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Black holes seem like they should have no entropy;
but, in fact, they hold most of the entropy in the universe.
Let's figure this out.
At first it seemed that black holes
were so simple they should have no entropy.
Well it turns out they contain
most of the entropy in the universe.
Let's see why;
because, this fact may force us to conclude
that the universe is a hologram.
Black holes are a problem.
They are the inevitable result of
extreme gravitational collapse.
At least, they are inevitable
according to the equations of
Einstein's general theory of relativity
That theory is one of the most thoroughly
tested in all of physics, which means
we should probably believe in black holes.
Also we've seen them,
in their gravitational effects on their
surrounding space, and in the gravitational
waves caused when they merge.
And yet
if black holes exist,
which apparently they do, they contradict other
theories in physics that are as sacred as
general relativity. They
cause all sorts of problems with quantum theory,
which we've talked about before
and will review in a sec, but they
also present an apparent conflict
with the notion of entropy and the second
law of thermodynamics.
It was while pondering that conflict
that Jacob Bekenstein realized an
incredible connection between black holes
and thermodynamics.
His insight was an entire new way of thinking
about the Universe in terms of
information theory
and ultimately led to the Holographic
Principle, which I promise
we're getting to and are almost there.
But first, you're going to need to know more
about why black holes contain
most of the universe's entropy.
Okay, I'm getting *way* ahead
of myself. Let's actually rewind back
to those episodes where we laid out the
black hole information paradox, because they're
going to be critical to a proper understanding.
We're also rewinding to the late sixties or seventies, when physicists
realized something odd about
black holes. What they realized is that
it doesn't matter what material goes into one;
from the point of view of
the outside universe black holes
can only have three properties: mass, spin and electric charge.
This is the so called "No hair theorem"
and it suggests that most of the information about
anything that falls into a black hole is lost to the outside universe
but a fundamental tenet of quantum mechanics
is that quantum information can never be destroyed
so if black holes evaporate
as Hawking discovered and we also covered,
this evaporation should destroy a black hole's internal quantum information
giving us the black hole information paradox.
Eventually, a possible resolution to this
paradox was found
by Gerard 'T Hooft. He described a
mechanism by which the information
contained by infalling particles
could be preserved on the event horizon
of the black hole. From there
it could be imprinted
on the outgoing Hawking Radiation
allowing the information to escape back into the universe.
Okay, problem solved.
But in our previous episode
we skipped the key insight
that started all of this.
It all began with Jacob Bekenstein
thinking about black hole entropy.
Okay, first;
entropy. Yeah, we talked about
that a lot recently, also you know
It's almost like all those episodes
are starting to come together.
Almost like we planned this.
Go and watch that background stuff
if you're behind, but of course for now
I'll give you a quick TL;DW
on entropy. So we can think of entropy
in two ways. One:
it's a measure of how evenly energy is spread out.
High entropy means
thermal equilibrium. So
energy is very evenly distributed, and can't be
extracted in a useful way.
And two: entropy measures
the amount of unknown information
that you would need to perfectly describe the
systems internal state, like all the particle
positions, velocities etcetera;
the higher the entropy, the more randomly distributed
it's particles; and the more possible
configurations lead to the same macroscopic
state. The higher the entropy
the less you can guess about
the properties of individual particles
based on the overall properties like temperature,
volume, pressure etc.
Okay, so, the second law of thermodynamics
states that entropy
of an isolated system must always
increase, which means energy tends
to spread out evenly and particles tend
to randomize, reducing our
information about their microscopic states.
How does this relate to black holes?
Let's make a black hole and
see what happens to entropy.
We start as usual by collapsing
the core of a dead star.
Now that's a high entropy beast.
Super hot and full of randomly moving particles.
We have almost no information about the
individual particles, but that
information still exists in the universe,
like, I guess the particles know where they are.
At the instant the star collapses
far enough to form an event horizon,
it becomes a black hole. We go
from knowing next to nothing about the object,
to knowing - everything.
We can easily measure it's mass,
spin and electric charge, and according
to the No Hair Theorem that's
all there is to know. The
region of space in which the black hole formed
appears to have gone from high
entropy, to zero entropy,
in an instant; shattering the second
law in the process, which
to put it mildly is a problem.
But if you paid attention to the whole information
paradox bit, you might be able to think of
a solution; if quantum information
is stored on the surface of the black hole,
can't we store entropy there also?
And, then
why not radiate the entropy back into the
universe as Hawking Radiation?
Actually, yeah; the resolution
to the information paradox
also saves the second law of thermodynamics.
Pfft. That was easy. I thought
physics was supposed to be hard.
Okay hang on. Let's think
about this a little bit more.
It was this seeming violation of the second
law that got Jacob Bekenstein
thinking about the connection between
black holes and information in the first place.
The breakthrough insight
was this simple observation:
the surface area
of a black hole event horizon can
never decrease.
At least not according to general relativity.
So you know how nothing can escape black holes,
ignoring Hawking Radiation for the moment.
That should mean that black holes
can only grow, that can
never shrink in mass or
radius. Well, that's not quite
true. If you merge two
black holes, some of their mass gets
converted to the energy radiated away
in gravitational waves.
There's also the Penrose Process in
which you can extract rotational energy
of a spinning black hole.
And by you, I mean not you, you,
I mean super advanced, far future
civilizations. Gravitational
radiation and the Penrose
Process reduce black hole mass and
radius. Or the sum of masses and
radii are from merging black holes. But
there's one property of black holes that no process other
than Hawking Radiation can decrease: that's
the surface area
of the event horizon. Do
anything to black holes, and their total surface area
can only grow or stay
constant. Bekenstein saw a close
correspondence between the always increasing
event horizon surface area and
the always increasing nature of
entropy. He also realized
that the equation relating the
change in black hole surface area to the
change in it's mass closely resembles
the original definition of
thermodynamic entropy.
Just replace change in entropy and internal
thermal energy with change in black
hole surface area and black hole
mass respectively. You could
also add the work done when you
extract energy from the black hole and
it looks the same as the equation
for the work extracted from a thermodynamic
system. Bekenstein had just
discovered black hole thermodynamics.
But, that didn't give him the
exact definition of black hole entropy.
For that, he turned to Ludwig
Boltzmann's informational
definition for entropy. So
entropy can be defined as the
information hidden in a system's
microscopic configuration times
the Boltzmann Constant.
Bekenstein estimated the amount of
information that would be lost
into a black hole as it grew.
Essentially, he built a black hole out of
idealized elementary particles,
that each contained a single bit of
information. And
guess what? The information content
of a black hole is proportional
not to its mass or radius or volume,
it's proportional to its
surface area. In fact,
the information content is very close
to that surface area
defined by the number of Planck
areas. It's as though
each of these minimum possible
quanta of area each contain
a single bit of information.
Now just multiply that information
content by the Boltzmann constant,
and you have the entropy of a black hole.
Which is gonna be directly proportional
to the surface area of the event horizon.
Bekenstein's connection between
surface area and entropy
could have been a coincidence,
at least until Stephen Hawking
came along. In 1974,
a year after Bekenstein's first
paper on black hole thermodynamics,
Hawking published his first Hawking Radiation
paper. He showed that
black holes radiate random particles exactly
as though they have a peak glow
for a particular temperature
that depends on their mass.
So, if black holes have a temperature,
then they also have entropy.
Good, old fashioned
thermodynamic entropy tells us that
change in entropy, is change in
internal thermal energy divided by temperature.
So Hawking just plugged
his Hawking temperature into that equation
along with black hole mass for
internal energy and figured out
the total entropy contained in a black hole.
He got an expression
almost identical to Bekenstein's,
with just a slightly different
constant of proportionality. So
you get the same result for black hole entropy when you
figure it out from the amount of information
that gets trapped building a black hole,
or, the amount
of heat that leaks, as it evaporates.
And it's proportional to the surface
area. How bizarrely
consistent. I'd
say that means it's right.
The second law of thermodynamics is saved,
because black holes do have
entropy. In fact, they have enormous
entropies. The maximum
possible. So much,
that black holes are now believed to
contain most of the entropy in the
universe. But the real importance
of this work wasn't the solution to
some obscure conundrum.
It changed our thinking about the
informational content of the universe.
Bekenstein's formula was derived
for black holes, but
it also gives the maximum amount of information
that can be fit into any
volume of space. In this
respect it's called the Bekenstein Bound,
and it's proportional to the surface area
of that space.
This is unexpected.
Surely the maximum amount
of information you could fit in some patch of
space depends on the volume of
that space? As in, one
bit per tiny volume element
inside that space. But
in fact the rule is, one bit
per tiny area element on the surface of
that space. That
also means that the information needed to
describe any volume of space
, no matter its contents,
is proportional to the area bounding that
space. I've hinted once or
twice that this simple idea
led to the Holographic Principle:
the idea that the entire 3D volume of the
universe, is just a projection of
information encoded on a 2D
surface surrounding the universe.
You just need to add a little bit
of string theory. It's a
hell of a conceptual leap, given it
started with Jacob Bekenstein noticing a
peculiar similarity between
some formulae. It might
also be true, and
obviously, we'll be back before too long
to talk about string theory
and the holographic nature
of spacetime.
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Hey guys, I am going to be travelling for a little bit, so we're going to have a
little bit of a backlog in comment
responses. I wanted to let you know
that we will be getting to them all,
and we'll be releasing episodes
on a normal schedule, so,
see you next week.