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Math Science Technology

At Long Last, Mathematical Proof That Black Holes Are Stable (quantamagazine.org) 75

Steve Nadis, reporting for Quanta Magazine: In 1963, the mathematician Roy Kerr found a solution to Einstein's equations that precisely described the space-time outside what we now call a rotating black hole. (The term wouldn't be coined for a few more years.) In the nearly six decades since his achievement, researchers have tried to show that these so-called Kerr black holes are stable. What that means, explained Jeremie Szeftel, a mathematician at Sorbonne University, "is that if I start with something that looks like a Kerr black hole and give it a little bump" -- by throwing some gravitational waves at it, for instance -- "what you expect, far into the future, is that everything will settle down, and it will once again look exactly like a Kerr solution." The opposite situation -- a mathematical instability -- "would have posed a deep conundrum to theoretical physicists and would have suggested the need to modify, at some fundamental level, Einstein's theory of gravitation," said Thibault Damour, a physicist at the Institute of Advanced Scientific Studies in France.

In a 912-page paper posted online on May 30, Szeftel, Elena Giorgi of Columbia University and Sergiu Klainerman of Princeton University have proved that slowly rotating Kerr black holes are indeed stable. The work is the product of a multiyear effort. 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. The new result "does indeed constitute a milestone in the mathematical development of general relativity," said Demetrios Christodoulou, a mathematician at the Swiss Federal Institute of Technology Zurich. Shing-Tung Yau, an emeritus professor at Harvard University who recently moved to Tsinghua University, was similarly laudatory, calling the proof "the first major breakthrough" in this area of general relativity since the early 1990s. "It is a very tough problem," he said. He did stress, however, that the new paper has not yet undergone peer review. But he called the 2021 paper, which has been approved for publication, both "complete and exciting."

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At Long Last, Mathematical Proof That Black Holes Are Stable

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  • by Petersko ( 564140 ) on Thursday August 04, 2022 @11:24AM (#62762208)

    ...he might just be getting tired of hearing, "Well, if this is true, then Einstein's work is wrong! Oh wait, no... he was right."

    • by splutty ( 43475 )

      Considering Einstein said many times "This can't be right", I'm fairly sure he'd be elated :D

      • by Z00L00K ( 682162 )

        What if there's a maximum mass that a black hole can have before it explodes because there's physics that we can't encounter in the lab?

    • by doug141 ( 863552 )

      As far as I know, Einstein never got tired of Bohr and others telling him he might be wrong. They dreamed up experiments together to prove who was right, but they didn't have the tech to run those experiments. Einstein did not live to see himself finally proven wrong by the Einstein–Podolsky–Rosen experiment. Also on the topic of the great minds being wrong, Tesla thought relativity was obviously wrong.

      • I'm not sure Bell's inequality would have been interpreted as a loss for Einstein.
        Bell's inequality tells us 1 of 2 things about the Universe.
        1) That spooky action at a distance is real.
        or
        2) That the Universe is deterministic.

        I.e., it didn't rule out local hidden variables. It only ruled them out if humans are "statistically independent", something Einstein would have argued to his death bed, as he was a strict deterministic like Spinoza.
  • I'm not sure this is 100% resolved yet. I looked at the equations but they didn't mention throwing your mom in there.

  • Finally! (Score:5, Funny)

    by CaptainLugnuts ( 2594663 ) on Thursday August 04, 2022 @11:33AM (#62762234)
    Some light beach reading before the summer is over.
    • I recommend listening to Einstein on the Beach [youtube.com] (by Philip Glass) at the same time.
    • by shanen ( 462549 )

      Mod parent funny. At least it's the joke I was looking for.

    • Man. This thing is extreme.

      800+ pages of postdoc level calculus.

      How do you even go about verifying the soundness of a proof this large?

    • Some light beach reading before the summer is over.

      I downloaded it for fun. Some observations:

      The freakin' Table of Contents is 19 pages long! I've read whole papers shorter than that.

      I didn't make it past the first page before I was hitting terminology I had never seen before and had no idea what it meant. Complete word salad to me.

  • Does anyone read an 800 page paper for a mathematical proof? I had an engineering professor once tell me that you're lucky if your advisor reads your thesis. If people don't read papers longer than a few pages in engineering, are mathematics academics actually reading something this long?

    • Look at proposition 4.3.3 - page 164/912. Not all the symbols are being rendered right in the PDF, or it's using non-embedded fonts that my machine doesn't have.
      So it's a challenge to fully review even if you can follow the mathematics.

      • by vux984 ( 928602 )

        Were you looking at the "squares"? If you look at the source (TeX); it looks like its "\square" in the source.

        \begin{proposition}
        \lab{LE:COMMTZSQUARE}
        The following commutation formulas hold true for a scalar $\psi$:
        \bea
        \lab{eq:commTZsquare}
        \bsplit
        \,[ \T, \square_\g] \psi&=\dk \big(\Ga_g \c \dk \psi\big)+\Ga_b \c \square_\g\psi ,\\
        \,[ \Z, \square_\g] \psi &= \dk \big(\Ga_g \c \dk \psi\big)+r\Ga_b \c \square_\g\psi.
        \end{split}
        \eea
        The following

      • by splutty ( 43475 )

        Most (all) mathematicians don't look at the PDF, they just render it from source (generally TeX or LaTeX).

      • don't know if being funny or not but this was typeset with Latex. it has a specific math mode for the math notation.

        Once you get use to it your brain naturally passes it, so it's like hearing/thinking the words and what is being formally stated (e.g. "field", "flow")

        They don't read it all, I assume they scan it. Generally you start around the conclusion where references to important parts are noted and then check those parts with you specifically following the referenced part till you find something that's

        • I wasn't joking. I wrote a 432 page book in latex, so I know how it goes.

          What happens every now and then is the that the unsupported symbols get replaced with an M sized square, which is what is happening there in the document. If you follow the links, you can download the latex source, so the information is available, but I didn't hunt that bit down in the latex.

          • I think it's intentional as the other commentator mentions. Some notion related to the physics.

            But of course, you can get all kinds of fun bugs, especially with complex templates. Honestly, I can follow most languages well and even learn to hack at them but LaTeX has always been magical to me.

            We seem like we might be in the same generation of nerdom. I hope the Young ones are still learning about this language and it's not "deprecated" to math and physics as a hand me down.

            Could a typesetting language ever

            • by imidan ( 559239 )
              I did all of my work (including homework) for a master's degree in stats using LaTeX. Way easier than doing it on paper and having to go back and fix a mistake early on. Just saying, I don't think it's going away any time soon.
            • by narcc ( 412956 )

              Some years back, I wrote a typesetting program for a company I was consulting with to help them typeset books. It had a gui and some simple markdown. Think of it like Lyx, but streamlined and special purpose. It generated LaTeX because there really wasn't anything better. Doing the typesetting myself was out-of-the-question at the time. (Typesetting isn't as easy as it might seem at first and it would have taken longer than was reasonable.)

              I can say with certainty that typesetting can easily be better.

              • Fair points. I think overall for most users these are minor. Thus to try to build a new language from the ground up would lead to more and greater issues because the logic behind LaTeXs design is deeply rooted in a history of find decent enough solutions to a wide range of issues. For instance, if you made a great piece of software for typesetting books then how well would it handed posters, post cards, or complex mathematical language.

                I believe Microsoft has a competing product different than Word but it's

                • There is a large incumbent effect. The effort to replace something like latex is huge and I don't expect to see it happen.

                  I would like to see language translators that take language with a saner syntax, better semantic set and library system and compiles it to latex. That seems doable. You can do most things in latex, but doing some of them feels troublesome and arcane. Following the normal latex ways leads you to some common styles you see in papers everywhere. It's not really my gig to write that sort of

                • by narcc ( 412956 )

                  You might be giving the design of TeX a bit too much credit. You could have the same system with different base language would be an immediate improvement.

                  Knuth had a different idea of what was "optimal" than most people. It reminds me a bit of someone building a Forth from an extremely minimal implementation. It can be a fun game, but it's not something I'd want to actually use!

    • Yes. Mathematicians will. No one is going to read this in one setting on the toilet, and it might take a team of people to look at all aspects.
    • I would speculate this kind of paper is not meant to be read in full as part of daily practice, only to be analysed until there is consensus that it is valid. When Andrew Wiles announced his proof of Fermat's Last Theorem in 129 pages, it was said other scholars would take some weeks or months to read through and ensure the proof was flawless.

      In engineering it's basically "we did it and it worked", or "we followed methodology of paper X to obtain equation Y" and you have to trust. That's why there can be fr

    • nothing insightful comes from 800 page proofs
      only the proof comes from it
    • by PLAST ( 416196 )
      That's the whole point of the proof: It's so massive that, once you get caught up in all the auxiliary definitions and lemmata, there's no way out, even at the speed of light, QED
    • Proof by induction:

      If the first n pages are OK, the next page is probably OK too.

      The first page looks OK.

      Therefore, the whole thing is probably OK.

  • by smooth wombat ( 796938 ) on Thursday August 04, 2022 @12:06PM (#62762304) Journal

    After rereading the synopsis, my question would be, wouldn't it be obvious if a black hole was bumped it would settle down at some point in the future? If gravitational waves were to hit a black hole, it should seem obvious the black hole might "jiggle" for a bit time, similar to a bowl of jello jiggling if the container is bumped, but as time progresses, the gravitational pull of the black hole would cause its mass to to stop jiggling.

    Maybe I'm looking at this with common sense and not mathematics. Everything eventually settles down after being bumped. It just takes time.

    • Re: (Score:2, Insightful)

      by iggymanz ( 596061 )

      No, "unstable" system would come apart. A cone balanced on its nose is unstable, it falls over. Everything does NOT settle down after being bumped, try smacking a puddle of trinitroglycerol (commonly called nitroglycerin) with a hammer and get back to us on how it settled back into a puddle, lolz

    • by gtall ( 79522 ) on Thursday August 04, 2022 @01:03PM (#62762442)

      Something "obvious" isn't necessarily easy to prove. When the real work of proving the conjecture started, they needed to show instability would not result from such a perturbation.

    • by elvstone ( 86513 )

      Nice proof by jello, but you could have saved them 900 pages of dense math if you submitted this a few years ago..

    • by Tablizer ( 95088 )

      "Honey, do these gravity waves make my ass jiggle?"

    • Or you could argue it's "obviously" true since there aren't unstable black holes careening all around the universe.

      I believe the issue here is that if the accepted mathematical models predict something different than what we actually observe (stability) then the models must be wrong, which would be a big deal.

      Of course one could make a simple model of a black hole as a simple attractor with newtonian physics that was stable - but that's not a fundamental model, it doesn't predict much else, and is known

    • by Anonymous Coward
      It sounds like we were already pretty sure black holes are stable. The proof is that assuming Einstein's model of gravity black holes are stable. In effect, that's really about increasing our confidence that Einstein's model of gravity closely matches the real universe.
    • After rereading the synopsis, my question would be, wouldn't it be obvious if a black hole was bumped it would settle down at some point in the future?

      To quote [goodreads.com] E.T. Bell, "Obvious is the most dangerous word in mathematics."

      ...Maybe I'm looking at this with common sense and not mathematics. Everything eventually settles down after being bumped. It just takes time.

      Only things that are stable. Poke a lump of uranium 235 with a neutron, and it does not "settle down."

    • by PPH ( 736903 )

      wouldn't it be obvious if a black hole was bumped it would settle down at some point in the future?

      I suppose that depends on what instability looks like. But aside from that, the number of black holes that we've detected combined with the probability of them having encountered gravitational waves in the past seems to indicate that they don't explode, evaporate or do something that we can detect and classify as "unstable".

    • Common sense would suggest that the energy would eventually dissipate, just as sound waves dissipates after encountering a wine-glass but strange things like them combining outside of the event-horizon to the point where they eventually form a separate singularity where also imaginable. That's the thing, you can never be sure if something is just highly improbable or indeed impossible until you have a mathematical proof.
      • To add one last though. Ultimately mathematical proofs are not really about proving something which is often quite obviously true but to hopefully gain some understanding of why is it that it works that way, insight which could then be used to further explore the object you are looking at.
  • by fredrated ( 639554 ) on Thursday August 04, 2022 @12:08PM (#62762310) Journal

    mathematics doesn't 'prove' anything about the real world.

    • mathematics doesn't 'prove' anything about the real world.

      It's supposed to be:

      "At Long Last, Mathematical Proof That Black Holes Are Stable (in mice)"

    • by Aliks ( 530618 )
      What is this "real world" of which you speak?
    • It's true, but then again nothing proves anything about the universe, since it's only based on what we have observed, and what might happen out of sight or in the future might be different.

      But it's not like mathematics renders prediction/extrapolation less reliable than it would be using logic or intuition or anything else.

      • As we've learned in the last hundred years... a lot of interesting things might be happening when you're not looking at them.

  • It'll be a Herculean effort to review a proof this long. Since it was generated by humans, and we have have a high error rate, I expect that will contain a lot of errors...
  • ... stable on all but the longest timescales. No mention of Hawking radiation slowly evaporating black holes; and which probably eliminated primordial micro black holes.

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