Theories of Everything with Curt Jaimungal - Frederic Schuller: The Physicist Who Derived Gravity From Electromagnetism

Episode Date: August 21, 2025

As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit https://www.economist.com/toe In this episode, I speak with Frederic Schuller, an award-winning theoretical physicist and professor, who insists the undergrad tale of energy sloshing between kinetic and potential is just talk unless the math says so. Borrowing Port-Hamiltonian thinking, he’s building probability ports to pull measurement talk into actual quantum formalism—no change to QM, just sharper math. He also flips gravity: start from the matter action and construct the compatible gravitational dynamics—Maxwell in, Einstein–Hilbert out. And if nature ever breaks our current causal picture, the scheme points to richer structures (and the gravity to match)—a modest idea, pushed hard. Join My New Substack (Personal Writings): https://curtjaimungal.substack.com Listen on Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e Timestamps: - 00:00 - Deriving Einstein from Maxwell Alone - 05:55 - Why Energy Doesn't Flow in Quantum Systems - 11:45 - How Modest Ideas Lead to Spacetime Revolution - 19:00 - Matter Dynamics Dictate Spacetime Geometry - 24:03 - Maxwell to Einstein-Hilbert Action - 31:00 - If Light Rays Split in Vacuum Then Einstein is Wrong - 38:04 - When Your Theory is Wrong - 46:10 - From Propositional Logic to Differential Geometry - 54:00 - Never Use Motivating Examples - 1:02:00 - Why Only Active Researchers Should Teach - 1:09:40 - High Demands as Greatest Motivator - 1:16:00 - Is Gravity a Force? - 1:27:00 - Academic Freedom vs Bureaucratic Science - 1:38:00 - Why String Theory Didn't Feel Right - 1:46:05 - Formal vs Conceptual Understanding - 1:54:10 - Master Any Subject: Check Every Equal Sign - 2:04:00 - The Drama of Blackboard Teaching - 2:13:15 - Why Physical Presence Matters in Universities Links Mentioned: - Frederic’s Papers: https://scholar.google.com/citations - Frederic’s Lectures: https://www.youtube.com/@FredericSchuller - Frederic’s Bio: https://people.utwente.nl/f.p.schuller - General Relativity Lecture Series: https://www.youtube.com/playlist - Quantum Harmonic Oscillator [Lecture]: https://youtu.be/s3I_MGfGm-w - Constructive Gravity [Paper]: https://arxiv.org/pdf/2003.09726 - Geometry Of Manifolds [Paper]: https://arxiv.org/pdf/hep-th/0508170 - Jacob Barandes [TOE]: https://youtu.be/7oWip00iXbo - Roger Penrose [TOE]: https://youtu.be/sGm505TFMbU - All Possible Paths [TOE]: https://youtu.be/XcY3ZtgYis0 - Neil Turok [TOE]: https://youtu.be/ZUp9x44N3uE - Space-Time Structure [Book]: https://www.amazon.com/Space-Time-Structure-Cambridge-Science-Classics/dp/0521315204 - Greg Chaitin [TOE]: https://youtu.be/PoEuav8G6sY - Ivette Fuentes [TOE]: https://youtu.be/cUj2TcZSlZc - Ted Jacobson [TOE]: https://youtu.be/3mhctWlXyV8 - Eva Miranda [TOE]: https://youtu.be/6XyMepn-AZo - Jonathan Oppenheim [TOE]: https://youtu.be/6Z_p3viqW1g - String Theory Iceberg [TOE]: https://youtu.be/X4PdPnQuwjY - Sabine Hossenfelder [TOE]: https://youtu.be/E3y-Z0pgupg - Leonard Susskind [TOE]: https://youtu.be/2p_Hlm6aCok - What Is Energy? [TOE]: https://youtu.be/hQk9GLZ0Fms - Claudia De Rham [TOE]: https://youtu.be/Ve_Mpd6dGv8 SUPPORT: - Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join - Support me on Patreon: https://patreon.com/curtjaimungal - Support me on Crypto: https://commerce.coinbase.com/checkout/de803625-87d3-4300-ab6d-85d4258834a9 - Support me on PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 SOCIALS: - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs Guests do not pay to appear. Theories of Everything receives revenue solely from viewer donations, platform ads, and clearly labelled sponsors; no guest or associated entity has ever given compensation, directly or through intermediaries. #science Learn more about your ad choices. Visit megaphone.fm/adchoices

I was stunned as a theoretical physicist. We now believe to have cracked at least two of the three most prominent problems. Professor Frederick Schuller has done something that should be impossible. He's derived Einstein's general relativity from Maxwell's electromagnetism alone. Not postulated, not assumed, but derived. So you start with electromagnetic fields on an arbitrary background, you demand predictivity, And out pops not only that must the geometry be Lorentzian, but the dynamics must be Einsteinian. However, Schuller's Maverick conclusions don't end there.

He's found a formalism from engineering, which may help shed light on the measurement problem from an extremely unlikely place. This podcast was tremendously impactful to me, which is not something I often say about the math and physics podcasts on this channel. the heartfelt impacts tend to come from explorations of meaning and even consciousness. But today, Frederick reveals for the first time his teaching philosophy and why starting from propositional logic and building all the way through set theory to differential geometry has resonated with millions of people. Professor Schuller shows physics as it actually is, a conceptual masterpiece where every piece interlocks like a da Vinci painting.

This podcast will not only give you, a new perspective on physics, but will fundamentally change how you think about physics. Welcome, Professor. Thank you very much, Kurt. This was a long time in the making, right? Yes, many, many months. Yes, yes. So tell me, what are you working on these days?

What's in your mind, research-wise? Research-wise, I am thinking about foundations of quantum mechanics these days, and I never wanted to do that because that's a very, very thorny subject. And, of course, there are obvious problems, the measurement problem above all that bother me, but they bother me from a new angle. And this new angle is that in engineering, I work a lot with engineers these days at the University of 20. Right. And they have a very strong robotics lab, many groups. And I learned something in 2020, actually.

I was invited to participate in a small conference, actually not that small, after all, in Paris. And I saw their engineers, and they talked about something that's called the Port Hamiltonian approach to dynamics. And this is, in essence, an extension of Hamiltonian theory. It's just that you do not only provide the formalism to talk about a closed system where no energy can flow out or in. you talk about open systems where energy can flow out, but you don't say how it flows out. There is an open port that gives you the possibility for it to flow out. And if you take two such systems, you can connect them via something is called the duroc structure.

That has been studied before. And that is a different decomposition of a physical system governed by a Hamiltonian, say classical mechanics, classical field theory. then we would usually have we would usually describe it as one big system and we would talk a lot about energy flowing from that part of the system

to that part of the system especially if you think about kinetic and potential energy even in the introductory lectures we talk about energy flowing from the kinetic energy to the potential energy back and forth such that all of energy is concerned we do a good talk

on this right but ultimately it's just talk it's not reflected in the formalism in how we describe the theory why is that what is that good for it's just a rewriting right but engineers have the need to control energy flows for instance if you have robots interacting with humans and the robots have you know joints and so on if the robot interacts there's of course energy being transmitted from from the human being and so on and how do you actually capture this, how does the robot react to this if you want to study this and you need to study this because that robot might otherwise decapitate you if there is an unwanted

energy flow, so to speak. Okay, so it's moderately important. It's moderately important if you want to survive, but who knows whether we do anyway. Joking apart, engineers understand the need of this, also huge networks like say the French electricity network. What happens if certain power plants all of a sudden shut down or their other problems, they need to get the energy routing through the system right. And this is done these days by Port Hamiltonian approach to all of this. And I was stunned as a theoretical physicist because I thought, oh yes, this is something is a good formism. You know, every time you reflect something in the formalism, you can then calculate with it. If you have just a good idea about it, you say, yeah, yeah, we know all about

that and of course of us, but if it's not reflected in the formalism, you can't really apply the mathematics to it. And so it's important or it could be important depending on what you want to do to reformulate theories like this. Well, long story short, in the classical domain, this has been developed over decades by, I think it originated with Aryan Fandershaft and Bernard Marshke and many other people work on this now in Germany. There's a big research center on this. So this Port Hamiltonian viewpoint is a very interesting one. It won't solve everything.

Maybe it solves nothing, but it gives you a new perspective. Okay, that's what it is. And I started thinking together with collaborators on how this could turn out or play out in quantum mechanics. And in quantum mechanics, we superficially, we also talk about energy, we have Hamiltonian, make energy measurements.

The Hamiltonian is still the generator of the time evolution, as long as you don't measure, right? So for the unitary evolution. Right. So could we talk about these energy flows in quantum systems? And the answer is, if you look closely at it, no, that doesn't seem to make much sense, but it's something else that flows there.

It's a probability that flows in quantum systems between subsystems. And that's the next thorny issue. How do you do with subsystems and quantum mechanics and so on? and we now believe to have cracked at least two of the three most prominent problems. How does this work? What is actually flowing there? How do you actually get this onto the street formally?

And also how do you deal with composed systems? Because quantum systems famously composed by tensor products. So the Hilbert spaces of composite system is the tensor product of the Hilbert spaces of what we think are the constituent systems. But of course, the 10 subproach contains many states that could never be understood in terms of the state of one subsystem and the other subsystem.

Those are the entangled states, of course, right? And so any idea of decomposition into systems and probability flows, all of this must work together. And we now think we made good progress. And one thing we haven't done yet, and that's, of course, the most difficult, thing, we would like to formulate

the measurement axioms as they are in quantum mechanics. As you postulate them, they're a little mysterious to say the least, but most of all, they contain a lot of talk. You say you conduct a measurement, for that you have a hamission operator. Let's talk finite dimensions, otherwise self-adjoint operator. Okay, let's think finite dimensions. Quantum information technology justifies us in doing that, right? You have a emission operator and then you find the eigenvalues and of course you can then calculate from the

spectral decomposition. You can calculate the probability with which a certain measurement will occur if you measure now. But what do you mean will occur and if you measure now and what does this all mean? It's just talk. It's talk that works spectacularly well as we know, right? Quantum mechanics in a sense, it works spectacular well, but much of this talk, especially around measurement, is not at all reflected in the formalism. And of course, many people have worked on that,

many very smart people have worked on that, trying to explain measurement, the idea that decoherence may play a role, I guess it does, but is that the whole answer to the measurement problem? No, it is not, you know, all of these things. But this, what we're trying to do,

is very modest, but maybe therefore it can be successful, we try to give an extended formalism, not deviating from quantum mechanics, but capturing much of the talk as much as we can in a formalism. And this idea of ports, but now not energy ports, but probability parts, play a big role. So it's a natural idea. Everybody can have it. We worked for a long time to really get this onto the street because there are many little things you can trip over

and now I think we made some good progress and well once we're convinced we went to a point that it's worth putting it out so that other people can start thinking about two if they like you'll hear from it that's it

but that's what I'm thinking about quantum mechanics and it's a very thorny issue because you know I mean it's also very easy to talk about the measurement problem because yeah yeah it has been talked a lot about it is a little bit quite dramatically

a problem, right? I mean, Roger Penrose talks about it, of course, is true. That's not a secret, so to speak. And we try to have a modest new approach to it. But the claim to fame, well, not fame, but you know what I mean, right?

The reason why I think it might be useful, it uses a new technique and a new formalism with a purpose. So that's what I'm thinking about on that side. Okay, so earlier you said that undergrads, well, you didn't use the word undergrad,

but undergrads are taught about kinetic and potential and how you can flow between them, as long as energy is conserved, but then he said it was all talk. What do you mean that it's all talk and not reflected in the formalism? And also port, because people keep hearing this word port. Port is spelled P-O-R-T and refers to the boundary ports. Yes, the boundary ports, the boundary ports, the other ports. It's very easily explained if we write it down. We can't do that right now.

They're simple examples. Well, the point is what is not reflected is you have a total energy that's conserved. And of course, you can define a potential and a kinetic energy. And you say it flows between them because the sum of them is preserved. Right. So, yeah, I mean, I'm not saying it's wrong talk. What I mean by its only talk is, is it built into the formalism that you make use of that insight. Now you're just saying you're looking at two observables, kinetic and potential energy,

yeah, okay, and they change their values as your system evolves classically. You can in a sense monitor that classically. But as I said, we're not making use of that in any way. We're not, and then the question is, is really the kinetic energy, does that constitute a conceptual subsystem from which something flows elsewhere. You know, this all sounds good. I think this also in some videos that you talk about where the mathematics

one confuses what physically happens with the mathematics and so on. It's of this type. I think it's really, I could now try to explain in words what a port is and it would sound very fancy with dual variables and the application to each other gives them the measurable quantity,

but it's really much better to actually write this down. Maybe I could write a small document after the podcast. That would be great. I think people would love that. Note, I'll be placing these notes from this podcast in my substack, which you can get by searching my name and the word substack, or by visiting kurtjimungal.com,

C-U-R-T-J-A-M-U-N-G-A-L.com. I know it's a mouthful. Yeah, yeah, yeah. But this is not my idea. So this has been done quite a while ago. It's just, I think it deserves, let me state it like this, it deserves some attention from theoretical physicists as well for application maybe in fundamental physics. We should at least think about whether this could help us with things we do.

So that's the, yeah. There are a couple of questions here that I have. So you said the phrase, it's a modest proposal, therefore it could be successful. Now that's interesting. Not it's a modest proposal and it could be successful. It's a modest proposal, therefore it could be successful. So I have a stickler for words and I noticed that. Yes, yes, yes. Please tell me, what did you mean by that? Why is the modesty connected to the success? Okay, at least for my means,

I always think if I have one idea about something, like measurement problem or something else, one idea, you know, ideas are cheap in our field. We can have many, many ideas, right? And we can then find the fifth and the sixth idea, and if we take these ten ideas together, I believe trying to bring one idea to success works because you have this one idea and then this idea, if it's any good, will lead you, if you try to implement it,

will lead you to more insights and then the problem dictates you what your next idea would have to be, something like that. Look, this is a very philosophical subject, right? And I claim no truth to this, of course. But if you asked, I don't know, you could view, say, the whole development up to general relativity starting from Maxwell theory that Einstein stared at the Maxwell equations. And the longer he stared and the more he thought about it, he realized there is something at odds with the idea of, you know, the speed of light being. in there. I mean the constant, the C constant by, what is it, epsilon

times mu is in there. How can that be in that equation? Because if you go to a moving system, shouldn't it be C plus V? Where we is the moving system, right? So Einstein looked at Maxwell theory. In a sense, you can see Einstein took Maxwell theory

very seriously. Okay? And taking Maxwell theory very seriously, he was prompted to change the idea of space and time or even the separate existence of something like space and something of time. There's only space time. Then there's no.

So it's a very simple idea. It's very modest to say there is a theory that tells you how electromagnetic signals. Now Heinrich Hertz, I think 1888, he proved that you can actually transmit by electromagnetic waves. You can send signals in the lab and stuff like this. And I think Maxwell was something like 1850, 1860, something like this, these equations. In 88, this was demonstrated so the predicted electromagnetic waves, they really exist and it works and so on. So in a sense, saying, well, that works and then we have a radio and the radio. So it's a modest idea to say, let's take Maxwell theory very, very

seriously. And then it leads to special relativity and then for the inclusion of gravity, by whatever way you want to take there, say the Einstein way, it leads to the curvature of space and time. and if you want to a prediction in that theory in that view in the prediction of the Big Bang, right? So in the sense, it's a very modest idea seen through. You see through this idea

that you say Maxwell theory is correct. Right. And so I think seeing through one idea can open doors. Yeah. So that's what I mean by, it's a modest idea one can make many

constructions and ideas and intuitions and so on I always think we need to rely on nature giving us a hint because theory space is infinite dimensional and if you tip with your finger

somewhere and you say oh the metric may be non-symetric Einstein did that right I mean Einstein in his later years he fanatically looked for the inclusion of Maxwell theory into the geometric framework of general relativity.

And of course we had Kalutzer Kline, all these nice things, all very nice. But ultimately it doesn't work so far, we think. Einstein didn't find it. And so once Einstein, let's take Einstein also as a counter example in the sense, once he started thinking in this formal way and I have an idea and we could include this here, that didn't work so much anymore. I think because he didn't have nature on his side, like he had. before with Maxwell theory.

Yes. I don't know whether I make myself clear. And obviously, this is not wisdom to follow by as a strict rule. It's a little bit my guiding principle in doing research for better or worse. Right. Well, I have the advantage that I've gone through your work. So I know that you're presaging constructive gravity as well. Yeah.

Yeah. And I'll place a link on screen. As a preview, we're going to talk about that, about how you direct. gravity from matter dynamics rather than seeing gravity in matter as separate postulates. We'll talk about that. Well, right, yes, yes. So I can tell you where this came from, this whole thing. So absolutely counter to what I just said in the past, I also looked at modified gravity theories for this reason and for that reason.

And of course, these are all reasons that are dreamt up. It's a lot of modified gravity ideas out there. how do we know which one is right? And at some point I came to the conclusion for myself. Of course, that doesn't disparage anybody else's attempts in any other direction. It's a little bit hopeless to just try to think that you think up something half formal, half motivated, and then write down a new Lagrangian. That's easy.

You can do that. Everybody can write down a Lagrangian, a modified Lagrangian for a theory. I thought, what could actually give us a hint at modified gravity if we don't just look at gravity itself? And, well, we do a lot of observations about matter, right? We do very precise observations about matter. And the question was, is it actually true that you have to postulate both the dynamics for gravity

and the dynamics for the matter? And at some point we stumbled across this that we thought, no, it's a crazy idea, but maybe if you're given a meta-Lagrangian on an assumed background, and I mean just the geometric structure, so to speak, of the background, but not the concrete, say, on a Laurentian background, on a background that allows for birefringens or something like this, could you actually determine from the matter action, how this background has to get its dynamics in order to be compatible with this matter action. And the only connection we saw is that the matter action and the gravity action,

they must both have, they must evolve together. Say, if you speak mathematically, from the same cushy surfaces, they must take the initial data. Those must be evolved to again a shared cocee source. surface or a whole family of shared cushy surfaces. This seems to be a minimal requirement if you want a predictive classical theory. Predictive being the key. And quantizable, not for the gravity, but for the matter. Yeah, quantizable.

This is that came in as a technical, but let's quantizable out for the moment. It became in, you know, to justify an apparently classical condition, technical condition to do that. that's why we let's leave it out for the clarity of the so and now you might think that's a very weak connection that they have to have the same cushy surfaces so to speak the metadynamics on the background you give me and what dynamics the background could get to evolve together with the metadynamics and it turns out it's a very strong connection actually in fact you can write down equations that you need to solve in order to get as a result as a solution to these equations,

you get the gravity action that you would have to give to that geometric structure. Yes. That is in the background of the matter theory, all right? Sorry, just a moment. I just want to make sure I'm understanding. So let's just make this concrete.

How do you go from Maxwell's equations to the action of Einstein Hilbert? Spell that out. So you know how in physics we like to reduce something that's complex into something that's more elegant, more efficient, more simple. Turns out you can do that with dinner.

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Again, that's hellofresh.com slash theories of everything, 10FM, all one word. HelloFresh. The best way to cook just got better. How do you go from Maxwell's equations to the action of Einstein Hilbert? Spell that out. Okay, so ask how does this bigger scheme work

in the special case, if you would say start with matter, that is Maxwell Electrodynamics, right? Yes. on some metric background. Let's not even say Lorentzian, any metric, any signature. Well, the first thing you would do,

you would find out for what background signature, if you start with a metric like this, would you have a well-defined Koshi problem for Maxwell theory? And then it turns out immediately only for a Lorentian signature, right? So, but then the question is, aha, so a Laurentian metric

is in the background of this theory and that's also clear from the phenomenology that we see. Aha, but what is now the dynamics for that background metric? That's the question. And of course, the textbook answer is Einstein's equation. Why? Because Einstein told us so, and there are good reasons, and it works out. But in fact, you can derive, if you solve these equations, we call them construction equations,

you set them up only with information extracted from the matter action. You set up these construction equations, you solve them, and their solution give you the action that is Einstein-Hilbert with a cosmological constant and, of course, a gravitational constant undetermined. So you can say, aha, well, we didn't learn anything new. And no, we didn't. But we learned new that if you just say Maxwell and you want the background to evolve together with it, it must be. Einstein. It must be Einstein Hilbert. Now, is that true with other fields, other standard model fields, not just Maxwell? Yes, it also, we also investigated with other standard model fields. Of course, if that hadn't

also yielded Einstein theory, there would be something terribly wrong, either with our construction or with the physics as we do it. No, indeed, everything works out, even if you take non-abille-engaged theories, actually had a very good master's student, Alexander Witts, but he worked on that and we figured out and know nothing new comes up. We also try to use the same method to break, because you see what it is, if you say you have the matter action, say you give the background constant like an ITERMU and Minkowski metric, right, flat space, then you could count these itamino as a constant, but the idea of gravity is to make it dynamic. And what we then did is say, you say, okay, but there are other parameters.

parameters in matter actions, maybe for them we can also predict dynamics. And that would, of course, be interesting, right, if some neutrino masses or something like this. But no, these are things we had a lot of hopes for once the actual thing worked, what we wanted to do to predict the gravity or to derive the gravity theory from a new meta theory. Okay, so look, the thing would be as far. Let me tell you a fantasy story, okay? Okay.

The fantasy story is that tomorrow we look into the universe and we see somewhere, either directly or indirectly, a light ray split, right? So like in a birefringent medium. Well, again, directly or indirectly, we could see it in the hydrogen spectrum and stuff like this. But let's say for simplicity, the light ray splits in space as it is bent in space time, you know. Yes. Before Einstein, if somebody would have said the light ray bends around some, something without there being a lens, you would say, no, in materials, this is possible. It's called glasses, right?

But without any material, how would it happen? Well, so again, it's a fantasy story. Assume the light ray would split in vacuum. So one photon is coming, one photon is moving. Let's say, let's say classical, a light ray, a light ray, a laser ray, something like this. I mean, it's fantasy anyway. But if that would be seen directly or indirectly, general relativity, And you can ensure it's vacuum and stuff like this, right?

General relativity would be dead because a Laurentian metric does not support the splitting of light rays into different polarizations, right? So with one observation, you can state the death of general relativity. Sure. I mean, with the usual caveats, is there another effect we don't take into account? Yes. You're talking to a theorist, right? So I can dream. So let's say something that is like, but if something that dramatic happened, or another case,

so many years ago when I developed this theory at the Max Planck Institute for Gravitational Physics, there was this group in the Grand Saso Laboratory. They ran experiments, so the renowned group is a renounced group, obviously, and they measured faster than speed of light neutrinos that was announced. And that was exciting and it was a serious group. I think something in the end somebody forgot to unplug something or plug something. That was the reason

so there are, in the end it was clarified. But I mean if that had been true and actually that day Herman Nicolai, the director of the Max Planck Institute there at the time, he came into my office and said can you do this with the theory you're working on

and I will of course he was my director and I say of course you can do this. We could do this we could do this but what we would need is a particle phenomenologist, right? And like, I mean, all physics is phenomenology, a particle phenomenologist to write down a really good model for it. And that good model couldn't be built on, if it was really in vacuum and so on, couldn't be built on a Laurentian metric. It needs a more refined structure. For instance, you could take a fourth rank tensor as a

geometric background with symmetries like the remand tensor, you know, and symmetric pairs and stuff like this. It's just the model. The point is we are prepared for something that hasn't happened yet, right? If somebody sees matter that cannot have, due to their behavior, a Laurentian background, but you would then phenomenologists would pretty quickly figure out what maybe the simplest background that could do that, then the question comes up, but what's the action for that background? It can't be Einstein, right?

Einstein is for a Laurentian metric or a metric in general. But then you would try to solve our equations, and they're hard to solve. In the Einstein, for the Maxwell case, we can solve them, we get Einstein. They're hard to solve, but the result would be the action for that geometry in the background that could support that new exotic, but the new required matter. So in a sense, we converted the physical question, what's the gravity theory that can support such matter into a mathematical question, solve these equations whose coefficients are constructed in various ways from the properties of the matter action.

That's the idea. That's the idea. And there are many technical issues. But, yeah, anyway, that was our idea. How can we actually say something about modified gravity that takes seriously some other physical observation we make about matter. And it's quite remarkable that this is possible in the first case. Yes. I mean, all of this builds on earlier work by Kukash and Title Boeim and ADM. I mean, there's lots of predecessors, but we kind of, in a sense, pushed it, pushed it there

because in a sense, I guess before people didn't have the motivation to look at this, but yeah. I have a quick technical question. So I can see how you can get the signature. You just talked about it with Maxwell, what makes a Lorentzian metric isn't just its, it's a zero-two tensor, it also has a certain signature, and then it's non-degenerate, and so on. But how do you get the condition that it is compatible with the connection? Is that a condition from the principal polynomial? Yeah. So this is also something iron. It's very elegant. Actually, Schrodinger did that, right? Schrodinger had a modified idea. It's again, Schrodinger had a idea of modified gravity, and he said, no, no, no, Einstein does this non-symmetric metrics,

but actually a deeper structural concept than the metric is the connection. And there are connections that come from a metric and they're more general connections, right? So that was Schrodinger's idea. Actually, a wonderful book by Schrodinger, a space-time structure, a very beautiful, thin book on general relativity

for a beginner. But there he follows this route and it's technically and conceptually wonderful. And then he assigns to this connection a deeper meaning than to the metric, you could argue. And he tries to make a theory for connections rather than for metrics as the fundamental structure. I think he even announced this in the New York Times or something like this, and Einstein was angry. I don't know whether it's historically correct.

This is the rumor. Anyway, it didn't work, right? I mean, we would know about it. And you can say, okay, all the idea with palatini action, all of these things come from there and relate to this. But it wasn't new physics. So the idea that a generalization of relativity goes via a connection if you go away from a metric is an old idea by Schroeninger, but it's not the idea here. Here the idea is, why do we talk about a space-time metric in the first place? Well, because of Maxwell.

Remember, the pluses and the minuses, say in the Minkowski metric are the pluses and minuses. Undergraduate students have to learn in the badly written three plus one decomposed Maxwell equations, right? but if we have matter that has a different causality than the regular cones, it would be cones that are a little bit more folded like this and so on. I'll place an image on screen at this point. Okay, okay, okay, okay, okay, no? Yeah, yeah, yeah, yeah, no?

And again, this is all theoretical considerations, right? Sorry, what did you ask before? Oh, I was saying, okay, so in the principal polynomial, I can imagine how you can get symmetry or anti-symmetry conditions. I can imagine how you can get signature. I can imagine how you get that is non-degenerate or that is degenerate. But I don't see how you get compatibility with connection as a condition of the principle polynomial.

But there is no connection at all. So, okay, what we call signature in the metric, if I look at it from the point of Koshy's surfaces, is to say that the theory, the metal theory that has a underlying background of a Lorentzian signature metric, that the theory is hyperbolic, globally hyperbolic. You have a cushy surfaces, so on. For a metric, it translates into this algebraic condition of having a certain signature,

1.3 or Lorentzian signature. For a non-metric structure, say a fourth rank tensor that could produce as a background to Maxwell theory, it could produce birefringens. There, still the global hyperbolicity of the matter theory is the condition we need, but it translates not directly into a signature condition. There are many other algebraic classes. So the point is, well, this is a whole thing

one has to work out. But it's the global hyperbolicity of the matter action tells you how to construct Koshi's surface is also for the matter action and remarkably also how this matter action must look like. So that's now. So at the end of the day, it's a very simple idea execute.

It has some technical hurdles in between, but it's a cute idea. And that's what it is. I mean, we have no claim that it's realized in nature. But in a sense, if really tomorrow somebody discovered something like I said, something that would be... Faster than light neutrinos.

Yeah, fast and light neutrinos could have been a possibility or by refringence in vacuum. Because some people say the whole theory of physics or Einsteinian physics is dead. We don't know what to do tomorrow if we saw faster than light particles. Exactly, exactly. I mean, general relativity couldn't be right, right? I mean, it builds with Laurentian metric.

This is just not possible. but if then again I'm not saying I could then immediately say how the gravity theory looks like but if some phenomenologists makes a really good model for this matter now so say standard model grade model right so to be incorporated the standard model let's fantasize and then we have a standard model with faster than speed of light neutrino say okay say then I would say now is the time to invest the time and money to set up our construction equations, well, that can be done over the weekend, so to speak, but then to solve them. And there you might really, I mean, you know, it's very complicated equations, but if you solve them, you get the gravity action

that would support the new full matter model. So we think, we think we showed that. Okay, I'd like to talk about your mind. When we spoke a couple months ago off air, we were speaking about how they're toy models in physics, and you were thinking, okay, well, what is it that compels a theorist to go in this direction? That's the question you would ask them. If you're getting too abstract, you just want to say, well, what compels you? Can you please talk about that? And how that guides your own research, maybe your teaching, maybe your philosophy. Yes, yes. Well, you see, in research, we're always tiptoeing the line between the known and the unknown, right? If we stay firmly in the known, we're not conducting research. We say, that's the

theory, that's how we look at it, nothing to change about it. Yes, there are problems, but who knows what that is. I'm simplifying here, but especially as a theorist, we're tiptoeing on this line where we're always with one foot in the certainty and one foot in the unknown. And so if we want to do research, we need to introduce a new idea into something. Otherwise, at least as a theorist, what are you doing? There needs to be one new element in it. As I mentioned before, I believe if you say, oh, I have five new ideas, how things could be different at once, I think it's not manageable to deal with this. At least I would claim that for myself. So let's say one idea. And you know, if you have two brilliant ideas and it works, my congratulations,

I just, you know, it's an indicator, right? So tiptoeing, tiptoeing the line. But But now you have a new idea and you have the old, first of all, are they compatible or not? Well, if they're not really compatible, you have a problem to solve because you have to incorporate this new idea without, and thereby you typically have to also change the old theory, right? Yes. If the new idea is not fully compatible. Okay. And typically, this leads to a contradiction in the old theory or to a prediction of the modified theory in three lines. I exaggerate, where you say, yeah, but nature isn't like that.

Well, what happens then? Well, you throw away your new idea, right? Hopefully. No, so if, no, hopefully, yeah, you should. You should. Oh, sometimes people don't, right? Sometimes people keep saying, oh, but the idea is right. It just hasn't been shown yet.

And it might be true, you see. But if you have a new idea and it, if you start incorporating or modifying the old theory or aspects of the old, old theory in that according to this idea or incorporating this idea or incorporating this technique, you get something slightly new, which is not immediately obviously false. And then the question is, what do you do next? And the answer is, well, I hope at least usually it is, what you're doing there, if you keep looking at it and keep working with it, you will be forced, you will be compelled to maybe

to do a next step. For instance, you could have a new idea, let's do this and that, and then it tells you, oh, but unless you choose this object to be of this and that class rather than the other class it could have, then there's an immediate contradiction. Well, you already have learned. You need this other class of object and so on, another algebraic class or whatnot, yeah. Can you be more concrete? Can you give a specific example?

Maybe you're trying to tiptoe and be diplomatic, a non-offensive. No, no, no, no, not really. You see, that's the problem of this. All I say are vague ideas of how to not run away in a theoretical direction that doesn't lead nowhere. Right? It's very easy, you see. And I think a lot of physics maybe always has been, maybe is, including some of my own work in the past, is having an idea and going here and going there and trying to implement it here and here. It's a little bit what guides us?

What is the guiding principles we use, right? And yeah, more concrete. Yeah, let's say what I said in the beginning with these Port Hamiltonian theories where you reformulate classical theories such that you have the energy flow, the fundamental variables immediately relate to the energy flow between subsystems.

It's a reformulation of Hamiltonian mechanics. And I said, let's apply to quantum mechanics. but then we played around for a few months with energy flows in quantum mechanics and in the end we discarded the idea well essentially because in quantum mechanics you don't have a continuous energy variable that describes the system.

You have a Hamiltonian but you see if you have a Hamiltonian, let's say you take two states that are two different eigenstates of that Hamiltonian right, two different energy eigenstates and you take their superposition. Well then you have a new state, right? superposition of two states is a new state. But for this

Hamiltonian, upon measurement, you would get either one or the other value, right? You would get one eigenvalue or the other one. You wouldn't get the average of the eigenvalues, right? So this state itself, this superposition state of two different energy eigenstates, I don't know how we would say

it has this and that energy. It doesn't. So that means most states, almost all states in a hill, Hilbert space, let's say a finite dimensional one, almost all states in a finite dimensional Hilbert space do not have an energy with respect to a given Hamiltonian operator, right? Yes.

That's a fact. So, and then this whole idea of construction of ports for reasons that are not clear now, but fails because you would need a continuously in time evolving or even differentially in time evolving. quantity, whatever it's supposed to be, you would need in order to create these parts. But energy is not the right thing. And while energy plays in quantum theory also the role of the generator of the time evolution, like in classical theory, energy is the generator of time evolution, it is not the right

quantity to introduce these ports in quantum theory, right? So what I say, you need to, you can't just push an idea, I want energy flows, I want to study energy flows, you need to react what the theory reports back to you if you try to modify it like that. That's the rough idea. Look, I think we're talking a little bit too much about how I think one could, should constrain oneself to make good progress without running away uncontrollably, but this is very personal.

You see, if I thought this had deep philosophical value, what I say, I would have written it up and published it. Okay. But it's something I discuss a lot with my research students. I tell them, well, ideas are cheap, very easy to have, and get rid of ideas if they don't seem to work out or put them to the side, right? and try to have some standards of how you push ideas forward. I think that's what it is.

And every researcher should have his own set of rules because otherwise we're all doing the same. That's not good, right? So variability is good, right? Yeah, yeah, yeah. So therefore, I want to qualify what I say. It's not out of not offending anybody, but I think it's the truth, right?

These are basic ideas of how to orient one's own research. but that's very personal. Yes. Okay, so these are yours. You're not advocating that if people aren't following it, then they're doing something incorrectly. Oh, no, no. Look, I mean, that would, how do you say?

I mean, look at the history of physics in the 20th century, right? I mean, the revolutions that were there and what people predicted about it before, it would be totally absurd to say, because it's not on my radar right now, or I can't imagine this to be a good way, to say nobody should do this, of course not. I tell it to my students, though.

I put this on my students, and my justification is, well, they don't have to listen to me, right? And they can take it as one element of what one could think about, and hopefully they add something else to it, or they reject it, or they take it over, and they learn from other colleagues too, from other sources too. So I think we should speak out about these things. These are subtle things, you know, probably somebody not doing research doesn't know the hell what I'm talking about here. But these are ideas and we should put them on our research students, but not to force them to take them, but just as, because otherwise we can't teach them. You can teach people by telling them your own ideas about something and they are less attached to them. They might change them a little and have more success, for instance. speaking of teaching

you're a world-class teacher you've won several awards some of the most prestigious awards yeah no I was very lucky because it was sort of history I tell you history of one of these awards I was very lucky to have been invited by the German physical society

they have a youth organization actually the German physical society is I think the biggest physical society in the world is a little bit funny but they have many many members and they have a youth organization and they invited me in 2015 to give a big lecture series on general relativity

and so I did and we did this in Austria and they gave us a fantastic place to do that and out of that came these gravity and light lectures on YouTube and we put them on YouTube and we

didn't think anything of it and they took off quite a bit and then there were other lectures taken in Erlangen University on the geometric anatomy of fundamental physics or whatever I call it, geometric anatomy of theoretical physics. Yes, yes, yes. It's about differential geometry. Yeah, yeah.

It was one of these things that they do in Germany, which is fantastic. Every now and then you can give a lecture that you want to give a lecture course, right? You give, of course, lecture courses that need to be given. They're part of the curriculum. But then there are optional courses where if you have an idea and you want to do it, you do it. and I just wanted to teach people differential geometry from the ground up as it's then used in theoretical physics

and so it's more of a applied mathematics course than it's a theoretical physics course and that kind of took off once the university put this on YouTube or first on iTunes or something like this and yeah that gave me some international followership and also national followership and then at some point I was proposed

for the Ars-Legendi Prize which is well it's probably the top German teaching prize for university teachers and yeah they were so kind

to find that in that year to meet the best candidate they liked the best at least yes yes so you have this geometric anatomy course yes

and that didn't have a curriculum before you came up with that that's right that's right yeah do you have other ideas for courses yeah all the time All the time, but the question is whether you can give them, right?

Whether the university gives the opportunity to give these courses or give these courses or when they say, no, we are already busy with the curriculum, right? But what's in your mind? I'm curious. So, for instance, for people who understand differential geometry, what's extremely interesting about your course is you started in the first lecture with propositional logic. Right, right. And then you built up to the empty set, I think, in the second or third lecture. No course on differential geometry starts with the empty set, let alone propositional logic.

You started from the ground up. That was extremely interesting. You're exceptionally clear, exceptionally clear. I absolutely love that. I think I've rewatched those. I may have rewatched that as many times as I've rewatched Seinfeld, which is many times. Oh, wow. Yes, yes.

Well, I mean, look, it was a course, because it was an extracurricular course, I had some mathematicians in it, some physicists and so on. And if you want to, I can tell you the reason it was very simple. If you want to tell people what a manifold is, you need to tell them a smooth manifold, need to tell them what the topological manifold is. They need to know what the topological space is. Well, the topological space is a very simple thing. If you know that you have a set and then the set has a power set, how do you know the set has a power set? How do you know the power set is a power set? aha, you need to have some set theory. Now, if you do naive set theory, you have all kinds of contradictions in two lines. If you say a set is a collection of elements, that sounds good, but that doesn't make any sense. First of all, I didn't tell you what a collection is. Second, I didn't tell you what an element is.

So to define a set as a collection of elements is not particularly insightful. And in fact, as we as well known, I mean, this is naive set theory is contradictory in line two. And so ultimately, if I want to, to tell people from a broader range of backgrounds, also in physics and mathematics, what are we actually talking about here? I have to tell them also about the axioms of set theory. Now, that's a thorny issue. It's a very kind of complicated issue if you really go deeply into this. But if you want to

do it, you can do it. Actually, if you wanted to explain what set theories, you need to write down the axioms. If you want to write down the axioms, you need a formalism in which you formulate these accents because if I then formulate them with other flowery words I'm as bad as I was before and so there was the idea, okay, we have to do some quantors and so on and some propositional logic

before. Well, that can actually be pushed even deeper and I didn't do that in this course to some first order logic and so on. So there are many steps. Ultimately, I mean, it's now known it was Hilbert's dream, but it's now known ultimately it's very difficult to find a really foundational beginning from nothing.

thing, even if you say you have an alphabet and symbols and so on. But what I wanted to do, at least I wanted to go beyond the usual undergrad or even master level idea about what a set is for these students. They're all very excellent students in Erlang back then. They had also this elite graduate program and so on. I mean, super students, top top students. So I could deliver that to them. It was an attempt at some type of completeness of the presentation

starting virtually from nothing. And then you see, rigor in mathematics, of course, extremely important. But for me, the best rigor is the conceptual rigor. I mean, of course, you can write down things with absolves and deltas and can make it very, very, very, very rigorous. Before that, actually, you need to be conceivable. If I say set, and we just have a vague idea about it, and then I build a big edifice on it. And then at every other juncture, I have to say, well, now you can show that, whatever, that a vector space always has a basis, even an infinite dimensional one.

Well, how do we know that? Well, ultimately, this comes from the axiom of choice, right? Well, why do you have the axiom of choice? Because at some point, I required it. You see, so I wanted to give it the full picture without claiming. that this is at the same time a foundational logic course a foundational set theory course

because you could probably spend your life if you want to fill in all the details but at least I want to be a bit more clear about what all the assumptions are and this is a general because you asked about teaching this is something that's very important in teaching and I always joke with people

when they ask me I say my assumptions in teaching the foundational assumptions are two A, students no matter whoever comes to you, beginners, master students, they know nothing, nothing at all. And second, they're infinitely intelligent. Okay. Okay.

So both assumptions are slightly wrong, right? Students do know things and they're not infinitely intelligent. But I present my courses a little bit like that. And I like to develop things from the beginning because I don't know what they know and I don't know in which way they know it. So I like to start from scratch. Yes. You even told me that instead of starting with a textbook, you'll go into a room, a blank room with blank paper,

and think, how can I teach this subject? Right. Well, I mean, I think that's, so that's another thing. I think, well, it's all not my original idea. It's a very old idea, you know, unity of research and teaching. I think at university only people who bring to the table some research grade thinking should actually

teach the important lectures at least for the students who take the subject very seriously. Let's say it like this. If some biology student has a physics course that can be excellently taught without much

ado. But if you say you want to educate the next generation of theoretical physics and maybe you hope that some of them might make groundbreaking discoveries or some. thing, well, we better give them our best and don't just repeat how we learned it.

And so how do you do this? And how do you not just follow what is written in the textbook? I think you say this in one of your videos where you say, oh yeah, we often just repeat what we have heard. Was it you who said that? Or somebody else you interviewed

and you can recognize this if you would never use this phrase somewhere else? Exactly. On an equal footing, I think it was. Yeah, there are many such phrases. I keep a catalog of them. All possible paths seems to be echoed due to doctrinal inheritance without thinking, just like the word equal footing. Time and space are relative and treated on equal footing. Time and space are supposed to enter on equal footing. What is equal footing? Have you seen a mathematical definition of equal footing? We're supposed to be rigorous. Yeah, yeah, yeah, space and time on an equal footing. Oh yeah, indeed. What the heck is that supposed to be? Okay. Nevertheless, we all do that.

But indeed, I would say one way to make your research much better is to try to detect where you're using such phrases in order to justify something. Well, they're of course placeholders for a better explanation. Interesting. Sometimes you have a much better explanation. Everybody knows the better explanation and then you refer to it on an equal footing. And then everybody, however, if pushed, would give you a brilliant explanation to it. Then you're allowed to use this short term.

Yes. But if it's just used to gloss over your own ignorance consciously or unconsciously, one should eliminate it. But we all do this. So first of all, if people ask me, how do I teach better or very well, I say something like that, right? And then say I, I mean, general relativity is one of my expertise.

So if I teach a course in general relativity, I can teach this in any number. And in two or three different ways. Okay, not any number. In two or three different ways. And then I first of all think, what is the best way for this group? And for instance,

I once taught, I had the task to teach physics to material scientists. They're not hardcore physicists, but they need some good quantum mechanics. And it was a quantum mechanics course. And I decided that because they need the energy bands, you know, in solids and so on, I teach them a tempered distribution theory, right? Schwartz spaces, tempered distributions and so on, a distribution theory, which you would say is a very advanced subject. I mean, most physicists don't hear this in their undergraduate.

graduate studies, right? Right. But I decided for the applications we want to do, we just need this. I taught this to them. It's not that difficult after all. They all did pretty well, right? So sometimes we must not shy away from using very advanced methods, of course, explained very well from the bottom up, also to people where you say, oh, normally they wouldn't use

this theory, but I think they should, right? And then I usually textbooks, many textbooks don't offer them precisely the line you want to take. And I think it's also good for the lecturing style. If I don't take a particular textbook, certainly in a subject I know very well, yeah, I take a stack of paper in the summer break and I start sketching what is a good storyline. but I mean scientifically, conceptually rigorous storyline as today one would have to present it

in order to get it accepted in a very good journal if this was a discovery, so to speak. Right? So, yeah. So I try to apply research-grade thinking to the design or redesign.

That's the better word. The redesign of also very established courses. And very often you change the order in which you teach subjects, what you think is an advanced subject is typically something you learned later. And a less advanced subject or topic is one you yourself learned earlier. But that's not a particularly meaningful classification of advanced and not advanced, right?

For instance, in Erlangen, I taught the classical mechanics course. I decided to teach it using half of the semester to develop differential geometry. Then I did the mechanics, and in the last lecture, I could tell them what general relativity is. Okay. That worked pretty well. Colleagues said, you're crazy, right? I mean, you have bad passing rates. Not true.

We have excellent passing rates, or very good passing rates, at least the usual ones. Yes. And some of these students did just absolutely spectacularly, because once you do it properly, you see, you should never do something because it looks fancy. Well, we did it with differential geometry. No, I need to tell you what a co-vector is if I want to talk about momenta, right? Because momenta, canonical momenta are co-vectors. They're not vectors. How do I tell you?

Do I tell you about this in vector space? I could, but then people think about the position vector. But position is not a vector. And you can't get away from this structurally, conceptually wrong idea unless you immediately put it in the setting of a manifold. And then, of course, if you do then Lagrangian mechanics or something, you anyway, generalize coordinates are nothing than what the differential geometer calls coordinates. They're not generalized. They're coordinates.

It's actually the Cartesian coordinates, which are very, very special coordinates, is existing under very special circumstances, and then you don't have to choose them necessarily, right? So you put the whole conceptual basis properly rather than giving people the wrong idea. Because once you taught something in a way that is out. ultimately not correct, ultimately doesn't carry you far and you have to replace it later on any way. You shouldn't teach people in the wrong way.

And that's why it's important to be a researcher because you know where it's going to be carried, whereas if you're just a general teacher, you don't know the forefront of the field? I think many, both, both somehow. Yes, it's because you need to know where you want to go. No, but the other thing is, you see in research you learn this oh could I explain this this other way oh this is a new take on let's let's explain it that way and but what if it's almost only slightly wrong if you do something in a new way you need to apply the criticism of a researcher to it to say is this really a valid derivation and is it really as general as the one that you get otherwise you know

So I'm a big skeptic of teaching methods as a one-size-fits-all method of how you teach better, whether you teach the piano or you teach general relativity. I do not believe that is true. I would never dare to recommend to a piano teacher my teaching methods because I think, oh boy, I mean, you're a concert pianist. I mean, the really good piano teachers that people study with, they're concert pianists. Right. they don't have just a fancy teaching method

they can play it at a really great level and then they have methods to teach them which you can't come up from general considerations so in physics is also like this teaching I think good teaching always springs from a deep understanding of the subject coupled with an awareness of what you already explained and what you didn't

Sometimes people say, oh, that person is a really brilliant scientist, but he doesn't teach well. That's often the case. In the University of Toronto, it's infamous for having great researchers, but they don't care about the teaching. Maybe they would be great if they... Okay, okay, that's, of course, one possibility you don't care about it or you hate your students. No, but I think a more benevolent view could be. They want to explain it well. Let's take those hypothetical ones who are...

are really great, who really know their subject, and who really want to explain well, but don't. I think the most likely cause in those, and I know such cases, okay, but not many, actually. And I think there it is, they're not fully aware what they already explained. They explain something. They say, oh, I forgot this. And then, okay, you forgot, okay, I put this as an aside. That happens to all of us.

But then it's the next thing. Oh, and I did this. And as we know, and as you could show, but, you know, then it's a mess. It's all over the place. Left as a homework exercise. There's a homework exercise. Yeah, yeah, yeah, yeah. Well, that's an easy opt-out, nah?

I say, you show this young man or young lady. Yeah. No, seriously, it's, I believe it's certainly the implication that goes in one direction. I think you need to know your subject really well, really deep, really far, in order to design a really nice lecture course on it. It's a necessary condition. It's a necessary condition. And yeah, and we all are fallible, right?

I mean, I think I eradicated these and those problems from this subject and teaching it and da-da-da and how it's strangely taught and things like this. But then do I know, right? I mean, then somebody else must come and do it better. But at least we do better and better. I think if I can improve 5% of how I learned a subject, and it's really solid, a solid 5% improvement, I think we. We can tap our own shoulders, right, and say, you know. But that's what we owe our students. That's what we owe our students to really improve it

because otherwise they must have a better starting point than we did, although my starting point was excellent. I had brilliant teachers everywhere. Some of the privileges in life, well, for me, I think for anyone is running water, air conditioning. And if you're a physics student, it's taking one of your courses. Oh, wow, immediately after water and air conditioning.

I'm not even kidding. It's an absolute joy. I don't know. So why do you care so much? Why do you care so much about teaching? That's a very good question. You see, I think, and then I think it's a good lesson to maybe young people out there. When I went to study to university, I was just so wanting.

to understand physics and to also make new discoveries and mathematics as well. I studied both in parallel. That was my aim. And if I now look back, even before that, when I was a younger man, I did lots of sports, and I actually started working as a trainer already at a very early time, myself being active, but also teaching youth groups and so on. And if I looked back in my life, I think from the age of 15 onwards, I have always in one or the other capacity in sports or elsewhere.

I have been teaching. I have been teaching. And more or less what I told you is my philosophy about teaching today was my teaching back then. I wanted to make them really good. I wanted to make them really good competition sports men and women and things like that. And I had this idea. And I had myself good teachers in all of these fields. And I passed this on, but not out of a.

reflected moral imperative or something, it's just in retrospect, I saw this. And when I went to university in the first semester, I had little meetings in the library where I told my fellow students things that we were currently discussing and that I understood before, and I taught them on the blackboard in the library, in the university.

So I always did this, and it was natural when I was at university and at research institutes. I was also invited. Well, Germany has a very sophisticated talent promotion program. We have something called the Scholarship Foundation. This is a state organization. They select 0.1% of the best university students in any subject.

And they get special summer schools by top professors and so on. And we have a similar thing for last year high school students. students where every high school in Germany can send their one best student to one of these academies and they can choose beforehand courses. And for many years, I've taught their course on general relativity to last year high school students. What did I do? Well, taught them differential geometry, taught them differential general relativity, right?

Wow. So I think, so I have always done that and my aim is always to make the people as good as can be. So I'm a little of an, I don't know, a little bit of an elitist this way. I think, look, you come here to study. I give you the real thing, but the real thing is hard. And I do my very best that I deliver to you so that you can live up to the demands I put on you. Right. So I put high demands, but at the same time, I know I'm responsible for teaching them. And in the end, if it fails, if not a significant proportion of them takes a big benefit, which from your kind words, I take it

many people do, right? It's now more obvious through YouTube than I'm at university too, of course I got always good evaluations or very good evaluations, not a doubt, but then I think I succeeded. I succeeded.

I once had a student who I met in Erlangen on the street. Years after he took my classical mechanics course I told you about that started with differential geometry and he stopped me and he was very kind and he said, because of you I stopped physics and I thought oh really he said no no it was the best thing ever your course was great but I just realized I'm not good enough and I said maybe you should have tried more

no no no absolutely not I'm so grateful to you because I realized I should have understood it so many in my course said it was great I thought it was great but I just couldn't do it so it's it's a weird how do you say compliment but but it was genuine I think it was genuine he meant it And I said, again, apologize, and I say, I hope I didn't do anything wrong and so on. He said, no, no, no, no. He's very happy now with what he's doing and so on. So anyway, that's it. I think putting high demands is important also to motivate people, right?

Today we have a lot of talk in teaching circles about people being demotivated by high demand. I think high demand, if it's justified from what you yourself, as a teacher deliver is the biggest motivator of all because the good students are pushed beyond what would be their comfort zone and the not so good students see that a good number of students succeed very well so it is possible yes maybe they need to up their game maybe they need to spend more time on it or be more courageous or be less disheartened or you know Anyway, that's my view and I So maybe that's the nicest outcome of a conclusion for me

From all these awards, I got others And the many, many, many, thousands of emails I got of course, not of course, but I got kindly on the And many I couldn't even read Because it was just too much at some time One year I got 6,000 emails on these YouTube lectures It was impossible for me to read they all go to some folder

but I feel very guilty about because some people write very nice things if I look at it I just can't cope with these masses of emails so maybe here my thank you to everybody who wrote

and addressed very kind words to me but the nicest conclusion of that for me is I think my method is at least justified right so to high demands and trying to deliver what it takes to satisfy them

one of the the reasons I resonate with you and your lectures is that I care about mathematical rigor, although you don't just care about rigor for rigor's sake or formality for formality's sake. You care about it because what is required to understand subject X, you have to drill down to point Y. So you don't care about point Z and A, B, C. You care about point Y as it relates to X, not just what's around Y. Yes.

Yes, although also you see the coherence. is actually what we're saying, well, true is it really conclusive what we're saying? Is it compelling, right? So we set up a theory. We start with assumptions. They could be wrong or false as far as physics is concerned. Wrong or false? No, yeah.

True or false as far as physics is concerned. But then once we set down our assumptions, is everything else we say, is it actually conclusive? Is that actually at least compatible with what we said? And there you need to be very careful. because many things are plausible, but just not correct. And if we start hand-waving, it's always very cool, right? It's a little bit, sometimes I blame it on Feynman,

who is, of course, quite a charismatic character, right? And physicists like me too, right? Someone's like, no, this is like this. You can think about it like this and that. Yeah, you can think about it like this, but is it correct? Does it fit with everything else? In the end, there is no way beyond the rigor, right? the rigor in how things are connected

and so that you know if you pull a little string in the theory here how does it move in the theory over there, you know. Yes. Why don't you be concrete? Why don't you give an example of what ordinarily is hand wavy, but then when you examine it, at least to something wrong? Oh, God, a million things, a million things. It's very easy to, for instance, talk about the center of mass,

of two particles, right? Two particles are flying around, center of mass. Now think of it relativistically and blah, blah, blah. For instance, relativistically, there is no center of mass. Because center of mass, of course, requires you to take the positions of it. It says they're flat space, not even curved space, flat space. You need to take the positions of the two particles. Say, Minkowski space, there could be position vectors in quotation marks, okay.

And then you find the middle. point, right? But for parties at the same time, you find the middle point, right? The position positions at the same time. But we know there is no similarity in objective similarity in special relativity

even. So in special relativity there is no center of mass. That's a concept. Kant, I think it's interesting because Kant says this is a what is it? a synthetic a priori or something like this,

something like this, or even analytic, an analytic truth. No, it's just not true. Nature isn't like this. Nature doesn't have similarity as a foundational thing that you can talk about that makes sense, right? So it's very easy to give arguments, hand-waving arguments, if you don't define it very precisely and then check whether it's well defined

under whatever yeah, you've got to be very careful. So the hand-wavy stuff is very, very dangerous. I mean, even the written down, the rigorous stuff is very dangerous. We all know this, right? We call it a side error or some other conceptual error. So it's not like because it's written in mathematics

is necessarily conceptually coherent or consistent. Of course, then there is ultimately a mistake. somewhere. But it becomes mere talk if we start hand-waving. Of course, if you do formalism and you don't occasionally bring it to life, say, to the students, right? That also doesn't work. But we really got to be careful in how we talk. I understand we don't have a blackboard and we don't have the ability for you to draw right now. But if you were to explain to the audience and they're educated in physics and math, what is the problem with quantum gravity? How is it? would you say it?

Oh, well, that's... Why is it so difficult to make gravity into a quantum theory? Should gravity be a quantum theory? Exactly, exactly, exactly. I want to hear your viewpoint. Well, I mean, okay, I have, I don't think I have anything new to add,

but I think if you ask me as a little examination, I'm happy to oblige. It's PhD defense. Okay, we have a nice theory of gravity, general relativity, is the best one we have. We use it. It's subject to the interpretation of the data we get and so on.

It has the following form. G. Mu Nu is T. Mu Nu, and G. Muneu is the space-time curvature, and T-Munu is the annoyingly metric containing energy momentum tensor of the matter. So the metric is also in the T-Munu, but very roughly speaking, the T-Munu, of course, is mainly determined by the matter content and distribution in the universe, say. Now there are many issues to be discussed. Well, this is already a godlike view on all of space time. You would actually have to go to 3 plus 1 to talk about evolution and so on.

But let's leave this all out. The problem is on the matter side, as long as you have classical fields like Maxwell feels classical. This T-Muneu is a classical field. It's a tensor on a smooth manifold. It can be beautifully equated to the Gmuneu, Einstein curvature tensor on the manifold. This is mathematically meaningful. and then you solve

that there's a big elephant there but you solve these equations and you get the prediction of how the gravity and the matter in the end work together you also need the matter field equations but unfortunately Planck got us the idea

and then people said oh yeah that's true there seems to be no classical matter all the matter light and particles and so on they're all quantum. I don't think this is to be doubted. That seems to be the case, at least the way we look at it, fine.

But then how do you build this T-Mu-Nu tensor on the right-hand side of this equation G-minu is T-Mu? Because now all of a sudden there are quantum fields there. Okay, big question. And then the easiest, and then you can say also,

in a sense, this T-MU is also more like an operator operating on some vector space Hilbert space, Fox space was not no matter how you, there are many, many issues perturbative theory and so anyway, you have this on the T menu side and then you say that doesn't fit you can't equate a classical tensor to some operator the equal sign makes no sense anymore.

The matter you want to put on the right inside is quantum but the right inside is formulated only for classical fields. and many things you can try oh we take the expectation value all of this doesn't work right all of this doesn't work obviously so then the

in a sense laziest idea and I'm unnecessarily provocative but the laziest idea is to say why don't we make the left hand side also an operator or a bit more seriously formulated maybe gravity

should also be quantized then we have both theories matter and gravity in the same formalism, and then we can write down a new equation, roughly speaking, in this new formalism, where now the matter and the space time speak the same language, or we speak the same language in looking at them. And that's the origin of the idea we should quantize gravity, at least from a formal point of view. There are other good reasons. what happens with black holes, what's with singularities, Big Bang, was there a Big Bang?

Is the theory actually to be extrapolated up down to the Big Bang? Many questions, there's Heisenberg's idea what does actually mean to talk about space time as a smooth manifold and the metric measuring so precisely. How would you measure? He would use higher and higher frequency to be more precise, say photons to check something. But then that would disturb. the space time, the very space time you'd like to measure. There are many, many reasons.

You can concoct a whole bouquet of reasons why you think quantum gravity, a quantum formulation of gravity might address, solve, enlighten us concerning all these problems. Okay. But this comes ultimately from the way we look at things in the first place. And I think there is no really compelling reason why gravity has to be quantized in the first place. Maybe something very else is going, something very different is going on, right? So that's the first thing. Do I know?

No. I don't know. I don't know better than anybody else. But it's not so clear that we want to quantize gravity. Look, I mean, there are many things one, many questions one could ask and probably none of them is original. If you do quantum theory, ordinary quantum three in three dimensions, I'd say in Euclidean space and you do, I don't know, quantum information to deal with qubits. What your quantum systems are, at least the constituents of it, the opposite, they must come from representations, if it's elementary, if it's elementary irreducible representations, of the universal covering group, of the symmetry group

of the classical space in which you think about this quantum system. So this is how you get the spin one-halfs in three-dimension. So they come from the three-dimensional Euclidean group and you look at all the unitary, well, projective unitary representations of the group, of this Euclidean group,

or technically you can also look at the universal covering group. that would be then something like say without translations SU2 right so if you just take SO3 universal covering group

is SU2 you find the unitary representations of those and then you know how you can build your quantum systems that live

that live in such a space sure okay and the same then works in with the Puanca Ray group and Wiener figured that out

what are actually the possible representations of the SL2C of SO-1-3 is SL2C of this universal covering group of the symmetry group of physical space.

Why do I say this? Well, the whole standard model is built on this. It has very clear conceptual reasons why it must be done like this. It's not an abstract thing. But that means classical space, be space time or space,

you need to have a concept of classical space in order to talk about quantum matter in the way we do it. Aha. If you now say, I want to quantize this classical geometry or

whatever ideas you might get, the dynamics of it, the germany, well, in a sense, undermining the very foundation of

what brought you to the quantum matter. Right. You see? So that's not immediately plausible to me. And of course, you can always say,

yeah, that was just the ladder we used to get up to the theory. and in the end it all works out and then we throw away the original idea of the space. But you know, the point is

I don't know, probably nobody knows, but these are also, I think probably to most of the ideas why gravity should be quantized, one could at least make intelligent counter arguments why that is a funny idea.

Okay, so maybe something else is at work. These proposals by Oppenheim and collaborators on this idea that gravity is not to be quantized but rather it's to be treated as a stochastical theory classical stochastic theory interacting with the matter

that's an interesting idea I find because the quantum axioms and never yeah the quantum axioms say something like this You say, you can't measure, you can't predict what the measurement outcome is, but you can give a probability distribution, a classical probability distribution. So the axioms tell you, classically, you're not connecting to a classical theory in a pure state, you connect into a classical probability distribution.

And I think that's a little bit the idea they have. I hope I don't misrepresent that. You should ask them. But you see, this is a priori, also a very plausible. way whether this works out and whether what that works. That's not the question. The questions, there are many other I don't know many. That for instance

is a plausible thing we say from the quantum axis. The quantum axioms at no point instruct us to quantize the space time behind it. Yes. And maybe this representation theory, construction accept construction speaks against it. So no, no, you need

this classical space idea. Otherwise, we don't know what we're talking about in the quantum theory either. But the quantum axioms talk about the contact to the classical word as a kind of a stochastic system contact. Maybe there's something in the idea, you know? So the, and I mean, it has failed for so long to bring this idea to quantize gravity to success. But, I mean, who knows? I mean, there's a lot of criticism of string theory these days and, hey, I share it. But you know what?

maybe they pull it off you know we don't know we should be not very not so judgmental about I mean very clever people spend a lot of time and very clever people in the past

were wrong about what they wanted to do although they were very clever and sometimes people are lucky I also wouldn't discard the string theorists who keep pursuing it I wouldn't

but who am I to say who is anybody to say I think that's the good thing about science that everybody can and should do what they think is right and it is enough if one of us at some point

is right about something and that's progress it's less personal less personal than we often take it of course yeah we all want to win a Nobel prize but we all won't right we all won't but it's a fantastic

the project of science is fantastic everybody does what they think they should do. And you see, I mean, in a sense, that has changed a little bit. There are also some commentators of that. Also in the German-speaking area, I think Sabina Hosenfeld and so on, she brings some very strong criticism of how the scientific system works these days and so on. And some of these points are just valid. Yes, absolutely valid.

Like what? Well, the old idea is freedom of teaching and research. There's a Humboldian idea. Humboldt had this idea when he was asked to, I think, to create a educational system for Prussia. Oh, God, my history. But I think something like that. And he said, education at university is not vocational training. It's not for you to find a job.

It's for you to learn to think. And if you learn to think and if you dig deep, well, I don't know whether he said it like this. Let me say what he wanted to say if he was alive today. Okay. If you dig deep in physics and you become a really good physicist and then you talk to a lawyer, a legal scholar who really dig deep on legal thinking, you recognize a good thinker because you learned thinking well in your subject and the other way around. And if you can do that and you go out in the world, you're infinitely more useful for a company who has a good thinker with a lot of skill and has learned to think deeply, to think critically. In the long run, that produces growth for the company, right, at least on average.

The idea that we educate people to conform with the expectations of current employers, well, the current employers do not know how the world looks like in 10 or 20. years or in 30 years, I don't know. Nobody knows. But for 800 years, the universities have actually provided us with thinkers who then took on the problems of their time. And you have to be educated to do that. And what is not going well, there is the idea of freedom of teaching and research. And the diversity to use that trigger word, But there it's actually appropriate. The diversity of different thinkers, because we're all naturally diverse, we're all different. We should be allowed to do what we want to do, and then society has to decide to who people, to which people this privilege is given.

What is, however, happening at the moment, say if you're in the European Union, universities, of course, they want you to gain grants. and the biggest grants are the European grants from the European Union and they decide on topics they're open grants to be fair they are open grants, you can bring your topic but some very big grants they decide on topics

what it would be important to have scientific research on and they are decided politically how does that work? Well would you think that in the current situation, energy situation and so on energy would be such a topic

of course such a energy is such a topic. Energy is such a topic. Okay, that means if you do energy research, you're good. You can apply for these big grants. I mean, I'm caricaturing the whole thing a little bit, but that's what it is. So ultimately, what we do research on is decided by bureaucrats and politicians who thinks that is plausible.

And of course, it's plausible to do energy research. But even if we look in the past, did nuclear power be invented by government programs looking for energy research, it came out of blue sky research, right? So if you look at the financial situation, it's no longer true that researchers really decide entirely freely on what they do research. That's a fact. It's a fact. And the problem with that is a monoculture.

Explain. Well, well, if people go for the big ground. either they're already famous, they've already done fantastic things. Then I would also say give them another grant. Very high probability they keep doing very good work. But new ideas are a little bit out of the box. And again, they're also funded. I'm just giving the general idea.

It's much harder because, look, research doesn't work by you telling me, I need you to research this and this. It works if I have to research a travel journey for you. I can do that for you. I can deliver that. But if you say, I need you to find the solution to the problem of quantum gravity, it's nice that you think it's important or anybody thinks it's important. I think it's important.

It's a good research field. But that doesn't mean that that is where the next breakthrough will be. So you think that the academics should be more free? Oh, yeah. Oh, yeah. Oh, yeah. Of course. Freedom, look, let's go back to teaching because it's a similar argument, I think every professor who has the right background educated in it, a specialist in it and so on, who should be given total freedom on how he teaches a say quantum theory course for audience X, total freedom.

this principle gets you some of the worst lecture courses you have ever attended. Well, that's unfortunate. But I also believe it gives you the most brilliant lecture courses you have ever attended. And I think it's better to have in your student life two or three or four or five would be luxury brilliant lecture courses. I think I had as many brilliant ones. Thank you to my teachers, right? But then there's some that are not that good. That also comes from the freedom.

But the freedom is necessary to develop new things, to do it better, to do it differently. You know, that's what I, there is no general commission that can decide. I mean, should we, very simple, curriculum, should we teach momentum vectors? Of course, we should we need it. There is no momentum vector. It's a co-vector.

So, you know, even if a committee agrees, this is important stuff. What if you say, yeah, but that doesn't really fit together at a deeper level of analysis? Should I teach it because the committee decided it? You know, the accreditation committee for the courses or whatever? I don't think so. I don't think so. Look, if we had a wild west of teaching and everybody would give terrible lectures

that everybody did what they wanted and nothing would ever fit together, I would advocate for some structure. at the moment we have at least in Europe the tendency of more and more structure of more and more ideas of centrally accredited causes and core systems and so on and what you get is an average good thing but if you want to get something really good you need to give freedom it's a belief I admit but it's an old Humboldt idea and it serves Certainly it served Germany very, very well for very long in educating an extremely broad, extremely broad part of the society, educating them very, very excellently in engineering and also in mathematics and physics and so on. I mean, early 20th century, I mean, Germany was quite the powerhouse of all these fields. Now, these, I mean, look, can I state this with sociological, how do you say, scientific certainty? No, but I think it's an outflow of the humble principles of how you teach at high level, at university level. Don't look at the vocational use first. The vocational use and that companies and the economy profits from these people, this is doubtless afterwards.

Don't be too short-sighted. And so, yeah, the diversity of the approach is important, and you have only diversity with freedom. I restrict the statement really to what I'm talking about here. The different researchers should follow their way. And then it's the question, who do you make a researcher, right? It's a fair question, you know, or a teacher at university. Yeah.

So let me ask you, I've already asked you about what you're pursuing, but I'm curious also, there are some fashionable subjects. the subjects that are in vogue like ADS-C-F-T or quantum information or black hole information string theory and so on, why are you not pursuing the more fashionable ones? If it merely is just you're not interested in it. What is it? That's not true.

Some of them are interested with some of them. I have some contact. Some of them I understand. Others I didn't look into. I mean, it's a lot of stuff out there. I like to follow the ideas. my collaborators have and I try to make my contribution that way

so I mainstream subjects on which already thousands of people work will I make the difference? Well maybe I would but I think it's I think it neither takes justification to work on a mainstream subject

or on a more on a sideline no I am so I was educated at Cambridge University by excellent people, Michael Green and Peter Goddard, John Stewart, Gary Gibbons, all prominent people, obviously string theory people and general relativity people. So I know certainly the foundations of many of these things quite well, and some attracted me

more than others. So string theory didn't attract me so much. At the time, there was the saying, oh, the best students go to string theory. and it was true the best students went to string theory I think it was a reasonably good student

there but I kind of didn't really feel it I didn't really feel it I'm not saying I was prescient and I knew it was not going to work

of course not of course but you know you make a pig it's also emotional to make it no way you say I thought that I find interesting

this direction I would go on and you weren't feeling it I wasn't feeling it. No, no, no. Well, I didn't understand. One thing actually I once discussed with, I think with Peter Goddard for a short while.

And I asked him, so why do we take the Minkowski metric in our, what was it, 26th dimensional space? I mean, basic string theory. And he told us, he told me, well, because Einstein told us so. And I say, yeah, but he told us for four dimensional space and didn't, isn't the reason for using it that you have. point particles that move there and you want to get the clock postulate by the length of the curve and stuff like this.

But this is a point particle-based idea. But now if you think about strings being the fundamental objects rather than point particles, wouldn't you rather have something that measures area fundamentally rather than length? Much later I wrote

a paper about these things. Interesting. It's a plausible idea. Okay, fine, it's a plausible idea. I also did something. I don't think it solves anything. But you see, the point is the setup, I found it too ad hoc in a way. In a sense, the only idea, at least as it

was, communicate to us back then, later on, of course, people change perspective. I'm not a specialist. It's like let's rather start from little strings close to open and quantize them, difficult enough, quantize them rather than point particles

as a first quantization, so on. And let's see what happens. there were some remarkable results and at least at the time I thought it was remarkable that the Einstein action kind of drops out if you look at it in a certain way

and so on. Later on I thought well it's not so remarkable because if you didn't do anything wrong from a differential geometry point of view or from a calculation point of view what else would you get but in lowest order generativity maybe the factor could be zero in front of the the richie scalar or something right

So, look, I mean, were these mature reasons to reject it? Of course, not. I was a student, right? But I didn't, I wasn't pulled in that direction. And so we all make this. And I'm happy that many other people did, right? So far it wasn't successful, which is maybe just very unfortunate. I would rather have that the people who did it were successful and would have an exciting new discovery in theory.

and that I would say, yeah, look, fool me, yeah. But, yeah, yeah, it's, it's, we don't know, right? It's like investing in the stock market. Will the stock go up and down? Nobody knows, nobody knows, right? And it's the same here, no? We can just try to do our best, and I think it's good in academic life. You do research, you really try to push something and to do something.

And but then on the side, you educate the next generation. And if you then educate them to a higher level, If you improve it by 5% what you present them with, we also did a great thing. So that's the nice thing about the academic job, right? I mean, you're not only relying on winning or doing Nobel or similar worthy work, of course, would be nice. But we have another very, very important task in society. Because if we don't at least pass on what we have and make it a little bit better so that more can be passed.

on, it was more efficient, then we lose that knowledge. That would be also catastrophic. So I sleep very well. I sleep very well. I think passing on stuff is very important. And trying to do new things. Yeah, trying to new things is an audible thing and it's exciting. And it's also training for students for them to do research.

I mean, you can do research which goes that far. But maybe your student learned many things contents wise. and method-wise, and they bring something else to success, right? So, so, I mean, that's the life of an academic. No, we're often talking about the holiest grail, right? There is the big, open questions. And, yeah, that's important. I'm also thinking about some of these things.

But I think the path there is more modest steps, more modest steps. So you just did something. psychologically interesting with the string theory case when they were extending to 26 or 10 dimensions with the Minkowski metric. They said it's because Einstein told us. And then you said, well, what was the motivation for Einstein? It was point particles in four dimensions and light clocks. Okay, even in your constructive gravity approach where you were saying Einstein tried to antisemitism the metric, or at least not make it symmetric, metric symmetric, that if you're doing so, then Raising and lowering isn't going to be the same, but you can only see that from your approach because there's a gauce map and there's a Legendre map, if I recall correctly. Oh, yeah, that's right. Okay, that's one of the... Okay, okay.

We can get into the technicalities of that, I would like to at some point, but the point is that what you just did there was extremely interesting. You said, look, we don't willy-nilly modify up here without understanding the reason from which here came from. Because if you modify here, maybe you don't... have to modify just this, you think you do, but it's all of this. Oh, yeah. Or it's only this. Why don't you talk on that then?

Well, I mean, that's, that's, that's, so, so what you're describing there is one could summarize it abstractly as follows. Formal generalizations typically fail. That's what I understand it. I have a symmetric metric. Now I do a non-symmetric one. There's a motivation because f-mu-new is not symmetric, right? Today, of course, we smile a little bit at that because the f-me-news the field strength and not the potential and stuff like that. But he had some reason for making it non-symmetric, right? But then if you formally say we generalized this formally, then you don't know what you're doing. You need to generalize conceptually. So you have to ask why was it symmetric there?

Why was it the metric? And then you might have a starting point. to say under what circumstances would the same why question have a different answer? Yes. All right. So, yeah, you have to think conceptually, not formally. Formal is good. Once, look, I mean, every equation in physics is only a secondary product. Let's take the first thing.

So what is energy? What does Einstein do? Einstein said energy is, what is it, MC squared? plus and then all the higher order terms in the P, now it's the square root of him. You have all these. And you can say Einstein, Newton wrote the P squared over 2M term

for say kinetic energy. But if you do relativity, you have also a P cubed, a P, well, at least a P square, a P quarter to the fourth power to the six pounds on. You have this whole expansion coming from the square roots, right, in relativity. So you say, oh, look

here, Newton didn't see or didn't consider the higher order terms in P. That's correct, right? They're there in relativity, if you write it in this form. But Newton also didn't see the 0th order term, the mc squared. You see what I mean? So the point is, if you say energy, it's a different concept in relativity than it is in

classical physics. And if you think you come from one theory to the next, by adding higher order terms because that is only the only thing that could happen measurement-wise or something. You make, I don't know whether there's a category error, but something like this, you want a new idea of energy,

then you have to first have a concept for that and don't do it purely formally. You see what I mean? So that's why you need to think conceptually. And once you have the conception right, I'm simplifying a little bit, then the equation flows out of the idea. It's not the equation first.

It's the concept first. Yes, yes. Okay, because otherwise the ambitious undergraduate would say, well, maybe there's a fifth order term in energy. Let me try that, but then you're wondering, well, where did that fifth order? Where was the impetus for the second order coming from? Something like that.

Something like that. You always, in order to generalize a theory, you must understand the to be generalized theory first at a deeper conceptual level and that's already ambiguous because you can look at things at different conceptual levels, right? You can describe, take gravity.

Okay, high school question. High school teacher asks a student here, final fail or pass will be decided on your correct answer to the following question. Students says, okay, is gravity a force? And we mean Newtoning gravity because high school, only Newtonian gravity. Is Newtonian gravity a force?

And a student will say, yes, it's a force. And he says, excellent, you passed. But if the student had said, no, it's not a force, he should also pass. Because you can rewrite Newtonian gravity as a curvature of space and of Newtonian space time. Not relative to this. Of Newtonian space time is also curved in the time direction, which there is now a unique time direction. fully equivalently, fully different conceptualization of what gravity is, not a force, but a curvature of the space. And you know free fall is duodesics, auto-parallels in that context, because you have to do it with a connection in Newtonian space time.

Sure. So that means the answer to the question is, is Newtonian gravity a force? Yes or no. Both answers are correct. If you conceptualize differently, predictions are the same. What does that tell us? Well, gravity is gravity, what it does, and our formalism for it could come in many different guises and it could give the same result. So we shouldn't confuse the formalism for the physics or the objects in the formism.

I mean, force is a concept in Newtonian mechanics, and gravitational force is then also there, but you could argue, well, gravity also in Newtonian theory should be made a curvature of Newtonian space time, and then the first axiom starts making sense. The first axiom sense says a particle under the influence of no force moves along a straight line. Okay, so you can say, do you know an example for that? No force.

And you would say, well, at least where we live on Earth, the first axiom is out of work, right? It's unemployed because there's always the gravitational force. it's just there. So what do you mean a particle under the influence of no force? If you ever say no, gravity, also in Newtonian theory, is the curvature of Newtonian space time, then it would mean, ah, any particle on which no force other than gravity, which we no longer consider a force axe, moves along a straight line,

that defines straight line in an operational way. And then the second axiom says, and should you ever see a particle not move along, the lines that particles on which no force acts, then those other particles are force acted. Yes, otherwise the first axioms, a special case of the second. Exactly, that makes no sense. Newton wasn't stupid.

He doesn't say, I first do it for the beginners. I say no force, then do it with the force for people who can't set the right inside of MA double dot or MA equals F to zero, right? I mean, no, no, no, right? He wasn't silly. No, the first one defines what a straight line is. Hi everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my substack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here. hit subscribe and let's keep pushing the boundaries of knowledge together thank you and enjoy the show

just so you know if you're listening it's c-u-r-t j-a-m-m-n-g-l dot org kurtjai mungle dot org how is it that you get to learn a subject how is it that you frederick schuller learn a subject now you said you had great teachers and so on but let's remove that self-studying how is it that you learn a subject so well what is it that you do to learn a subject conceptually well absolute beginner you do need a teacher because in the in the flood of of textbooks and lectures you need a guide you need a good guide but if you now say forget about the teachers how how do you learn it well it's different at different times as a student i i i learned the lecture notes i learned the textbook in the sense i've tried to understand every equal sign

I still tell this my student. Every implication arrow, I must be able to say why precisely this is a valid step. It's a passive approach, right? Great people set this up. If it's properly presented, I can check. I can check that what is said makes sense, fits together. This is very far away from finding it, right? And then the quality of material reveals itself if you can start

on page one, and you're properly equipped, you know, if you can start on page one and you really try hard to understand it, but again and again, there are jumps and gaps and things you don't understand. It might be you, but I always, yeah, but actually good material goes step by step. And I always tell my students, my quality guarantee or my hope is that if you don't understand something in lecture 23 then the answer should be in lectures 1 to 22 otherwise

I need to up my game look do I satisfy this in the strictest sense of course not I'm human like everybody is human right we all make mistakes as we said before we all use ways to think about it which

one could would better think about it twice when this is really a good argument and so on but at least we should try to eliminate that as much as possible. But as you ask the question, from a student's side, I judge material, can I understand every step in detail? Okay. And then you have a lot to do, this as a student. Now, of course, with so many years of experience,

I look at things, I read over them, I read the gist of them, and I say, okay, what do they actually do? and then I apply all my knowledge as far as it's existent and I try to redevelop it as I think

what they actually wanted to say but might not have said and half of the time at least let's make that more modest 80% of the time I realize I just didn't understand they did the right thing and they did the right

I just didn't understand silly me but I don't know whether it's 20% I'm making up these percentages, obviously. But I would think 20% of the time I say, oh, come on, that's such a naive argument. No, no, no, no, no. Oh, that's more like, and then I try to set it up new. And so in some cases, I really succeed in this.

And then I bring this into my lectures and I redevelop a course entirely from there. There's, of course, a difference of how I did it as a student and how I did it later. However, what I did as a graduate student, as a PhD student, I often set in the cafeteria on Friday, and I also took only white paper so that's a thing that started there and I thought okay now I'm here doing my PhD in theoretical physics and let me

write down for myself what is actually the basic subjects classical mechanics quantum mechanics maximal electromagnetism what is it really how would I write it down now if I had to write some summary of it and then I realized all my gaps

and all my lack of understanding and sometimes this lack of understanding was simply because it wasn't properly said and taught. And so I confronted myself with what happens if I have no textbook with me

and I'm supposed to write up a subject, a defined subject you know, a lecture course type subject as we know, classical mechanics, Lagrology mechanics, from scratch including its justification, the whole setting, what is the scope of it, and so on.

That's very humbling. It's very humbling, but it's a very good training. And I think meanwhile, I can make more out of it. I can then make it better. Okay, so let me see if I can formalize that. You are in a room, you're alone, you have blank paper, and you're thinking you pick a subject arbitrarily? Or is it just whatever is next up?

Well, back then, no, back then I thought, let me go through the four elementary things, sometimes classical mechanics, electronics, quantum mechanics, statistical physics. You know, these are standard big lecture courses, say, in Germany, but anywhere, right, in theory, theoretical thinking, a theory. And, yeah, and then I said, okay, how was this again, the whole Lagrangean business, what's actually the setting, what's the idea, what is the formalism, how does it start, what are the concepts, and how do I bring this into an order that makes sense, A, B, C, D, E, not in another order, right? and if one really tries this and even as a very good student

and very good graduate student I realized oh my gosh I mean there's more holes than anything and then you recognize to go the holes then you can start to fill them and in a sense I'm still filling them on some points right

now I also notice that when you're lecturing sometimes you'll pause at the blackboard and you'll derive you'll make sure that you can derive it right then and there, even in front of the students. Oh, yeah. I wasn't sure if that was for you, for your own pride,

that you need to be able to derive it, or if it's for your own intellectual satisfaction, or if it's to role model for the students, because it seemed clear to me, you have lecture notes behind you. You could actually look, well, what's the final formula? Where am I trying to get to and use that as a hint?

But then you don't. Yeah. Well, the reason is all of the above, plus a sanity check that what I, with all my experience, of doing this for, what, 30 years? Can't do freely after I, of course, had the, thought about it for four hours in the morning.

I always prepare my lectures four hours earlier in the morning, and then I go and I write them up in nitty-gritty detail with headlines and everything in order to give them structure. Then I put the paper on the big desk, and then I only look at the headlines typically. On the rarest occasion, I need to look at a, formula again. I get confused. But I essentially really think, A, what I cannot freely develop after having thought about in the morning again, and of course it's a big thing, a whole lecture course,

just one lecture of those, right? So everything needs to make sense in the end. What I can't lecture freely and precisely and convincingly and step by step on, how can I expect the students at the end of the term to do it. Yes. They should be able to do it. So I must be able to do it. So it's not only, it's pride as well. I want to be able to do that, but it's a sanity check.

If I can't do it, what am I asking the students to do? I ask too much of them. Okay, number one. Second, only if I develop this life, does a student, can a student follow? It slows me down as opposed to, I copy from my paper. If I take, sometimes I did that on very, very occasion. I took the sheet I had there because it was a bit complicated.

And I started writing what was on the sheet. My teaching quality goes down 70%. It's not a good thing. It is as if you had sent me your questions before. I would have thought about your questions and would now have paper where I have very intelligent answers to your questions, at least intelligent sounding answers. and I would now read them off and I say oh yeah Kurt that is a very interesting we wouldn't have a conversation anymore right it would feel strange only the best speakers can read from a sheet and speak well okay so it also serves what I want to say is it also serves the communication with the students it's a real conversation it's like a story I tell you over a campfire it's very real I might not complete every single sentence ideally and so on, but your focus is on it.

Yes. That has to do with yet another thing. What do you have to do if you have 200 people sitting there and you want them to not open their laptops? They all have laptops. You don't want them to open the laptops. You want them to listen what you do, to write what you write, to see or copy maybe what you write and have their eyes where you want them to have their eyes, where you use.

a blackboard. And on the blackboard there is a drama developing with actors, my fingers, the chalk, sometimes I tip on the blackboard, and then here, and so on. People look there, nobody opens their blackboard. I think in the last six or seven years, I once had to ask

a young lady who was probably for very good reasons on her mobile phone. I told her, please take it away. I can't lecture like this. And it's really true. I have an inability. If people don't pay attention, it makes me nervous. Okay? Yeah. But of I can't force people to pay attention.

I have to play my game such that people don't want to look away. And if you go to the cinema and you watch a well-made movie, you don't look away. So I have to present a well-made movie, which has to be scientifically sound as well. And one of the things is use the blackboard, speak to the people, speak to them in real time, and let them participate in your thought, you know, roughly speaking. So all of this, I see no alternative to using the blackboard. Nobody follows a little laser dot on a screen, you know, on a projection wall. If you project your, and then you can have your little laser pointer and then you point at things, you point at the next thing.

Honestly, do you want to follow this little jumping red dot? You don't. It's very difficult to focus. But a blackboard lecture is natural. the equation evolves and the nicest thing is the lecturer makes mistakes and then the student can say

your second equation there's something wrong it must be a minus I say no and then I honestly I'm honestly surprised or honestly secure about it not confident about it

and then still wrong and shouldn't happen too often right in every lecture you make a mistake in every lecture you make a mistake where the student knows better they think you're a fool Maybe that's correct. But occasionally, you can't get it, and I say, that's right.

Then I look at it, and sometimes, very, very rarely, but sometimes I spend 20 minutes in a lecture recovering an error, which I made, oh, that's not how it, oh, do you write? And then the first thing I say to the student, very honestly, you saw something that I didn't. Very good. Now let's see whether I can save this. And that is pride, then it's pride. But it's also to a student's a statement like, you know, I don't say, yeah, yeah, look this up at home. And then you will see, next time I say, oh, there was a little mistake. No, a mistake is a mistake.

Mathematics and our subject in general makes you very humble, you know. If you have made a mistake, you can be the smartest and the cleverest or the most arrogant or whatever. If it's a mistake is a mistake, you better immediately admit it. You immediately admit it and try to see whether you can repair it. and sometimes you can't. And so all of this is only this is the life. I think, and it's remarkable that in the YouTube videos, I think some of these things still come through.

So that's students who attended my lectures and watched the videos as well say that still comes through somehow. Okay, good. Wouldn't have predicted that, I must say. I thought it needs the immediacy of the classroom. But maybe, yeah. So maybe that's also part of why, why maybe

it might be more than not enjoyable to watch them so these all you see these are little ideas

none of them per se as a principle that's set in stone and everybody who does it different

does it wrong I'm not saying this there are people who give lectures in a very different way

I say oh god I would never do this I recently had this a very

excellent colleague Pim van Jav he gave a lecture in a course we gave

together and he explains something and he said oh and what if you did it like this and I thought oh no you're leading them on the wrong track don't do that I mean I thought I didn't make this gesture but internally I did no don't do this no this is such a bad idea and then he turned this around

in such a brilliant manner I thought well done respect I would never do it like this but respect so if as a teacher you can make your method fly wonderful wonderful now and I have my method to make it fly.

So what was the bad method or the topic that he had that you wouldn't have taught like that? I think if I remember correctly, we talked about quantum mechanics to high school students, highly talented high school students who every year we have at the university, we invite 100 top talents from Germany and the Netherlands for a talent course, where in three days we teach them quantum theory from nothing up to the teleportation protocol. but including the foundations, I mean, the axioms and the techniques and all of this.

Anyway, let me not talk too much about this. Anyway, and there on the first day we do complex numbers, but we don't say complex numbers, you know, I squared is minus one. We do them as tuple of real numbers with a very special additional multiplication because then they're totally demystified, okay, the complex numbers. And he said, okay, you add them like this, here, you know, component-wise, so to speak. And then the multiplication, of course, is not component-wise. It needs to mix. And then he asked them, so how would you multiply them? And then, of course, he got them on the wrong track

because they would also say component-wise, you multiply the components of these two pairs of real numbers. And I don't do that as a principle because you can't guess how you multiply tuples of real numbers if you want to make them into complex numbers because it's a definition and only after you made it, you can then start investigating what it is.

Yes. Okay, and here the point was to define the call. So I didn't, I wouldn't ever do that. It's a micro thing. You see, I thought about all these things very careful. Interesting. I would never do that. I think, ah, no, don't do that

because then they have this bad idea, which you then say is wrong. And no, no, not this way around. But he did it so skillfully. excellent teacher. He did it so skillfully. Damn, well done. I'll still not do it. But you did. Well, mad. I mean, I thought I was sitting there and watching. Okay. And so I want to say these are all, for instance, another obsession I have is no motivating examples. Well, that sounds very strange, right? I mean, isn't a motivating

example a great thing? I mean it in a strict sense. If I say I'm to teach you a vector spaces or vectors. We'll never teach you vectors because there's no such thing. There are vector spaces, but vectors. And then people say, well, look, you hear it one and then you can add them up by moving this one like here and then you have this one.

That's the addition and you can scale them. As soon as you start with little arrows or fingers or whatever, you don't have a very special type of vector space. Right? And why is addition by moving things around? What the heck are you talking about? about. If you give people this kind of motivation, which almost every one of us got upon seeing vectors the first time in high school or something, then you get the idea. A, there are objects that are vectors. And if I see a vector, I recognize a vector. Well, this is all not true. Because the only mathematical object in the game is a vector space. It's a set with two operations, plus and the scaling. And this set, and this set, together with the operations, only if the following eight axioms hold,

then we call that entire structure a vector space. And yes, we can then nickname the elements of that set if the set carries with it as companions these two operations. We can nickname the elements of such a set vectors, a vector, and we mean an element from that set. But the correct word would be an element from the set that underlies the vector space.

The vector space is a triple v plus dot. But there is no such thing as a vector. If I write on paper a tuple 1-2, write 1-2 as a tuple brackets, is that a vector? You can't answer that question. Do you know that there are others which can be added to this one

and scaled by which addition to which scaling? You might think that is clear what it is. No, it isn't. can make beautiful vector spaces out of, for instance, positive real numbers, they can be made to R plus to the end, it's kind of like an octon or something, that can be made into an R vector space where the addition is by pointwise multiplication of the components, and the scaling is by taking the components to the power of the scaling factor.

That vector space, the zero in that vector space, so the neutral element of addition is one, one, one, one, one. It's not zero, zero, zero, zero, zero, stuff like this. But if you start with this little vector, you never get the general. And if you give a motivating example, you always introduce special situations, special structures that people will never forget. And they will always somehow refer to these. maybe in extreme cases only,

why not, instead of giving a motivating example, which by its nature is not understandable for anybody who doesn't know the concept yet, is made more vague, and then you give the definition, but the definition doesn't follow from the example. Of course it doesn't.

It's a more general thing. It cannot follow. Why not make a little commutation, give the definition first, and then give 15 examples afterwards. Very, very different examples. That's a much better teaching. Because that's the object we want to talk about. It's very mathematical thinking. And then I show you what variety of crazy situations are all covered by this general structure. So we better study the

general structure than all these crazy examples. You see, so I have obsessions. One obsession is no motivating examples. Sometimes you start a subject, especially in physics, and it's too good. I think it's too good to not give this motivating example. But I always regret it. I always regret it. I mean, I can pull this off. It sounds very good. But ultimately, I always realize it down the line,

it caused confusions in the students. Motivating examples are of the devil. Anyway, and you see, and whoever skillfully, like my colleague I mentioned before and another principle of mine, whoever skillfully and masterfully violates these principles bravo

but these are my principles I try to not do that so and I think I've thought about many of these things of course a lot of experience and trying it this way and trying it this way and being dissatisfied and the highest judge is always

was it correct what I said or was it mumble jumble and almost by construction motivating examples are mumble jumble unless it really follows one from what you did before and leads to a question not based on the previous lectures

I do this calculation or this construction or something and everything can be understood in detail and then I say but here's a question and that motivates this lecture that would be good that would be more a derivation than a motivating example

to introduce a new idea now so so I have probably I don't know whether I have 50 of these principles, but many things I think very strongly about. And they're extremely personal. Of course. That's one of the reasons why

many people have asked you, hey, can I interview you about how you teach and you're not so keen to do so? That's true. I gave some interviews. Usually if my own university asks, I don't want to say no, I was once introduced. It's interesting.

It's an interesting, I think even audio snippet on the internet where I was asked whether this one, these online lectures took off like crazy internationally. They asked me from a research center for education in Erlangen whether by doing these videos I also want to promote the new online teaching media or something like this to the social media or something like this.

And I said very undiplomatically I said that doesn't interest me one bit. So no, I don't want to promote. I don't want to do YouTube videos. I don't want to promote distance teaching, learning. None of this, I want to do good physics. And back then, they actually, in Erlang, the students could always choose whose lectures are being recorded. And that's when a number of these Erlangen lectures that are on the internet have been recorded by the university. So I just try, as I said, I try to give something really valuable and try to make them really good as much as I can.

That's my only aim and the other things are side effects. And of course, I like it. Of course, I like it that these lectures are so well received. So, wait, I'm confused. You're not a fan of your lectures being online or you're not a fan of distance teaching? What do you mean? No, I'm not a fan of neither a fan nor an enemy of distance teaching. No, my lectures, well, first of all, a little anecdote, none of the lectures online have been put online by myself.

So there is a YouTube account called Frederick Schuller that features these lectures That's not me Somebody took my picture Somebody took my name And downloaded these lectures I think mainly from

Erlangen University And put them on YouTube Great service I thought okay It's a fan Whatever Only a few years later When there were really

Views went into the millions And total and so I thought okay I got a little bit scared What if this person One day post something inappropriate, right? I mean, it could be anything. And it's my name. And then I wrote via YouTube to them. I never received a reply. And then meanwhile, then all of a sudden it was

a name I didn't recognize. And then it was back my name. And I decided, come on. I mean, I could now tell YouTube about this. And then maybe then they take the lectures down. And come up, but people are watching them at the moment. I don't want that. And, you know, So I also, then at some time there was some commercials in between. I thought, okay, somebody is making money of it. But I thought, okay, fine, okay, good for them. So these are not my accounts.

The gravity and light, I gave my okay to the gravity and light lectures, but the others have all been taken from however public. It's not kind of copyright infringement or anything, public service of the university. I see. Where they were recorded, no? And no, I mean, look, I think, of course, it's great that there are so many online resources. I mean, it's the positive side of the internet.

What we can all look up is brilliant, okay? And if people like the stuff and I think it's not wrong or terribly wrong, then I leave it on. Everything is fine. But I think ultimately, I'm psychologists, would tell you that you can only be formed children for sure, but also young adults in the presence of another human being on whom you can

in the weakest sense model yourself. And so this goes back to my teachers in high school, at impressive teachers in high school and on all these people without trying, oh, I want to be like him or her,

but in a way you model you, you imitate, you model yourself. So I think we need the personal contact. And then of course in the tutorials to all these courses. I have of course tutorial teachers, assistants now very good people as well

with themselves very good researchers. I go to many of these tutorials and I sit there in the row and if I see the students don't engage enough because they're maybe too shy or too now. I start pushing the tutors on say, well, how can you say this

and stuff like this? And that then mixes up the whole atmosphere a little bit right. And sometimes I confront students, right? I didn't come prepared. And I said, oh, do you play basketball? Anyway, I pushed them a little bit and I say, well, I mean, it's a professional enterprise here.

I mean, you want to become a professional? What do you mean? You come unprepared. I don't understand. Do you have a big business running on the side? Because otherwise, I mean, come on, right? So sometimes confront people a little bit, right? I mean it well.

I always mean it well. But sometimes I'm a little, I try to confront them with what it means to, to be there and sit there and do things and but not do it seriously. Well, then don't do it at all and face that you're not doing it, right? Because where does that get you? So anyway, so I mean, I try to be

nice most of the time. No, I think I'm nice, but I push people. I challenge people sometimes a little bit in the tutorials. All of this is not on the videos. But it's also because I believe, yeah, there is some you see, it's like if I was an arts professor, I would take my students to the gallery and I would show them the

well, the Picasso's and all others and I would discuss with them about these pictures and would say, isn't they, look at it. First of all, look at it. Just look at it. So it's a little bit like this. I say, look at classical mechanics. I mean, look at this. And of course, in order to make them look, I have to expose the theory, right?

I have to go step by step. But it's really, I think the classical subjects we teach, I show. them, how do you say, like a masterpiece of an artist, like of a justifiably world known art, world known artist. Yes. Look at this thing, this theory. Look what marvelous cultural achievement this is. I don't use ever these words, right?

I just do whatever do. I do what you see all the videos. But essentially I do that. I show them. And then I showed them a theory. which is one of the most developed, one of the classical mechanics, and all the theories,

most developed theories we have. How can I expect them to later on make their own theory, that's asked a lot, or modify liberty, if I didn't show them the masterpieces. How does an end product of research look like? If 10 geniuses,

saying classical mechanics, let's say 10 geniuses developed it, maybe it was five, 10 super geniuses and a thousand almost geniuses who kept shaping it. What you get today is the product of whatever, five or 10 geniuses and a thousand almost geniuses who shaped it into this form and this you get. This is my present to you, the thought of genius and you understand it. Is that something? I think that's something.

I think that's something. It's quite different from, oh, I can calculate this. no, it's brilliant. It's a brilliant thing we have the privilege to have done and others are younger, generativity, quantum field theories, studies, and others are still developing. But we want to make the masterpieces

that don't look like the old masterpieces, like a Picasso doesn't look like a Rembrandt, right? But new masterpieces that stand next to the old masterpieces and all are recognizable as masterpieces, right? And so that's also one part of teaching at a high level in theoretical physics or mathematics, the same logic. But I think you need to be present. So that's your question.

Look, it might be that one day only very few, very rich people can afford to go to the remaining few universities who offer in-person teaching and everything else is economized into. to online teaching. Would that be the worst? I don't think it would be the worst, but it would probably, in the long run, wouldn't be so good. I think every one of us has some, I don't know, spark of maybe, maybe.

I don't know, that's not a good statement. A little, the tiny spark of genius, I think, is in everybody. And everybody should go to places where he sees other people participate in lectures. As I said before, right? Students need to see you as a teacher to model themselves on you.

They need to see there are many other people in the classroom who struggle with this but they also see so I'm not the stupid one but they also see people who apparently

fly through the material and excel so you say it can't done if you can do it are apparently very special but others can't do it either but some actually make it

I'll give it a try you know you need both sides you need to see it can be done to see. It's not so easy. I'm not stupid because I haven't gotten it yet so well. All of this only happens in presence. And during corona, we had beginners starting and I gave, I think it was a lecture, I think it was a tutorial. I think it was a lecture. I gave lectures. And I didn't see the

students, of course not, right? It was online. They saw my face and I had some things. They were beginners, right? It didn't work well. It didn't work well. I didn't work well. I did my best. I think I gave a good lecture. But it's too remote. I need to see their faces. They need to see mine. They need to see my gestures. And they need to see other human beings doing the same thing. I think they were mainly disoriented because they didn't see who succeeds and doesn't or that there are other people who succeed and don't. We need this. So no, I am not a proponent of online or remote learning. But of course, it's not a bad thing as such. But none of that is. my intention to and I'm happy that people else who don't have many people wrote to me it's wrote many positive things and then they said

it's nice that these lectures are available for free online I would never have access to such teaching where I live okay I believe that for those people we write very kind emails and I'm very happy it makes me very happy that they take so much out of it

not only joy but also maybe future development and so on but I have no agenda on this. I think scientists shouldn't have an agenda anyway. I think the only thing we should do is do our stuff well. Well, you do your stuff extremely well. Speaking of masterpieces, your lectures are a masterpiece.

I'll place a link to all of the lectures, all of your playlists, and they'll be in the description. It's a privilege for me to be able to speak with you. It's a privilege for the audience. The audience is, well, I'm great and I'm sure the audience is grateful as well. Thank you so much, Professor.

Thank you very much, Kurt, for the invitation. It's very pleasurable to talk about this because indeed we talked about many things I usually don't talk about because they're not science, their opinions on science and views. And I think it's very important to talk about this too. I really very much like your podcast.

I watched a good number of episodes. So I think this is also going to be a masterpiece. I think you said this in some places, you like to at least occasionally delve into technical detail or at least pseudo-technical detail for people who know more who know about the real background of things. Thank you very much. Thank you. Yes, in parts, I think what it is, it's like overhearing two colleagues in the office talking with the door open. And I think your podcast does this particularly well.

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substack on Google. Oh, and I've received several messages, emails, and comments from professors and researchers saying that they recommend theories of everything to their students. That's fantastic. If you're a professor or a lecturer or what have you, and there's a particular standout episode that students can benefit from or your friends, please do share. And of course, a huge

thank you to our advertising sponsor, The Economist. Visit Economist.com slash Toe, T-O-E to get a massive discount on their annual subscription. I subscribe to The Economist, and you'll love it as well. Toe is actually the only podcast that they currently partner with, so it's a huge honor for me, and for you, you're getting an exclusive discount. That's economist.com slash tow, T-O-E. And finally, you should know this podcast.

is on iTunes. It's on Spotify. It's on all the audio platforms. All you have to do is type in theories of everything and you'll find it. I know my last name is complicated, so maybe you don't want to type in Jymongle, but you can type in theories of everything and you'll find it. Personally, I gain from re-watching lectures and podcasts. I also read in the comment that Toll listeners also gain from replaying, so how about instead you re-listen on one of those platforms like iTunes, Spotify, Google Podcasts, Whatever podcast catcher you use, I'm there with you. Thank you for listening.