Theories of Everything with Curt Jaimungal - Curt Jaimungal: General Relativity Is NOT Deterministic (Here's the Proof)

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Starting point is 00:01:00 into with Shopify today. Sign up for your $1 per month trial at Shopify.com slash tow. That's Shopify.com slash T-O-E. You've probably heard that Einstein's theory of general relativity, the theory of gravity, is a deterministic theory. Technically speaking, this is false. The reason is quite subtle, but many physicists like Penrose and others that have spoken to

Starting point is 00:01:30 know about this intimately. There's a severe failure of determinism in general relativity that cannot be taken lightly. Einstein's general theory of relativity is not a deterministic theory. There is a lack of predictability. And these go on and on as you'll see and here over the next few minutes. Today we'll learn what precisely is GR, so general relativity, beyond the bowling ball on a rubber sheet, which is actually a circular argument, if you think about it, we'll also learn what is determinism precisely, which also has two subclasses, a local and a global determinism. And then also, what is this global hyperbolicity that you've heard about or may have heard about? It's not just over-scrupulous, abstruse math. It's actually extremely important. And we'll talk about

Starting point is 00:02:15 what these three have to do with one another, of course. General relativity allows close time-like curves, and you could say in some sense circumstances, it even encourages them to happen. So I'll speak at two levels simultaneously. One is for the person who wants a rigorous technical definition, that's where all the sources come in, and another is for those who just want the crux of the argument. For that latter person, you also have to keep in mind that almost any claim, when simplified, is replete with nuances that challenge it. That's why on this podcast I attempt to be as technical and rigorous as I can to ameliorate some of the predicaments of this compressed message. I'll put a link on screen about my opinions against this whole,

Starting point is 00:02:56 hey, explain it like I'm five, bro, or you don't understand it. So why the heck is saying, general relativity is deterministic, is technically a false claim that needs to be caveat. And I'm not speaking about false in some punctilious sense that only philosophers care about. I mean false in that there exist perfectly valid solutions to Einstein's field equations, where specifying the complete state at one time doesn't uniquely determine the future. I'm not talking about singularities, although there's that as well. I'm talking about regular, smooth regions of space time where your future simply isn't determined yet.

Starting point is 00:03:35 I find this super interesting because unlike quantum mechanics, where you at least get probabilities, in some sense, the indeterminism in gravity in GR is worse. You've heard of Einstein's field equations, but what is the relationship between that and GR? Well, GR, general relativity,

Starting point is 00:03:55 is a five-fold package. It comprises some theoretical principles like the equivalence principle. You may have heard it stated informally that physics in a free-falling frame is locally going to be reduced to special relativity, or that physics doesn't depend on your coordinate choice. Another is the mathematical scaffolding

Starting point is 00:04:13 of certain types of Romanian geometry called pseudo-Romanian geometry on a four manifold, and of course there's the field equations of Einstein, and then a geodesic equation that talks about how free particles move in that straightest possible path you've heard of in curved spacetime. And then there's also the physical interpretation that curvature of space time is gravity. Not that gravity causes curvature, but that they're identical.

Starting point is 00:04:37 This is why that Wheeler's statement is a bit misleading that space time tells matter how to move and matter tells space time how to curve, because it gives the sense that there is some causal path that ticks forward. But actually, as you can see, these are dynamically coupled equations. It's not like there are time steps of causation here. But anyhow, these form a package deal. This is all part of general relativity. You're not just referring to one element here.

Starting point is 00:04:59 I have substack notes on this here, which is what you're seeing on screen, and this is the coordinate free notation, one that I prefer, because I don't like my physics with arbitrary choices, and not these coordinate index gymnastics. You probably have some intuitive sense of what determinism is. It's something like, look, if you know everything about the universe right now, whatever that means, then you can predict everything about the future. But let's be precise about what physicists and philosophers actually mean by that term.

Starting point is 00:05:28 A theory is deterministic if the complete state of a system at any given time, combined with the laws governing it, uniquely specifies all future states. Now notice the key words, complete state uniquely specifies all future states. Each of these matters. So then, what counts as the system? And what if there is no such thing as the complete state at a given time? This brings us to distinction that most discussions gloss over. Local determinism is if you specify initial data in a small region of space time.

Starting point is 00:05:59 So technically some open set. The equations uniquely determine what happens in the immediate future of that region. I'll put some mathematical jargon on screen here. Again, this is written in my substack, in more detail, and there are other sources in the description. Most of my best ideas don't happen during interviews. They come spontaneously, most of the time in the shower, actually, or while I'm walking. Until I had Plaud, I would frequently lose them because by the time I write down

Starting point is 00:06:25 half of it, it's gone. I tried voice capture before, like Google, home, and it just cuts me off in the middle. It's so frustrating. Most of my ideas aren't these 10-second soundbites. They're ponderous. They're long-winded, and I wind around. They're discursive. They're five minutes long. Apple notes, even Google keep. The transcription there is horrible. But Plod lets me talk for as long as I want and there's no interruptions. It's accurate capture. It organizes everything into clear summaries, key takeaways, action items. I can even come back later and say, hey, what was that thread I was talking about regarding consciousness and information? In fact, this episode itself has a plot summary below, and I'm using it right now over here. My personal workflow is that I have their

Starting point is 00:07:06 auto flow feature enabled, so it sends me an email anytime I take a note. Look, the fact that I can just press it and it turns on instantly like right now it's starting to record without a a delay is extremely underrated. This, by the way, is the Note Pro, and then this is the Note Pin. I have both. Over 1.5 million people use Plod around the world. If your work depends on conversations or the ideas that come after them, it's worth checking out.

Starting point is 00:07:32 That's Plod.aI slash T-O-E. Use code T-O-E for 10% off at checkout. Global determinism, on the other hand, is if you specify initial data across the entire universe at one moment, then the equations uniquely determine the entire future of that universe. And then there's some more mathematical jargon about kosci surfaces, but it is important. So I'll just say it's a space-like slice that every causal curve hits exactly once. The hope is that this uniquely determines the space time thereafter. Now, you might ask, are these two the same, the local and the global? Isn't the global just made up

Starting point is 00:08:11 of many locals taken as a totality? In flat space-time, this is super-trival. This is super-trival. yes, but it becomes tricky with curvature. There are solutions to the field equations of Einstein where you literally can't define a moment across the entire universe. And it's not because you're not clever enough, it's that you can prove that they don't exist. In general relativity, there may not be a way to slice space time into all of space at time T. And without that, you can't even formulate what global determinism even means. Though you can retain local determinism even means. Though you can retain local determinism, which brings us to our third concept. You're likely thinking, okay, Kurt, who cares?

Starting point is 00:08:56 Some space times don't have this property that you say about slicing, whatever slicing means, into a space at a certain time. Big deal. Let's just work with the ones that do. Great. Okay, that's what physicists often do. So this condition is called global hyperbillicity. It means a space time is globally hyperbolic if it admits, so a mathematician's word for, allows for the existence of, a Koshi surface, which is a space-like slice, that every inextendable causal curve intersects it exactly once.

Starting point is 00:09:31 Again, I'm speaking at two levels, one technical and then to the other probably sounds like gibberish, so let me unpack that. A koshi surface is like a snapshot of the universe at one time. And what's interesting, when you even just take a Polaroid photo of something, that's a spatial slice. You're not looking at a time slice of something, provided your Polaroid was instantaneous. You're looking at space. So this is a spatial slice. And then you can recall those particle whirl lines that you've seen, every one of those whirl lines, every light ray, they all cross the surface exactly once. And I say surface in a slightly more abstract sense. Now, in the Polaroid case, that's a 2D surface, that's fine. But when it comes to

Starting point is 00:10:11 space time, you have four dimensions, so a surface is a hyper-surface, you're just minus one dimension. That's our three dimensions of space. Anyhow, there's no looping back of these world lines. They cross it exactly once. They don't miss it entirely. The existence of this guy, this 3D spatial guy, is what even allows you to say the state of the universe at time T. Now, if your space time is globally hyperbolic, you're golden, yo. There's a theorem that says if you have initial data on a kosci surface like that, then you uniquely determine the maximum globally hyperbolic development. Try saying that three times fast. So what's the problem? Well, not all solutions to Einstein's equations are globally hyperbolic. And I'm not talking about exotic mathematical

Starting point is 00:10:59 curiosities that only sadists who read counter-examples in topology would find prepossessing. Some of the most physically interesting space times violate global hyperbolycility. So charged black holes, for instance, rotating black holes. Antideocidder space has a time-like infinity that you can reach in finite time. Your world line just ends. Girdle universes actually contain closed time-like curves where you can travel to your own past. This is something I referenced in this video I made about misconceptions about Girdle's incompleteness theorem that somehow went viral. In these space times, knowing everything about the now actually doesn't tell you everything about the later. And not because you're missing information, but because that information literally does not exist yet.

Starting point is 00:11:43 So what happens? In some of these, it seems like your future would just stop, but the Einstein equations actually have multiple incompatible slash inequivalent solutions beyond that surface. It's as if the surface reaches a point and says, I have no idea what's going to happen next, pick any of these infinite options. There's no probability distribution. There's no select, selection principle, there's just ambiguity. See, quantum theory is famous or infamous for its uncertainty principle. It at least gives you probabilities, though. GR, which is supposedly a paragon of determinism, can leave you with genuine ambiguity. In other words, Schrodinger's cat doesn't know if it's alive or dead, but it at least knows the odds. An observer crossing a

Starting point is 00:12:33 koshe horizon on the other hand, God doesn't even play dice. He just shrugs. Now you can see how these three concepts play together. General relativity has these field equations and they're perfectly deterministic locally, and there's a theorem in PDEs about this, which means any small patch of space time. If you know the conditions there, then you can evolve them forward uniquely. However, when you zoom out to a global picture, there can be problems. So without caveatting by restricting yourself to global hyperbolicity, you can't even define what the state of the universe at time T meet. And when global hyperbolicity fails,

Starting point is 00:13:14 such as when there exists koshe horizons, which are different than kosci surfaces, by the way, then the equations give you various inequivalent answers for what occurs beyond that surface. Let's take a specific example. Let's say the charged black hole. An observer would see the entire future history of the outside universe compressed into a finite time.

Starting point is 00:13:35 Beyond that point, the Einstein equations become ill-posed as an initial value, problem. Now, you can extend the space time, yes, but the cost is that there aren't just many ways to do it. There are infinitely many ways to do it. All of them are equally valid mathematically. So then you wonder, what the heck would this feel like physically? I don't know. I'll let you know when Musk sends some minions there. As far as I can tell from the safety behind my LCD screen here, information would emerge from nowhere. And I'm not talking about quantum uncertainty again,

Starting point is 00:14:14 where at least you get these born probabilities. Instead, it's just new information, appearing without a cause. You'll often hear some physicists dismiss these examples as pathological or unphysical. Only some do this, though. Most relativists that I know are sharp enough to realize that there's no rigorous definition of pathological. It's basically saying, I don't like this solution, and when I look out my window and astrophysically, I don't see this. Now, recall, black holes were said to be pathological before.

Starting point is 00:14:48 Even the Big Bang is a solution. Einstein didn't want that. We can't a priori dismiss something as being unphysical. Now, there are some attempts at rigor. So, for instance, some of these solutions are unstable, and maybe we just say unstable solutions are pathological. Although, as I mentioned, pathological seems to be more of the times slash an opinion. girdle universes are unstable under perturbations, under small changes. You can think of this as,

Starting point is 00:15:15 yes, a pencil upside down is a solution to classical physics, but it's just extremely sensitive. However, even this objection about unstable as a synonym for pathological has some problems, as you can have koshe horizons, which are stable in certain contexts, like charged black holes with the cosmological constant. This is work by Cardoso. To be clear, a Koshy surface is that initial data that you evolve forward, whereas a Koshi Horizon is where that data breaks down. They both happen to have the same name as that guy that you've heard from differential equation courses. The Koshi Horizon is the boundary of the region that the Koshi surface can predict.

Starting point is 00:15:54 So beyond the Koshi Horizon, determinism fails because new information can, quote, leak in, end quote, in a sense from somewhere. It's all subtle and quite odd. I talk much more about various sorts of space times and indeterminism. with Professor J.B. Manchak here, so subscribe to get notified for that. It may already be out, and the link is in the description if so. Then you could also say, well, if there's a solution that violates energy conditions, then is pathological, except quantum fields violate these routinely. Dark energy violates the strong energy condition. So double oops. Then you could say,

Starting point is 00:16:30 well, cosmic censorship saves us. This is Penrose's conjecture that nature's sensors, naked singularities behind event horizons. It's unproven and has potential. counter examples. He knows this, of course, and I spoke to him about this personally here. The truth is that we have no principled way to exclude these non-deterministic solutions. The space of solutions of Einstein's field equations is infinite dimensional. As far as I know, we don't even have a natural measure to say something like most solutions are globally hyperbolic. But then, who cares about global determinism? Local determinism is all we need, right? Well, the problem is that in a space time with closed

Starting point is 00:17:11 timeline curves, even local determinism becomes suspect. You can have a region where the future loops back to influence the past, and this creates a consistency condition that constrains your free initial data. The Gertil Universe is mentioned before I have closed time like curves

Starting point is 00:17:27 through every point. It's quite trippy to see visualizations of this. Here's one from Busser and Kajari and Schleck. Hopefully, I'm not mispronouncing their names. In this Gurdial universe, both local and global determinism fail. Another problem is that if nature allows naked singularities, then information can appear from the singularity with no prior cause. This violates determinism in a finite region,

Starting point is 00:17:53 not at some abstract infinity. Think of it like this. Quantum mechanics, it has indeterminism, yes, but it's domesticated. It's random, but it's predictable in distribution. It's like a good boy. Einstein's general relativity, on the other hand, has genuine indeterminism, which is feral. It's either unmeasured or it's unknowable or both. It's like a Torontoian raccoon. There's no rules. There's no remorse. When I'm deep in research reading papers on, say, the philosophy of mind or quantum gravity,

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Starting point is 00:19:27 by clicking the link in the description box below ExpressVPN, VPN.com slash theories of everything YT. That's expressvbn.com slash theories of everything YT. Look, we've spoken about quantum mechanics and quantum field theory, so perhaps the hope here is that quantum gravity saves determinism. After all, these classical solutions wouldn't survive quantization. Or would they? Well, let's think about this. Quantum gravity will still have quantum indeterminacy, so you'll still have indeterministic gravity. But even disregarding that, The irony is that many approaches to quantum gravity assume hyperbolicity from the start. Canonical quantum gravity, loop quantum gravity, even many formulations of string theory, but not all

Starting point is 00:20:12 of them, require kosci surfaces to even define the theory. Technically, yes, world sheet amplitudes can be computed on non-globally hyperbolic backgrounds like girdle spaces or even orbifolds with CTCs. It's specifically S-matrix formulations and unitarity requirements that typically demand global hyperbolycity, not the world's cheap consistency conditions themselves. I did a three-hour iceberg into string theory explaining the math in case you were wondering what the heck that was. Links are in the description. You're correct if you notice that this is like assuming determinism to prove determinism, much like how John Norton shows that Newtonian physics actually has indeterminism

Starting point is 00:20:52 in it as well, unless you assume Lyft's continuity, a special condition, which amounts to assuming the very determinism you're attempting to prove. My view is that by the strict definition of what a deterministic theory is, namely that we always have the future being entailed uniquely by the past, then the answer is no. GR is not a deterministic theory as such. Of course, a more comprehensive answer would be that GR is a theory whose solution space contains both deterministic and non-deterministic equations.

Starting point is 00:21:29 Furthermore, the physical realisability of these nondeterministic solutions is empirically underdetermined. It's an open empirical question. Manchak would say, even when you try to formulate what determinism is in GR, the phrase, complete state at any given time, has many non-equivalent rigorous translations. So what can we definitely say? Number one, the Einstein equations are locally deterministic. Great. Number two, global determinism requires global.

Starting point is 00:21:59 hyperbolicity. And number three, many physically interesting solutions, perfectly valid, lack global hyperbolicity. So perhaps the better way to define determinism isn't a property of theories, but a property of specific solutions. And in a universe described by general relativity, whether your future is determined could depend on where you are in space time. Einstein said, God doesn't play dice. Turns out in Einstein's own theory, God sometimes doesn't even show up to the table. The economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments

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Theories of Everything with Curt Jaimungal - Curt Jaimungal: General Relativity Is NOT Deterministic (Here's the Proof)

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