This entire time is a bit of a daze in my mind, it was so exciting; I think I slept maybe two hours a night for weeks. The calculation came together over a period of three weeks, I want to say. I was at Princeton at the time. We just had a meeting on campus. I have a very distinct memory of driving home, and I was thinking to myself, wow, this could be it.
The crux of the matter was, there’s more than one quantum extremal surface in the problem. There’s one quantum extremal surface that gives you the wrong answer — the Hawking answer. To correctly calculate the entropy, you have to pick the right one, and the right one is always the one with the smallest quantum-corrected area. And so what was really exciting — I think the moment we realized this might really actually work out — is when we found that exactly at the time when the entropy curve needs to “turn over” [go from increasing to decreasing], there’s a jump. At that time, the quantum extremal surface with the smallest quantum-corrected area goes from being the surface that would give you Hawking’s answer to a new and unexpected one. And that one reproduces the Page curve.
Let me try to intuit a little bit what a classical, non-quantum extremal surface feels like. Let me begin with just a sphere. Imagine that you place a light bulb inside of it, and you follow the light rays as they move outward through the sphere. As the light rays get farther and farther away from the light bulb, the area of the spheres that they pass through will be getting larger and larger. We say that the cross-sectional area of the light rays is getting larger.
That’s an intuition that works really well in approximately flat space where we live. But when you consider very curved space-time like you find inside a black hole, what can happen is that even though you’re firing your light rays outwards from the light bulb, and you’re looking at spheres that are progressively farther away from the bulb, the cross-sectional area is actually shrinking. And this is because space-time is very violently curved. It’s something that we call focusing of light rays, and it’s a very fundamental concept in gravity and general relativity.
The extremal surface straddles this line between the very violent situation where the area is decreasing, and a normal situation where the area increases. The area of the surface is neither increasing nor decreasing, and so intuitively you can think of an extremal surface as kind of lying right at the cusp of where you’d expect strong curvature to start kicking in. A quantum extremal surface is the same idea, but instead of area, now you’re looking at quantum-corrected area. This is a sum of area and entropy, which is neither increasing nor decreasing.
Recall that when the Page curve turns over, we expect that our ignorance of what the black hole contains starts to decrease, as we have access to more and more of its radiation. So the radiation emitted by the hole must start to “learn” about the black hole interior.
It’s the quantum extremal surface that divides the space-time in two: Everything inside the surface, the radiation can already decode. Everything outside of it is what remains hidden in the black hole system, what’s not contained in the information of the radiation. As the black hole emits more radiation, the quantum extremal surface moves outwards and encompasses an ever-larger volume of the black hole interior. By the time that black hole evaporates altogether, the radiation has to be able to decode everything that way.
Now that we have an explicit calculation that gives us a unitary answer, that gives us so many tools to start asking questions that we could never ask before, like where does this formula come from, what does it mean about what type of theory quantum gravity is? Also, what is the mechanism in quantum gravity that restores unitarity? It has something to do with the quantum extremal surface formula.
First, we can’t really get around the fact that our universe contains both quantum mechanics and gravity. It contains black holes. So our understanding of the universe is going to be incomplete until we have a description of what happens inside a black hole. The information problem is such a difficult problem to solve that any progress — whether it’s in a toy model or not — is making progress towards understanding phenomena that happen in our universe.
Now at a more technical level, quantum extremal surfaces can be computed in different kinds of space-times, including flat space like in our universe. And in fact there already have been papers written on the behavior of quantum extremal surfaces within different kinds of space-times and what types of entropy curves they would give rise to.
We have a very firm interpretation of the quantum extremal surface in AdS space. We can extrapolate and say that in flat space there exists some interpretation of the quantum extremal surface which is analogous, and I think that’s probably true. It has many nice properties; it looks like it’s the right thing. We get really interesting behavior and we expect to get unitarity as well, and so, yes, we do expect that this phenomenon does translate, although the interpretation is going to be harder.
A full resolution of the information paradox would have to tell us exactly how the black hole information comes out. If I’m an observer that’s sitting outside of a black hole and I have extremely sophisticated technology and all the time in the world — a quantum computer taking incredibly sophisticated measurements, all the radiation of that black hole — what does it take for me to actually decode the radiation to reconstruct, for instance, the star that collapsed and formed the black hole? What process do I need to put my quantum computer through? We need to answer that question.
This algorithm that decodes the Hawking radiation is coming from the process in which quantum gravity encodes the radiation as it evaporates at the black hole horizon. The emergence of the black hole interior from quantum gravity and the dynamics of the black hole interior, the experience of an object that falls into the black hole — all of that is encoded in this reverse algorithm that quantum gravity has to spit out. All of those are tied up in the question of “how does the information get encoded in the Hawking radiation?”
It’s one thing to ask how can you decode the Hawking radiation; you also might ask, how complex is the task of decoding the Hawking radiation. And, as it turns out, extremely complex. So maybe the difference between Hawking’s calculation and the quantum extremal surface calculation that gives unitarity is that Hawking’s calculation is just dropping the high-complexity operations.
It’s important to understand the complexity geometrically. And in 2019 there was a paper by some of my colleagues that proposed that whenever you have more than one quantum extremal surface, the one that would be wrong for the entropy can be used to calculate the complexity of decoding the black hole radiation. The two quantum extremal surfaces can be thought of as sort of constrictions in the space-time geometry, and those of us who have read Le Petit Prince see an elephant inside a python, and so it has become known as a python’s lunch.
We proposed that multiple quantum extremal surfaces are the exclusive source of high complexity. And these two papers that you’re referring to are essentially an argument for this “strong python’s lunch” proposal. That is very insightful for us because it identifies the part of the geometry that Hawking’s calculation knows about and part of the geometry that Hawking’s calculation doesn’t know about. It’s working towards putting his and our calculations in the same language so that we know why one is right, and the other is wrong.
I like to think of this as a puzzle, where we have all the edge pieces and we’re missing the center. We have many different insights about quantum gravity. There are many ways in which people are trying to understand it. Some by constraining it: What are things that it can’t do? Some by trying to construct aspects of it: things that it must do. My personal preferred approach is more to do with the information paradox, because it’s so pivotal; it’s such an acute problem. It’s clearly telling us: Here’s where you messed up. And to me that says, here’s a place where we can begin to fix our pillars, one of which must be wrong, of our understanding of quantum gravity.
How stupid does Apple think we all are? I mean, seriously. The announced changes this past week to the App Store rules are maybe the most absurd attempt to placate bad PR that I’ve ever seen. Instead, they helped clarify something pretty fundamental: Apple no longer wishes to compete.
I feel as if I’ve been dancing around this realization for a few months now. But reading over Steve Jobs’ statements in the original press releases around in-app subscriptions and Marco Arment’s recent post on the matter, mixed with these changes, really drove it home. Again, Apple clearly — I mean this: clearly — does not want to have to compete with the best offering and experience any more. They want to leverage their user base to bend the will of developers.
I’m not an antitrust lawyer, nor will I play one on the internet. And there’s obviously a lot going on here at the moment.¹ But simply as a diehard user of Apple products, I find this extremely disappointing.
I want an Apple that wins not by obfuscation, but by offering the best experience. Make everyone want to use the in-app purchasing system across the board because it’s the best. Because users demand it. Because it’s so seamless for developers. Not because if you don’t, it’s no app for you.
Look, it’s Apple’s App Store. To date, they’ve earned the position of power they’re in. They do and should benefit from this. But I also want them to aspire to be better than using that strength to prey upon the weakness of others. Again, I want them to win on the field. To compete. These days, they seem more interested in some Sun Tzu shit.
But it’s more like the CliffsNotes version of Sun Tzu. It’s hilariously obvious what they’re doing. They’re now re-writing rules on the fly based on specific exceptions they wish to grant. And they’re doing so because they’ve gotten bad press around those deviations. It’s ridiculous.
Apple, revisit and rewrite the App Store rules. Don’t Frankenstein the rules to try to please certain developers (and the public) with some one-off changes. This is a Band-Aid on a dam. Blow it up, and start again. And more generally, just go back to competing for business by offering the best products and services. The ones both users and developers love.
I know this is easier said than done. It may not be as lucrative at first. And that’s hard as the most valuable company in the world trying to appease the public investors.² But this is the definition of short term “pain” for longer term gain. Think about this differently.
¹ This is where I disclose that I’m a partner at a fund whose LP has been in the news around such topics recently. My thoughts and opinions on these topics are my own, obviously!
² Of which I’m one, in a small way.
Compete! was originally published in 500ish on Medium, where people are continuing the conversation by highlighting and responding to this story.
Google Drive may have a way for hackers to trick you into installing rogue code. System administrator A. Nikoci has told The Hacker News about a flaw in Drive’s “manage versions” feature that could let attackers swap a legitimate file with malware. The cloud storage service reportedly doesn’t check to see if a file is of the same type, or even enforce the same extension. An innocuous cat photo may be a program in disguise.
The online preview doesn’t hint at any changes or raise alarms, so you might not know there’s a problematic file until you’ve already installed it. Chrome seems to “implicitly trust” the Drive downloads even when other antivirus programs detect something amiss.
The approach could be used for spear phishing attacks that trick users into compromising their systems. You might get a notification of a document update and grab the file without realizing the threat.
Nikoci said he notified Google about the issue, but that it was still unpatched as of August 22nd. We’ve asked Google for comment.
This would mainly be useful for attacking companies that rely on Google Drive for sharing documents, but that’s increasingly common. The description also suggests that this would require a significant change to Drive’s version control. For now, the best solutions may be to use antivirus software and be wary of Google Drive file update alerts, especially if you weren’t expecting them.