MATT O'DOWD: Thanks to the Great Courses
Plus for supporting PBS Digital Studios.
I've got a physics joke for you.
A proton, an electron, and an antineutrino
walk into a black hole.
That's it.
[MUSIC PLAYING]
We've established by now that black holes are weird,
the result of absolute gravitational collapse
of a massive body, a point of hypothetical infinite density
surrounded by an event horizon.
At that horizon, time is frozen and the fabric of space
itself cascades inwards at the speed of light.
Nothing can travel faster than light,
and so nothing can escape from below the event horizon--
not matter, not light, not even information.
These ideas are pretty mind blowing, but as crazy
as black holes are, they're also kind of simple.
Don't get me wrong, the math is complicated,
but the objects themselves are simple.
In fact, every black hole in the universe,
no matter how it formed or what happened to it afterwards,
can be perfectly described with only three properties.
Those properties are mass, electric charge, and angular
momentum, or at least this is the proposition
behind the famous no-hair conjecture or no-hair theorem.
Today, we're get to see why black holes are bold,
and in an upcoming episode we'll combined the no-hair conjecture
with everything we learned recently
about the conservation of quantum information.
The result will be the black-hole-information paradox,
one of the greatest unanswered questions in modern physics,
and also the gateway to black-hole thermodynamics
and the holographic principle.
Jakob Bekenstein was a graduate student at Princeton
when he proposed that no other properties besides mass,
electric charge, and angular momentum
should emerge from beneath the event horizon.
Or as Bekenstein put it, black holes have no hairs.
This is why you don't let graduate students name things.
Bekenstein's advisor, the legendary John Archibald
Wheeler, took up the phrase, and so now we're stuck with it.
The basic reason for the no-hair conjecture
is straightforward enough.
The interior of a black hole is cut off
from the external universe.
It's causally disconnected.
Nothing beneath the event horizon
can influence the exterior universe
because no signal can escape the event horizon
to carry that influence.
The black hole could have formed from a collapsed star
or entirely out of antimatter or photons or monkeys,
but the only thing we can know about the material that
went into the black hole are its mass-energy content,
electric charge, and angular momentum.
Given the one-way nature of the event horizon,
the loss of this information makes some intuitive sense.
Perhaps more mysterious are the three exceptions,
the three remaining hairs around our otherwise bald black hole.
After all, the forces involved, namely the gravitational
and electromagnetic forces, are only
communicated at the speed of light.
So how does mass, electric charge, and angular momentum
communicate their influence across the uncrossable horizon?
Let's think about this in terms of fields,
and we'll start with the gravitational field.
Gravity may be caused by mass, but a gravitational field
is a very real thing all on its own.
In Einstein's general theory of relativity,
we think of the gravitational field as curvature
in the fabric of spacetime.
In the good old rubber-sheet analogy,
a heavy ball stretches the sheet downwards,
affecting the motion of objects moving on that sheet.
We can think of the gravitational field
at any point as being caused by the gravitational field
at surrounding points.
Each point on the rubber sheet doesn't actually
see the source of the gravitational field.
Instead, it stretches and curves in response
to the behavior of the surrounding patches.
The Earth orbits the Sun, but more directly it
orbits the Sun's gravitational field.
You could change anything about the Sun other than its mass
and the Earth would continue in the same orbit.
If the Sun were to suddenly vanish,
Earth would continue to orbit the existing
gravitational field for 8 minutes.
The spacetime at the location of Earth's orbit
would remain curved until the elastic fabric straightened
itself out at the speed of light.
In the case of the event horizon,
the outside universe can't see the mass inside the black hole,
but that mass is remembered in the gravitational field,
the curvature of spacetime above the event horizon.
In fact, the space at the event horizon
is already falling inwards.
It doesn't need to be told to continue doing that,
and it drags the layer directly above it,
continuing the cascade even after the central singularity
is lost to all causal influence.
There's an extremely important law
in physics that describes how the universe remembers
the contents of a region of space.
It's Gauss's law, which applies to both gravitational
and electric fields.
Gauss's law of gravity states that the total gravitational
field added up over an enclosed surface
is proportional to the amount of mass and energy
contained by that surface.
The distribution of matter inside that surface
is irrelevant.
The mass could all be located at a single point
within the surface or could be evenly distributed
across that surface.
The resulting sum of the gravitational field
would be the same.
In the case of a black hole, at least a nonrotating one,
the event horizon is a closed spherical surface
with a singularity at its center.
That surface remembers the internal mass
as though it was spread evenly across the surface.
Now, the original Gauss's law actually
applies to the electric field.
In fact, it's one of Maxwell's equations,
which together give the classical description
of electromagnetism.
Gauss's law for the electric field
says that the total electric flux passing through a closed
surface depends on the amount of charge interior
to that surface.
The universe remembers the total charge
that exists in the region of space enclosed by the surface.
This means that the electric field above the event
horizon of a black hole remembers
all of the electric charge that fell through that surface.
Black holes act as though their charge is spread
across the event horizon.
If you've studied some introductory physics,
you might remember that the gravitational and electric
fields have something in common.
Both obey an inverse-square law.
Newton's law of universal gravitation
and Coulomb's law for electrostatics
say that the strength of the force produced by these fields
drops off with the square of the distance
from the source of that field.
This comes from the fact that if you
draw increasingly large spheres around a pointlike field
source, the intensity of those forces
gets spread out over an increasingly large area.
You can think of lines of force getting further and further
apart for larger spheres.
The area of that sphere is proportional to the square
of its radius, so the density of the lines of force decrease
with distance squared.
In fact, Gauss's law is a more general form
of these inverse-square laws.
It applies to any shaped surface surrounding any shaped mass
or charge.
But both Gauss's law and the inverse-square law
work because of a key similarity between gravity
and the electric field.
They are infinite in range and they
arise from conserved quantities, namely mass and charge.
The lines of force never fade, never vanish, and never
merge into each other.
They only spread further apart.
That means all lines of force created within a closed region
must eventually cross that region's surface.
OK, so the gravitational and electromagnetic forces
have infinite range, and so Gauss's law
demands that the mass and charge content of any region of space
are remembered in the gravitational and electric
fields on the surface surrounding that region.
Because mass and charge are fundamentally
conserved quantities, the only way
to change the contents of a region of space
is for more mass or charge to cross its surface.
In the case of the black hole, nothing can get out,
but new infalling material will adjust the black hole's
external gravitational and electric fields on its way in.
By the way, it's worth mentioning
that real black holes out there in the universe
are never going to have a net electric charge.
A black hole with nonzero charge will quickly attract particles
with the opposite charge until positive and negative charges
within the black hole balance out
and the black hole becomes neutral.
OK, so we've covered gravity and the electrostatic part
of electromagnetism.
But what about the magnetic part of electromagnetism,
and what about angular momentum?
Well, you get magnetism for free.
A changing electric field produces a magnetic field.
And so if the black hole is spinning or racing past you,
you'll see that magnetic field.
In a similar way, you can see a black hole's rotation
in its gravitational field.
In Einstein's general relativity,
a spinning mass drags the fabric of space time
around with it in a phenomenon known as frame dragging.
The incredibly subtle frame dragging
due to Earth's rotation has been detected,
but it required the incredible precision and ingenuity
of Gravity Probe B.
The frame dragging around a rotating black hole
is rather stronger.
It dramatically changes the shape of the event horizon
and the orbit of anything nearby.
If material with angular momentum
falls into a black hole, whether it's
a spinning star or a whirlpool of gas,
it will either add or subtract from this flow of space
above the event horizon.
Its angular momentum is remembered
in the frame dragging as though the entire black hole was
spinning.
I hope I've given you a sense of why mass, charge, and angular
momentum are remembered by the space outside a black hole.
It's because these properties are conserved and are
communicated by gravity and electromagnetism, which
have infinite range.
As a result, they are remembered by the fields surrounding
the black hole.
However, there are some important conserved properties
that are entirely local.
If matter with these properties falls into a black hole,
information about those properties
is lost to the outside universe.
An example would be the number of particles
of different types, like the balance
between quarks and antiquarks represented by baryon number.
Baryon number is a conserved quantity,
but there's no way to know the baryon number of a black hole.
Now this doesn't sound like a big problem.
So what if the universe forgets what type of particles
a black hole is made of?
Actually, it should care a great deal.
We discussed recently why quantum mechanics
demands the conservation of quantum information.
It's fundamental to quantum mechanics
that the universe keeps track of its quantum states, which
also means the types of particles it contains.
Yet Stephen Hawking showed that black holes
may break this rule, revealing a conundrum that we now
call the information paradox.
The solution to the information paradox is highly speculative,
but it may reveal that black holes are
more hairy than we thought.
We will delve into all of that in an upcoming
episode of "Space Time".
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In our last episode we looked at what
might be the most fundamental rule
in quantum mechanics, the conservation of quantum
information.
You guys had a lot of questions.
Let's see what you had to say.
qbslug asks a tricky one.
How is probability conserved when an electron and a positron
annihilate each other?
In our discussion during the episode,
we talked about the unitary evolution
of a one-particle wavefunction.
We said that there's always a 100% chance, for example,
that the particle position always has some value.
So what if the particle is destroyed?
Well, particle creation and annihilation
is described by quantum-field theory,
and unitary evolution and the conservation
of probability and information still works there.
QFT describes the evolution of quantum fields
in which particles are excited states.
Now it's the evolution of the fields, not the particles,
that conserves probability.
In the specific case of electron-positron annihilation,
information continues to exist after the event in the products
of that annihilation.
It's encoded in the energy, phase, polarization,
et cetera of the two gamma-ray photons that are created.
Peter K. asks how the universe can be deterministic
given the fundamental probabilistic random nature
of the quantum world.
Well, the key here is that that randomness arises
at the level of observation.
A wave function prescribes the probability
of observing a given value for a property
when the observation is made, but the wave
function itself evolves in a unitary manner,
preserving all information.
Exactly what happens at the moment of observation
is still debated.
In the Copenhagen interpretation,
the wave function collapses and unitarity is not preserved.
But that's increasingly disfavored.
More likely is that the observer and the observation
are a small part of a global wave function that continues
to evolve in a unitary manner.
John asks whether the only way to truly erase a hard drive
is to throw it into a black hole.
Well first, John, I assume you're asking for a friend.
Second, as we'll see next week, even black holes
probably preserve information.
Whatever terrible secrets your friend
has on that drive most likely get
radiated back out over countless of billions
of years as Hawking radiation where
it can be collected and giggled at by a future super-advanced
civilization.