Transcript:

What’s your opinion on wormholes?

Brian Keating: What’s your opinion on wormholes?

Sergiu Klainerman: From my understanding, they extremely unstable. If it’s unstable mathematically, as with very few to small perturbations, it disappears. Then it’s like a ghost. The part inside is pure mathematics now, right? Because we cannot go there.

The AI is going to become more intelligent and it’s going to reproduce itself at a huge speed and then kill us off. It’s just absurd. It’s ridiculous.

Brian Keating: Sergiu Klainerman is a mathematician at Princeton University. More than 30 years ago, he and Demetrios Christodoulou proved that empty spacetime is stable. That sounds simple, like almost a technicality, but it isn’t. Every exact solution of Einstein’s equations has to survive that test before it even can count as physics. Slowly rotating black holes pass. Wormholes, they just don’t. And at the end, he takes the same scalpel to the AI apocalypse that we’re all confronting these days. And it all started with one question after a talk he gave.

Sergiu Klainerman: I was in Berkeley. I came as a postdoc and Yau invited me to give a colloquium at Stanford, which I did. So I talked about my thesis. At the end he came to me and said, but have you considered the stability of Minkowski space? For some reason he knew about it. Even he, his own interest is also [inaudible]. It’s not always why. Why would somebody think, I mean, we don’t think of Newtonian [inaudible] been unstable, so why would we think that Minkowski space would be more difficult, right? Because you could say that there’s a positive energy, the positive mass theorem, which, if he actually, yeah, contributed. This was maybe a little bit later, or about the same time, that he and Schoen proved the positive mass. And indeed, from a point of view of physicist, you could say, well, okay, that proves stability. And in fact, actually, I remember later on, when I was at Courant, after we proved the stability of Minkowski space, it was [inaudible] Ed Witten gave a talk in which he said that he had another proof of the positive [inaudible] in general relativity, and he claimed that this proved, that’s the way to do Minkowski space. So I went to him, said, what do you mean? And, well, you have the dynamical systems, you have positive energy Hamiltonian, that of course shows. Of course I mentioned to him the Burgers equation, right? You hear Burgers equation, it’s an equation, you should also have positive energy, and you had shock forms. And he just smiled superior, I remember, and he didn’t really answer the question. That’s the sort of things. For finite dimension systems, Hamiltonian systems, having positive energy is good enough. But for infinite dimensional systems, like the case of gravity, it’s not enough.

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What does it mean fundamentally that something is stable versus unstable?

Brian Keating: What does it mean fundamentally that something is stable versus unstable? Starting with classical mechanics, and then we can go into quantum mechanics.

Sergiu Klainerman: Yeah. So that means that you have a solution and it starts with some initial conditions. You perturb the initial conditions a little bit. You’d like, overall, asymptotically, for all time, the new solution to be either converging to the one you perturb or to stay nearby, right? So stability means it stays nearby. It doesn’t change very much. The profile looks similar. Which is of course extremely important, because you never have exact solutions in anything, right? In order for that solution to have some kind of physical meaning, it has to be stable.

Brian Keating: In terms of perturbing it, if you were to perturb it, you could put mass in there, or you could send in radiative excitations, like gravitational.

Sergiu Klainerman: That’s a good question. Obviously, that depends on the problem. So stability can mean lots of things. But within the context of a specific equation, say I’m interested in compressible Euler equations, right? So I expect that I have shock waves. If I perturb the trivial initial condition, stability would mean that you have solutions which stay nearby, which stay close to constant, or they even converge to the original one, let’s say zero solution, right? So the solution will converge back to zero, or at least stay nearby. But of course that’s not the case, and the best example is the Burgers equation, right? Which is the simplest possible compressible equation where shocks form. So you’re starting at constant, let’s say u_t, so this is the derivative of u with respect to t, plus u times derivative of u with respect to x. You have one space and one time dimension, so two dimensions. It’s not a finite dimensional dynamical system. And shocks form. No matter how small perturbations you make, no matter how small, shocks will form, right? So that means the derivative of the solution blows up in finite time. You’re very far from stability.

Brian Keating: If we think about shock waves, and sometimes associated with turbulence, or Mach number, Reynolds numbers and things like that, in an empty spacetime, what is a shock wave in that case?

Sergiu Klainerman: In Minkowski space, you mean? In the context of an equation?

Brian Keating: Yeah, the Einstein equation.

Sergiu Klainerman: What?

Brian Keating: No, physically, what does it represent? What would happen to an observer? What does it do to observers in [inaudible] coordinates? Let’s say you have a population of observers and there’s a shock. What is being shocked, effectively? Because this is classical, right?

Sergiu Klainerman: Right, this is right. So, okay, the Burgers equation, that’s the simplest possible example, right? So you take initial conditions and the particles follow characteristics, which are very simple, right? They are just straight lines, but the straight lines can change depending on where you start from, right? So it can be a straight line at an angle. If I start here, so this is the line, I start at this point, I get one straight line at this angle. But if I go here, the same initial condition, then I get a straight line in this other direction. And the solution remains constant along the straight lines, right? So obviously at that point I have a solution with two distinct values, right?

Brian Keating: Depending on conditions.

Sergiu Klainerman: Yeah, depending on the initial condition. So a value from the first line and the value from the other line. So obviously this doesn’t make any sense. Something wrong just happened, right? So what happens is that, in fact, first derivative of the solution blows up, becomes infinite, and then you have to continue the solution. So all the shock wave theory is a way of continuing the solution past this kind of [inaudible]. And Riemann is the first. People may not know, but Riemann, among many things he did, he also played a very important role in the development of the shock wave.

Brian Keating: I didn’t know that.

Sergiu Klainerman: Yeah. We associate him with geometry, and then eventually with relativity, general relativity, but he also had a major impact.

Brian Keating: That’s very interesting. Yeah, it sort of makes, I mean, Mach was also very interested in shock waves, turbulence, sonic phenomena, and he came up with Mach’s principle, and maybe the last of the philosophical kind of physicists, where you could do philosophy. Maybe Einstein was the last, think philosophically and then derive something interesting in the physical universe. We physicists typically look down with some derision at philosophers. Oh, they don’t do anything, they don’t produce anything. Although now they’re having a renaissance. Your fellow professor at Princeton, Tom Griffiths, and others have written, basically, philosophers are getting a renaissance now with AI, which we’d definitely have to talk about in the context of how do we actually know these 800-page proofs that you generate, and that is kind of the Klainerman brand, are valid, and without tools like Lean and other things. I want to get your impressions of that, but we’ll get there.

Add some mass to the problem and you get a black hole

Brian Keating: I think the first thing I want to do now is take the reader through something more realistic. Add some mass to the problem and you get a black hole. You get a Schwarzschild solution, which was first represented by Schwarzschild soon after the creation of GR, and it really kind of lay dormant, and Einstein didn’t really know if it would ever have applicability. But what happens when you add mass, you add some point mass, you add something that you can still solve exactly, and then it also becomes unstable, correct? So talk about stability in the context of general relativity.

Sergiu Klainerman: Are you talking about just with the vacuum equations, or you…

Brian Keating: Well, now I want to add a Schwarzschild measure.

Sergiu Klainerman: Yeah. So the stability of a Schwarzschild solution. Oh yeah. So the Schwarzschild solution, of course, is still a solution of the vacuum equation. So you are talking about the vacuum equation with, in other words, no sources.

Brian Keating: Physicists would say there is a source. It’s a point mass of some mass capital M, and that’s, otherwise you don’t generate…

Sergiu Klainerman: Yeah, it’s not perfectly [inaudible].

Brian Keating: But it’s a delta function, right? I guess the point I’m trying to make is the black hole solution has a mass. It’s a point mass, the Schwarzschild solution, and then you generate things like the event horizon and so forth.

Sergiu Klainerman: Right, this point of view of mass and so on. And so of course I found it not realistic. The way we think about the black hole, and I’m more interested in the exterior of the black hole. So a black hole is separated. It has a part which looks like Minkowski space, right? So at least far away from the black hole itself, it looks like Minkowski space. But as you approach the black hole, if you are unlucky to go in the direction of the black hole, at some point you pass the event horizon. So the event horizon is a mathematical boundary between the visible part, let’s say the Minkowskian part, in which we live, we believe, right, and the part which is in the black hole. And of course, as you pass the horizon, nothing happens. You don’t even know, right? There’s nothing.

Brian Keating: Like passing the South Pole, which I’ve done many times. Nothing happened to me.

Sergiu Klainerman: But once you are in, that’s it, right? You can never come out. And then at some point there will be a singularity. The issue of that singularity is more murky. So the visible part of the black hole is the sort of thing that makes sense astrophysically, makes sense mathematically. The part inside is pure mathematics now, right? Because we cannot go there.

Brian Keating: Cannot access.

Sergiu Klainerman: Cannot access. So if you accept that the Einstein equations are correct, and that you have these exact solutions of the Einstein equations, then you can speculate what happens inside the black hole. And one thing that you can easily see mathematically, you can see that at some point there will be a singularity. That singularity, you cannot describe it as a point. It’s not a point. The singularity is more like a spacelike effect, if you draw it geometrically. You have to be very careful once you are in this kind of extreme regimes. You have to be very careful with your geometric description. So in reality it’s a spacelike hypersurface, right? So that corresponds to, I don’t know, r equals 0, let’s say, in coordinates, let’s say t, r, theta, which are called the Boyer-Lindquist coordinates. So in those coordinates, r equals 0, the curvature becomes infinite. So obviously something [inaudible] has happened, right?

God doesn’t know about coordinates

Sergiu Klainerman: The problem with general relativity, which is something that really puzzled Einstein, I think Einstein had problems with it, but many people had, is coordinate dependent, right? So any kind of description, if you want to describe a solution, you have to pick up some coordinate system. But the coordinate system itself can create problems, right?

Brian Keating: Right, yeah, the Kruskal coordinates.

Sergiu Klainerman: Kruskal coordinates, for example, that’s the best example, right? It took people, I don’t know, 40 years, maybe more. In 1965. It took 50 years to actually understand. I’m sorry. [inaudible] was puzzled. I remember that he was in Paris, actually, where people were asking him about it, and he looked very [inaudible]. They had a lot of issues with this, right? So that’s why I’m saying, the geometric description is very much coordinate. If you want to visualize things, you have to pick up a coordinate system.

Brian Keating: That’s right. And then that’s implicit. He’s sort of, philosophically at least, committing a hate crime against the notion that all the laws should be coordinate invariant, right?

Sergiu Klainerman: So you’d like to think so, but in the universe as a whole you can’t create such coordinates, right?

Brian Keating: It’s a cosmological, there’s no such thing, right?

Sergiu Klainerman: And there is, God doesn’t know about coordinates, you could say. But we as humans, in order to solve a problem that has to do with general relativity, we actually have to look at the coordinates. So, for example, the stability, both the stability of Minkowski, and even more, stability of Schwarzschild and Kerr, the coordinates play a fundamental role in the proof. The choice of coordinates play a fundamental role. You couldn’t do anything without them. In fact, actually, I would say maybe the principal role. There are many things that go into the proof, but one of the fundamental issues is a problem of gauge. Your description of stability is based on a choice of a gauge condition, or coordinate condition.

Brian Keating: And there’s no unique way [inaudible], because you cannot impose it.

Sergiu Klainerman: No, actually any proof [inaudible] will have this issue. You cannot impose it from the initial conditions. You cannot say what the correct coordinate system should be. You also find it dynamically, right? As you solve the problem for long time, you have to solve the problem for infinite time. At the same time as solving for the solution of the equation, you actually solve for the gauge in which the solution is described. This is what makes it very hard.

Brian Keating: It’s not that you choose the gauge and then that enforces the form of the solution.

Sergiu Klainerman: It’s not, that it will change. The choice is dynamical itself.

Brian Keating: In other words, you constantly change the gauge as you evolve. Oh. So that’s unlike any other approach. In particle physics and electromagnetism, you’ll choose a gauge, you’ll choose [inaudible].

Sergiu Klainerman: Exactly.

Brian Keating: Whatever. You don’t continually dynamically change it. So that’s unique.

Sergiu Klainerman: That’s, at least as far as I understand, I’m not an expert, but there’s a modulation theory that people have used also in describing other kinds of nonlinear equations. It’s a nonlinear effect. In linear, you don’t have to worry about it. You choose a gauge and you stay with it, right? But for some nonlinear problems, if you want to describe some non-trivial solutions, you have to also do some modulation, which means sort of picking up the correct mathematical description of the solution. The solution usually depends on some parameters. In this case it depends on coordinates, and in other cases it depends on other things. In that case you also have to find the solution together with a correct modulation to describe it. But this was done usually for equations which are much, much simpler than the Einstein equations, in which the choices are finite dimensional. The gauge condition is infinite dimensional. It’s a full coordinate. The diffeomorphism group, right?

Brian Keating: [inaudible] a monster. You perturb this as you perturb spacetime. You send in gravitational radiation. You look at its behavior, stability in space. What if you sent in other excitations, of electromagnetic gauge?

Sergiu Klainerman: So that would be now, right. So when you talk about stability of Kerr, right, stability, that’s the next level. When you are talking about Einstein equations in vacuum, right, you can talk Einstein equations coupled with, say, Maxwell field, and then you have the Kerr-Newman instead of Kerr. You have the Kerr-Newman problem, which is another problem where you would like to prove stability. There’s been some progress, but there is not yet a solution like, say, [inaudible] solution. And Jérémie Szeftel and Elena Giorgi and another student of [inaudible], there are five papers. Anyway, it’s a long project, but the ideas from that one can of course be used also for the case of the Kerr-Newman solution.

It’s like a ghost

Brian Keating: But you talked about these problems of ghosts, or artifacts. What does that mean to a mathematician in a solution? Is it what Einstein was saying, subtle is the Lord but he’s not malicious? What is a ghost? What is an anomaly? What are these?

Sergiu Klainerman: I used it as a sort of metaphor for instability, right? So in the case, again, let’s go back to stability of Kerr. You have an exact solution of the equations in vacuum, right? So mathematically, if you accept, mathematics always accepts something, right? It’s always in the context of some system of axioms. In this case the axiom would be that you accept the Einstein equations, and therefore we are in the realm of general relativity.

Brian Keating: Yes.

Sergiu Klainerman: Right? Classical GR. We don’t talk about quantum, right? We talk about classical GR. So in that context, you solve the Einstein equations, you find an exact solution, which is the Kerr solution.

Brian Keating: [inaudible] you’d say it’s a rotating massive black hole.

Sergiu Klainerman: It’s a massive black hole, which you can write down explicitly. You can check, right, either by hand, or you can use AI nowadays. You can check that it’s indeed an exact solution. So there’s no issue. The mathematical reality of the Kerr solution is uncontestable, right? Okay. But then comes the question: is it physically relevant?

Brian Keating: Yes.

Sergiu Klainerman: So my metaphor about that is that if somehow it’s unstable mathematically, in other words, relative to small perturbations, it disappears, then it’s like a ghost. It’s a physical ghost that doesn’t have any manifestation. It doesn’t have manifestations. So that would be the…

Is a wormhole a real object?

Brian Keating: So let’s talk about the Kerr black holes, because it’s going to dovetail into a conversation I had when Juan Maldacena, your fellow Princeton dweller, was sitting in that chair, and it has to do with wormholes and sort of the work that he’s done, which does branch away from, I would say, what a physicist would call real, or important even. So talk about wormholes. Let’s stop there for a second. To build a wormhole, it seems almost impossible. It seems actually logically contradictory, even though they appear in movies. And my friend Kip Thorne won the Nobel Prize for his work with black holes, but he’s the one who popularized it for movies like Contact and Interstellar, right? So tell me, is a wormhole not just possible? Of course things are possible, we go into the impossible here. But is it a real object in the sense that a black hole could be considered, whatever extent you consider a black hole real, is a wormhole? Because to build a wormhole you need to build multiple black holes. You need to sort of have them be traversable. You need to have them be stable. You need to have them emit certain dynamical properties. So, but you could think of a unicorn also, right? And just because you can think of it doesn’t mean it exists. So what’s your opinion on wormholes? Are they constructed out of Kerr black holes? Are these even possible?

Sergiu Klainerman: From my understanding is that they’re extremely unstable, right? So there are probably zillions of solutions of the equations that are unstable, mathematically unstable. But of course, in the context of what I said before, classical general relativity, they’re clearly unstable, as far as I can see, right? So you can construct a solution. Even Gödel constructed, in some cases, rotating universe, right? Clearly it’s unstable. In other words, if you want to understand it dynamically, in the context of the [inaudible], it’s clearly unstable. So classically it doesn’t make any sense. Now, I think what Maldacena will be saying is that quantum mechanically they do make sense and they are useful. But of course it’s way beyond my grade, right? I can’t say this, but what he really wants to do mechanically, it’s not that I would be… if I had another life maybe I would be, but unfortunately we are all limited in terms of what we can do.

The freedom of mathematics to follow its own logic

Brian Keating: Well, does your taste say what makes a problem interesting to Sergiu Klainerman versus me, Brian, an experimental physicist? When do you get interested in a problem?

Sergiu Klainerman: The part of mathematics that I think is interesting is the one that has something to do with the physical world, right? But it’s maybe a shadow of the physical world, or there’s some complicated, intimate relationship with the physical world, right? I think all beautiful mathematics started like this. Even, say, number theory or geometry, right? They all start from numbers and shapes, which are physical. Numbers, of course, are abstractions of physical things. One, two, three, right? And shapes are also [inaudible]. There’s no triangle.

Brian Keating: I always say, give me a triangle. You can’t do it.

Sergiu Klainerman: All right. No, but geometry started, however, with this need somehow to describe things which are physical, but then it went to different levels, and good mathematics started to follow its own logic. And you can start with triangles and circles and so on, which have some connections with the physical world, but then of course mathematics takes over, and after a thousand years or more you get to Riemannian geometry, and then Lorentz and Minkowski and special relativity, and all of a sudden it has an enormous impact back in physics. So what I want to say is that, on one hand, I like the kind of mathematics that has its origins in the physical world, but I also like the freedom of mathematics to follow its own logic and its own path.

Brian Keating: And hey, do me a favor. I know you’re enjoying this on YouTube, but make sure you follow me on Spotify, too, and leave a review. That helps out more than I can tell you.

Why are extremal black holes the final event horizon of black hole stability?

Brian Keating: But you’re not immune to it either. Looking at your career, it would seem to me I could predict with almost 100% certainty what you’re going to work on next, because there’s these unanswered questions from the real physical universe. I’ll just tell you, I think maximal black holes, I think extremal black holes and Kerr black holes, you haven’t done that yet. You’ve done an incredible job. I’m not going to say that you guys did. You did it for slowly rotating black holes. Explain why it’s natural, not just because, well, fast is better than slow, but why are extremal black holes kind of the final event horizon of black hole stability theories? I think they’re stable. To start with, the [inaudible] like them. They seem to be relevant. They seem to power black hole jets, accretion discs, quasars, and so forth. Not truly maximal. I don’t know if truly maximal black holes can even exist.

Sergiu Klainerman: Close. Yeah.

Brian Keating: Oh, yeah.

Sergiu Klainerman: No, large a, yeah. This is one of the major problems that many people are still working on, right? It’s the stability of Kerr for large a. So in principle you could go all the way to what is called the sub-extremal Kerr, right? And your theory would still hold, your proof would still hold. I think there’s been a lot of progress. My former collaborator Jérémie Szeftel has done some recent work which indicates that the problem would be solved. In other words, that you could prove the stability for the sub-extremal [inaudible].

Are all mathematical objects manifest in the physical world?

Brian Keating: In classical mechanics, we have the notion of the Poisson bracket. We can make a Poisson bracket. We can take the conjugate variables, momentum and position, and we can form mixed partial differential relations between them, and we get Lagrangians, and we get Hamiltonians, and we get their dynamics. And if you do that in classical mechanics, it works out and you get equations of motion. And you can’t do it in quantum mechanics without adding the square root of negative one. And it’s just so bizarre to me that adding in the square root of negative one, now you get the Heisenberg equations, you get quantum mechanics. Why should that be? And then, if that’s true, Sergiu, are there other levels of abstraction, analogues of the square root of negative one, quaternions, octonions, or whatever comes beyond them? Are they all manifest? In other words, are all mathematical objects manifest in the physical world?

Sergiu Klainerman: I wrote actually an essay about this, which [inaudible]. So, yeah, we’re doing it. I don’t have a solution of the Wigner mystery, as I say in the essay, but I try to somehow understand it on my own terms. One conclusion that I got, sort of reflecting on these things, is concerning what we call reality. What is reality, right? You can start with things that you can see, touch and hear, and we can share information about it, right? I can see these flowers, you, right? So there’s objectivity, direct objectivity associated with it, which is given by the senses, right? So we call that real, right?

Brian Keating: Yes.

Sergiu Klainerman: But then you go further, right? And you start talking about things that are far away, and you need a telescope to see them, or black holes, which are not even bright, so you have very little direct information, or microscope. And so those we consider real also. Why? Because they are consistent with everything else that we started with. So it’s like having sort of circles of reality. You have the first circle, which is the most direct, connected with our senses, and then the next one, which is further away, and it’s not direct already, but you can relate to a microscope.

Brian Keating: It’s reliability. You’re bringing up a good, excellent point, that when you have something that is trustworthy, to an experimentalist, it’s always subject to falsification. But we don’t have Gödel in physics. We don’t have an incompleteness theorem. The best we have is falsification, which means the violation of reliability, doesn’t it?

Sergiu Klainerman: That’s exactly my point. So my point is that there’s these circles of realities which are further and further removed from the center. They’re connected by, as you say, reliability or consistency, or what I call objectivity. As they go further and further away, they become more and more mathematical. Surprise, surprise. The more abstract it becomes, the more mathematics. Okay, so this is on one hand. On the other hand, mathematics is also very objective, right? We deal with things which, again, are very consistent. So do I have any right not to call them real, right? Even have concepts, real numbers, imaginary numbers, rational numbers. And then mathematics develops in this very strange way, right? It starts with numbers and very basic properties of numbers, commutativity, associativity, and so on and so forth. Then somehow algebra comes out, equations, and then we start solving equations, and you get the [inaudible]. Mathematics evolves through this. It starts again from basic objects like numbers and shapes, but it builds up levels of concepts which seem apparently very far away. We talk about an equation. An equation seems like something very abstract. It has nothing to do with something that we see. Not to speak about the Einstein equations.

Brian Keating: No, it’s true. E = mc², even simpler. But the Pythagorean theorem seems much more real than any physics equation, right? Even though we can’t touch a triangle, right? Hand me a triangle, you can’t do it. You can think of it only in your mind. Which maybe segues to the next point, which I want to talk about, AI and computation.

Sergiu Klainerman: Yeah, but just to finish this sense. So, anyway, for me mathematics is real. Not any kind of mathematics is directly connected to the physical world. Some people say that everything that is consistent mathematically is real, right? It has physical significance. I’m not going to go so far, but everything that has developed naturally, starting from the basic physical objects through the natural developments of mathematics, which I talk about in this essay. I had a clear idea, I shouldn’t say clear, but I had some idea about how this works out. So that’s the kind of mathematics that should not surprise us, that it has a huge impact back into physics.

Brian Keating: But it does shock one when you think that you’re using Fourier’s theorems, he described originally to describe heat flow in a metal plate in 1800, 1820, whenever he was working, and then you’re sending Fourier modes into a black hole. That’s kind of shocking, that these things couldn’t be more removed on a physical scale, and yet they have identical pedigree mathematically.

Sergiu Klainerman: It is shocking, and I don’t think anyone can really, Wigner didn’t answer it either, right? And actually he makes, in my opinion, mistakes that many physicists make, which is to claim that somehow mathematics is kind of an invention of the human mind. He says it very clearly, and I completely reject it. It’s not an invention, it’s discovery. Mathematics is discovered, right? Just like the physical world, discoveries, we also have mathematical discovery.

Who could possibly check the proof?

Brian Keating: Let’s talk about how we know what we know in the case of these proofs, some of which are in excess of 500, 800 pages, and it’s not uncommon in mathematics. First of all, it reminds me of what Einstein said when they made the 1919 eclipse expedition, the Eddington expedition, to see if the sun’s mass bent rays of light from distant stars, and of course he confirmed it. And Einstein was asked, well, what would you have done if they couldn’t confirm it? He said, “Well, I would have felt sorry for God.” Because, on the other hand, four years earlier, he was wrong. He had the factor of two wrong. So it was a good thing that the eclipse of 1914 was eclipsed by war in the Crimea.

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But tell me, Sergiu, when you look at this, who could possibly check the proof? And how is that possible? And was that sort of the last human-checked proof of that magnitude, or are we just going to Lean? We’ll talk about AI proof checking, but who could referee such things?

Sergiu Klainerman: Yeah. Yeah. Okay. So it’s a long paper for many reasons. Because we wanted to be very thorough, very thorough and complete, right? Many papers refer to other papers, results from other papers. We want to have, essentially, you can read the whole paper from beginning to the end. It’s completely self-contained. So that’s one thing. And also we try to put all the calculations. Again, many papers, people don’t really put the calculations, right? You just write down the conclusions. So anyway, we want it to be very complete. That’s one thing. The second is that these proofs are very conceptual. The only reason you can do a proof like this is because it’s very conceptual, right? You can split it into various parts. Now, the Einstein equations are nonlinear, which means that every part influences another part in some way. So you have to develop a language of how to put these pieces together, but it’s very conceptual. At the end of the day it’s very conceptual. That’s the second answer. The third answer is that many of these long proofs, if a result is important, then the proof will simplify in two ways. First of all, some of the things that are used in the proof will become standardized, right? So they are part of the field, right? That every student that works in that field would have to know. So in that sense, a paper later on will be simplified, and instead of having 2,000 pages will have only 400 pages, right? Which is what I expect to have. So that’s my answer. Things simplify in many, many ways. But if the result is important, people will try to simplify, will try to go beyond, right? Other people will take over and do something else with the methods which are used. And that’s how, in time, that’s the best way for something to be verified. In the short run, if you are asking how the paper was accepted, obviously a lot of people looked at it. Different people read different things. In the future it will become much easier, as you say, because of AI. I think AI is going to presumably be very good, maybe not to deal with a whole proof, but let’s say somebody who wants to check the proof would ask AI to check. First of all, they’ll have to have some idea about the whole, and then if that person would be unsure about one thing or another, will ask AI to help. So I think AI is fantastic from this point of view. I think it would be a great…

What’s the most complex proof that AI has reproduced?

Brian Keating: What’s the most complex proof that AI has reproduced? Because we hear a lot about these Erdős problems. I was pouring your coffee earlier and I said, we have to pay homage to Erdős. You do fewer amphetamines than he did. He said to Ron Graham, who was my colleague here, as you know, when Ron asked him to give up taking amphetamines for a month, that he did it, and then he collected the $500 bet and told Ron, you set back mathematics for one month. So you don’t use meth, but other mathematicians have. But you like coffee. But we hear about the Erdős problems falling every day now from different models. But to my knowledge, I talked to Terry Tao, as you know, you gave me a couple of hints, questions, to prep for that interview last year. He told me that it can’t, quote unquote, even do things like reproduce Andrew Wiles’s proof of Fermat’s Last Theorem.

Sergiu Klainerman: That’s not something simple.

Brian Keating: But what is the most complex, can it do the…

Sergiu Klainerman: It can or it cannot?

Brian Keating: It cannot. He said it cannot. If it’s changed since then, it could have, with the progress rate that’s happening. But tell me, what is sort of the most impressive, just reproduction of, like, the Peano axioms? What has it done that is beyond, like, helping you like a good graduate student can help you, I’m sure, and you must have brilliant ones? But tell me, can it reproduce things? We call it in the art world, it’s called copy work. Before you want to be the next Picasso, show me you could paint everything he did, or Rembrandt or whatever. Don’t just start off with, I’m the great. And we both know mathematicians that think that they have theories of everything, and before you do that, show me that you can reproduce it. And I don’t know, is there anything that AI can do that just straight up reproduces a proof that a genius mathematician, not prove the Riemann hypothesis, but prove some things that have been landmark proofs in mathematics, without human intervention?

Sergiu Klainerman: I would argue that that’s probably not possible. Peter Sarnak is my colleague. He has this, he called it the AlphaZero test, which is, so he connects this with chess. So in chess there are two ways now to automatize chess. There’s a procedure which is based on creating large libraries of openings, movements, sorry, that AI knows and then is going to use, and of course it’s so fast that it would beat everybody. But then there is this AlphaZero model that starts from nothing, right? So you just give it the rules, and then it plays by itself, and then after a while it beats everybody. In fact it beats even the best computers, apparently, from what I understand. So then the AlphaZero test is: can you give AI just the Peano axioms, let’s say, and then develop whatever you ask, if you want [inaudible]? Can AI do it? Most probably not. I have reasons saying, AI intelligence, it’s remarkable. As remarkable as it is, it is impressive. I’m using it more and more also myself. And it’s just pretty stunning what it can do. But still, to claim that somehow LLMs are at the same level with what happens in the human mind, it seems to me a huge stretch.

The Einstein test

Brian Keating: Yeah. I’m trying to do what I call the Einstein test. If you take an LLM now, if you take Claude or OpenAI or whatever, and you say, tell me about the multiverse, or tell me about this and that, it knows so much, not just from the physics but also from the culture. In other words, there’s movies about wormholes, like we just talked about. There’s movies about Stephen Hawking and Hawking radiation, right? So you can’t even trust the training data, which is a large fraction of what makes these things unique. So we tried asking the following question: what if you gave an LLM a lobotomy? What if you took everything about the physical world, just trained it on stuff before 1900? Could it come up with quantum mechanics and relativity? And of course there are other people working on this, and there are people that have actually made these LLMs and actually denatured them, denuded them of any pop culture references after 1930. But that’s even too late for quantum mechanics, right, and relativity. And we tried to do it with just giving it data from the planet Mercury, and just its orbital history over thousands of years that JPL has, and it gave us back, it couldn’t derive the Riemann curvature, it couldn’t arrive at the Ricci tensor. What it gave us back is this very discretized, hyper fine resolution, and it matched the planet, but it was just copying. It was just mimicking it. And I think until that’s passed, I’m not going to be… Because what Einstein said was the happiest moment of his life was that in free fall I wouldn’t feel any gravitational field. So that led to the, that’s a kind of exact, how does it…

Sergiu Klainerman: That’s about, I wrote an article [inaudible]. Yeah, I want to talk about the AI thinking, and that’s exactly my… So I made various points about the fact that I could very well imagine that AI can do all sorts of things, but it doesn’t change itself, right? Humans are able somehow to design something new. Can it design another version of itself based on different principles? Like, for example, when we invented the [inaudible], right? It was an invention that completely changed the world. You cannot say that it was continuous of something else. Or the same thing with Kepler’s laws, for example, right? You could imagine, based on periodic approximation, the epicycles, you could approximate everything perfectly. It was better. Epicycles, you grab absolutely everything, right? So, hey, but would you discover Newton? Kepler’s laws, even more important Newton’s law, then Einstein. Come on. Give me a break. It’s just not even close, based on what we know so far. And also you can go to more specific things. For example, it’s pretty clear that our thoughts, I’m not talking about all thoughts, but the ones that matter, predate the language, right? We have an intuition about something and then we find the words to explain it, right? It’s like a proto-language, right? That’s something that AI doesn’t have. AI is completely based on guessing the next word, the next token. In other words, it’s completely based on words.

Brian Keating: Yeah. Language on language, right?

Sergiu Klainerman: But as I said, I think our thoughts predate the language, right? We have intuitions about things.

Brian Keating: Which we have a physical, embodied emotion. Like Einstein falling in space. I can’t describe that to you, but you know exactly what it feels like to go on a roller coaster, or falling in love. Is it embodiment?

Sergiu Klainerman: Embodiment, probably.

Brian Keating: How, I was like, well, maybe we blow a capacitor in the computer when it gets the wrong answer. How do you train it to have pain or loss or poignancy? What does that even mean?

The AI apocalypse is absurd

Sergiu Klainerman: And then these absurd notions that people have, incredibly, very smart people, right? They claim that somehow we passed the event horizon, or the…

Brian Keating: The singularity, they call it.

Sergiu Klainerman: Yeah, you are going to singularity. The AI is going to become more intelligent than everybody, and it’s going to reproduce itself at a huge speed and kill us off. It’s just absurd. It’s ridiculous. Even this business of killing ourselves, we can stop it at every moment. If it becomes dangerous, we can stop it.

Brian Keating: We can’t destroy it?

Sergiu Klainerman: We could. We just stop the internet and that’s it.

Brian Keating: Exactly. And another thing, I always joke, what’s the first thing you do every morning that you use AI? Push here to update to version 1.974. And then they go to Claude, push update to 4.32. Every single day they’re updating. They’re responding to attacks. It’s kind of like the grandfather paradox. If a time machine was possible, you could go back and kill your grandfather, which would prevent you from inventing a time machine. Here, let’s assume artificial superintelligence, the most powerful intelligence that ever could be made. It wouldn’t make itself. It wouldn’t allow itself to be made, because the next thing after it, version 1.9, is going to be more powerful than it and turn it off. And so, yeah, I think these things are overblown, but they do have a very important purpose.

Sergiu Klainerman: Yeah. Oh, no. Certainly.

Brian Keating: No, no, the purpose is to generate revenue for these closed-model AI laboratories. They do it, but they also scare everybody. They scare everybody to get more money, so the government will protect them as a national resource, like the uranium stockpile.

Sergiu Klainerman: No, but the problem is, now lots of people are calling for the destruction of data centers.

Brian Keating: That’s a separate issue, I think.

Sergiu Klainerman: Yeah. I think Musk has taken it very far.

Brian Keating: I see. Yeah. But does Musk really believe…

Sergiu Klainerman: Well, if he does want to build… He changes his mind.

Brian Keating: Yeah. More frequently than my toddlers. But it went from going to Mars, now then it was going to the Moon, now it’s not important anymore because we can have AI reproduce everything. But sticking with this thought about where does humanity come in. Your Tablet essay, I think it does stress the kind of uniqueness of humanity, but it’s not some, like, Pollyanna, everything’s going to be okay, everything’s wonderful.

Sergiu Klainerman: No, I think there’s huge benefits.

Is it really worth being a professor in the age of AI?

Brian Keating: And you and I know how powerful it is and how fast science can change. I guess we don’t have that much time. We both have other things to do later on today. But I do want to talk about being a professor nowadays in the age of AI. Is it really worth being a… do students need to pay, what is your tuition? $70,000. Our out-of-state tuition is about that now. I think Princeton’s about that, one of the highest levels. But with all these tools, with all these wonderful things, with [inaudible], with all the available things on the internet, including conversations with you and your writing, and the podcast and stuff that you do and I do, do we need to pay $70,000? What are you really getting for your money?

Sergiu Klainerman: No. No. Of course. I think it’s ridiculous. Universities…

Brian Keating: How would you restructure [inaudible]? If you were…

Sergiu Klainerman: I used to be more optimistic. That’s how I got more involved in reforming. And so I’m becoming actually quite pessimistic that universities can be reformed, except by creating new universities. Creating completely new institutions that will have to compete with these other ones.

Brian Keating: Mhm.

Sergiu Klainerman: In science it’s not so bad, of course. But I’m a firm believer that a good education should involve also humanities and social sciences. I don’t think you can be a good scientist without sort of a firm foundation. So you have at least some understanding of basic philosophical issues. So, yeah, to be a good scientist you also have to have a broad view.

Brian Keating: Yeah, I say Galileo is one of the best writers, period, in human history, and he happens to be a scientist as well.

Sergiu Klainerman: Einstein was phenomenal. Newton wasn’t quite as good. He was much more dry.

Students should have a right to protest

Brian Keating: I think Newton, in terms of university politics and so forth. When you and I started university, you’re older than me, but when you started university, it was very different. It was sort of the best job I could possibly imagine. And nowadays I don’t know if I could advise my children to become a professor, an academician. It’s not necessarily only the life of the mind anymore. It’s become so much more. I want to give you credit for inspiring me personally, but also doing incredible work as a citizen, but also a scientist, and speaking out on behalf of people that you don’t agree with. You’re unique in that. I don’t know if I have the same courage as you, or maybe the same intellectual honesty. After October 7th, you said that students should have a right to protest, and the anti-Israel factions on campus were very vocal there and here. We had an encampment here that lasted a couple days, and I testified in Congress about how evil and destructive and anti-intellectual it was. But you wrote that they should have a right to protest, but it should have limitations. So let’s talk about what is your view on students participating in protest. I agree they should. Students should. I get concerned when I see my colleagues doing it, for different reasons. But where do you fall? What rights do students have to kind of advocate, as they did, that your university is racist, or it should be more pro-Palestinian? What level of impact do you seek to have in these debates?

Sergiu Klainerman: On one hand, yeah, the freedom of thought is extremely important. On the other hand, what do you do with situations where universities have become so unbalanced in terms of what is being taught, what is being accepted, kids really becoming more and more expressing radical ideas, anti-American ideas? I think the universities have lost their ability to frame what their mission is. They’re completely confused. The mission is to find the truth and to educate the kids in the best that Western civilization, or the world civilization, has to offer. None of this is true anymore. Kids are educated all sorts of crazy things about human nature, lies. Truth doesn’t seem to matter. There’s a lot of relativism, a lot of political advocacy for certain causes which the left considers sacrosanct, right? Equity, diversity, inclusion, Israel. This business with Israel is just ridiculous, right? So, singling out one country versus [inaudible], huge deal about situations where Israel was attacked, right, by Hamas, and Iran essentially surrounding Israel by factions that really want to destroy it. Anyway, yeah, I have a huge problem. So I don’t know where I am. I do think that this is probably going to happen, that universities will become more and more, the university that we have today will become more and more irrelevant. Another thing that really bothers me is that you would think that in a capitalist system like the United States…

Brian Keating: Yeah.

Sergiu Klainerman: That if a university goes very much to the left, then there will be another one that would get an advantage, right? Will get an advantage by going the other way, right? But it just doesn’t happen. How do you explain that? How do you explain the universities go in the same one direction?

Brian Keating: Yeah. It was capitalism. You’re seeing, like, Palantir, and you’re seeing corporations that will take the people that were too unwoke for Google and hire them, and there are also some of the people that are supporting new institutions. But yeah, I’m also pretty pessimistic. The thing that made me most pessimistic: I thought COVID would be the end of the current university system. Going through that, charging the exact same tuition for Zoom lectures that you could basically get online for free from past years. The fact that it was so resilient to that kind of just horrible product delivery and service provided for the vast amount of money that far exceeds inflation. The amount of tuition that’s being raised at all of our colleges vastly dwarfs it, and then the debt you take on can’t be discharged in bankruptcy. It’s a cartel. Basically, we in the university system operate in a cartel system.

Sergiu Klainerman: No, I think there’s a lot that’s troubling about the university system. And like I said, I don’t know if I’d want to return to this life.

What would you tell that 19-year-old version of yourself?

Brian Keating: There are other things you could be doing. There are other things I could be doing. But let’s assume that there’s a 19-year-old, maybe not in Bucharest like you were in 1969, but let’s say there is some young person, maybe they’re living in a slightly repressed state. I like to ask the following question. What would you tell that 19-year-old version of yourself to do, to give him the courage to do what you did, to go into the impossible, the namesake of this podcast? 20 seconds with a 19-year-old Sergiu Klainerman. What do you tell him?

Sergiu Klainerman: Would you change anything? In science, it’s still the best place to go, to go to the top universities. It’s still true, right? It’s true in astrophysics.

Brian Keating: Mhm.

Sergiu Klainerman: Even though, even there, you can see signs of decline, right? But if he comes to something else, if you really want to do humanities or social sciences, I presume you have to do it on your own. I see no advantage to go to university. You can read the best books. Now you have AI. AI will direct you. And the same thing with school, right? Even more radical, right? What do you do with the kids, right? Do you have kids? Should you send them to school, or, I don’t know, maybe frame some kind of educational system for them?

Brian Keating: I think it’s both. I think you have to do both. You have to socialize them. They have to network. They have to understand how to interact, and maybe do sports or athletics or arts and things. And then there’s stuff that, yeah, I’m not going to let them only learn in a public school the basic mathematics. One of my kids is taking advanced calculus now and he’s not even able to drive. So, but that’s on us.

Sergiu Klainerman: All the great scientists, they learn by themselves. They’re autodidacts.

Brian Keating: Yeah.

Sergiu Klainerman: Right. The same thing with art, the great artists. And so, I don’t know, maybe AI will help us to go back to situations where people have personalized tutoring and personalized things. Right.

Brian Keating: Exactly. Yeah. I think that’s a wonderful thing. That’s a good place to stop. And I just want to thank you again for being the scientist you are, the citizen that you are, and the citizen scientist. Those don’t always go together. And I think it’s rare, and it’s refreshing, that you take all these roles so seriously, and that you’re not afraid to go after these monstrous, challenging, difficult problems. And it just seems like you have no bounds, no limits to what you’re willing to take on. I think that’s a sign of your courage. So, Sergiu Klainerman, thank you so much. From Romania to the heights of academia and beyond, thank you for coming on.

Sergiu Klainerman: Thank you. Thank you. Thank you very much.

Brian Keating: Sergiu just told us that wormholes don’t survive Einstein’s own equations, and that the AI apocalypse is, quite frankly, absurd. If that changes how you think about what’s real in physics, subscribe, turn on notifications, and tell me in the comments whether you think an unstable solution counts as physics or only mathematics. And for the man who put wormholes into the movies, here’s my conversation with Nobel laureate Kip Thorne, the man who worked on Interstellar with none other than Christopher Nolan. Link right here. Thanks for watching.