And now, finally, armed with all of the 
understanding that we could gear 
ourselves up with, the general theory of 
relativity were going to try to talk 
about what might happen to the core of a 
star that is too massive to be stabilized 
by neutron degeneracy, and collapses 
completely. 
And then, there, as far as we know, is 
nothing to stop the collapse, and, the 
star collapses completely. 
We won't talk much about what happens to 
the star, but we'll talk about what 
happens outside the star. 
The collapse itself is, of course, a very 
highly turbulent and energetic process 
releasing tons of, gravitational 
potential energy, and creating a wild and 
violent and turbulent supernova. 
There are people who model such things. 
We will not, discuss what they find. 
We will discuss what's left behind 
outside the star. 
And we'll start this a little bit 
obliquely by talking about our 
gravitational red shift. 
Remember, from the principle of 
equivalence, 
we found that, near Earth, 
if you shoot light up at height h, 
you find a gravitational red shift given 
by gh over c2. 
squared. 
This has to do with the con, the 
acceleration of gravity being G. 
We've already, played with that game 
before. 
And away from Earth, 
we expect that, because the force of 
gravity is weaker as you move farther 
away. If you're not near the earth's 
surface, we expect GH to be replaced by 
something like minus GM over R up to a 
constant. 
And, in fact, you can do the general 
relativistic calculation. 
And you find that if you're observing 
from a great distance from Earth. 
And way off in infinity, and someone is 
emitting light at distance R from earth. 
So the situation is that here is the 
earth, 
you are sitting a distance R from earth 
emitting a light beam and is traveling 
off, and this light is being received way 
far off from earth essentially at 
infinite distance. 
Then it will be reddened because your 
emitting from the ground floor from the 
top floor, but the amount is not GH of 
course. 
there is the weakening of the 
gravitational field the correct 
relativistic expression, is this one. 
The observed frequency will be the 
emitted frequency divided by the square 
root of this ratio. we can use the 
Newton, Newton's approxiamation to write 
this as lambda zero, times 1 minus 2GN 
over RC squared to the minus a half and 
then, That's the same as lambda zero 
times one minus times two times minus a 
half is plus, GM over r c squared and 
that's the expression I wrote there. 
And so, 
to a good approximation, we get exactly 
what we expected. 
Here is the GM over R that we expected. 
This is valid so long as this expression 
here in the square root is much smaller 
than one. 
This means that R needs to big, bigger 
than what? 
Well, such that 2GM over RC squared is 
much less than 1or R is much bigger than 
GM over C squared. 
This is called Schwarzschild radius, and 
for Earth the Schwarzschild radius 
plugging in, 
and earth over here is about nine 
millimeters. 
You're not likely to be broadcasting from 
less than nine millimeters from the 
center of earth, so I'm not worried about 
that too much in the case of earth. 
But what happens if R is not much bigger 
than the Schwarzschild radius. 
Then this redshift factor can become 
quite significant and it's deviation from 
the Newton approximation can be big. 
because the X here is not less than one. 
And in fact, when you get near to not, 
the, the, Schwarzschild radius for some 
given mass what happens is that this 
denominator vanishes and the redshift 
becomes infinite. 
So can you even do such a thing? 
Well the Schwarzschild radius for earth 
is nine millimeters, you can't get there. 
You need something where mass is more 
compactly concentrated. 
if you take the Schwarzschild radius for 
any object you can compute, it's directly 
proportional to the mass. 
And it can compute the schwartzfield 
radius for the sun to be about three 
kilometers. 
So the Schwarzschild radius for an object 
of solar mass is three kilometers, the 
Schwarzschild radius for an object of ten 
solar masses is 30 kilometers and so on. 
Neutron stars remember have a mass of 
about one and a half solar masses, and 
their radii are ten kilometers, so 
they're not too far from the 
Schwarzschild limit you can get to 
within, 
Three, maybe four Schwarzschild radii of 
a neutron star. 
At that point, when light is being 
emitted from the surface of a neutron 
star, the gravitational redshift due to 
GR effect is very significant. 
So neutron stars are pretty close to the 
Schwarzs, to be inside their own 
Schwarzschild limit. 
But if you can imagine an object that 
collapses to its even denser than a 
neutron star, then there will be a point 
in space, a ball around this object of 
radius determined by its mass, this 
Schwarzschild radius and when, you 
approach that 
distance from the center of the object, 
the gravitational redshift diverges and 
becomes infinite. 
What does this mean? 
This means that if I'm watching from very 
far away and, and watching something fall 
onto this massive, dense object, which 
I'm going to call a black hole. 
then when I look at it from afar, I will 
never see the thing that I'm watching 
getting there because, remember, 
gravitational redshift is really time 
slowing down. 
As I see, as I observe from far away, 
something falling onto a, the black, a 
black hole or a black hole's horizon, 
this ball around the massive compact 
object who's radius is the Schwarzschild 
radius is called the horizon for reasons 
that will become apparent. 
But as something approaches the horizon, 
if you're observing it from far away what 
you're going to observe is that it slows 
down and never gets there. 
It basically freezes as it approaches the 
horizon. 
Moreover any light, when I say you see it 
you would be seeing it by light say as a 
bright. 
luminous object, its light will be 
redshifted in the amount of energy per 
photon, remember, is proportional to the 
frequency so decreases with increasing 
wavelength. 
There's less energy in a red photon than 
there is in a blue photon. 
So the amount of energy this thing is 
emitting becomes less and less what you 
would see as this object slowed down and 
dimmed and then it will basically become 
dark as it approached the horizon, but it 
would take an infinite num, amount of 
time as you see it from far away for 
something to actually get to the horizon. 
In particular since any light is coming 
from the horizon is infinitely red 
shifted. 
That means it carries zero energy. 
No light comes out from this ball of 
radius Schwarzschild radius around 
whatever this compact massive object is. 
This is why we call it a black hole, 
because absolutely no light can leave. 
There's this area, region called the 
horizon. 
It's a ball of radius Schwarzschild 
radius. 
Any light from that ball or within it, 
but from that, sphere, no light can 
emerge. 
And so of course not from the inside now 
because of the way gravitational 
redshifts work light that is falling on 
to the horizon is blue shifted so if you 
happen to be sitting near the horizon you 
can look back and light falling on you 
will be. 
Intensely blue shifted and time will 
appear to you to be moving faster so your 
clock will run slow, you can look back 
and watch you know, the end of the 
universe happening behind you as you are 
sitting near the horizon. 
It will take a mere instance of your time 
because of this general relativistic red 
shift. 
this is what the horizon is the horizon 
of a black hole is the point at which the 
gravitational red shift is infinite this 
is why it is that no light comes out this 
requires an extremely intense 
gravitational field. 
typically we observe things from 
infinity. 
Notice that when I say you have to be 
infinite far away, the gravitational 
redshift will be infinite compared to any 
finite distance. 
All the infinity is coming out and the 
region where R is very close to RS. 
So if you're out at five Schwarzschild 
radii, you will still see someone, at the 
Schwarzschild radius have an infinite, 
red shift. 
So things slow down. 
Nothing ever goes through a hurr-, a 
black hole's horizon. 
So, if you ask what happened to the star 
as it collapsed. 
The answer is, I never saw that. 
In as well see it very far from the star 
the stars collapse stopped and froze and 
dimmed and darkened and all we see is no 
more starlight coming from there we'll 
see later what the other astronomical 
signatures are but we've never actually 
observed let alone see 
Star collapse into a black hole for the 
simple reason that by our clocks, it 
hasn't happened yet. 
It takes an infinite amount of time. 
Now okay that's looking from far away 
what happens if you get more, close to a 
black hole, then you can see some 
interesting effects as you get to within 
a few Schwarzschild radii relativistic 
effects become interesting so stable 
orbits exist. 
you can draw nice, stable circular orbits 
around an object like the sun, 
and as you get closer, to a black hole, 
relativistic effects become more 
important. 
You have to orbit rather fast. 
And, for radii, 
less than three times this Schwarzschild 
radius, 
No stable orbits exist. 
there exist circular orbits inside that 
radius, but they're all unstable. 
What this implies is that, 
any small perturbation, rather than 
knocking you into an elliptical orbit, 
will knock you into an orbit that either 
falls into the black hole or away from 
the black hole. 
So, that means that when we are looking 
at objects orbiting a black hole. 
None. 
Because there's always a britter basion 
from somewhere none will be orbiting near 
than three svoutshield radieye from the 
black hole now if the black hole has the 
mass of a few solar masses that's ten 
kilometers out that's not really very 
impressive but if a black hole has could 
stand the mass of 1,000 solar masses 
that's 3,000 kilometers will be the 
closest that anything can orbit. 
And by that time maybe you can make some 
hay of it. 
because of the deflection of light, at a 
radius, you can compute of one and a half 
times the solar ray, the, the, 
Schwarzschild radius. 
You find what's called the photosphere. 
There, light can actually orbit in a 
closed, circular orbit. 
Light is deflected by this massive star. 
It's deflected enough that you can 
imagine photons running in circles, 
around at the speed of light, around this 
massive object. 
Unfortunately, they don't stay there. 
That orbit is unstable, but what it means 
is that by the time you get to within one 
and a half Schwarzschild radii, the 
lensing, the optical effects of the black 
hole are going to be very extreme, and 
even farther out they are quite marked 
and we'll see it back in a bit. 
And 
the other thing, of course, is that 
because, so what is the deal? 
Why is gravity so intense in a black 
hole? 
It's got nothing to do with the fact that 
it collapsed. 
It, gravity is intense for the same 
reason gravitation of a one solar mass 
neutron star is so intense. 
It's because you can get so close. 
It's the GM over R squared becoming huge 
when N is the solar mass, but R is ten 
kilometers. 
In this case there is an, general 
relativistic enhancement, and in the 
neutron star case by the way. 
Similarly, the M over R cubed tidal 
forces become huge because you can get so 
near to such a large mass. 
And so, tidal forces can become extreme 
when you get so close to, when you get 
close to a black hole. 
However, I note that since the closest 
you're ever going to get is the 
Schwarzschild radius, and if your size is 
D, this is our estimate for the tidal 
acceleration between, say, your head and 
your feet. 
You can stick for D, I don't know, two 
meters if you're a tall person. 
it's twice g times the mass divided by 
whatever your distance is cubed, 
we'll plug it in as the distance the 
Schwarzschild radius. The Schwarzschild 
radius, remember, RS, is, 2GM over C 
squared, and so plugging R cubed here, we 
see that one G and one M and one two 
cancel, leaving us with four G squared M 
squared on the denominator, and a D times 
C to the 6th on top. 
These accelerations can be huge, 
but because, the larger, the more massive 
the black hole, the less extreme the 
tidal effects are. 
So a solar mass black hole will 
completely rip you to sheds if you 
approach its horizon. A four billion 
solar mass black hole will not. 
In fact, you might not even notice that 
you crossed the horizon. 
We'll talk about that in a second. 
Okay so this is what you'd see, a few 
Schwarzschild radii, I want to go and 
look at the horizon and beyond it, 
How can I see beyond the horizon? 
I thought you could never get there? 
Well no, no, you can never get there, as 
seen by a clock, ticking far away here on 
earth, but if you were to fall into a 
black hole, believe me, you would fall 
in. 
I never said the star never collapsed. 
I said we have never seen it. 
It hasn't happened in our time. 
This is the weirdness of gravitational, 
redshifts. 
the star, if you were near the star, you 
would have been collapsed. 
The star itself indeed collapsed, and 
we'll see what happened to it, but, 
We, from far away, have not seen it. 
So what we need if we want to, study 
things closer into the black hole is we 
need to use a better clock. 
Let's use the clock of an observer that 
is freely falling in. 
And then the shape that you see of the, 
neighborhood or vicinity of the black 
hole is given by this sort of diagram 
here to the right. 
And I have to elaborate a little bit on 
what this diagram means. 
So these, this diagonal, 
this region over here is, the space 
outside the black hole, 
and out here, very far away, is where we 
are. 
now, this line over here, this thing over 
here, and this thing over here, this is 
the horizon. 
Now, it's very strange, this is a space 
time diagram. 
This is the radial direction and this is 
the time direction. 
And what you notice we've ignored the 
fact that you can go around a black hole, 
we've pretended only distance from the 
black hole and time, are relevant. 
And what you notice here is something 
very strange. 
The horizon is not a place at all, it is 
tilted at 45 degrees. 
The horizon is the position of a light 
beam, when you use suitable coordinates 
to be able to distinguish that. 
In the coordinates we got using clocks 
from far away, you couldn't really tell 
that because the light cones in those 
coordinates were squished down to 
nothing. 
But in these coordinates where you can 
see what's going on near the horizon, you 
see that the horizon is in fact, could be 
the world line of a light beam. 
And this immediately explains why light 
cannot leave. 
So, 
if you are an observer near the horizon, 
you can still broadcast a signal, and it 
will at some point arrive far away, but 
once you cross the horizon, and you're 
inside, a signal even traveling at the 
speed of light, will never get out of the 
horizon. 
To get out of the horizon, a signal would 
have to, would have to travel faster than 
light. 
And that is why, no kind of proportion 
will get you out, once you cross a 
blackhole's horizon. And this is very 
strange, because we said the horizon was 
a place, but suddenly it appears to be a 
light-like thing. 
It's a, 
it's different places at different times. 
How did the horizons move with time? 
this again has to do with the fact that 
it moves as seen by a clock that is 
relevant near the horizon. 
Remember, this clock ticks infinitely 
slowly, as seen from far away. 
So from far away, the horizon looks like 
a place. 
This is a more accurate description. 
This is why light cannot escape. 
Over here is the interior. 
This is where the black hole is. 
And if you try to relate, this 
description, for the description in terms 
of distance from the black hole, 
then there's an animation that sort of 
tries to exemplify this. 
These lines are lines of constant value 
of r. 
The, sort of, punitive distance from the 
black hole. 
And far away from the black hole, you see 
that the constant r surface might look 
like this. 
And so this coordinate could be thought 
of as distance from the black hole. 
But as you move closer in, lines of 
closer distance to the black hole look 
more and more like this. 
and then, the line, r equals, 
our Schwarzschild, is exactly the 
horizon, and after that we see something 
funky happening. 
The line, changing R now is no longer a 
motion in space, changing R becomes a 
motion in time. 
And indeed this region over here is 
something called the singularity. 
Once you are inside, and we'll talk about 
its. 
significance, once you are inside the 
black hole, anything you do even if you 
move at the speed of light in either 
direction, you are bound to land on this 
boundary here. That is your future, so 
the singularity of a black hole is not a 
place, it looks from infitinity like a 
place. The singularity at the center of a 
black hole is in fact a time. It's a 
when, not a where. It's everybody's 
future once you enter the black hole. 
So now, 
okay, 
penetrate through the horizon. 
Note that this is quite possible. 
You can be an observer here. 
And, there are definitely, 
inertial trajectories that take you 
through the horizon. What happens when 
you cross the horizon? 
Well, once you cross the horizon, here is 
what happens to you once you'er inside 
the horizon, you can make an easy 
calculation within a finite proper time. 
You reach the singularity, that crazy 
contour at the top of the diagram. 
What is a singularity? 
A singularity is a place where the tidal 
acceleration, remember this is the actual 
gravitational effects, go to infinity. 
And the verges literally becomes infinite 
that means that first your feet get 
ripped off from the rest of you but then 
eventually your atoms get ripped apart 
mattin matter gets decomposed by the 
title forces. 
And then what? 
As you get even closer to the 
singularity. 
Well, then things become infinite and 
this divergence, this infinity in the 
calculation, signals a breakdown of the 
equations. 
The truth is that I told you, at some 
point that Einstein's theory of 
relativity was incomplete. 
This infinity is a signal of its 
incompleteness. 
It cannot predict. 
We do not know. 
We do not have a theory that predict what 
happens when you fall onto the 
singularity or what happens to the 
material of the star. 
Remember the star in it's own, time, 
receded inside its Schwarzschild radius 
and then every bit of the star. 
In finite proper time as measured by 
Europe by its clock crashed into the 
singularity and then, 
because we don't have an equation that 
describes it, and hopefully some day we 
will.