
Title:
8. Introduction to Black Holes

Description:
Frontiers/Controversies in Astrophysics (ASTR 160)
The second half of the course begins, focusing on black holes and relativity. In introducing black holes, Professor Bailyn offers a definition, talks about how their existence is detected, and explains why (unlike in the case with exoplanets where Newtonian physics was applied) Einstein's Theory of Relativity is now required when studying black holes. The concepts of escape and circular velocity are introduced. A number of problems are worked out and students learn how to calculate an object's escape velocity. A historical overview is offered of our understanding and discovery of black holes in the context of stellar evolution.
00:00  Chapter 1. Introduction
02:38  Chapter 2. Escape Velocity
12:12  Chapter 3. Defining Black Holes and the Schwarzschild Radius
18:50  Chapter 4. Gravity and Pressure in the Evolution of Stars
28:06  Chapter 5. From Electron Degeneracy Pressure to the Chandrasekhar Limit
37:59  Chapter 6. Neutron Stars
42:38  Chapter 7. Conclusion
Complete course materials are available at the Open Yale Courses website: http://open.yale.edu/courses
This course was recorded in Spring 2007.

Professor Charles
Bailyn: Okay,

welcome to the second part of
Astro 160.

This is going to be about black
holes and relativity.

And just to give you kind of a
preview, the whole point of

black holes is that,
of course, they emit no light,

so you can't see them directly.
And so, the question arises,

"How do you know that they're
there?"

And the reason you can
demonstrate that black holes

exist is because they're in
orbit around other things and

you can see the motion of the
other things that interact

gravitationally with the black
hole.

This concept should be
familiar, to a certain extent,

because it's exactly the same
thing we've been doing for

discovering exoplanets.
You don't see the exoplanet

directly.
What happens is that there's

something else that you can see
that's affected by the presence

of the exoplanet.
So, exactly the same thing

happens with black holes.
And so, we're going to use the

same equations,
the same concepts,

to explore this very different
context.

So, black holes can't be seen
directly.


And so, instead of detecting
them directly,

you use this combination of
orbital dynamics and things like

the Doppler shift to infer their
presence,

and more than just inferring
their presence,

to infer their properties.
Now, the context is more

complicated.
And, in particular,

we're no longer going to be
using Newton's laws – Newton's

Law of Gravity,
Newton's Laws of

Motionbecause there is a more
comprehensive theory that

replaced Newton,
which is necessary to

understand these things.
That more complex theory is

Einstein's Theory of Relativity.


So, we're going to be using
some relativity rather than

Newtonian physics.
This gets weird very fast, okay?

And so, I'm not going to start
there.

I'm going to start with a kind
of Newtonian explanation for

what black holes are,
we'll do that this time,

and then the weirdness will
start on Thursday.

So, the first concept and the
easiest way, I think,

to understand black holes is
the concept of the escape

velocity.
This is a piece of high school

physics.
Some of you may have

encountered it before.
And it just means how fast you

have to go to escape from the
gravitational field of a given

object.
If you go outside and you shoot

up a rocket ship or something
like that, how fast do you have

to shoot it up so that it
doesn't fall back to the Earth?

And so, you can define an
escape velocity for the Earth,

or for any other object for
that matter,

which is just how fast you have
to go to escape its

gravitational field.
There is an equation associated

with this.
It looks like this,

V_escape,
that's the escape velocity,

2GM / R all to
the 1/2 power.

And this is the speed required
to escape the gravitational

field of an object;
supposing that the object has

mass equal to M and
radius equal to R.

Oh, one other assumption,
I'm assuming here that you're

standing on thethat you start
from standing on the surface of

the object.
If you are on the surface.

Okay.
This equation should look

vaguely, but not a hundred
percent familiar to you,

because you derived something
that looked a lot like it on the

second problem set,
where you worked out the

relationship between the
semimajor axis of an orbit and

the speed and object had to go
in to be in that orbit,

and whatthe way that
calculation worked out it was

the velocity equals GM
over the semimajor axis,

a.
So that 2 wasn't there in that

derivation, but otherwise,
the form of this is actually

quite similar to that.
And so, let me explain why that

is.
Here's some object.

It's got a radius of R
and a mass of M.

And imagine that you've got
something that's in orbit around

this object, but it's in orbit
right above the surface.

It's just skimming the surface
of the thing.

That would be impossible in the
case of Earth,

because the friction from the
atmosphere would slow you down,

but in a planet without an
atmosphere, this would be

possible.
Imagine you're just sort of

skimming over the tops of the
mountains, or whatever.

So here you are in orbit,
this is something in a nice

circular orbit right above the
surface of this object,

and it's going around.
And it's got some velocity,

which I'm going to call the
circular velocity.

And as we calculated in the
previous section of the course,

that's GM / a,
which is, in this case,

GM / R, because
you're skimming the surface,

to the 1/2 power.
So that's how fast you have to

go to stay in orbit.
If you go slower than

thatlet's take a different
color, here.

If you go slower than that,
you crash into the surface

right away, because you
don'tyou're not going fast

enough to stay in orbit.
So, that's what happens in the

caseif you have a velocity
less than the circular velocity.

What happens if you're going
faster than the circular

velocity?
Well, you're going so fast that

you move further away from this
object.

You don't stay skimming the
surface.

So, here you are,
going a little bit faster.

And now you're not in a
circular orbit and your orbit

ends up looking something like
thisnice elliptical orbit.

And this is a velocity greater
than the circular velocity,

but still less than the escape
velocity because you haven't

escaped the gravitational field,
because you're still in an

orbit.
It's an elliptical orbit now,

but you're still in an orbit,
and it'll come back to the same

place.
Now, if you're going at

somewhat faster than that at the
escape velocity,

then you never come back.
You continue all the way out to

infinity.
Your orbit continues to change

direction a little bit due to
the gravitational force of this

thing.
You continue to slow down,

but in the end you never come
back.

So, it's a nonrepeating orbit.
And, if you're going even

faster than the escape velocity,
then you get to infinity even

faster.
And, more importantly,

when you get there,
you're still moving pretty

fast.
This, you gradually slow down.

So, all of these different
kinds of orbitsthe one that

crashes into the planet,
the one that skims the planet,

the one that's an elliptical
orbit, the one that escapes

altogetherthose are all within
a factor of the square root of 2

of each other.
Because, remember,

the escape velocity here is
equal to (2GM /

R)^(1/2).
So, just by increasing your

orbital speed by a factor of the
square root of 2,

you go from being in a nice
circular orbit to escaping the

gravitational field of the
object altogether.

So, you can calculate escape
velocities of things.

Let's do that once.
The escape velocity of

Earththat would bethe escape
is (2GM /

R)^(1/2).
2G = 7 x 10^(11)

M is, for the Earth,
is 6 x 10^(24) R for the

Earth is something like 7 x
10^(6).

This all has to be taken to the
½ power.

The 7s cancel.
What do we have here?

Let's see, 12 x 10^(13).
24  11 over 10^(6) 1/2let's

see, that's 1.2.
14  6 = 8, times 10^(8),

to the 1/2.
That's something like 10^(4)

meters per second,
because I've used the value of

G that's appropriate for
meters per second.

So, that's about 10 kilometers
a second.

So, that's the escape velocity
of the Earth.

If you go outside,
you throw a football up into

the air at 10 kilometers per
second, it's not going to come

back down.
Try itgood exercise.

Okay, you could calculate the
escape velocity of any

particular object.
You could calculate the escape

velocity of the human beingof
a human.

A human has a mass of 100
kilograms, a radius of about 1

meter;
this is kind of a big human,

but we'll go with it.
Let's see,

V_escape is
equal to 2GM overa

halfthere's a very famous book
about physics with the title,

"Assume A Spherical Cow," and
thisso, this is the kind of

thing that physicists like to
do.

It's an idealized situation.
We have this perfectly

spherical human being andokay
so let's do the calculation,

2 x 7 x 10^(11) x 100,
that's 10^(2),

over 1, to the square root.
That's 14 x 10^(9).

Call it (1.4 x 10^(8))^(1/2).
That's something like 1.2,

let's call it 1.
1 x 10^(4) meters per second.

Orlet's see,
that's 1/10 of a millimeter per

second.
That's less than 1 meter per

hour.
And that's why we don't go into

orbit around each other.
Okay?

Because you're always moving
way, way, way faster than the

escape velocity of the people
you're interacting with.

The escape velocity of the
Earth is 10 kilometers a second.

So, you're not moving that fast.
So, you're bound to the Earth.

But when you're hanging around
with your friends,

you're probably going faster
than 1 meter per hour,

and therefore you're moving
faster than the escape velocity.

You don't feel a great effect
of the gravitational force of

other people.
This is kind of

antiValentine's Day
calculation, right?

Because it proves that human
beings are not attracted to each

other;
at least not by the force of

gravity.
And so, you can do this

calculation for any given
object.

Fine, so what's a black hole?
Black hole has,

actually, a very simple
definition.

A black hole is simply
something in which the escape

velocity is greater than or
equal to the speed of light.

c is the expression for
the speed of light.

This is 3 x 10^(8) meters per
second.

And, if you've got something
where the escape velocity is

greater than the speed of light,
that's what a black hole is.

And it makes a certain amount
of sense.

If the escape velocity's
greater than the speed of light,

then light won't escape this
object, so you won't be able to

see it, right?
Because, the way you see

something is you see the photons
that come off it.

And, there's nothing
particularly extraordinary about

this, as far as it goes.
In fact, the middle ofthis

was already talked about and
worked out in some detail in the

middle of the eighteenth century
by an otherwise obscure English

clergyman named,
I think, John Michell,

and he did the following
calculation.

He said, okay now,
if this is going to be true,

how big is such an object?
How dense does it have to be?

And, you can work this out.
Supposing the escape velocity

is equal to the speed of
lightso c = (2GM

/ R) ^(1/2).
You can square both sides and

regroup, and you have R =
2GM / c^(2).

This is nowwasn't called this
in the eighteenth century,

but it's now called the
Schwarzschild radius.

Schwarzschild was a
contemporary of Einstein.


And, this is the size something
has to be in order for its

escape velocity to be equal to
the speed of light.

And a black holeanother
definition of a black hole is

something in which the radius of
the object is less than the

Schwarzschild radiusbecause if
the radius is less,

then the escape velocity will
be even greater.

So as I say,
this was worked out in

obscurity in the middle of the
eighteenth century,

and nobody thought anything
much about it.

Michell sort of pondered for a
little bit what such an object

would look like,
and he decided it would look

dark.
And true enough, but who cares.

And nobody thought anything
further about it for 150 years

until Einstein came along at the
start of the twentieth century,

and came up with the Theory of
Relativity.

And one of the pieces,
important pieces of relativity

is that the speed of light,
c is a very,

very special velocity,
and has rather bizarre and

extremely profound properties.
And it was only at that point

that the concept of the black
hole was recognized as something

that was in any way out of the
ordinary.

And so, for 150 years this idea
kind of lay dormant until it was

resurrected by a profound change
in the thinking about the

underlying physics.
So, this is one of these kinds

of historical fables that we've
hit a couple of times.

This is the fable of Michell's
discovery of black holes.

And the moral of this little
fable might be that the

importance of a result changes
depending on the context;

result changes,
sometimes dramatically,

with context.
So, something that looks quite

unimportant for a long time can
all of a sudden become a very

big deal.
And so, it happened in this

case.
Okay, so, one question you can

ask is that's all fine,
how big is the Schwarzschild

radius for some bunch of
objects?

Let's try the Sun;
here, let me start a new piece

of paper.


So, how big is the
Schwarzschild radius of the Sun?

Schwarzschild radius,
again, 2GM /

c^(2) = 2 x 7 x 10^(11).
Mass of the Sun,

now, 2 x 10^(30).
Divided by c^(2).

c is 3 x 10^(8) and
we'll square it.

Okay, so let's see,
2 x 2 = 4, times 7 is 30.

30 x 10^(19) 30  11 = 19
Divided by3 x 3 = 10.

(10^(8))^(2) = 10^(16).
So, that's 3 x 10^(3),

because 19  16 = 3oh,
in meters, because we're still

in MKS units,
because that what we're using.

That's the units of G
that we've chosen.

3 x 10^(3);
that's 3 kilometers.

That's pretty small for the Sun.
And so, you might imagine that

such objects are rare,
or perhaps even nonexistent,

because it would have to be
incredibly small,

and therefore,
incredibly dense,

to have a strong enough
gravitational field to prevent

light from escaping.
And so, you might have thought

that this is an entirely
theoretical concept,

and that no matter how
interesting it is,

there's no point in really
studying it further,

because you're unlikely to ever
encounter one of these things in

real life.
But one of the strange things

that washas been known for
quite some time is that black

holes really should exist,
and that they are predicted to

exist.
This has been known for quite a

whileknown for at 70 years,
that black holes should exist.

And the reason they should is
that they are one of the

possible end points of stellar
evolutionthe evolution of

stars.


And so, now I want to summarize
how stars evolve.

I should say,
this is the subject of whole

courses.
You can take Astro 350 and they

will talk about this for the
entire semester.

You can take Astro 110 and then
they'll talk about it for a

month.
But we're going to do it in

about twenty minutes,
save you time and effort,

here.
So, a star's life is determined

by the competition between two
forces;

so, star's lifetime and
evolution is determined by two

forces.
One is gravity,

which has the tendency to hold
the star together.

And now, the thing about
gravity is, it ought to work.

But stars have no solid surface.
So, if you can imagine gravity

pulling on an atom on the
surface of the star,

why shouldn't that atom just
fall all the way down to the

bottom of the star?
Because there's no solid

surface to prevent it from doing
so.

And the answer is that gravity
isn't the only force operating,

there's also pressure.
So, there's gravity,

which pulls stuff in,
and pressure,

which has the tendency to push
out.

And these things balance in
most stars.

In the Sun, for example,
these two forces are in balance

at all points in the Sun,
and this balance goes by the

technical name hydrostatic
equilibrium.


Hydro, because it's a fluid,
it's not a solid surface.

Staticnothing's moving,
and equilibrium is just

balance.
And, to be a little more

precise, the way this
workshere's the surface of

some star.
Here's some point within the

star, and there are two kinds of
forces acting on this point.

There's gravity,
which is pulling the thing

toward the center of the star,
and then there's pressure

forces in two different ways.
The outer regions of the star

exert a pressure inward.
So, there's an inward pressure.

And the inner regions of the
star exert a pressure outward.

And the outward pressure has to
be greater than the inward

pressure by exactly the right
amount to counteract gravity.

So, it basically looks like
P_out 

P_in = gravity.
And this holds true at all

points.
Now, in orderit's the other

way around right?
Thank you.

P_in 
P_outyeah,

right.
P_in minus

P_out has to
equal gravity.

And that requires that the
pressure on the inside has to be

bigger than the pressure on the
outside, because you don't have

negative gravity;
at least, not until the third

part of this course.
But, at the moment,

you don't have any negative
gravity, and so the pressure on

the inside has to be greater
than the pressure on the

outside.
Okay, so what is pressure?

Cast your mind back to high
school chemistry.

Remember high school chemistry?
It doesn't matter if you don't.

I'll tell you everything you
need to know.

There's something called an
ideal gas, and there's something

that the ideal gas does,
which is to exert gas pressure.

Your high school chemistry
teacher probably wrote down

something that looks like this:
PV = nRT. And

here's the thing about this.
V is volume in this case;

P is the pressure;
n is the number of

particles per volume.
And so, the key thing here is

that n divided by
V is equal to the

density, basically,
times a constant.

Because, remember,
density is mass per volume.

If you take the number of
particles and you multiply by

the mass of each particle,
that'll give you the total

amount of mass in a given
region.

You divide by the volume,
that equals the density.

And so, this also is a
constant, this R thing.

And so, what you get is
P is equal to a constant,

times the density,
times the temperature.

So, this is how physicists
think of the ideal gas law,

because we prefer to work in
terms of the density.

Okay, so here's the pressure.
And the pressure on the inside

had better be bigger than the
pressure on the outside,

or this isn't going to balance,
which means,

either the density or the
temperature,

or both, had better be larger
in the middle of the star than

it is in the outer parts of the
star.


So, inside of the star,
the T and/or the ρ has

to be bigger than it is on the
outside.


Now, it turns out that if you
just keep the temperature

constant all the way through the
star, you never achieve this

balance.
So, if only the density varies,

then inner regions do have
higher pressure but the increase

in density also increases the
force of gravity,

because gravity is dependent on
how much mass there is.

And, if you increase the
density, you also increase the

amount of mass.
So, you have higher pressure,

but you also have higher
gravity.

And it turns out that you can
prove, mathematically speaking,

that no balance is possible,
because it always ends up being

the case, for gas pressure,
that the amount you have to

increase the density by,
if you're only increasing the

density, will also increase the
gravity, and you'll never get a

balance.
So, the consequence of that is

that the inner parts of a star
must be hotter than the outer

parts.
Otherwise, the star wouldn't

exist;
it would collapse.

And this is true.
The inside of the Sun turns out

to be something like 10^(7)
degrees.

The surface of the Sun turns
out to be something like 6 x

10^(3).
So yes, indeed,

the inside very much hotter
than the Sun,

that is very much hotter than
the outside,

in the Sun, and that's what
keeps the Sun in balance.

And there's a problem with this.
And the problem is that there

is something called
thermodynamics.

And one of the laws of
thermodynamics is that heat

tends to flow from places where
it's hot to places where it's

cool.
This is evident in everyday

life.
If you take a little piece of,

I don't know,
molten lead or something,

and you drop it in a bucket of
water,

the heat from the lead will
spread into the water.

The water will increase in
temperature very slightly.

The heat will come out of that
piece of lead.

The lead will solidify,
and everything will come into a

kind of temperature balance.
Similarly, if you put a

snowball in some hot place,
it'll melt.

Why?
Because the heat from around it

will go into the snowball.
The temperatures will try and

equalize each other and
they'llit'll come out even.

So, this law of thermodynamics
is why the snowball has no

chance in hell.
And so, this happens in stars

too, right?
So, the heat in the center of

the star flows out.
And when it gets to the

surface, it's radiated.
At the surface,

it radiates,
and that's the energy that we

see coming from the star.
It's this heat that was in the

center.
It's gotten to the surface.

It's now radiating away out
into the cold depths of space,

and that's what we see.
But, that means that the

temperature in the center of the
star, which is holding the star

up, decreases,
and then the star wouldn't hold

itself up.
So you requirein order for

the star to hold itself outan
energy source at the center of

the star.
And this does two things.

It replaces all that lost heat,
and it preserves the

equilibrium of the star so it
doesn't collapse.


Okay, so this is all pretty
abstract.

And it was known that this had
to be true before they figured

out what the energy source was.
It was known for people just

sort of thinking in very general
terms about how the Sun could

existunderstood that there had
to be some kind of large source

of energy down in the middle of
the star.

And, notice it has to be in the
middle of the star.

It does no good for the energy
to be created all the way

through the star,
because if it's created all the

way through the star,
then the temperature is

distributed throughout the star
and you don't get a situation

where it's much hotter in the
middle than it is on the

outside.
So, everybody knew for quite a

while that there was energy
being created in the center of

the star.
They just didn't understand how

that was done.
And then when people invented

nuclear physics in the 1930s and
'40s and '50s,

it was understood that this
comes from nuclear

reactionsnuclear fusion,
in particular.

In the case of the Sun,
it's the fusion of hydrogen,

atoms together to make helium,
that does this.

And that releases energy in the
same way that a nuclear bomb

does.


The problem with this is that
eventually you run out of

hydrogen, or whatever your
nuclear fuel is,

because you've got only a
limited amount of it in any one

star.
So, eventually,

the nuclear fuel runs out.


And then the star has many
adventures before it settles

down.
And for these,

I will have to refer you either
to a textbook or to some other

course,
because I'm not going to take

you through the whole exciting
life of a star once its nuclear

fuel is exhausted.
Suffice it to say that you know

in advance what the outcome has
to be, because there's no way it

can hold itself up,
in the long run,

because it doesn't have an
energy source down at the center

of the star.
So, the consequence of this has

to be that the star collapses.
Now, it doesn't necessarily

collapse all the way down to
being a black hole,

because there are other kinds
of pressure besides the pressure

exerted by an ideal gas.
So, at very high densities,

you get other kinds of
pressure.


In particular,
there's something called

"electron degeneracy pressure."


This is sometimes called Fermi
pressure, after the guy who

thought it up.


Degeneracy is another one of
those words that means something

different to physicists than
they do to ordinary,

normal people.
I remember the first time I

taught Astro 110,
right after I came here,

about fifteen years ago.
I started talking in the middle

of a class about degenerate
white dwarfs.

And you could feel this sort of
wave of something between

confusion and anxiety permeate
through the class,

and I had no idea what was
going on.

Some teaching assistant had to
pull me aside afterward and

remind me that these words mean
different things.

So, I apologize if this sounds
like a sort of adult version of

a Grimm's fairy tale,
you know, with degenerate white

dwarfs wandering around and
stuff, but such are the words we

have to work with.
Okay.

So you have this electron
degeneracy pressurethat's a

different kind of pressure –
and that can stabilize the star.

So it stabilizes a
starstabilizes the star

ataround the radius of the
Earth.

So, that's very high density
material.

We can calculate the density.
You'll remember our equation

for density: mass over volume.
And so let's see,

how does this work?
This will be the mass of the

Sun, let's say,
divided by the volume of the

Earth.
That's 4/3πr^(3).

Radius of the Earth is (7 x
10^(6))^(3).

So, let's see,
(2 x 10^(30)) / 4.

7^(3) 7^(2) = 57,
cubed is 350.

Times 10^(18).
That's 6 x 3yeah,

okay, we're doing all right.
And let's say that that's 2,000

x 10^(27) over,
I don't know,

1,400 x 10^(18).
Let's cancel those.

10^(9) kilograms per meters
squared.

That's about a million times
denser than water.

Water, you'll remember,
is 10^(3) kilograms per meters

squared.
And water is defined to have a

density of 1 gram for a cubic
centimeter.

So, if you were to pick up one
gram of this white dwarfone

cubic centimeter of this white
dwarf,

it would be a million times
more massive than a gram.

That's about a ton.
So, a thimbleful of this stuff

weighs about a ton,
very dense.

And the Sun will end its life
as a white dwarf with this

electron pressure balancing the
gravity.

Thesesuch stars are called
white dwarfs,

and there are many of them
known.

White dwarfsthis is the end
point of the Sun.


And then, in the 1930s,
what happened to make black

holes inevitable,
was that one of the great

theoretical astrophysicists of
the twentieth century,

a man named Subramanyan
Chandrasekhar,

discovered that electron
degeneracy pressure doesn't

always do the job.
So, in the 1930s,

Chandrasekhar discoversproves
that if the mass of an object is

greater than 1.4 times the mass
of the Sun,

electronthis kind of electron
pressure is insufficient.


And the star continues to
collapse.

Now, Chandra was a graduate
student in England at the time

he figured this out,
and he presented this rather

dramatic result at a big meeting
of the Royal Astronomical

Society in London.
And then, every graduate

student's worst nightmare took
place.

Chandra's thesis advisor was a
very famous man named Arthur

Eddington, one of the great
physicists of the early

twentieth century.
And after Chandra had presented

his results to all these
assembled scientific

dignitaries,
Eddington got up and denounced

his own student,
and said, "This can't possibly

be true."
And Eddington made the famous

remark, "There ought to be a law
of nature to prevent stars from

behaving in this foolish
manner."

And the consequence of that is
that a lot of people didn't

follow up on Chandra's idea.
Chandra got miffed,

as you might understand.
He got on the boat to the

United States.
He spent the rest of his career

at the University of Chicago.
On the boat,

he wrote a great textbook still
in use on the structure of

stars,
in which he laid out in detail

all of the arguments that the
Chandrasekhar limit must

actually exist.
And fifty years later he got

the Nobel Prize for it.
But thatthere was this kind

of time lag there,
and it is, I think,

important for those of us,
say, on the faculty to recall

that if Eddington had listened
to his student instead of to his

intuition,
the study of black holes would

probably be forty years more
advanced than it is.

So, another fable for our times.
Chandra was always very

gracious about this,
actually.

He would praise Eddington to
the skies as a wonderful

advisor, and then with this
little asterisk.

So, fable: Chandrasekhar's
limit is the title,

and the moral here is,
"believe your student,

not your intuition."
And actually,

the story of Eddington was
actually kind of interesting.

As Eddington got older,
he became more and more

convinced that,
you know, he could guess the

right answer.
And so, it wasn't just the

Chandrasekhar limit,
it was other things.

He got a little weird toward
the end of his life,

and he started believing his
intuition.

Einstein did this too, right?
Einstein famously discovers all

this great stuff,
and then the second half of his

life is completely useless,
scientifically,

because he becomes convinced
that his gut is telling him that

quantum mechanics is wrong.
This famous remark,

"God does not play dice with
the universe"but it turns out,

that isn't true.
And so, there is this

probabilistic nature of reality
that quantum mechanics shows.

And so, Einstein spent the
second half of life trying to

prove that his intuition was
correct,

that quantum mechanics couldn't
really be true and thus,

did no physics worth doing for
about thirty or forty years.

So, you have to watch out for
this.

If you're too smart and start
believing yourself,

you can get into trouble.
All right.

Now, having said all that,
Eddington was partly right.

There is, sort of,
a law of nature that prevents

stars from behaving in this
foolish manner.

So, to understand that,
what happens when white

dwarfswhen the white dwarf
collapses?

It's got to get rid of its
electrons, because the problem

is, the way this electron
degeneracy pressure worksyou

can't squeeze electrons any
closer together than they are in

a white dwarf.
So now, you have to get rid of

electrons.
And so what do you do?

You combine the electrons and
the protons and you turn them

into neutrons plus neutrinos.
These are neutrinos.

They stream out.
And so, you end up with

something that's made entirely
out of neutrons.

So, the whole star turns into
neutrons.


A chemist would think of this
as essentially turning the whole

star into one atom,
into one atomic nucleus.

An atomic nucleus with noan
atom with no protons,

no electrons,
and 10^(57) neutrons.

And you could imagine putting
that somewhere on the Periodic

Table.
Astronomers call these things

neutron stars,
and they exist.

They were discovered in the
1960s.


And a typical neutron star,
a couple times the mass of the

Sun hasmass equals 2 times the
mass of the Sun.

Radius of about 10 kilometers.
And you can work out the

density for that.
Density is a billion times

greaterI'll leave this as an
exercisegreater than for white

dwarfs.
So, instead of a cubic

centimeter of the stuff weighing
a ton, it now weighs a billion

tons.
And you're having a tough time

moving it around.
But 10 kilometers – that's

getting close to the
Schwarzschild radius.

Remember, we calculated the
Schwarzschild radius of the Sun.

It was about 3 kilometers.
And, in fact,

if you calculate the
Schwarzschild radius of a star

in terms of the Schwarzschild
radius of the Sunlet's see,

you get 2GM /
c^(2),

where M is the mass of
the star,

divided by 2G mass of
the Sun, over c^(2).

So the Gs and the
cs all cancel,

here.
And you get

M_star /
M_sun.

So, if the Chandrasekhar mass
of the Sun is equal to 3

kilometers, as we calculated,
the Schwarzschild radius of a

star withwhose mass happens to
equal 3 times the mass of the

Sun,
is going to be 3 x three

kilometers, or 10 kilometers.
That's equal to the radius of a

neutron star.
So, a neutron star with mass

greater than 3 times the mass of
the Sun has a radius less than

its Schwarzschild radius.
And that's a black hole.

Remember?
And the key thing here is that

there are lots of stars with
mass more than 3 times the mass

of the Sun.
We don't see them as black

holes because they're still in
hydrostatic equilibrium.

But eventually,
they're going to run out of

nuclear fuel,
and they're going to collapse.

Now, in fact,
during the course of the star's

life, one of the things I
glossed over is stars tend to

lose mass as they live.
And so, they don't end up with

the same mass they started with.
But stars with initial masses,

at the beginning of their
lifetime, greater thanI don't

know,
something like thirty times the

mass of the Sun,
will end up with masses greater

than three times to the mass of
the Sun.

And then, there's nothing to
stop their collapse.

What happens is,
they turn into neutron stars,

but they turn into neutron
stars whose Schwarzschild

radiuswhose radii are smaller
than their Schwarzschild radius,

and that is a black hole.
So, they collapse down into

black holes.
And so, you expect a large

number of black holes to
actually exist.

This is what happens to massive
stars at the end of their life.

And so, we expect there to
bethat there are many black

holes.


And so, the question we'll be
exploring in the rest of this

segment of the class is,
how can you find these things?

What are the properties of
these things from a theoretical

point of view?
What does Einstein's Theory of

Relativity suggest that these
things are going to behave like?

And then, the big question is,
once you've found some,

and you have a theory for what
they behave likethen you can

ask the question,
does the actual behavior of

these objects conform to the
theoretical expectations?

Another way of saying that is,
was Einstein right?

Is general relativity the
correct theory to describe these

very exotic objects?
And so, that's what we'll be

talking about in the rest of
this section of the course.

Now, let me turn back to the
previous section of the course,

which waswhich culminated
last time in this little test.

And I think we're ready to hand
these back.

Is that true?