn't allow me to do this.
If you do that,
you're going end up
violating the conditions
that define weak spacetime.
You can't do that in
linearized gravity anyway.
So in practice, we can
always choose the gauge
that puts in linearized
gravity my Einstein
tensor in this form.
This form is exactly
analogous to the Lorentz gauge
condition that is used
in electrodynamics.
And so we call this Lorentz
gauge in linearized gravity.
Once we've done that, here's
what my Einstein field
equations turn into.
In the next lecture, we
will solve this exactly.
See, what you want
to emphasize is
this is one of those
situations where the answer is
so easy and simple for
us all to work out,
we don't actually really need to
even do that much calculation.
I'll remind you that
in electrodynamics,
if you work in Lorentz gauge
of electrodynamics, the wave
equation that governs the
electromagnetic potential
turns out to be--
could be factors of c
and things like that,
depending on which
units you're working in.
But we find an equation
that has exactly
the same mathematical structure.
Possibly there's a plus sign.
I should have looked that up.
Wave operator on my vector
potential is a source.
And this is very
easily solved using
what's called a radiative
Green's function.
I will discuss this
in the next lecture.
You can look up the details in
any advanced electrodynamics
textbook.
Jackson has very nice
discussion of this.
I have an extra index.
I have a different coefficient.
But the mathematical
structure is identical.
So as far as linearized
theory is concerned,
we're basically done.
So I'm going to talk about
the exact solution of this
in the next lecture.
To wrap up today's lecture, to
wrap up this current lecture.
Let me look at the solution
of this in a particular limit.
So I'm going to
take my source to be
a static, non-relativistic,
perfect fluid.
The fact that it is static means
that all of my time derivatives
will be zero.
And if that's true
for my source,
it has to be true for the
field that arises from it.
Non-relativistic tells me
that the fluid density greatly
exceeds its pressure.
And as a consequence, I
can write my stress energy
tensor as approximately density
four velocity four velocity.
And when you go and you look at
the magnitude of these things,
I sort of looked at
this a little bit
in the previous lecture.
T00 is approximately rho.
All others will be negligible.
Probably there's a small
correction to this,
but we can neglect
that on a first pass.
So my field equation is
dominated by the zero zero
component.
That's going to be the most
important piece of this.
Since this is static,
I can immediately say--
I can change that wave operator
into the plus operator.
And we now notice this
is exactly the equation
governing the Newtonian
gravitational potential,
modulo a factor of four.
Pardon me, factor of minus four.
And so we see from this
that h bar zero zero
is just negative 4 times
the Newtonian gravitational
potential.
At this order in
the calculation,
all other contributions to the
trace reverse metric are zero.
OK, so let's go from the
trace reverse metric back
to the metric.
We use the fact that h mu nu
is the trace reversed h mu nu.
If we trace reverse it, we'll
get the original metric back.
Basically, we trace
reverse twice.
So the trace of this guy.
OK, and so putting all
these ingredients together,
what we see are that the only
non-zero contributions here
are 8 0 0.
Let's do this one carefully.
This is minus 4 times Newtonian
potential minus 1/2 times 8 0 0
and 4 times Newtonian potential.
I have a c of minus sign here.
This turns into minus 2 phi n.
And h1 1 equals h2 2.
h equals h3 3.
This is going to be 0 minus
1/2 times 1 times 4 phi n.
We put all these together.
And what we get is--
And I'll remind you.
This is a metric that I
quoted in a previous lecture
that I said we would
prove in an upcoming one.
Well, here it is.
This is the Newtonian limit
of general relativity.
And it's worth remarking
that this thing--
we are now in the
very first lecture
after having derived the
Einstein field equations.
20 years ago, almost all
laboratory tests, laboratory
and astronomical observational
tests of general relativity
essentially came
from this based on.
This ends up being
the foundation
of gravitational lensing.
This is used to look at
post-Newtonian corrections
in our solar system.
To a good approximation, it
describes a tremendous number
of binary systems that
we see in our galaxy
and in a few other galaxies.
You really need to look for
a much more extreme systems
before the way in which
the analysis changes
due to going beyond linear order
starts to become important.
There is an upcoming
homework exercise.
And for students taking this
course in spring of 2020,
it remains to be
determined how we
are going to do problem
sets at this point.
I will be making a decision
on that in coming days.
But I want to tell you
about an exercise on P
set number 7, in which you do a
variation of this calculation.
So instead of just
having a static body
with a body who has
massive density rho,
consider a rotating body.
And the thing which
is interesting
here is that in
general relativity,
all forms, all fluxes
of energy and momentum
contribute to gravity through
the stress energy tensor.
So if I have a body that
is rotating about an axis,
there's a mass flow.
There are mass
currents that arise.
And what you find if you do
this calculation correctly
is that there is a correction
to the spacetime that
enters into here, which reflects
the fact that a rotating
body generates a
unique contribution
to the gravity that is
manifested in this space time.
Now, when one looks
at the behavior
of a body in a
spacetime like the one
I've written down right here.
It's very reminiscent of--
well, it's just like
Newtonian gravity.
It's the Newtonian limit.
And Newtonian gravity looks
a lot like the Coulomb
electric attraction.
So this is often called
a gravito electric field.
People use that term,
particularly when
they're talking about
linearized general relativity.
If I have a rotating body,
I now have mass currents
flowing in this thing.
And the correction
to the spacetime
that arises from this, it's
qualitatively quite different
from us.
It doesn't have that simple
gravito electric Coulombic type
of form.
It, in fact, looks a lot
more like a magnetic field.
And in fact, when
you ask, how does
this new term that is generated
affect the motion of bodies?
You find something that looks
a lot like the magnetic Lorentz
force law describing its motion.
So this is a very,
very powerful tool.
But it's already not enough.
So we can go a lot
further than this.
We have done so far, the
simplest possible thing
that we can do with this toolkit
that we have derived so far.
In the next lecture
that I will record,
we're going to return to my
linearized Einstein field
equations.
And I am going to explore
general solutions of this.
This is going to lead us
into a discussion of how
things behave when my
gravitational source is
dynamic.
I do not want to lose the
time derivatives that are
present in that wave operator.
And so this is going to lead
us quite naturally, then,
to a discussion of
gravitational radiation.
And so after the
next lecture, we'll
spend a lecture or two
discussing the nature
of gravitational radiation.
And there will be an
upcoming homework assignment
or two in which you
explore the properties
of gravitational radiation.
