So black holes exist and they are an important part of the constitution of
galaxies and they are the end point of stellar evolution for mass of stars and
most of them we don't see. But if they happen to have a bindery
partner we can detect the accretion discs around them.
Okay, what can we say about these objects from understanding GR.
What do people's studies tell us. the black holes
we've been considering have been the simplest, spherically symmetric,
non-rotating, simple objects. how complicated can black holes get?
Well, they can get complicated, but it turns out not very.
there's a beautiful theory known colloquially as the Black Hole No Hair
theorem, or a black hole has no hair. it turns out that in the collapse from a
star to a black hole, all of the properties of the stars are lost.
The chemical composition makes no difference.
The temperature is gone, the shape of something that collapsed is
gone. All of those features are radiated away
as gravitational waves or energy. All that information is gone in the
collapse. Black holes are completely characterized
by three numbers. Tell me three properties of a black hole,
and I can exactly describe, that is all there is to say about it.
We said that stars are by and large described by mass, but the stars also
have magnetic fields, and rate of rotation, and [INAUDIBLE], chemical
composition, and metallicity, and age. No such thing for a black hole.
A black hole is completely characterized by its mass,
its angular momentum and its electric charge.
Now, electric charge is expected to be zero.
Why would a black hole carry an electric charge?
Everything in the universe is pretty much neutral.
we know that its mass. We can measure if something's orbiting
it. Angular momentum is expected to be rather
large. Remember, this is a collapsed neutron
stars. These neutron stars we saw are in a
collapsed core. When it collapsed to a neutron star, it
carried enough angular momentum to drive a pulsar.
We expect most black holes to be rotating rapidly.
the Schwarzchild example that was the topic of the simulations we've been doing
was a simplified case with vanishing angular momentum.
Angular momentum makes it a lot more complicated to understand,
but that's all there is. There's only that one extra parameter.
The second thing is, we discussed this very special, spherical symmetric case,
and we found the singularity. Maybe that's a artifact of the symmetry
of the situation. This was a real consideration when the
black hole solutions were first discovered.
work of Hawking and Penrose shows that no, this is real.
general relativity is a theory which when you start with an initially smooth and
reasonable un, non singular space time you can prove that within a finite time
in the future, or possibly in a finite time in the past, you had actual
singularities. General relatively produces singularities
starting from non-singular initial conditions, and then it doesn't know what
to do, which means that relativity as a theory
is intrinsically incomplete. We know that we don't have the final word
on gravity, and some of us are trying to do better, but at the moment this is what
we have. Do we need to care about what happens at
the singularity? Well, again we will never see what
happens at the singularity, at the center of the black hole, because we don't see
anything inside the horizon. There's this light like-surface
surrounding it, and light cannot escape, so in some sense maybe we don't care.
there is a conjecture called the cosmic censorship conjecture,
which is our model of base, which says that all singularities are hidden inside
horizons, which means, things that are not
described by the equations might be going on in your theory.
But they're going on in places, that have no causal effect on anything that's
happening outside. We could go there and investigate them,
though we'd never come back to tell, but they will never influence anything
about us. The inside of a black hole has no impact
on the exterior of a black hole. Nothing can escape,
not even light. so this is the cosmic censorship
hypothesis. there are indications that it may fail
and that there maybe such things violating it, as singularities not hidden
behind horizons. Those would be naked similarities, and
there is recent evidence that such things are in fact possible in reasonable
gravitational theories, but we of course have not observed any.
in the realm of fantasy, let's look again at this image of the black hole that I
showed you, where you remember that I described this is the outside of the
black hole. This as the inside of the black hole.
This was the singularity, and this was the horizon.
This was the line r equals r Schwarzschild, which was lake like, in
fact, and yeah well this is the black hole, this is the outside.
What is the rest of this? and, this region, and I should note,
this, description, is a description that is valid for a region just outside the
horizon. That's why I was using it.
But it's obtained by ignoring completely the fact that there's a star.
you're basically using the solution to the part of Einstein's equations where
you set the energy and momentum bit to zero.
Which is true, everywhere outside the star.
So those are the equations you solve to describe everything except where
materials are. Most of space is free of matter and
energy. It's mostly vacuum.
And so we use those equations and then you, you get that solution, you can
extend that solution. So in real life there would be a big fat
star over here and while you could hit the singularity, just like most of the
matter in the star is, all of this stuff down here and to the left would be
complete fantasy. But let's imagine, we've drawn a valid
solution to Einstein's equations anyway. we don't know how to create such a thing,
but what does this look like. Well so there's this region out here
which is the outside. There's this region out in here which is
the black hole. There's this region in here which is
interesting. Notice that you can't get into that
region no matter what you do. Remember time moves this way.
you would have to be moving faster than light.
To tip through this light cone. This is what's called a white hole.
Everything can only come out of it, nothing can ever fall into it.
It's a time reversed version of a black hole.
And this region out here, it turns out, looks a lot like this region out here.
So there is another asymptotic region. And away from the black hole, it looks
quite reasonable. Like normal space time.
And so, remembering that time runs vertically.
You now have this great description of two regions, which,
outside, the outsides, that look like the outside of a black hole,
which are disjoint, right? They, they don't touch each other.
Except at this, one point. they actually intersect, and this one
point is just, too short. The, the, length of time that these two
universes are connected, if you want, is just too short for a light beam to get
from here to here. A light beam sort of that can travel the
boundary from one to the other. but, this is the idea for what is called a
wormhole. A small deformation of this leads to
something called the Einstein-Rosen Bridge, where there is a sort of finite
time where the space time on the right, space time on the left are connected.
that is the origin of the idea of worm holes.
You can imagine that somewhere in our space time is a horizon, and there is a
sort of a, a length of time during which by going through that region of strong
curvature. You could come out and go completely
different universe or into a part of our own universe that is very far you could
imagine sort of our universe bending over and a wormhole cutting through or some
such nonsense. And at the moment
we don't have any, working solutions. So the Einstein-Rosen Bridge has been
shown to be unstable, Any perturbation anywhere, would,
collapse it down, People have been working on various ways
to modify the solutions to get wormholes that are both stable.
And last long enough for, at least a, a photon or a particle to get through and
maybe something macroscopic. that has not happened, but there is no
theorem that I know of, that precludes its existence,
and so maybe there are solutions to the equations that generate wormholes, and
maybe, maybe, there are actually wormholes out
there. I cannot exclude that, and then what's on the other side could
be a different universe, could be a different part of our own universe.
Going through a wormhole would be an exciting thing to think about.
I didn't want to leave you without them. And then the last thing I wanted to
discuss is quantum issues of black holes. And so it's a result again due to Hawking
that quantum effects, the quantum fluctuations near a horizon
violate this idea that a black hole produces no energy and in fact a black
hole does emit radiation. It's called Hawking radiation.
It's not a property of general relativity which is a classical theory.
It's a property of the quantum fields, electromagnetic fields and electron
fields and whatever other fields you have in your theory fluctuating near, in the
presence of the large curvatures at black holes horizon.
And That means that a black hole radiates,
and Hawking's calculation of the black hole radiation led him to, later figure
out that a black hole radiates black body radiation.
A black hole has a temperature. The thermodynamics of a black hole was
later developed. It's, quite a developed art, but it's a
weird object. The temperature of a black body,
decreases with its mass, of a black hole, decreases with its mass.
It's inversely proportional to its mass. Bigger black holes are cold.
Smaller black holes, less massive black holes, are hot.
So, smaller black holes, that means, radiate more.
This is very strange because when a black hole radiates of course a the energy came
from the black holes mass so the black hole become loses mass, becomes a little
bit smaller and therefore a little bit hotter.
A black hole is one of those weird objects with negative specific heat.
As it loses heat, it heats up. Typically, objects lose energy and they
cool. A black hole heats as it loses energy.
So as it loses energy it heats more, it loses, it radiates more.
Remember T to the fourth, it heats more, it radiates more.
So the end of a black holes life would be a very dramatic explosion.
In fact the title of Hawking's paper was Cosmic Explosions, because he was
thinking about, maybe we should go look for these black holes that at the end of
their life produce these very explosive, high temperature, large amounts of
radiation and black holes can completely disappear, evaporate.
That's not a big worry it turns out. the expected lifetime of a five solar
mass black hole is on the order of the ten to the 62 years, given a universe
that is ten to the ten or so years old. That's not yet a major concern.
But if there are tiny black holes around, that therefore radiate more.
We know about million solar mass black holes.
Those last even longer. We know about stellar mass black holes.
Those last too long, but if some mechanism produced two kilo
black holes or proton mass black holes, those would
evaporate much more rapidly. And the possibility of microscopic black
holes, and their evaporation is something that people are studying in various, both
theoretical, and at the LHC, even experimental level, to try to see if they
can form a black hole and watch it evaporate essentially as part of their
high energy collisions. So, this is the theory bit of the black
hole.