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SCOTT HUGHES: I'm going to
record two lectures today.
These are the final two lectures
I will be recording for 8.962.
I actually have notes on
an additional two lectures.
I have notes for an
additional two lectures.
But those additional
two lectures
are somewhat advanced material.
It's sort of fun to go over them
in the last week of the course,
for certain students.
It can be a great
introduction to some
of the most important
topics in modern research.
But I am not going
to come into campus
and present those lectures,
OK, given everything that's
going on right now.
To be blunt, they are
kind of bonus material.
And this isn't the time.
This is not the semester for
us to go through our bonuses.
I will make those
notes available.
I will be happy to discuss
them in a saner moment
with any student who
is interested in them.
But this semester, let's just
focus on the core material.
So that brings us to the topic
of what we are studying today.
Pardon me while I
correct my handwriting.
So if I can do a recap of
what we discussed last time,
we took a look at a spacetime
that has this Schwarzschild
solution written in
the Schwarzschild
coordinates everywhere.
In our previous
lecture, we looked
at a spacetime that described
a fluid body, a spherically
symmetric fluid body.
And it had a surface.
This was the solution that
we used for its exterior.
This ends up describing
a solution that
has zero stress energy tensor.
So it describes a
vacuum situation.
OK?
So if I imagine a spacetime
that looks like this everywhere,
well, what I end up finding is
that this is a vacuum solution.
It has T-mu-nu
equals 0 everywhere.
However, it also has a mass m.
So this is the vacuum
solution with mass,
which is, well, that's weird.
We examined some of its
curvature properties,
and we found that at r equals
0, there is a tidal singularity,
OK, in invariant
quantity that we
constructed from the tidal
tensors, blows up at r
equals 0.
R equals 2GM looks
a little bit funny.
And it turns out tides
are well behaved there.
OK, there's nothing pathological
in the spacetime there.
But there is a
coordinate singularity.
Our coordinate t
is behaving oddly.
And where we left
things last time
is I did a little
diagnosis of this
by imagining a body that
falls into spacetime
from some starting radius R0.
And I looked at the
motion of this thing
as a function of time.
What we find is that if we look
at the motion of this thing
as a function of the proper
time of that in-falling body,
it crosses 2GM in
finite proper time.
And shortly afterwards,
again, in finite proper time,
it reaches the r equals
0 tidal singularity.
If I look at that same
motion as a function
of the coordinate
time t, it never
even reaches, never even
reaches r equals 2GM.
We found a solution,
what you can
see in notes that
I've put online
and that are presented
in the previous lecture.
But we found a solution
in which it just
asymptotically
approaches r equals 2GM,
only reaching that radius in
the t goes to infinity limit.
These are two starkly
different pictures
of the kinematics of
this in-falling body.
So the question is,
what is going on here?
And as food for
thought, I reminded us
that these coordinates, if
we sort of look at the way
that spacetime behaves for
a very, very, very large M--
Excuse me, very,
very, very large R,
not M, but for a very large R,
R much, much greater than 2GM,
this turns into the line
element of flat spacetime.
Flat spacetime is what we
use in special relativity.
And our time
coordinate there, it
is designed using this
Einstein synchronization
procedure, which means that
the properties of light
as it propagates
through spacetime
are built into the
coordinate system.
So that suggests
what we might want
to start doing in order to try
to get some insight as to what
is going on with this
weird spacetime is
to think about light as it
propagates into spacetime.
Let's think about what happens
to radiation as it propagates.
So let's imagine as the body
falls that it emits a radio
pulse with a
frequency as measured
according to the in falling.
Let's say there's
an observer who
is falling in who's got a
little radio transmitter that's
beaming this message out.
And according to that observer,
this thing is emitted,
the pulse has a
frequency omega--
very far away, we can
describe the momentum
of this radio pulse like so.
Imagine the thing is propagating
out radially and very,
very far away where the
spacetime is approximately
flat.
It's simply the
four-momentum that
describes a null geodesic moving
in the radial direction, OK?
Minus sign here because with the
index in the upstairs position
and this asymptotically
flat region,
that would be the energy.
So a couple of facts
to bear in mind.
The energy measured by an
observer with four-velocity
U is given by, let's call it
E sub U, the energy measured
by observer U. This
is minus P-dot-U.
We developed this in
special relativity.
But remember the way that we
use the equivalence principle.
We're gonna use the Einstein
equivalence principle.
And any law that holds in a
freely falling frame, if I
can write it in a
tensorial way that
works in that freely falling
frame, it works in any frame.
So this tensorial
statement holds true,
even though we are now
working in the spacetime that
is distinctly different
from special relativity.
We also know that in a
time independent spacetime,
the downstairs T component
of four momentum is constant.
So what this basically means
is if I think about this radio
pulse with its light
propagating out
through the spacetime, the value
P0 associated with this thing's
four momentum, it's
the same everywhere
along its trajectory.
So let's consider a
static observer sitting
in the Schwarzschild spacetime.
OK, I'm going to require
G-alpha-beta U-alpha
U-beta to be equal to minus 1.
And I will require that this
have some timelike piece,
that this observer is static.
So their spatial components
of their four-velocity
are equal to 0.
It looks like, by the way--
just pause one moment here.
It looks like the projector
is on because of--
I should probably
just leave it as is.
All right, I'm not going
to worry about that.
Just leave that as it is.
Sorry, let's go back to
what I'm talking about here.
So I have a static
observer, so an observer
who is not moving in space.
They're only moving
through time.
And I need to normalize
their four-velocity.
Bear in mind, this is not
a freely-falling observer.
OK, this is an observer who must
be accelerated, in some sense.
And there must be some
kind of a mechanism that
is allowing this observer to
hover at the fixed location
in spacetime where they are at.
So when I put these two
constraints together
and I tie it into
that spacetime,
what I find is that the timelike
component is 1 square root
of 1 minus 2GM over r.
So let's compare the energy
that is emitted at radius
r to the energy that is absorbed
at some radius big R. So
I'm going to imagine--
And so the reason I formulate
it in this way, what
I want to do now is
ask myself, well,
what is the energy that
an observer at little r
would measure here?
What is the energy
that is observed
by the person at
radius capital R?
And now, we know, if I
imagine that these are both
being measured by
static observers,
because P sub T, P
sub 0 or P sub T,
because it is a constant as
the light propagates out,
I can take the
ratio of these two.
And the energy observed at
capital R versus the energy
emitted at little r--
Hang on a second.
As I was looking
over my notes, there
was a result that made no
sense because I had a typo.
So for those of you
following along,
the four-velocity component
should have been a 1
over the square root of
that quantity I had earlier.
Suddenly, what I'm
about to write down
makes a lot more sense.
So this becomes--
Let's imagine that
the observer is
so far away that they are
effectively infinitely
far away.
So the question
I'm asking is, if I
imagine that the
light pulse is emitted
at some radius little r, what
would a very distant observer
measure the energy
of that pulse to be?
OK, so my light
pulse or radio pulse
is emitted at some
finite radius.
And we'll call e infinity,
the value that is measured
by a very distant observer.
This becomes square
root 1 minus 2GM over r.
Notice that the
energy, no matter
how energetic the light
is when you emit it,
I described it as
being a radio pulse,
but it could be a laser pointer.
It could be ultraviolet.
It could be a gamma ray.
You know, you could
just hawk whatever
massively-powerful source
of photons you want
and you're pointing it out
in the radial direction
and hoping that your distant
friend can measure it.
No matter what energy you
emit as you are falling in,
as you approach r
equals 2GM, the amount
of energy in that beam that
reaches a distant observer
goes to 0.
No matter how energetic
my pulse of light,
no energy reaches
distant observers
as the emitter
approaches R equals 2GM.
In a similar way, imagine
that as you are falling in
and you've got this little
beacon that you are sending
messages out to your
distant friends,
imagine you send them out
with pulses that are separated
by delta T. So the person
in the falling frame
turns his beacon on for
a moment every delta
T. Let's say delta
T is every second.
The interval between pulses,
as measured far away,
one can show using a
similar kind of calculation.
You pick up a factor of 1 over
square root 1 minus 2GM over r.
This goes to infinity
as r goes 2GM.
So remember, the time
coordinate was originally
defined by imagining that I can
synchronize all of my clocks
using light pulses that
are bouncing around.
But what we can kind of see
here is that light pulses that
are emitted in the
vicinity of r equals 2GM,
they're kind of
going to hell, OK?
So if I imagine, let's just
say for the sake of argument,
I'm using a green laser
pointer as the thing
that I use for my Einstein
synchronization procedure.
And suppose that I communicate
what time is on the clock
by a modulation of the signal.
So suppose that I
modulate it by putting
little spaces on it that tend
to be about a second long.
Or let's just say it's
like a microsecond long
so I can really pack some
information into that.
Well, the laser pointers that
are emitting from r equals 2GM,
their signal infinitely
redshifts away.
They lose all their energy.
So they're not green.
If they're really close
to it, maybe they'll
be red when they get
to the next thing.
Instead of being a
pulse every microsecond,
it'll be every two microseconds.
And the closer we get to
it, the more redshifted
it becomes and the more extended
the interval between pulses
becomes.
I'm going to post
to the 8.962 website
a set of notes that
sort of cleans up
this calculation a little
bit, just goes through this
in a little bit more detail.
It was something I
wrote in the Spring 2019
semester in response
to a good question
that a student had
asked me about this.
And I think they do a nice
job of just going over
this calculation.
And they give a
couple of examples
of the way this is behaving.
The key thing which
I want to emphasize
is that as r goes to 2GM,
what we see is that--
well, let's just
put it this way.
In fact, let's
remove the word "as."
The surface r equals
2GM corresponds
to infinite redshift.
OK, we've talked already
a little bit about,
if I have light climbing out
of a gravitational field,
the light gets a little bit
redder as it climbs out.
Well, at this
particular surface,
the sphere of radius
2GM in this spacetime,
if you can get down to
there, that redshift
then becomes infinite.
All of the energy is drained
out of it as it climbs out.
So what this
basically tells us is
that this surface breaks
the Einstein synchronization
procedure and it renders
that time coordinate bad,
at least if we are concerned
about understanding things
right at this surface.
OK, as long as we're
concerned with the exterior
of the surface, that's
not a problem, OK?
Everything works fine
as long as we move away
from this coordinate
singularity,
sort of in the
same way that many
of the pathologies
associated with the spherical
coordinate system here,
the north and south pole,
they are fine as long as
you're not trying to do things
like measure the longitude
angle corresponding to the North
Pole.
So this is not
even well-defined.
In the same way,
this is basically
telling us that the coordinate
T in which we wrote down
this spacetime isn't really
well-defined at r equals 2GM.
So what we need to do, if we
want to try to get some-- hey,
I'm over here, camera!--
if we want to try to get
a little bit of insight
as to what is going on here, we
need a better time coordinate.
So I'm going to
talk about a couple.
And the way that we're
going to formulate
these is all the
pathologies are revealed
when we look at the behavior
of light propagating
in this spacetime.
So let's play around with light.
Let's look at null
geodesics in this spacetime.
So to begin with, let's stick,
for just the next couple
of moments, with the original
Schwarzschild coordinates.
OK, so I'm going to
look at null geodesics.
So I'm going to set 0,
and I'm gonna ask myself,
how does it move through
an interval at dt
and an interval of dr?
OK, so we can solve
this to find how,
if an object is moving on
a radial null geodesic,
how do dt and dr behave?
How does dt dr behave?
Put it that way.
OK, so that's what
my dt dr looks like.
Plus, the plus sign
corresponds to a solution
that is moving the
outward in all direction.
Minus sign is an inward
directed null geodesic.
These define what we consider
to be the opening angle.
So dt dr defines the opening
angle of a light cone.
So if we go to very,
very, very large r,
OK, we get dt dr equals 1.
And this corresponds in
units in which c equals 1
to light moving on a 45 degree
angle in a spacetime diagram.
This is familiar behavior
from special relativity.
But as r goes to 2GM,
we see dt dr going to 0.
Let's make a sketch
and see what,
sort of, dt dr, what the
tangent to a null geodesic
looks like in the t-r plane
as a function of radius.
So this little
dash here means I'm
sort of imagining that I'm
going to stretch my r axis so
that, out here, you're
in the asymptotically
flat region where things
look like special relativity.
So here we are out in this
asymptotically flat region.
My outward-going
light ray goes off
at 45 degree angle
in the r-t plane.
Inward one goes at a 45
degree angle pointing inside.
Down here in the
stronger field, it's
going to be a little
bit steeper than this.
And so the opening angle of
my light cone is closing up.
Here, we'll have
closed up a lot more.
Here, it's closed up
practically all the way.
Now, as I approach r
equals 2GM, both the inward
and the outward
direction in these
coordinates go parallel to 2GM.
So you can see the
collapse of the light cone
in these coordinates
as you approach
this coordinate singularity.
So we need a healthier
coordinate system.
One thing that we can do is
we can move the pathology out
of our time coordinate and
into our radial coordinate
with the following definition.
Suppose you choose a radial
coordinate r-star such
that dt equals plus or
minus dr-star everywhere.
OK?
So if I replace my
horizontal axis with--
pardon me-- if I replace
my horizontal axis
with the r-star, this will be
45 degree angles everywhere.
OK?
But to make this work,
what you find is r-star
must look like this.
What you basically see is
that this coordinate system
takes r equals 2GM
and it moves it
to r-star equals minus infinity.
So the way that this
coordinate representation,
that this different
radial coordinate,
the way that this makes
the light cones always
have 45 degree opening--
really 90 degrees opening--
it makes the light rays always
go off at 45 degree angles
is by essentially constantly
stretching the radial
axis so that this guy just
gets stretched out so that
it's opening at 90 degrees.
This guy is stretched
a little bit less.
This guy is stretched
a little bit less.
Basically, not stretched
when you're really far away.
But as you approach
r equals 2GM,
you're infinitely stretching
at these coordinates.
These are known as tortoise
coordinates, basically,
because you start
walking, taking ever
slower and slower steps.
Even steps, if you imagine
you're stepping evenly
in r-star, you're taking ever
smaller and smaller steps
in r, as you approach the
infinite redshift surface.
So with that tortoise
coordinate defined,
you use that as an
intermediary to define
a couple of new coordinates
for your spacetime that
are adapted to radiation.
So we're going to define
v to be t plus r-star.
And the importance
of this is that this
is a coordinate that's not
hard to convince yourself
this is constant on an
in-going radial null ray.
I'm going to define a coordinate
U to be t minus r-star.
And this is constant on
outgoing radial null rays.
OK?
So this basically
means that if I'm
working in this
coordinate system, r-star,
if I want to know the
behavior of this guy,
well, an outgoing
radial null ray,
you just might say, ah,
that's the null ray that
has U equals 17.
OK?
And that will then
pick out, basically,
a whole sequence of
events along which
that null ray has propagated.
And, you know, as you
can see, it vastly
simplifies how we describe it.
So once you've defined
these two coordinates,
you can rewrite the
Schwarzschild spacetime
in terms of them.
It is generally best to choose--
So we're going to take this
another step in just a moment,
but we'll start--
you choose either V or U, and
you replace the Schwarzschild
time.
So let's use V to replace time.
So when you go and you look at
what your new coordinate system
looks like--
and remember, the way you
do this is the usual thing,
you're going to make your
matrix of partial derivatives
between your old
coordinate system.
So your old coordinates
are t, r, theta, and phi.
And then they go over
to v, r, theta, and phi.
OK?
So you make your matrix of
partial derivatives describing
this.
And here's what
you find when you
change to the metric in
the new representation.
Notice, there's no dr
squared term at all.
OK?
We do still see something
that's, you know,
at least a coordinate
singularity at r equals 2GM.
We haven't quite gotten
rid of it entirely here.
But we've definitely
mollified the impact
of this coordinate singularity.
So in this coordinate
system, you can then
set ds squared equal to 0,
and you find two solutions
describing radial null curves.
So dv dr equals 0 for in-going.
And, you know, by
definition, these things
are constant on an
in-going radial null ray.
And so as you move along
it, V remains constant.
OK, and you get something a
little bit more complicated
for the outgoing one.
Let's redraw this using
my new coordinates, OK?
So I'm going to leave
my horizontal axis as r.
So I'm going to make my
time axis be v. So out here,
here's my in-going null ray.
And here's my, eh--
let's make that
a little closer to 45 degrees.
Here's my outgoing null ray.
OK?
As I move in to smaller
and smaller values of r,
notice in the limit,
as r goes to 2GM,
the slope becomes infinite.
OK, so this thing gets steeper.
This one keeps pointing in, so
this guy gets steeper, steeper.
Right here, it lies pointing
exactly straight up.
What's kind of cool is
that in these coordinates,
I can actually look at what it
looks like inside this thing.
And so inside, this guy tips
over and gets a negative slope
and looks like this.
Now, bear in mind,
these two things,
they denote the null rays.
All radial timelike
trajectories,
and indeed all
timelike trajectories,
not just the radial ones,
all timelike trajectories
must follow a world line that
is bounded by these two sides.
OK, so if I'm out here,
everything in here
describes trajectories that a
timelike observer can follow.
Everything in here
describes a trajectory
a timelike observer can
follow, everything in here,
everything in here.
Notice, when I am at this one
here, right at r equals 2GM,
every allowed timelike
trajectory points towards r
equals 0.
At best, I can
imagine an observer
who's very close to the speed
of light who sort of skims
along inside of this thing.
But they are
timelike, so they will
have a little bit of a slope
that points them inward.
OK.
As you move inside r equals
2GM, it's even more so.
OK, you can't even sort
of go parallel to the r
equals 2GM line.
They all point
towards this thing.
Once you get to r equals
2GM, all trajectories,
as they move to the future,
must move to smaller radius.
What this tells
us, in particular,
is that once you have
reached r equals 2GM,
you are never coming back.
All allowed
trajectories, everything
that is permissible by
the laws of physics,
moves along a trajectory
that points towards r
equals 0 once you hit
that r equals 2GM line.
Because of this, this
surface r, this surface
of infinite redshift
is given the name--
Let me come back to
the point I was making.
So the surface r equal
2GM, nothing that crosses
it is ever going to come back.
This surface of
infinite redshift,
we call an event horizon.
OK?
No events that are on
the other side of r
equals 2GM can have any
causal influence on events
on the outside.
If you have a spacetime with
an event horizon like this
and this one that we are
talking about right here,
I'm going to talk at the
end of my final lecture
that I record about--
Maybe, actually, I'm
gonna do it in this one.
Yeah, I am.
So at the end of this
lecture that I'm recording,
I'm going to go over a
couple of other spacetimes
that have this structure.
Such spacetimes are called--
wait for it-- such spacetimes
are called black holes.
They are black because light
cannot get out of them.
And they are holes because
you just jump into 'em,
and you ain't never coming out.
So let me pause for
just one moment.
I want to send a quick
note to the ODL person who
is helping me out here.
I just want to let her
know that this lecture may
run a tiny bit long,
since I spent a moment
chatting with a police
officer who checked in on me.
OK, if you're
watching, Elaine, hi!
So I just sent you a
quick note, letting
you know that I'm likely
to run a little bit long.
All right, let's get
back to black holes.
So to sort of call out
some of the structure
of this spacetime, I want
to spend just a few minutes
talking about one final
coordinate transformation that
is very useful, but
looks really weird.
So just bear with me
as I go through this--
a very useful, but
unquestionably somewhat
obtuse coordinate
transformation.
What I'm going to
do is I'm going
to define a coordinate v-prime.
This is given by
taking the exponential
of the in-going coordinate
time V, normalized for GM.
U prime will be
the exponent of U,
a time that works well for
the outgoing coordinate system
divided by 4GM.
I'm then going to
define capital T
to be 1/2 V-prime
plus U-prime, capital
R to be 1/2 V-prime
minus U-prime.
It's then simple to
show, where simple
is professor speak for
"a student can probably
do it in an hour or so."
It's sort of tedious,
but straightforward,
just hooking together
lots of definitions
and slogging through a
couple of identities.
It's simple to show
that capital T relates
to Schwarzschild time T and
Schwarzschild radius r like so.
There's two branches.
OK, so this is how one
relates capital T and capital
R to Schwarzschild t and
Schwarzschild r in the region
r greater than or equal to 2GM.
You find a somewhat
different solution
in the region r less than 2GM.
So if you're looking
at this and kind
of going, "what the hell
are you talking about here,"
that's fine.
Let me just write down
two more relationships.
And then I'll describe
what this is good for.
So a particularly
clean inversion
between TR and the
original Schwarzschild tr,
both the r greater than 2GM
and r less than 2GM branches
can be subsumed into this.
And you find T over R looks
like the hyperbolic tangent
T over 4GM when
you're in the exterior
and the hyperbolic
cotangent in the interior.
So these rather
bizarre-looking coordinates,
these are known as
Kruskal-Szekeres coordinates.
I'll just leave
that down like so.
So when one goes into this,
I'm not going to deny it,
this is a bizarre looking
coordinate system.
OK, but it's got
several features
that make it very
useful for understanding
what is going on physically
in this spacetime.
So first, if you
rewrite your metric
in terms of capital T and
capital R, what you get
is a form that has
no singularities.
It's well-behaved everywhere,
except at r equals 0.
So you do get a
singularity there,
things blow up as r goes to 0.
There's no other
coordinate pathologies.
And then you get sort
of an angular sector.
It's actually cleanest in terms
of the Schwarzschild radius r,
so we'll leave it
in terms of that.
One thing which
is nice is notice
that radial null
geodesics, they simply
obey dt equals plus or
minus dr everywhere, OK?
The only place where you run
into a little bit of problem
is as r goes to 0.
And that's special, OK?
So I got that just by setting
ds squared equal to 0.
It's radial, so my
D-omega goes to 0.
And then just dt equals
plus or minus dr.
So that's really nice, OK?
I'm gonna make a sketch
in just a moment.
And the fact that I know
light always moves along
45 degree lines in
this coordinate system
is going to help me to
understand the causal structure
of this spacetime.
The causal structure is
what I mean by which events
can influence other events.
What can exert a causal
influence on what?
So as I move on, I'm going to
make a sketch in just a moment,
I want to highlight
a couple of behaviors
that we see that are
really sort of called out
in this mapping between
the two coordinate systems.
So notice that a surface
of constant Schwarzschild
radius, constant r
forms a hyperbola
in the Kruskal-Szekeres
coordinates.
Notice that surfaces
of constant time,
they form lines
in that they form
lines of slope t over r
equal to some constant.
So they are lines with t over
r equaling, on the exterior,
let's just focus
on the exterior,
they have a slope that's given
by the hyperbolic tangent of t
over 4GM.
On the interior--
replaced with cotangent,
hyperbolic cotangent.
The last thing which I'd like
to note before I make a sketch
here is let's look at
the special surface
of infinite redshift,
this event horizon.
So if I plug in r equals
2GM, plug this in over here,
I get t squared minus
r squared equals 0.
This is the asymptotic
limit to those hyperbolae.
They just become lines t
equals plus or minus r.
So the event horizon in this
coordinate representation
is just going to be a
pair of lines crossing
in the origin of these
coordinate systems,
OK, a pair of 45
degree lines crossing
into this coordinate system.
Notice, also, that t
equals plus or minus r,
this corresponds to
Schwarzschild t going to
plus or minus infinity.
So this, indeed, is a
weird, singular limit
of the Schwarzschild
time coordinate.
So you can find much prettier
versions of this figure
than the one I'm about
to attempt to sketch.
Let's see what I
can do with this.
So horizontal will be the
Kruskal-Szekeres coordinate
r, vertical will be
the coordinate t.
Here is the event horizon,
r equals t or little r
equals 2GM.
Some different surface of r
equal to some constant value
greater than 2GM will live
on a hyperbola like so.
Some value of r equals
constant, but less than 2GM,
lies on a hyperbola like so.
In particular, there is
one special hyperbole
corresponding to r equals 0.
And this is where my artistry
is going to truly fail me.
This is an infinite
tidal singularity.
Now, the thing which
is particularly
useful about this
particular coordinate system
is, remember, light always
moves in the capital R,
capital T coordinates.
It always moves on lines that go
dt equals plus or minus dr. So
what you can see is that
imagine I start here
and I send out a
little light pulse,
OK, a radially outgoing
light pulse, it will always
go away and go to larger and
larger values of r, just sort
of constantly moves along
this particular trajectory.
Let me write out again
what I'm doing here.
So a radial outgoing light ray,
it will follow dt equals dr.
But notice that this line goes
parallel to the event horizon.
If I am on the
inside of this guy
and I try to make a light
pulse that goes outside,
points in the radial
direction, all it does
is, in these coordinates,
it moves parallel
to the event horizon.
It can never cross it.
And in fact, because
this is a hyperbola,
one finds that even
though you have
tried to make this guy as
outgoing as outgoing can be,
it will eventually intersect the
r equals 0 tidal singularity.
Since I cannot have
any event, here,
that communicates with any
event on the other side of this,
this region, everything
at r less than 2GM,
these will be causally
disconnected--
not "casually," pardon me.
They are causally
disconnected from the world
that lies outside of r is 2GM.
So in my notes and in
Carroll's textbook,
there is another
coordinate system
that you can do which
essentially takes
points that are
infinitely far away
and brings them into a
finite coordinate location.
And that final thing,
it puts it in what
are called Penrose
coordinates and it allows
you make what's called
the Penrose diagram, which
displays, in a very simple way,
how different events are either
causally connected or causally
disconnected from the others.
Fairly advanced stuff, not
important, but many of you
may find it interesting.
Happy to talk
further, once we all
have the bandwidth to have
those kinds of conversations.
So let me summarize.
So the summary is that this
spacetime, so earlier we
were looking at the spacetime
of a spherically symmetric fluid
object.
We found a
particularly clean form
for the exterior of that object,
where it was a vacuum solution.
If we imagine a spacetime
that has this everywhere,
then we get this solution
that we call a black hole.
So I emphasize this
is vacuum everywhere,
but sort of the analysis kind
of goes to hell at r equals 0.
So there are some singular
field equations there trying
to describe the stress energy.
Its behavior as
you approach there,
let's just say we're not
quite sure what's going on.
In this coordinate system,
we see weird things
happening as we
approach r equals 2GM.
This is simply a
coordinate singularity.
There's really nothing going
bad with the physics here.
But our attempts to
use a time coordinate
that's based on,
essentially, the way light
moves in empty space, it's
failing in this region.
And all of this lecture is
about uncovering this and seeing
that, in fact, this is a surface
of infinite redshift beyond
which things cannot communicate.
So this is one of the big
discoveries that came out
of general relativity,
OK, this creature we
call the black hole.
It is not the only solution of
the Einstein field equations
that we call a black hole.
Let me talk briefly
about two others.
So another one has a spacetime
that looks like this.
So where did this come from?
Well, suppose I bung this
through the Einstein field
equations, what I find is that
it comes from a non-zero stress
energy tensor.
In fact, it's a stress energy
tensor that looks like this.
Pardon me.
This is a stress energy tensor
of a Coulomb electric field
with total charge q.
This represents a
charged black hole.
It turns out if you analyze
this thing carefully,
you find it has
an event horizon.
It turns out to be
located at G quantity that
involves the square root of
m squared minus q squared.
Now, if q-- don't even ask me
what the units are that this is
being measured in, they're
pretty goofy units--
but if, in these units,
the magnitude of q
is greater than m,
there is no horizon.
There is still, however, an
infinite tidal singularity at r
equals 0.
So such a solution
would give us what is
known as a naked singularity.
I have a few comments and I
have a couple notes on these.
But they're not as
important as other things
I'd like to talk about.
You might wonder, where
does that r horizon actually
come from?
OK, that is what you get when
you find the route, where
you look for the place where
the metric function vanishes.
OK, you notice there's only
one metric function that
appears in there, 1
minus 2GM over r plus q
squared over r squared.
So in general, if
you have what's
known as a stationary
spacetime, you can find
coordinates such that
surfaces of constant r
are spacelike surfaces.
If you look for the place
where a surface of constant r
makes a transition from being
a spacelike surface to being
a null surface, that
tells you that you
have located an event horizon.
OK, this is discussed in
a little bit more detail
in my notes.
There's also some
very nice discussion
in Carroll's textbook.
What it boils down
to is that if you
can find a radial coordinate
that allows you to do this,
then the condition G
upstairs r upstairs r
equals 0 defines your
event horizon, OK?
It just so happens
that it's also
equal to G downstairs t
downstairs t equals 0,
in this case and in
the Schwarzschild case.
But it's not like that
for all black holes
that you can write down.
In particular, let me write
down the final black hole
spacetime I want to
discuss in this lecture.
This is gonna take a
minute, so bear with me.
OK, so in this spacetime, the
symbol delta I've written here,
this is r-squared minus
2GM r plus a squared.
Rho squared is r squared
plus a squared cosine
of the square root of theta.
This thing turns
out, if you compute
the inverse metric
components, you
find that g upstairs r upstairs
r is proportional to delta.
And so there is a horizon
where delta equals 0.
Sorry.
That turns out to be located at
a radius that looks like this.
This represents the spacetime
of a spinning black hole.
It was discovered by Roy Kerr, a
mathematician from New Zealand.
I think this was actually
a big part of his PhD work.
So it is known as
a Kerr black hole.
The parameter a is related to
the angular momentum, the spin
angular momentum of the
black hole in the units
that we measure these,
normalized to the mass.
So notice that in order for
this to actually have a horizon,
you need that a to be
less than or equal to GM.
If it saturates
that bound, then you
get what's known as
a maximal black hole.
So it has a couple of
noteworthy features.
First, it is not
spherically symmetric.
If it were
spherically symmetric,
we could write the d theta
squared d phi squared piece,
we could find some
radial coordinate such
that there was
some simple radius
such that G theta
theta was simply sine
squared G theta theta.
This is the condition that
defines spherical symmetry.
And there is no
angle-independent radial
coordinate that
allows you to do that.
Notice also that
there is a connection
in this coordinate
system between t and phi.
Gt phi is equal to minus 2 GM
a r sine squared theta over rho
squared.
Why minus 2 and not minus 4?
Well, remember what I have there
is Gt phi dt d phi plus G phi
t d phi dt.
One can show that this
term, it reflects the kind
of physics in which the
spin of the black hole
introduces a spinning,
almost magnetic-like element
to gravitation.
If you guys do the
homework assignment
I have assigned in which you
compute the linearized effect
on a spacetime of
a spinning body,
you'll get a flavor of this, OK?
This ends up giving you,
that calculation gives you
a similar term in the spacetime
which further analysis
shows leads to bodies.
Essentially, what you find is
that if you have an orbit that
goes in the same
sense as the body's
spin versus an orbit that
goes in the opposite sense
of the body's spin,
there's a splitting
in the orbit's
properties due to that.
OK?
So it breaks the
symmetry between what
we call a prograde orbit
and a retrograde orbit.
This is one of the most
important solutions
that we know of in general
relativity because of a result
that I'm going to discuss now.
Oh, first of all, I should
mention that, in fact, you
can combine charge with spin.
I'm not going to write down
the result because it's just
kind of messy.
But it does exist
in closed form.
If you're interested
in this, read
about what is called the
Kerr-Newman solution.
So if you're keeping score, we
have this spherically symmetric
black hole, which
only has a mass,
you have the charged black hole
whose name I forgot to list.
Ah!
Sorry about that.
This guy is known as the
Reissner-Nordstrom black hole.
One of those O's, I
believe, is supposed
to have a stroke through it.
Those of you who speak
Scandinavian languages
can probably spell it and
pronounce it better than I can.
So we have
Schwarzschild, which is
only mass, Reissner-Nordstrom,
which is mass in charge,
Kerr, which is mass and
spin, and Kerr-Newman, which
is mass, spin, and charge.
You might start thinking,
all right, well,
does this keep going?
Do I have a solution for a
black hole that's got, you know,
northern hemisphere bigger
than the southern hemisphere?
You know, every time
you think about adding
a bit of extra sort of
schmutz to this thing,
do I need another solution?
Well, let me describe
a remarkable theorem.
The only stationary spacetimes
in 3 plus 1 dimensions with
event horizons are the
Kerr-Newman black holes--
completely parameterized
by mass, spin, and charge.
If you take the Kerr-Newman
solution, you set charge to 0,
you get Kerr.
If you take Kerr and
you set spin to 0,
you get Schwarzschild.
So the Kerr-Newman
solution gives me
something that includes these
other ones as sort of a subset.
And what this theorem
states is that the only-- so
stationary means
time-independent.
So in other words,
the only spacetimes
that are not dynamical, but
that have event horizons,
at least with three space
and one time dimension,
are the Kerr-Newman black holes.
Once you know these, you have
characterized all black holes
you can care about.
And in fact, in any
astrophysical context,
any macroscopic
object with charge
is rapidly neutralized by
ambient plasma that just sort
of fills all of space.
And so this Kerr
solution, in fact,
gives an exact mathematical
description to every black hole
that we observe in the universe.
That is an amazing statement.
OK?
Of course, as a physicist,
you want to test this.
And this, in fact,
is much of what
my research and research of
many of my colleagues is about.
Can we actually formulate
tests of this Kerr hypothesis?
And many of us have
spent our careers
coming up with such things.
Suffice it to say, in the
roughly negative 5 minutes
I have left in this lecture,
that the Kerr metric has
survived every test that
we have thrown at it.
So this metric, like I
said, was essentially
derived by the mathematician
Roy Kerr as his PhD thesis.
And it has really earned him
a place in physicist Valhalla.
Let me just conclude
this lecture
by making one comment here.
An important word
in this theorem
is the statement that the
only stationary spacetimes
are the Kerr-Newman ones,
stationary spacetimes
with event horizons, so the
Kerr-Newman black holes.
What this means is that
when a black hole forms,
it may be dynamical, it
may not yet be stationary.
And so the way that
this theorem, which
is known as the No-Hair theorem,
the way that it is enforced
is that, imagine I have
some kind of an object that
due to physics that we don't
have time to go into here,
imagine that this thing, its
physics changes in such a way
that its fluid can
no longer support
its own mass against
gravity, and it
collapses to a black hole.
OK?
Initially, this could be
a huge, complicated mess.
So we have a mass, charge, spin,
magnetic fields, who knows?
The No-Hair theorem guarantees
that after some period of time,
it will be totally characterized
by three numbers-- the mass,
the spin parameter
a and the charge q.
What goes on is that during
the collapse process,
radiation is generated.
What this radiation
does is it carries away
gravitational waves, carries
away electromagnetic waves.
Some of this is actually
absorbed by this black hole.
And it does so in such a way
that it precisely cancels out
everything in the
spacetime that does not
fit the Kerr-Newman form.
You wind up-- so this is one
of these things where we really
can only probe this either with
observations that sort of look
at things like black
holes and compact bodies
colliding and
forming black holes
and looking at what the
end state looks like.
Or we can do this with
numerical experiments
where we simulate very
complicated collapsing
or colliding objects
on a supercomputer
and look at what
the end result is.
And what we always find is
that the complex radiation that
is generated in the collapse
and the collision process
always shaves away
every bit of structure,
except for exactly what
is left to leave it
in the Kerr-Newman
form at the end.
Really, when we do
these calculations,
we generally wind up
with a Kerr black hole
because we tend to study
astrophysical problems that
are electrically neutral.
So this is a result
that is sometimes
called Price's
theorem, based on sort
of foundational
calculations that were done
by my friend Richard Price.
He did much of this right around
the time I was born in the days
of being a PhD student and
looking at the behavior
of highly distorted black holes
and seeing how the No-Hair
theorem--
You can imagine making
a spacetime that
contains what should
be a black hole,
but you somehow distort it.
What you find is it
becomes dynamical
and it vibrates in such a way as
to get rid of that distortion.
And you leave behind
something that
is precisely Kerr or
Kerr-Newman if you have charge.
Price's theorem is a
semi-facetious statement
that tells me everything in the
spacetime that can be radiated,
is radiated.
In other words, any bit of
structure, any bit of structure
in the spacetime
that does not comport
with the Kerr-Newman
solution radiates away
and only Kerr-Newman is left.
So that concludes this lecture.
My final lecture, which I will
record in about 15 minutes,
is one in which we
are going to look
at one of the ways in which
we test this spacetime, which
is by studying the
behavior of orbits
going around a black hole.
So I will stop here.
