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.