Do Wormholes Solve the Black Hole Information Paradox? (Script) (Patreon)

Black holes are very real, but are also a theoretical nightmare. It turns out that in order to make sense of their paradoxical nature, every black hole has to be thought of as a multitude of imaginary black holes, all connected by 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’s 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 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 the paradox is in terms of entropy. Think of entropy as the amount of information hidden by a system - information not observable by 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 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 and giving all of its past Hawking radiation a permanent and unrecoverable non-zero von Neumann entropy.

Von Neuman entropy of the radiation should increase over time as the internal information of the black hole evaporates and loses its internal storage space, which is proportional to its shrinking surface area. Meanwhile the black hole’s entropy decreases. Actually, the black hole information paradox arises as soon as these lines cross.

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 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. 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 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 don’t depend on its significant assumptions.

The tidal wave of math in these papers pulls ideas from string theory, holography, quantum field theory, and quantum computing to name a few. 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 analyse some patch of spacetime changing from one geometry to another. It does that 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. In other words, 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 to be 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’ 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 set n equal to 1, you end up with a new equation for the entropy of the Hawking radiation.

Since the wormholes seem to disappear when there’s just one single black hole, you’d expect their effect to disappear, but amazingly, the math says 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 a gravitational path integral!

So, what’s the result? Did these virtual, wormholey topologies solve the information paradox? Um, sort of. At least it looks like they might. Using the gravitational path integral 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 escapes from the black hole. But what’s the real physical picture here? Uh, no one really has any idea - yet. 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? 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.