Black holes are very real, but are also a theoretical nightmare.
It turns out that in order to make sense of their paradoxical nature,
we need to consider that each real black hole has the potential to be connected
to multiple imaginary black holes via wormholes. And you thought the universe couldn’t get any weirder.
Some of the most profound leaps in our understanding of the universe
have come when we noticed inconsequential seeming inconsistencies in our theories. The fact that
Maxwell’s electromagnetism didn’t square with Galileo’s relativity led Einstein to his special
theory of relativity, from which followed our modern description of gravity, space and time in
general relativity. Now, there’s an uncomfortable conflict between general relativity and quantum
mechanics when we try to describe the tiniest scales and the highest energies. This drives our
quest for theories of everything. A quest that seems to have stalled after a century of work.
But there is another less well known glitch between GR and quantum theory
that might provide a way forward. I’m talking about the black hole information paradox.
Efforts to resolve it have led to stunning realizations about the nature of entropy,
quantum information, and even to the holographic principle. Now,
the latest attempt to solve the black hole information paradox are pointing to a bizarre
picture in which each black hole behaves like many parallel black holes connected by wormholes.
Before we jump into that particular wormhole, let’s remind ourselves what the black home
information paradox is all about. Feel free to have a look at our original video on the paradox,
but it’s not essential for understanding this new angle. A little review from me, should be enough.
Stephen Hawking discovered that black holes aren’t quite as, well, black and inescapable
as we thought. They radiate, and this Hawking radiation causes them to slowly evaporate.
But that means that all the information that went into making the black hole
is erased from the universe. This conflicts with the law of
conservation of quantum information, which is a non-negotiable constraint of quantum mechanics.
Perhaps the most powerful way to think about this paradox
is in terms of entropy. Think of entropy as the amount of information
hidden by a system - information not observable in the system’s gross properties. Black holes
have huge entropy because every black hole looks the same no matter how it was formed.
The quantum version of entropy is called von Neumann entropy. This is the entropy of
entanglement. If two particles are entangled then they share mysterious correlations - you can learn
about and even influence the properties of one particle by measuring its entangled partner. Some
of the quantum inimformation of each of the pair is stored in its partner. The von Neumann entropy
of an entangled particle or system of particles is a measure of how much quantum information
is not stored locally in the system itself, but rather in whatever it’s entangled with.
If you can entangle a particle you can entangle a black hole. One way to think about Hawking
radiation is that the black hole is swallowing and emitting virtual particles. We can think about the vacuum
of space as being filled with a boiling flux of particle/antiparticle pairs that constantly appear
and annihilate each other. If these particles get separated by a black hole event horizon
before they can annihilate, one particle escapes and becomes real, while the other is swallowed.
But those virtual particle pairs are entangled, which means that Hawking radiation and the
interior of the black hole are entangled. The black hole interior contains quantum information
about the radiation. But when the black hole eventually evaporates, its internal quantum
information seems like it should disappear, violating the law of conservation of quantum information and giving
all of its past Hawking radiation a permanent and unrecoverable non-zero von Neumann entropy.
The Von Neuman entropy of the radiation should increase over time as the internal information
of the black hole evaporates and the black hole loses its internal storage
space. Meanwhile the black hole’s entropy
decreases. Actually, the black hole information paradox arises as soon as these lines cross each other.
You might wonder if the Hawking radiation itself can serve as an escape route for that information.
But according to Hawking’s original formula, and according to the no-hair theorem,
Hawking radiation should be completely random. It should contain no information about anything
besides the black holes gross properties. Efforts to resolve the black hole information
paradox have largely focused on ways to encode Hawking radiation with that quantum information,
so that each new particle is entangled with all previously emitted particles.
If information leaks out this way, then the von Neumann entropy of Hawking radiation
should rise over time as more and more radiation is produced,
but then at some point the entropy starts to drop again because the information from past
radiation is increasingly leaked out in newly emitted radiation. This is the Page curve,
figured out by physicist Don Page. It’s actually the exact evolution of von Neumann entropy
that must occur if Hawking radiation is our information escape route from the black hole. Any theory trying
to solve the black hole information paradox has to exactly reproduce the Page curve.
I mentioned that the information paradox inspired the holographic principle. The most
advanced form of this is AdS/CFT correspondence, a branch of string theory that reveals that a
particular type of universe with three spatial dimensions is encoded on its own 2-D surface.
The power of AdS/CFT correspondence is that some calculations that are horrendous in the 3-D universe become
possible on its boundary, and vice versa. Physicists have managed to explore black
hole entropy using these holographic methods, and even managed to produce the Page curve. The
problem with this approach is that we have no idea whether string theory is right, and forget
about the fact that some of the assumptions of AdS/CFT definitely do NOT apply to our universe.
And that brings us to the new, intriguing discovery. In 2020, two teams managed to predict
the Page curve using only general relativity and accepted quantum mechanics - no string
theories attached. To be fair, these US teams, dubbed the “east-coast” and “west-coast” teams,
used some of the insights of the AdS/CFT solution, but they don’t depend on its significant assumptions of AdS/CF.
The tidal wave of math in these papers pulls ideas from string theory, holography,
quantum field theory, and quantum computing to name a few fields.
We’re going to focus on the one main concept behind this innovation - the gravitational
path integral. This is the general relativistic analog of the Feynman path integral. To remind
you, the Richard Feynman’s path integral calculates the probability of some quantum
particle traveling between two points by adding up all ways the particle could make that journey.
And that includes paths that are impossible according to the laws of physics. In a way,
the “classical” and sensible path that we observe is the sum of infinite paths, many nonsensical.
With gravitational path integral you analyze some patch of spacetime changing from one geometry to
another. And you do this by adding up all possible geometries that spacetime could go through
in that transition. For example, you could use this to study the evaporation of a black hole,
as its geometry changes with each outgoing particle of Hawking radiation.
When Stephen Hawking did his original calculation, he didn’t think it was necessary to consider
a lot of different geometries - after all, black holes are pretty simple objects - it’s
not like there are many geometries they can go through. But it turns out that, in this respect,
Hawking was wrong. The new papers from 2020 argue that to get a proper calculation of the von Neuman
entropy of Hawking radiation, you need to consider all transitional states of the evaporating black
hole - even those that seem impossible - just like with the Feynman path integral.
Specifically, they argue that different topologies should contribute to the path integral.
New geometries should be included which include spacetime folding into itself in a way
required to build wormholes. Even though these wormholes don’t really exist in a classical sense,
the fact that they can possibly exist in between measurements is enough to alter the outcome.
Let’s go into a little bit more detail. What do these crazy topological spacetimes look like?
It’s very difficult to properly picture these geometries since they’re so weird,
the image wouldn’t fit in our primate brains. They’re actually complex geometries,
distances between some points are complex-valued rather than real valued. But for the purposes
of this video we can take some artistic license to get an idea of what’s going on.
When computing the von Neumann entropy, there is a neat mathematical trick you can do.
For some reason, it turns out that it's easier to compute the entropy of many evaporating black
holes rather than just one. The entropy of many black holes is an example of a quantity
called the Rényi entropy. You can find the von Neumann entropy around a single black hole
by computing the Rényi entropy for an arbitrary number of identical black holes
n, and taking the limit of n going to 1. The black hole copies are called replicas.
The Renyi entropy is found using the gravitational path integral. For a spacetime geometry where none
of the black holes interact with each other, the entropy doesn’t change from Hawkings’ original result.
But there’s another geometry that turns out to totally change the result:
one where all of the black holes are connected by a network of wormholes.
These replica wormholes connect regions of the interior of each replica, called “islands”.
When you take this spacetime topology into account, and then you set n equal to 1 single black hole,
you end up with a new equation for the entropy of the Hawking radiation.
Since the wormholes seem to disappear when there’s just a single black hole,
you’d expect their effect to disappear, but amazingly, the math stays different. The mere
possibility of a wormhole connection between our black hole and a bunch of its clones
is enough to leave an imprint on the entropy. This results in a new equation for the entropy of the
radiation, which has been called the island rule. This was actually not the first time
physicists had seen this equation. The island rule had already been derived in string theory.
The researchers were surprised to find that you don’t need string theory to
derive the island rule: all you need is the gravitational path integral.
So, what’s the result? Did these virtual, wormholey topologies solve the information
paradox? Well, it sort of looks like they might of. Using the gravitational path integrals and
the island rule, the physicists found that the von Neumann entropy of Hawking radiation
exactly follows the Page curve. Somehow these imaginary transitional geometries
allow the radiation to leak quantum information from the black hole interior.
OK. so we have this quite elaborate and technical calculation that tells us that
information can escape from the black hole. But what’s the real physical picture here? Well,
no one really has any idea. More work is needed to translate
this crazy math into a physical picture - if that’s even possible. But the prediction
of the correct Page curve tells us there’s something right with this picture.
So, has the paradox really been solved? Well the community is divided. Some of the mathematical
tricks that were used have invited skepticism. Regardless, these breakthroughs have clearly
resulted in a new chapter in the story of black holes and information. Since replica wormholes
hit the scene, hundreds more papers have been published on the subject. Scientists are
performing a sort of theoretical path integral - exploring all possible explanations of the result.
In time one of these seems sure to land on the true path - a path that leads to deeper
understanding of our universe, past an infinity of strange topologies and imaginary spacetime.