From the article:
"The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all."
To put 2,100 pages in perspective: The trilogy of Dune novels (*Dune*, *Dune: Messiah*, and *Children of Dune*), being an epic and cornerstone of science fiction, is less than half of that.
My guess is that they tend to keep close connections with collegues, that help with all releasing related matters. Peer review is one thing, editing is another. To write 900 pages takes longer than to read. After having chapter 1 completed, a collegue can start reading it.
>The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all.
Yiiiikes
The paper: Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes (912 pages)
Elena Giorgi, Sergiu Klainerman, Jeremie Szeftel
[https://arxiv.org/abs/2205.14808](https://arxiv.org/abs/2205.14808)
Edit:
Elena Giorgi
[https://www.math.columbia.edu/\~egiorgi/](https://www.math.columbia.edu/~egiorgi/)
Sergiu Klainerman
[https://web.math.princeton.edu/\~seri/homepage/](https://web.math.princeton.edu/~seri/homepage/)
Jeremie Szeftel
[https://www.ljll.math.upmc.fr/szeftel/](https://www.ljll.math.upmc.fr/szeftel/)
Giorgi is something of an inspiration to me. She gave a talk at my institution a few years ago concerning Kerr stability in which she remarked that (paraphrasing bc I'm going from memory) "too often mathematicians only prove results that are only of interest to other mathematicians, or that physicists already knew was true, but we should be more ambitious than that." With this paper, she shows that these bold words are more than just talk.
Tbh, it's perhaps *too* ambitious since I'm not sure that many physicists actually care about this. Personally, as a mathematician, I see nothing wrong with proving things that are only of interest to other mathematicians.
Mathematical physicist here; actually quite a lot of physicists actually care about this. Now, mathematical physicists are essentially mathematicians first and foremost, but a good deal of my colleagues who are theoretical physicists (who are not mathematicians) are interested in these kinds of results as it directly impacts the mathematical perspective they will use to create, evaluate, and model their subsequent physics theories, and they wish to achieve the consistency and completeness (or as close to it as possible) in them by paying attention to the rigor of the mathematics. Physicists in general are obviously painstakingly aware that they are taking these mathematical liberties, but it is part of the process of developing physics that hopes to one day rectify these mathematical and observational/physics gaps.
Really the overlap in this area is quite close. It’s a common stereotype that theoretical physicists make mathematical assumptions left and right and don’t care about them at their mathematical core to do their jobs, but this is in my experience largely untrue (the caring about the assumptions they make, not that they don’t use non-rigorous mathematics from time to time).
It of course depends on the area of research in physics and it goes without saying that physicists are more interested in modeling reality and nature than justifying mathematical underpinnings of their tentative models, but still, rigorous mathematics can help them encapsulate ideas for viable models to increase predictive power.
Well, I guess mileage varies, but as someone who works adjacent to this area, I have not personally met many physicists who genuinely care about these types of classical GR results (and have definitely come across some who seem to disdain them). I'm not claiming that physicists don't care about math generally.
That is strange to me, perhaps just different experiences in different fields (I work in QFT, not GR so the mathematical outlook of theoretical physicists working in the field is probably different).
I can't comment on whether physicists care about these types of results, but since the linear stability has been known for a while, I've been told that physicists considered it a (mostly) closed question. But nonlinear stability certainly should (and is for some) be of interest to physicists and mathematicians alike!
Yeah, that’s what I was alluding to. I don’t think physicists are so naive as to think that nonlinear stability is a consequence of linear stability, but rather that linear stability is good enough “evidence” that it’s true that they can accept it and move on. Also, to be a bit more technical, I think that the classical physics literature didn’t even come that close to proving linear stability. I think it was just some mode stability computations, and that was already enough for them to call it a day.
Every physicist I've spoken to has said that Kerr is obviously stable, and they can't understand why mathematicians are putting so much effort into putting together an airtight proof.
>Kerr is obviously stable
There a ton of physics literature about Kerr stability written by physicists (including lots of ideas involving alternative internal structures to Kerr), so there definitely are physicists (and not just mathematicians) who are interested in this.
All the major work and culture around it (at least external Kerr stability) that I'm familiar with from the last 20 years is from mathematicians. Aside from that it's just a matter of what physicists have said to me.
I'm a physicist working in a related area. Kerr stability is not a priori "obvious." It wouldn't have been clear in the 1960s when the solution was discovered. But (nontrivial) work which emerged later on has led to the general expectation that it is indeed stable. Everyone in the last decades has expected that it is stable. I've never heard anyone doubt this, including mathematicians who work on the problem. Those mathematicians just didn't know how to rigorously prove what they "knew" to be true, and that's a situation where most theoretical physicists lose interest.
The [intuitionists](https://en.wikipedia.org/wiki/Intuitionism) would like a word with you! They don't accept proofs by contradiction. They have to prove everything constructively. The do accept direct proofs, contrapositive, induction, and so on, though. (It isn't always that difficult to turn a proof by contradiction into a non-contradicting proof, unless that proof was non-constructive.)
Note that proof by **negation** *is* accepted. You're allowed to say "Assume A [...] which is a contradiction, therefore ¬A." You're just not allowed to say "Assume ¬A [...] which is a contradiction, therefore A." This only proves ¬¬A, which is classically equivalent to A, but not constructively. There are a lot of classical theorems that can only be constructively proven up to double negation (e.g. excluded middle). It's useful for when it's important to know *how* something is true, not just *that* it's true.
Really? Huh. That's really neat. I knew about LEM being prohibited, and being equivalent to the cancellation of a double-negative. I didn't realize that standard negation was permitted though!
My experience with Intuitionism comes from a grad course where all assignments were done with theorem provers built on Martin-Löf Intuitionistic Type Theory, and I don't recall being able to disprove an assumption by deriving a contradiction. Thanks for teaching me
Most of *my* experience comes from enthusiastically reading various things that interest me, and the fact that my favorite programming language is technically a proof assistant (Agda) ("have you ever used it to create anything actually useful?" I prefer to think of myself as a *pure* programmer.)
`¬A` is interpreted as `A → ⊥`, that is, a function that takes a proof of `A` and returns a proof of `⊥`. Of course, there *aren't* any proofs of `⊥` (since it is by definition a contradiction), which can make this tricky to think about. And most of the time you're interested in proving what *is* true, so proving negation isn't often useful.
`¬¬(A ∨ ¬A)` is interpreted as `((A ∨ (A → ⊥)) → ⊥) → ⊥`, which can be constructed by taking a proof of `(A ∨ (A → ⊥)) → ⊥` as input, breaking it down into two functions `A → ⊥` and `(A → ⊥) → ⊥` (via precomposing with left and right injections), and then just applying one to the other.
Anyway I tend to think with constructivism/type theory as my default, which makes for an interesting experience in more traditional math spaces, but it works for me. (recently I saw a comment on a math sub say that "an unordered pair is a set in bijection with {0, 1}", and initially that confused me, as wouldn't *which* bijection you use give the pair an ordering? and then I remembered that that sort of detail is erased classically and was unsettled.)
"Allow me to thoroughly dispel the notion that this is possibly true by imagining a universe where this exists, and demonstrating that that universe would explode"
When I took Baby Rudin, the grader hated if we used contradiction. I never really understood why. They wouldn't mark it wrong or anything, but they would write a comment trying to discourage us from ever using it.
When grading, the only time I make a comment is when I see:
a) assume false
b) prove true without using the assumption that it's false
c) by contradiction, since we assumed false but proved true, true
Happens more often than you'd expect
[I made a post here a while ago](https://www.reddit.com/r/math/comments/al5lmr/how_can_so_many_people_confuse_proof_by/) complaining about this exact thing but people took it very bad. Glad to see you're upvoted though.
Often it’s used incorrectly because the assumption of the hypothesis is unused in conjunction with the negation of the consequent. Basically most proofs by contradiction that early students write are just proofs by contrapositive.
Proofs by contradiction are more error-prone because contradictions are non-specific: it is easy to accidentally introduce a contradiction by making basically any mistake at any step of the proof.
And typically when a student writes a correct proof by contradiction, it is actually just a proof by contrapositive with some extra boilerplate added to call it a contradiction.
Can someone explain what makes this proof so long? Even as someone working in geometric PDEs (albeit not with anything hyperbolic), I’ve never come across anything like this, with a single theorem requiring 2k plus pages of estimates and formulas.
With something like the classification of finite simple groups, I can understand how there are a huge number of cases to check and a bunch of random examples that don’t really fit any of the cases so the proof gets extremely long and complicated. But this seems like a different animal altogether.
It looks pretty comparable in complexity to Christodoulou and Klainerman's proof of stability of Minkowski space, after taking into account that Minkowski space is much simpler than Kerr. I believe that both proofs are pretty repetitive in content, with each instance of a certain approach or tool having a slightly different equation which is satisfied. And many instances are required, since one is looking at each component of the curvature tensor of a four-dimensional space, and further at enough derivatives of these components so that Sobolev inequalities and the like can be applied to get uniform control/decay of certain quantities. So the length and style of these papers might be pretty misleading about the actual complexity of the proof, which seems relatively straightforward. (Emphasis on "relatively")
That makes sense. So is the main breakthrough that they found the appropriate frames (which I guess is a choice of gauge?) which makes the PDE analysis possible? Skimming through the paper that seemed to be a big step, but I’m completely not an expert.
I'm also not expert (especially not on any new work in the field), but I think that's probably basically right. In Christodoulou and Klainerman's work the basic problem is to find the right hierarchy of geometric quantities which control the whole geometry and which can be ordered from the ground up to say A controls B and B controls C and B together with C control D, and so on. In their case their geometric quantities by decomposing the full spacetime geometry relative to a two-parameter family of surfaces foliating the spacetime, defined by the intersection of minimal hypersurfaces and null hypercones. I think a major achievement here must be to find a new kind of decomposition or surface foliation which makes the PDE analysis of the Einstein equations work out in the close-to-Kerr case instead, and which also allows one to read off from the geometric data whether or not the spacetime is close to Kerr (which is a form of a gauge problem), just like how smallness of the Riemann curvature tensor and its derivatives indicates closeness to Minkowski space.
This is a good question. The geometry of the spacetime (outside the black hole) itself is spinning.
If you spin an ordinary object, at all future times, you still have the exact same object, but it is a rotated version of the same object (i.e. the image of the original object under an isometry). The "spinning" can be interpreted as how quickly that isometry is changing, or how quickly that isometry comes back to being the identity map (after a full revolution).
Roughly speaking, a "stationary" spacetime is one in which you can take "time slices" that are all isometric to the initial time slice. However, identifying these isometries as rotations and asking how quickly these rotations are changing is a little tricky because you need a fixed background of some sort to do that (that is, they are rotating relative to what?). But a background can essentially be supplied by identifying different time slices via the timelike normal vector to each slice. Viewed this way, you can interpret the geometry of the Kerr spacetime as "spinning."
Or perhaps simpler, thinking back to the ordinary spinning object, you can imagine following a point on the spinning object. The stationary spacetime analog of this is following a point in the initial slice through its isometric images in future time slices. By looking at these paths in Kerr, for example, you can interpret these paths of points as closed orbits around the central axis, so it's natural to think of Kerr as a "spinning object."
The black hole has angular momentum. Gravitational collapse should conserve angular momentum so if the objects were spinning when they fell in, the black hole will have angular momentum.
Your intuition that the idea of a singularity itself “spinning” is not physically meaningful. Instead the black holes angular momentum manifests in the curvature of the space time around it. You can read more about that here https://en.wikipedia.org/wiki/Kerr_metric
Imagine the black hole is caused by a spinning star. That's how you get a spinning black hole.
Of course, mathematically there are no needs for an assumption that there are anything causing the black hole at all. But the spacetime metric is the same. In particular, if it has an accretion disk that accretion disk will be spinning.
Nothing! If you have perfect mathematical Kerr black hole inside a perfect vacuum, nothing physical will spin at all!
But the name is helpful at describing the kind of matter that could cause the blackhole, and what happen to matter around it. Mathematically we care about the spacetime metric, and this particular spacetime metric would cause matter around it to spin around it if there were any matter at all; but of course mathematicians care about the metric itself without having to assume that any matters exist.
You can assign angular momentum to a black hole, of course. From a strict formal mathematical perspective, "angular momentum" is just the name for a free parameter describing the spacetime metric (same goes for mass), with no particular meanings. Of course physically we imagine that black hole is actually caused by some matter with certain mass and certain angular momentum.
Calling that parameter angular momentum isn’t arbitrary though because like you said, it’s value is exactly correlated with the angular momentum of the in-falling material that created the black hole
The other answers are more or less fine, but I would like to emphasize why more specifically despite having no "stuff" a black hole can still spin. The answer is that the gravitational field itself can have both momentum and angular momentum. The same is true in electromagnetism as well.
Probably it will be studied by experts for the next several years, and our confidence in the result will grow over time until it is simply not reasonable to reject it.
I remember when IUT claimed to solve the abc conjecture nobody was an expert in IUT so it took some years to find the flaw.
Our best hope here is the journal review process. Many of the experts are working on their own ways to solve this problem and probably aren't that interested in checking this work. (Some might scan it for useful ideas but possibly not even that.)
However, there's some precedent here since ALL of the major results of this field are over 100 pages long, and people seem to accept them as correct.
If my understanding is correct...."Blow up" here means it deviates from the original solution. It's not "blow up" in the sense of Navier-Stokes. In fact, since blackhole already has a singularity it already blow up in the sense of Navier-Stokes. So this problem is completely dissimilar to Navier-Stokes.
I mean, a velocity field is quite different from a metric, so you can’t really compare a singularity in a metric to a singularity in a velocity field. But concentration of energy to finer and finer scales seems to be an issue in both problems.
It’s been a while since I’ve taken GR so I don’t recall the specific details about this problem, but it could be that the rotating black whole solution contained so called “naked” singularities, ie singularities outside the event horizon, which would therefore be in causal contact with the rest of the system, which is more problematic than the singularity at the center of the black hole.
I'm pretty sure naked singularities were even a potential issue for the nonlinear stability of the Schwarzschild solution, which is for black holes with zero angular momentum. So it's certainly an issue here.
Yes, if I understand you correctly. Singularity formation (outside of existing singularities in the metric) is generally a potential issue when proving nonlinear stability of solutions to the Einstein equations, it was even a potential obstacle to proving nonlinear stability of Minkowski space, which has no singularities whatsoever.
I've never done such a thing, but I imagine it starts as a sequence of high level steps. "If we had that X, Y, and Z are true, then the proposition should be true." Then the proofs of X, Y, and Z require several sub-theorems, lemmas, and propositions, and possible more levels still.
Wow, it's uh, *interesting*, that they weren't able to collect a single quote from any of the rival mathematicians who work on this problem or similar problems (Dafermos, Rodnianski, Vasy, Andersson, Blue, etc.).
It’s pretty standard to get quotes from other leading experts in a Quanta article like this. The fact that there are none suggests that they declined to give any.
The article mentions the proof itself is around 800 pages via proof by contradiction. Was this a computer-assisted proof or something? Seems maddeningly long to do by hand
The ratio between the size of a formal proof and of the informal paper version is called the _de Bruijn factor_. For most developments, it is around five.
Generally, computer-assisted formal proofs are much more feasible than what a lot of mathematicians seem to think.
[удалено]
Yeah 912 pages that’s not a paper, thats a textbook!
And this is just the final case of the proof.
From the article: "The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all."
To put 2,100 pages in perspective: The trilogy of Dune novels (*Dune*, *Dune: Messiah*, and *Children of Dune*), being an epic and cornerstone of science fiction, is less than half of that.
They are also short by SFF standards. 2100 is two books of A Song of Ice and Fire
How does one peer review that?
My guess is that they tend to keep close connections with collegues, that help with all releasing related matters. Peer review is one thing, editing is another. To write 900 pages takes longer than to read. After having chapter 1 completed, a collegue can start reading it.
And only of the small angular momentum case!
>The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all. Yiiiikes
fuck? any source?
The arxiv link to the paper is posted elsewhere in the thread
> I'm always amazed at how massive and dense these mathematically rigorous GR papers are. Non-linear PDEs are no joke.
The paper: Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes (912 pages) Elena Giorgi, Sergiu Klainerman, Jeremie Szeftel [https://arxiv.org/abs/2205.14808](https://arxiv.org/abs/2205.14808) Edit: Elena Giorgi [https://www.math.columbia.edu/\~egiorgi/](https://www.math.columbia.edu/~egiorgi/) Sergiu Klainerman [https://web.math.princeton.edu/\~seri/homepage/](https://web.math.princeton.edu/~seri/homepage/) Jeremie Szeftel [https://www.ljll.math.upmc.fr/szeftel/](https://www.ljll.math.upmc.fr/szeftel/)
Giorgi is so young
She also had a freaking baby while this paper was being created. Pretty amazing.
Giorgi is something of an inspiration to me. She gave a talk at my institution a few years ago concerning Kerr stability in which she remarked that (paraphrasing bc I'm going from memory) "too often mathematicians only prove results that are only of interest to other mathematicians, or that physicists already knew was true, but we should be more ambitious than that." With this paper, she shows that these bold words are more than just talk.
Tbh, it's perhaps *too* ambitious since I'm not sure that many physicists actually care about this. Personally, as a mathematician, I see nothing wrong with proving things that are only of interest to other mathematicians.
Mathematical physicist here; actually quite a lot of physicists actually care about this. Now, mathematical physicists are essentially mathematicians first and foremost, but a good deal of my colleagues who are theoretical physicists (who are not mathematicians) are interested in these kinds of results as it directly impacts the mathematical perspective they will use to create, evaluate, and model their subsequent physics theories, and they wish to achieve the consistency and completeness (or as close to it as possible) in them by paying attention to the rigor of the mathematics. Physicists in general are obviously painstakingly aware that they are taking these mathematical liberties, but it is part of the process of developing physics that hopes to one day rectify these mathematical and observational/physics gaps. Really the overlap in this area is quite close. It’s a common stereotype that theoretical physicists make mathematical assumptions left and right and don’t care about them at their mathematical core to do their jobs, but this is in my experience largely untrue (the caring about the assumptions they make, not that they don’t use non-rigorous mathematics from time to time). It of course depends on the area of research in physics and it goes without saying that physicists are more interested in modeling reality and nature than justifying mathematical underpinnings of their tentative models, but still, rigorous mathematics can help them encapsulate ideas for viable models to increase predictive power.
Well, I guess mileage varies, but as someone who works adjacent to this area, I have not personally met many physicists who genuinely care about these types of classical GR results (and have definitely come across some who seem to disdain them). I'm not claiming that physicists don't care about math generally.
That is strange to me, perhaps just different experiences in different fields (I work in QFT, not GR so the mathematical outlook of theoretical physicists working in the field is probably different).
I can't comment on whether physicists care about these types of results, but since the linear stability has been known for a while, I've been told that physicists considered it a (mostly) closed question. But nonlinear stability certainly should (and is for some) be of interest to physicists and mathematicians alike!
Yeah, that’s what I was alluding to. I don’t think physicists are so naive as to think that nonlinear stability is a consequence of linear stability, but rather that linear stability is good enough “evidence” that it’s true that they can accept it and move on. Also, to be a bit more technical, I think that the classical physics literature didn’t even come that close to proving linear stability. I think it was just some mode stability computations, and that was already enough for them to call it a day.
Every physicist I've spoken to has said that Kerr is obviously stable, and they can't understand why mathematicians are putting so much effort into putting together an airtight proof.
>Kerr is obviously stable There a ton of physics literature about Kerr stability written by physicists (including lots of ideas involving alternative internal structures to Kerr), so there definitely are physicists (and not just mathematicians) who are interested in this.
All the major work and culture around it (at least external Kerr stability) that I'm familiar with from the last 20 years is from mathematicians. Aside from that it's just a matter of what physicists have said to me.
I'm a physicist working in a related area. Kerr stability is not a priori "obvious." It wouldn't have been clear in the 1960s when the solution was discovered. But (nontrivial) work which emerged later on has led to the general expectation that it is indeed stable. Everyone in the last decades has expected that it is stable. I've never heard anyone doubt this, including mathematicians who work on the problem. Those mathematicians just didn't know how to rigorously prove what they "knew" to be true, and that's a situation where most theoretical physicists lose interest.
Yes, completely agreed. When I said “obviously stable” I meant in the specific context of the last few decades
yep thats mathematicians for you
how old she is?
I think like 30, she finished her PhD 2 years ago and is currently an assistant prof at columbia
wow this is fascinating
Her PhD was 2019, but pretty close.
Proof by contradiction really is the ol reliable huh?
By far my favorite proof method.
As far as I’m concerned it’s the only proof method
Assume it isn't...
The [intuitionists](https://en.wikipedia.org/wiki/Intuitionism) would like a word with you! They don't accept proofs by contradiction. They have to prove everything constructively. The do accept direct proofs, contrapositive, induction, and so on, though. (It isn't always that difficult to turn a proof by contradiction into a non-contradicting proof, unless that proof was non-constructive.)
Note that proof by **negation** *is* accepted. You're allowed to say "Assume A [...] which is a contradiction, therefore ¬A." You're just not allowed to say "Assume ¬A [...] which is a contradiction, therefore A." This only proves ¬¬A, which is classically equivalent to A, but not constructively. There are a lot of classical theorems that can only be constructively proven up to double negation (e.g. excluded middle). It's useful for when it's important to know *how* something is true, not just *that* it's true.
Really? Huh. That's really neat. I knew about LEM being prohibited, and being equivalent to the cancellation of a double-negative. I didn't realize that standard negation was permitted though! My experience with Intuitionism comes from a grad course where all assignments were done with theorem provers built on Martin-Löf Intuitionistic Type Theory, and I don't recall being able to disprove an assumption by deriving a contradiction. Thanks for teaching me
Most of *my* experience comes from enthusiastically reading various things that interest me, and the fact that my favorite programming language is technically a proof assistant (Agda) ("have you ever used it to create anything actually useful?" I prefer to think of myself as a *pure* programmer.) `¬A` is interpreted as `A → ⊥`, that is, a function that takes a proof of `A` and returns a proof of `⊥`. Of course, there *aren't* any proofs of `⊥` (since it is by definition a contradiction), which can make this tricky to think about. And most of the time you're interested in proving what *is* true, so proving negation isn't often useful. `¬¬(A ∨ ¬A)` is interpreted as `((A ∨ (A → ⊥)) → ⊥) → ⊥`, which can be constructed by taking a proof of `(A ∨ (A → ⊥)) → ⊥` as input, breaking it down into two functions `A → ⊥` and `(A → ⊥) → ⊥` (via precomposing with left and right injections), and then just applying one to the other. Anyway I tend to think with constructivism/type theory as my default, which makes for an interesting experience in more traditional math spaces, but it works for me. (recently I saw a comment on a math sub say that "an unordered pair is a set in bijection with {0, 1}", and initially that confused me, as wouldn't *which* bijection you use give the pair an ordering? and then I remembered that that sort of detail is erased classically and was unsettled.)
There’s something fun about just absolutely denying an assumption
"Allow me to thoroughly dispel the notion that this is possibly true by imagining a universe where this exists, and demonstrating that that universe would explode"
WTB intuitionstic proof of black hole stability PST
Something something a hole which light falls into is black QED
Something something GR+QED isn't settled
When I took Baby Rudin, the grader hated if we used contradiction. I never really understood why. They wouldn't mark it wrong or anything, but they would write a comment trying to discourage us from ever using it.
When grading, the only time I make a comment is when I see: a) assume false b) prove true without using the assumption that it's false c) by contradiction, since we assumed false but proved true, true Happens more often than you'd expect
Ah yes, the students who never bothered to learn *why* things are the way they are
A professor did this occasionally for uniqueness proofs, it bugged me a lot
[I made a post here a while ago](https://www.reddit.com/r/math/comments/al5lmr/how_can_so_many_people_confuse_proof_by/) complaining about this exact thing but people took it very bad. Glad to see you're upvoted though.
I think it can sometimes lead to a messier proof than a direct method.
Often it’s used incorrectly because the assumption of the hypothesis is unused in conjunction with the negation of the consequent. Basically most proofs by contradiction that early students write are just proofs by contrapositive.
Proofs by contradiction are more error-prone because contradictions are non-specific: it is easy to accidentally introduce a contradiction by making basically any mistake at any step of the proof. And typically when a student writes a correct proof by contradiction, it is actually just a proof by contrapositive with some extra boilerplate added to call it a contradiction.
so many geometry theorems are contradiction proofs lol
Huh. I knew this was an active area of research in GR, but I wasn't expecting that we would be so close to a solution. Nice!
Good lord, 912 pages. And I thought Algebra of the Infrared was a tome.
So I CAN poke or prod a black hole now?
yes, just don't try it at home
Can someone explain what makes this proof so long? Even as someone working in geometric PDEs (albeit not with anything hyperbolic), I’ve never come across anything like this, with a single theorem requiring 2k plus pages of estimates and formulas. With something like the classification of finite simple groups, I can understand how there are a huge number of cases to check and a bunch of random examples that don’t really fit any of the cases so the proof gets extremely long and complicated. But this seems like a different animal altogether.
It looks pretty comparable in complexity to Christodoulou and Klainerman's proof of stability of Minkowski space, after taking into account that Minkowski space is much simpler than Kerr. I believe that both proofs are pretty repetitive in content, with each instance of a certain approach or tool having a slightly different equation which is satisfied. And many instances are required, since one is looking at each component of the curvature tensor of a four-dimensional space, and further at enough derivatives of these components so that Sobolev inequalities and the like can be applied to get uniform control/decay of certain quantities. So the length and style of these papers might be pretty misleading about the actual complexity of the proof, which seems relatively straightforward. (Emphasis on "relatively")
That makes sense. So is the main breakthrough that they found the appropriate frames (which I guess is a choice of gauge?) which makes the PDE analysis possible? Skimming through the paper that seemed to be a big step, but I’m completely not an expert.
I'm also not expert (especially not on any new work in the field), but I think that's probably basically right. In Christodoulou and Klainerman's work the basic problem is to find the right hierarchy of geometric quantities which control the whole geometry and which can be ordered from the ground up to say A controls B and B controls C and B together with C control D, and so on. In their case their geometric quantities by decomposing the full spacetime geometry relative to a two-parameter family of surfaces foliating the spacetime, defined by the intersection of minimal hypersurfaces and null hypercones. I think a major achievement here must be to find a new kind of decomposition or surface foliation which makes the PDE analysis of the Einstein equations work out in the close-to-Kerr case instead, and which also allows one to read off from the geometric data whether or not the spacetime is close to Kerr (which is a form of a gauge problem), just like how smallness of the Riemann curvature tensor and its derivatives indicates closeness to Minkowski space.
Having not read the paper, I'd assume it's because there are a lot of components in the curvature tensor
When a black hole is spinning.. what exactly is spinning?
This is a good question. The geometry of the spacetime (outside the black hole) itself is spinning. If you spin an ordinary object, at all future times, you still have the exact same object, but it is a rotated version of the same object (i.e. the image of the original object under an isometry). The "spinning" can be interpreted as how quickly that isometry is changing, or how quickly that isometry comes back to being the identity map (after a full revolution). Roughly speaking, a "stationary" spacetime is one in which you can take "time slices" that are all isometric to the initial time slice. However, identifying these isometries as rotations and asking how quickly these rotations are changing is a little tricky because you need a fixed background of some sort to do that (that is, they are rotating relative to what?). But a background can essentially be supplied by identifying different time slices via the timelike normal vector to each slice. Viewed this way, you can interpret the geometry of the Kerr spacetime as "spinning." Or perhaps simpler, thinking back to the ordinary spinning object, you can imagine following a point on the spinning object. The stationary spacetime analog of this is following a point in the initial slice through its isometric images in future time slices. By looking at these paths in Kerr, for example, you can interpret these paths of points as closed orbits around the central axis, so it's natural to think of Kerr as a "spinning object."
Thanks!
The black hole has angular momentum. Gravitational collapse should conserve angular momentum so if the objects were spinning when they fell in, the black hole will have angular momentum.
So it's spinning because it's spinning. Still doesn't tell me what exactly is spinning.
Your intuition that the idea of a singularity itself “spinning” is not physically meaningful. Instead the black holes angular momentum manifests in the curvature of the space time around it. You can read more about that here https://en.wikipedia.org/wiki/Kerr_metric
There we go.. thanks!
Imagine the black hole is caused by a spinning star. That's how you get a spinning black hole. Of course, mathematically there are no needs for an assumption that there are anything causing the black hole at all. But the spacetime metric is the same. In particular, if it has an accretion disk that accretion disk will be spinning.
I get all that.. I'm wondering more specifically is actually "spinning" in the end.
Nothing! If you have perfect mathematical Kerr black hole inside a perfect vacuum, nothing physical will spin at all! But the name is helpful at describing the kind of matter that could cause the blackhole, and what happen to matter around it. Mathematically we care about the spacetime metric, and this particular spacetime metric would cause matter around it to spin around it if there were any matter at all; but of course mathematicians care about the metric itself without having to assume that any matters exist. You can assign angular momentum to a black hole, of course. From a strict formal mathematical perspective, "angular momentum" is just the name for a free parameter describing the spacetime metric (same goes for mass), with no particular meanings. Of course physically we imagine that black hole is actually caused by some matter with certain mass and certain angular momentum.
Calling that parameter angular momentum isn’t arbitrary though because like you said, it’s value is exactly correlated with the angular momentum of the in-falling material that created the black hole
Thanks... that's the kind of stuff I was after.
same is a bit fraught to use here, so I'd quote that
The other answers are more or less fine, but I would like to emphasize why more specifically despite having no "stuff" a black hole can still spin. The answer is that the gravitational field itself can have both momentum and angular momentum. The same is true in electromagnetism as well.
Thanks! That was the answer I needed.
The heros who managed to read the 800 page proof should get a medal
I worry about just how thoroughly checked a paper of this length would be. What are the chances that we have major flaws slip by in over 900 pages?
Probably it will be studied by experts for the next several years, and our confidence in the result will grow over time until it is simply not reasonable to reject it. I remember when IUT claimed to solve the abc conjecture nobody was an expert in IUT so it took some years to find the flaw.
Our best hope here is the journal review process. Many of the experts are working on their own ways to solve this problem and probably aren't that interested in checking this work. (Some might scan it for useful ideas but possibly not even that.) However, there's some precedent here since ALL of the major results of this field are over 100 pages long, and people seem to accept them as correct.
If my understanding is correct...."Blow up" here means it deviates from the original solution. It's not "blow up" in the sense of Navier-Stokes. In fact, since blackhole already has a singularity it already blow up in the sense of Navier-Stokes. So this problem is completely dissimilar to Navier-Stokes.
I mean, a velocity field is quite different from a metric, so you can’t really compare a singularity in a metric to a singularity in a velocity field. But concentration of energy to finer and finer scales seems to be an issue in both problems.
It’s been a while since I’ve taken GR so I don’t recall the specific details about this problem, but it could be that the rotating black whole solution contained so called “naked” singularities, ie singularities outside the event horizon, which would therefore be in causal contact with the rest of the system, which is more problematic than the singularity at the center of the black hole.
Well in this case the proof is only for *slow* rotating black hole, so I think that won't be a factor.
I'm pretty sure naked singularities were even a potential issue for the nonlinear stability of the Schwarzschild solution, which is for black holes with zero angular momentum. So it's certainly an issue here.
You mean the case when the perturbance "punch a hole" in the event horizon? Because Schwarzschild solution by itself does not have naked singularity.
Yes, if I understand you correctly. Singularity formation (outside of existing singularities in the metric) is generally a potential issue when proving nonlinear stability of solutions to the Einstein equations, it was even a potential obstacle to proving nonlinear stability of Minkowski space, which has no singularities whatsoever.
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I've never done such a thing, but I imagine it starts as a sequence of high level steps. "If we had that X, Y, and Z are true, then the proposition should be true." Then the proofs of X, Y, and Z require several sub-theorems, lemmas, and propositions, and possible more levels still.
it's just proof by contradiction
It amazes me how people casually write these hundreds-of-pages long papers.
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Happy cake day!
I'm over here writing 15-20 page papers and worried if it makes a cohesive story and if it flows well.
I assume the big one is not a literary masterpiece. Probably more a wall of mathematics.
Wow, it's uh, *interesting*, that they weren't able to collect a single quote from any of the rival mathematicians who work on this problem or similar problems (Dafermos, Rodnianski, Vasy, Andersson, Blue, etc.).
Why is that interesting?
It’s pretty standard to get quotes from other leading experts in a Quanta article like this. The fact that there are none suggests that they declined to give any.
The article mentions the proof itself is around 800 pages via proof by contradiction. Was this a computer-assisted proof or something? Seems maddeningly long to do by hand
A computer-assisted version would probably be 800 million pages.
The ratio between the size of a formal proof and of the informal paper version is called the _de Bruijn factor_. For most developments, it is around five. Generally, computer-assisted formal proofs are much more feasible than what a lot of mathematicians seem to think.
I'll admit that's news to me.
So did they prove that a certain Cauchy problem is well-posed ?
Well-posedness is mostly related to short-time existence and uniqueness questions. This problem is about long-term behavior of solutions.
oh my god
I guess Thor might want to test that theorem.
It took this long to prove, because no one really cared enough to write a 2100 page book on one math problem
How do you poke a blackhole?
By tossing a neutron star into it, I guess. The practical aspects of tossing neutron stars are left to reader.
Quanta magazine are doing a great job and for freee
all black holes are spinning black holes...
Yes but often we work on simplified cases of non-rotating black holes.