Okay, so in the last class, we had discussed
the thermodynamics of radiation.
So we found several things where we were able
to derive the black body spectrum, which was
first guessed by Planck by looking at the
experimental data.
So, we derived this using a combination of
statistical mechanics and Einstein's quantum
hypothesis of radiation.
So, the other thing we did was, we derived
what is called a Stefan-Boltzmann law, which
basically tells you the power that is radiated
by black body as a function of temperature.
So, the total power over all the frequencies
that is radiated by the black body as a function
of temperature and that is proportional to
the fourth power of the absolute temperature
of the black body.
So now, we are going to use these ideas, that
is the thermodynamics of radiation is going
to be used in the next topic that I am going
to discuss, which is a very interesting subject
of the thermodynamics of black holes.
So now, this is a very hot topic nowadays,
because recently, the black hole, which is
at the center of the galaxy M87, so it is
the 87th galaxy in the Messier catalog, and
it is at a distance of 54 million light years
from earth and this is one of the most massive
black holes ever seen in the universe.
In fact, it is 6.5 billion times the sun's
mass and compare that with the black hole
at the center of our own galaxy, which is
the Milky Way, and that is only a few million
times the sun's mass.
So this is actually 1000 times heavier, so
it is 6.5 billion times the sun's mass.
So as you very well know, a black hole is
basically the end result of the stars, just
like people, stars are also born, they grow
older, and then finally they die.
So depending upon how heavy they were when
they were living, so when they die, their
final resting state depends upon how heavy
they were before they died.
So if they were sufficiently light, they end
up becoming what are called white dwarfs,
but then if they were heavier than the Chandrasekhar
limit to begin with, then they collapse further
and they typically become what is called a
neutron star, so where the matter is very
dense, but it is mostly made of neutrons,
you know.
There is another limit that comes into play
for neutron stars and if the star is even
heavier than that limit, then gravity takes
over and the star collapses and becomes a
black hole.
So the matter just becomes a point, so that
means it compresses itself to a point and
the space time around it becomes extremely
curved and generally, you will have to use
general relativity to describe the space time
around it.
So in fact, as Chandrasekhar was one of the
main persons who thought of this idea that,
you know, stars may not be stable after all
and he made this very beautiful remark that,
you know, black holes of nature are the most
perfect macroscopic objects there are in the
universe.
The only elements in their construction are
our concepts of space and time.
So you see, it is such a beautiful idea that
Chandrasekhar expressed and you can see why
people are excited about this, and till now,
it was not possible to study black holes properly,
first of all because they are far away from
us and secondly because by definition, they
are black, so they do not emit anything, the
only way you can study them is by looking
at the effect of these black holes on nearby
objects which are luminous like stars and
other objects.
So people have been indirectly recognizing
the existence of black hole, but it is only
in the last couple of months that we have
been able to actually see a black hole, actually
image a black hole using radio astronomy and
this was accomplished using what is called
the Event Horizon telescope, which is an array
of radio telescopes, located at various corners
of the globe.
So now, this is a beautiful picture of that
Messier 87 galaxy’s black hole, which is
at the center of the galaxy.
So you can see that there is this region where
it is black here, the central hole there.
So, you cannot really tell where the black
hole starts because first of all, you know,
black holes have something called an event
horizon.
So an event horizon is an imaginary surface,
so if light exists below that surface, it
cannot escape out of that surface.
So, this is roughly of that order, the size
of this is of that order of the event horizon.
So the bright gas here that you are seeing
here, so the brightness here is actually the
radiation that is emitted from the surroundings,
especially the background that is from behind
the black hole, that is in some sense lensed,
so you see the intense gravity of the black
hole causes space time to bend and it acts
like a gravitational lens.
So all the radiation gets collimated and concentrated
in these lumps and then you are able to see
the light that is coming out.
So, it is all lensed light.
So first of all, it is not visible light,
it is in the radio wavelength, so it has been
converted to optical wavelengths for human
beings to see, alright.
So, this is all fine, but what I want to discuss
in this class is something slightly different.
I am not going to discuss radio astronomy,
but I am going to discuss the thermodynamics
of black holes because surprisingly, black
holes not only have mass, they also have an
entropy.
So, in fact, this was first pointed out by
Bekenstein and later on by Hawking that the
entropy of a black hole is maybe thought of
as being proportional to the area of the event
horizon.
So, you can motivate this idea.
So you might be wondering how do you know
that?
So, how do you know that the entropy of a
black hole is proportional to the area of
the event horizon?
So you can motivate this idea in the following
way.
So see, entropy after all is a measure of
the number of microstates.
So, we may assert that these microstates are
on the event horizon, because after all, see
for a black hole, like Chandrasekhar said,
the only thing going for a black hole is the
nature of space and time itself.
So the event horizon is the only structure
that survives in the black hole.
So the matter itself has shrunk to a point.
So the only structure the extended object
that survives is the event horizon.
So it stands to reason then that these microstates
live on the event horizon.
So what you can postulate is that the information,
so that means that entropy is in some sense
if you think of the information theoretic
version of entropy is just a number of bits.
Entropy is just a number of bits.
So you just want to ask yourself, how many
bits can I store on the event horizon?
So the idea is that the area comes in discrete
lumps because what happens in quantum gravity,
that means if you look at, there is something
called the Planck length, so which tells you
the smallest length that is possible, you
know, smallest length that has physical meaning.
So you can combine the fundamental constants
of nature, which is Planck's constant, gravitational
constant and the speed of light in this way
and get a square of length.
So lP is called the Planck length.
So this is called the Planck length, and the
square of the Planck length is related to
the fundamental constants.
So now you can think of this as some kind
of smallest area that is possible to exist
in nature.
So if that is the case, then you can accommodate
one bit on this area, so is the number of
bits, so it is area divided by the area of
this smallest piece of the event horizon,
which is lP squared.
So this factor of 4 cannot be derived by this
argument.
So this is one-fourth of A by lP square.
So other than that, you can kind of guess
that this is what it should be.
So now, we have vaguely convinced ourselves
that black holes should have entropy, but
it is more obvious, of course, that black
holes have mass and therefore energy.
So because energy is mass according to special
relativity, energy is Mc squared.
Now, anything that has entropy and energy,
also has temperature.
So that is what we have been telling ourselves
till now in this course.
So now that we have established that a black
hole has entropy, by virtue of the fact that
it has an event horizon, which has an area,
and the entropy is proportional to that area
and we also know that the black hole has mass
and therefore it has energy, which is Mc squared.
So now, we have a situation where the black
hole has energy and it has entropy.
So if it has energy and entropy both, then
you know that such an object has temperature.
So write down this, this constant is called
Schwarzchild radius.
So the reason why it is called that is because
this is the radius of the event horizon basically.
So below this, light cannot escape.
So the entropy of the black hole is now expressible
in terms of the energy because now you can
rewrite.
So you can rewrite the Schwarzchild radius
in terms energy and 4 pi r squared is the
area of the event horizon, and you can therefore
express entropy in terms of energy.
Now once you know entropy in terms of energy,
you can go ahead and differentiate this and
you get an inverse temperature, so the absolute
temperature, the thermodynamic temperature
of a black hole can be derived as the slope
of the entropy versus energy of the system
and that comes out to be this expression.
So, this is the temperature of a black hole,
1 by temperature of the black hole.
So, you can ask some interesting questions
now.
So suppose I have 2 black holes, say solar
mass black holes.
So the question is, of course, see you might
be wondering how can it be a solar mass black
hole because Chandrasekhar limit says that
it has to be greater than 1.4 times the solar
mass and even then you just do not get a white
dwarf, you get something more dense than a
white dwarf, which may not be even be a black
hole.
So of course, this is an order of magnitude
calculation.
When I say solar mass black hole, I mean,
maybe three times the solar mass, that is
good enough to make it a black hole.
So this is a rough back of the envelope order
of magnitude calculation.
So please do not take it too literally.
So the question is suppose you have 2 solar
mass black holes and they collide and merge
into each other, so the question is what is
the change in entropy?
So, you expect entropy to be lost.
In other words, you expect information to
be lost, and so you can actually see that
happening here because now you see the entropy
is proportional to the square of the mass.
So, the final entropy is basically the 2M
whole squared, which 2M is the mass of the
combined black hole and M squared into 2 is
the mass of the 2 separate black holes.
Then if the difference is something positive,
so these many bits have been destroyed.
So if entropy is positive, that means there
is an increase in disorder as it were and
so the change in the number of bits, so you
can express this in terms of the change in
the number of bits because 2 raised to number
of bits is e raised to entropy, which is the
number of microstates.
So the number of microstates is 2 raised to
number of bits, and the number of microstates
is also e raised to S. So by combining these
two, you can convince yourself that the change
in the number of bits is entropy divided by
log 2, which is of the order of 10 raised
to 77 bits.
So this is nice to know that if two black
holes combine with each other, this many bits
are destroyed.
Now, what is more interesting is that, see
by virtue of the fact that black holes have
temperature, so we can imagine that there
should be some kind of a black body radiation
near the event horizon.
The reason is because what is going to happen
is that see in elementary particle physics,
there is something called pair production.
So that means if energy of the vacuum is more
than twice Mc squared, where say Mc squared
is the rest energy of, say for example, the
electron, so what is going to happen is that
energy is going to disappear and reappear
in the form of electron and a positron.
So that is called a pair production.
So, what will happen is that you can have
these kinds of virtual processes will become
so you get a particle and an anti-particle
and you may imagine roughly that one of them
falls into the black hole and the other escapes
to infinity.
So that is the proposed mechanism for what
is called Hawking’s radiation.
So black holes lose mass through this process,
so this is called Hawking radiation.
So I would not go into too many details about
how this exactly happens, but roughly this
is what happens.
So now you know that the rate at which radiation
is being emitted by a black body of temperature
T is sigma times T to the power 4.
So this is power emitted per unit area.
So if you multiply that by the area of the
event horizon, then you get the rate at which
energy is being lost by the black hole.
Sorry, this is not capital T, it is small
t, so this is time.
So the rate at which energy is being lost
is d by dt of E where E is Mc squared.
So the rate at which energy is being lost
is, because it is losing energy there is a
negative sign there, and it is minus sigma
T to the power 4 times the area of the event
horizon and we know that the area of the event
horizon is nothing but it is proportional
to a square of the mass of the black hole.
So because of this and we also know that the
temperature of the black hole is inversely
related to the mass, when you put them all
together, you will be able to derive this
equation how the mass kind of evaporates with
time.
So that means, the black hole actually evaporates
slowly with time and the rate at which it
evaporates is determined by this constant,
and by solving this equation, you show that
the time dependence of the mass of the black
hole is given by this formula.
So if you start with a solar mass black hole,
you can ask yourself, how long does it take
before it all disappears?
So the answer is basically this quantity and
you can see that it is much larger than the
age of the universe, which is of the order
of 10 raised to 18 seconds.
So, black holes are going to live forever
once they are created, so long as the astrophysical
variety, which are mandated to be solar mass
or above, but then it is possible to have
a situation where you can create black holes,
which are much tinier through processes, which
are not related to astrophysical processes
and those very tiny black holes actually evaporate
very quickly.
So in fact, when the Large Hadron Collider
was going to be built, there was a lot of
excitement that these tiny black holes would
be observed, but for some reason, they were
not observed finally, but nevertheless, this
is the story of the thermodynamics of black
holes.
So what we have learned in this lecture is
that, firstly black holes are exciting because
they have finally been imaged by a telescope
on earth and it was a worldwide event, which
was televised and streamed online and so that
is the excitement about black holes.
Black holes have long known to have these
basic features, namely, that it has an area
which is basically the event horizon, it has
a mass, and Hawking and Bekenstein said that
entropy of the black hole is proportional
to the area of the event horizon and anything
which has mass, has same as energy.
So anything which has energy and entropy has
temperature, and anything that has temperature
is going to behave like a black body and radiate.
So the moment it starts radiating away its
energy, its mass is going to slowly decay,
and then finally, it is going to disappear.
So, what we have been able to calculate is,
actually been able to calculate how long that
takes and for astrophysical black holes, we
are quite safe that it is going to last forever.
So if once it is created, it is going to be
around forever.
So, that is the story of the thermodynamics
of black holes, I hope you enjoyed this.
So in the next class, we will discuss something
more down to earth, which is basically the
physics of the van der Waals fluid.
The van der Waals fluid would be simple prototype
of a non-ideal gas.
So I am going to revert back to the more classical
topics of statistical mechanics and discuss
van der Waals story.
So, I hope you enjoyed this lecture.
Hope to see you next time.
