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Mathematicians Disprove Conjecture Made to Save Black Holes (quantamagazine.org)
227 points by bainsfather 9 months ago | hide | past | web | favorite | 59 comments



Ok, yes, this is how you promote scientific research to a lay audience:

1) They don't bury the lede. The first paragraph says what the result is immediately, if you understand it already, you're done reading.

2) Inverted pyramid structure. After they explain what happened, they break apart the historical context of the problem itself and give copious examples and metaphors to give the gist of what the problem is about and why it matters that it was solved.

I can't tell you how many of these popsci articles start out with "When Mary was a 3 year old, she used to look up at the stars and ... blah blah ... Now, she's taking on the scientific establishment and daring to do the unthinkable..." etc etc. I just dread skimming through the fluff to try to pick out what the hell was actually done.

Thank you Kevin Hartnett (the author of this piece) for not attempting to turn scientific papers into a human interest story.


It's not just science reporting. One of the reasons I hate the Olympics is that I like to watch sports on TV: Not heartfelt stories of overcoming adversity to become one of the world's elite. Not teary eyed medal ceremonies with semi transparent backdrops of national flags blowing in the wind. Not endless medal count standings. Not interviews of people with three medals around their necks, with insets of proud parents in the upper left hand corner. I just want to see the sporting events.

Today's society values drama above everything else. It's a shame (either that they do, or that I don't fit in ;-) ).


I really liked watching the original Ninja Warrior (the Japanese version). I even liked (to a lesser extent) the first US-heavy version.

I stopped watching for exactly the reason you describe.

The exact same thing happened with esports. Five minutes of actual play, 25 minutes of fluff.


I largely agree, but to add another perspective I have always viewed sports themselves as simply action-driven drama. It can be entertaining to watch, but ultimately the end results seem inconsequential to me compared to things like scientific discoveries.


They're just different types of articles--straight news reporting vs feature writing. It's fine of course to not like one or the other style, but they are both effective ways of connecting with audiences.


It depends on the audience though. In this case, I think it's correct to say there's a strong chance that the reader is not interested in prose. The sad thing about the human interest style is that, if that's not what I'm reading for, I begin skimming as soon as I detect it and much of the author's work ends up being for nothing.


Yeah, plain info versus story. Both have their uses.


Amen. I am so tired of trying to read these articles that spend so much time on random personal backstories because they don’t have faith that the core subject matter is interesting to read. The worst are the ones that jump back and forth between A/B/C plots, jumping away before any satisfying conclusion is reached on any of them, as a way of tricking your brain into reading the article ... like a really bad TV show, or Westworld.

All those articles can go burn in a hot, hot fire.


Just one correction, that's the classic pyramid structure, not a reverse pyramid.


Their work is subtle — a refutation of Penrose’s original statement of the strong cosmic censorship conjecture, but not a complete denial of its spirit.

I wish somebody out there could cover social science research and politics with this kind of attitude. This is really good science writing.


Einstein and Penrose theory totally crushed by mathematicians -- that makes a better headline.


"Einstein and Penrose were wrong; this is why"


"10 crushing proofs Einstein and Penrose don't want you to know" : Best headline.


What color is this proof: Einstein or Penrose?


So spacetime exists beyond the Cauchy horizon, but it's discontinuous?

What on earth would discontinuous spacetime involve? It sounds like a sort of shattered chaos of torn-up bits of space.


I need to read this more carefully, but a discontinuity in the derivatives of a system of equations at an interface is fairly common. If there are two regions obeying Maxwell's equations with a difference in the derivatives of the E/B fields at the boundary, this means the boundary has some delta function of charge or currents on the interface.

Likewise in GR. There is a whole sub-field of joining solutions to Einstein equations together and determining the material at the interface. Wormhole solutions can be made with this cut-and-paste approach. (The best book on this is Eric Poisson's "A Relativist's Toolkit").

The paper referenced here is 200+ pages. The abstract states in part "We prove that for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric.", which is what disproves strong cosmic censorship.

I have no grasp of the details (my PhD in GR junction conditions is from the 90's and I left the field).


It's not discontinuous, its derivative (a function describing the rate of change of space time) is apparently discontinuous, or infinite, or something else equally hard to picture.

Plenty of strange things can happen with derivatives, e.g. https://en.m.wikipedia.org/wiki/Cantor_function


I don't find failing to be differentiable all that strange, even if the failure is on a perfect nowhere dense set.

What I find weird is that the derivative, if it exists, cannot have a jump discontinuity. That means that the only other kind of discontinuity it can have is infinite oscillation like sin(1/x) near zero.

This one is a bit of an obscure property of derivatives, corollary to theorem 5.12 in Baby Rudin.


"What I find weird is that the derivative, if it exists, cannot have a jump discontinuity."

You mean the derivative of a specific function you have in mind, like perhaps the field equations? Or do you mean something other than what I understand by a jump discontinuity in a derivative, such as one gets for f(x) = {-x for x<0, x for x>=0}?

Tone: Clarification request for my own understanding, not a "gotcha" post; I strongly believe you are saying something true but there's just too many details elided because they are trivial to you for me to quite follow, and I'm intrigued enough to want to be able to follow up, if you'd be so kind as to indulge me.


What is f'(0), the derivative of f at 0? It doesn't even exist, therefore it has no discontinuity at 0.

Darboux's theeorem says that there is no way to create a jump in the derivative, in part because a derivative at a point is defined in terms of limits from both sides, so the limits must be the same.


> What is f'(0), the derivative of f at 0? It doesn't even exist, therefore it has no discontinuity at 0.

This is definitely wrong. The derivative of |x| is -1 where x < 0, and 1 where x > 0, and doesn't exist where x = 0. That is a perfect match to the definition of a jump discontinuity -- the limit from the left is not equal to the limit from the right.

It's not at all necessary for the function to exist at x = 0 in order for it to have a discontinuity at x = 0.

But hey, don't take my word for it; why not check the definition on Wolfram?

http://mathworld.wolfram.com/JumpDiscontinuity.html

The original claim was "the derivative, if it exists, cannot have a jump discontinuity." This is badly stated. You're defending the idea that if the derivative exists at a particular point, then there is no jump discontinuity in the derivative at that point. But there can be a function f which satisfies both of these properties:

- f is the derivative of some other function F. ("The derivative of F exists.")

- f has a jump discontinuity, somewhere. ("The derivative of F has a jump discontinuity.")


The definition you link states that a function has a discontinuity at a point /in its domain/ if yadda yadda. 0 is not in the domain of f'. See for example: https://math.stackexchange.com/questions/1431796/if-a-functi...


That is a question of your personal focus. For example, I'd expect a theorem that applied to "functions from ℝ to ℝ" to apply to f(x) = 1/x unless a specific qualifier was given.


Ah, I see. This is a definition other than the one I understood, which is most assuredly less precise than the one understood by people who have extensively studied analysis. (This being the internet, let me be very clear that I'm basically saying my definition was wrong.)

(I'm pretty decent in mathematics in the general sense, reasoning from axioms, proofs, etc. But as I came up on the computer science side, I'm very lopsided into discrete mathematics, which is a bit unusual. Almost every other way to become a good mathematician makes you lopsided into real analysis and the fields that build on that.)


There's a certain cultural thing in mathematics that is kind of hard to convey unless you've been around other mathematicians about what is a definition and what constitutes proof. And these things definitely change culturally with time. Our current standards of analysis are quite modern. From a modern viewpoint, nobody really bothered to define continuity and limits until the 19th century, even though calculus is from the 17 century. The 19th century cultural fixes had a purpose: actual misunderstandings and/or errors crept up in previously published works.

So when we get pedantic about what things and use early 20th century formalism, we do so kind of as a reaction to historical misunderstandings. When we ask, "what is the derivative of f at 0?", we're trying to show holes in understanding that have been patched by more modern frameworks.


What s/he is saying, if I understand correctly, is that two and only two kinds of discontinuities in derivatives (and second- and higher-order derivatives) can exist: jump derivatives and those exemplified by the pathology inherent in sin(1/x) at the origin thereof; given that spacetime continues beyond the Cauchy horizon but the derivative does not, the first kind (the abrupt ”vertical step” kind) must be ruled out, leaving only the second variety as a candidate. Graph that function, and it’s first and second derivatives around the origin, and you will begin to realise how bizarre this behaviour is and what strange implications it may have. (Then again, this is deep within a black hole, in an area where time and the radial dimension of space are switched, so it’s just a new kind of devilry in an already fraught area.)


No. The relevant theorem is called Darboux's Theorem. It states that if f(x) has a derivative f'(x) everywhere, then f'(x) satisfies the intermediate value property, meaning: If f'(A) < 0 and f'(B) > 0 then there exists a C between A and B such that f'(C) = 0. Why this is interesting is because f'(x) can be a discontinuous function.

f'(x) being discontinuous is not the same thing as f(x) not having a derivative at some point; for example, the absolute value function |x| is simply not differentiable at x=0. The following function IS differentiable at x=0 but its derivative at x=0 is discontinuous:

      f(x) = x^2 sin(1/x) if x != 0 else 0
You can verify that this function satisfies Darboux's theorem.

Darboux's theorem implies in particular that if f'(x) exists at some point x and is discontinuous at that x then the discontinuity is not a jump discontinuity.

Link to Darboux's theorem: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)


I stand corrected.


Thank you.


Sorry, I was hoping the reference to Baby Rudin would be ok. I had forgotten that the theorem had a name.

A jump discontinuity is a discontinuity where intermediate values are not attained, such as in the heaviside or signum functions. If a derivative exists at a point x, then it cannot have a jump discontinuity at x. However, it can have a discontinuity like the one exemplified by f(x) = 2x sin(1/x) - cos(1/x), with f(0) = 0, as that's the derivative of g(x) = x^2 sin (1/x)


That’s what I thought you meant.


Discontinuous derivative. If you want a visual analogy, think of the transition from a smooth fluid flow to turbulence where fluid goes from moving in a simple direction to going every which way at once. See https://www.ethz.ch/content/specialinterest/mavt/fluid-dynam... for some pictures of that.


Is this just another implication that general relativity is incomplete? We already know it must not be because it does not work at the quantum level.


Anything that has singularities coming out of general relativity is pretty much guaranteed to be incomplete. But we can ge close to some workable solutions in particular cases.


The non-deterministic nature of atomic decay by itself makes the universe unpredictable. And this is a thing that happens at STP. I think the author of this article might have forgotten that. There was never really any determinism to "save".


General relativity is classical, so there is plenty of determinism to save.


Uh, this is why I tried to study pure maths (it's less physics and more weird differential geometry) I can recommend this paper [0] that talks about simulations that show a black hole just implodes in a weird way and kind of does not stop imploding, due to space-time getting stretched with a speed above that of light, as measured within a frozen moment in time, and summed over some line from inside to outside. A working analogy could be a 2d-spacetime, represented in 3d-space as a soap bubble film. Imagine the traditional visual funnel shape the space time around a black hole is often depicted as, compared to the downwards bump normal stars/planets are depicted. So, now, the thing is that the effect of gravity, e.g. gravitational waves, are bound by the speed of light. They can not escape a black hole. In the soap example the gravity waves would be film thickness waves, e.g. longitudinal waves in the thin soap sheet. Those are bound by the speed of sound in their medium. Imagine a stream of air with significantly higher speed than the sound in the soap, getting blown downwards this funnel. Also imagine the funnnel still having a closed tip made from soap at the start. Thing is, this air will hit the tip, propell it downards, and suck the part close to the center down just by itself, without the center indirectly pulling on it. Due to the supersonic nature, the ripple created from the initial impact of air onto the center will _never_ get out of there, just because the medium the waves travel through, when measured over the distance from where the wave is right now, to where the outside world with neglegible space-time (or soap-film) curvature is, expands faster than the wave travels. This does not mean the wave does not travel at all, just that once the distance you want it to travel increases enough, the propagation medium's expansion results in weird effects.

If someone is willing/able to point me to some research or possibly even wants to use existing skills with the related differential geometry maths, I'd really like that.

Edit: I might add that anything that falls into the black hole will, even in it's own reference frame, _never_ reach the center, and the only reference frame that possibly sees a steady state field curvature in finite local time could be the center of the collapse.

[0]: https://arxiv.org/abs/1402.1524 (Which was published about half a year after I initially and timestamped communicated the idea to a physics teacher who was willing to explain me the differential maths used in Einstein's field equations.)


> https://arxiv.org/abs/1402.1524 (Which was published about half a year after I initially and timestamped communicated the idea to a physics teacher who was willing to explain me the differential maths used in Einstein's field equations.)

Maybe I'm misunderstanding your point, but your parenthesis seems to suggest that you would like to claim some credit for the idea. If not, then you can just ignore what I'm about to say.

If so, then your comment seems to suggest that you came up with the idea, but weren't sure about the mathematics of it. In modern physics of this type, where experimentation is not practical, the math is the physics; that is, I think the problem is not so much coming up with ideas—my impression is that there are hypotheses and to spare—but rather being able to back up those ideas with rigorous calculations.


No, I'm fine with the credit I got. I'm not going to try to reach the level of understanding of differential geometry to confirm my theory with a simulation. I just hope someone who can might get an idea and do the work. A side note of me bringing the idea to this person would be all I'd hope for.


> In modern physics of this type, where experimentation is not practical, the math is the physics

I think Feynmann would disagree with you.

http://www.youtube.com/watch?v=obCjODeoLVw


I'm not sure that's the case. He's not saying that the mathematics doesn't represent the physics. He's saying that in order to apply the mathematics we have to have some empirical model to which the axioms of the mathematical system applies. As long as that's the case, the mathematics can provide insight into the physical system. But where the mathematical tools break down are in the areas where we don't have any data.

From a physical perspective the math, in this case, is moot. We have no data about what happens past the event horizon, so I could just as easily say that it encompasses a space filled with tiny pink unicorns which exhale confetti and it would be just as meaningful.

What we know is that something different happens. So there's really no room for either the mathematicians or the physicists to speak with authority on what happens. What we're left with is very well-informed speculation about a really interesting region of space-time.


> I think Feynmann would disagree with you.

I cleverly added so many qualifiers that the comment is unfalsifiable:

> In modern physics of this type, where experimentation is not practical ….

Anyway I am a mathematician and not a physicist, so of course I am biased in my evaluation of the ability of mathematics to model reality.


feynman died 30 years ago. it's plausible the culture of the field has changed (nm that feynman wasn't the arbiter of culture to begin with)


> it's plausible the culture of the field has changed (nm that feynman wasn't the arbiter of culture to begin with)

To be fair to monocasa's objection (https://news.ycombinator.com/item?id=17101892), the rebuttal was not of a claim that the culture of physics was mathematical but literally of my claim that (certain) physics was mathematics.


> anything that falls into the black hole will, even in it's own reference frame, _never_ reach the center

Are you saying that the standard GR model of black holes says this? It doesn't; it's simple to show that the proper time, according to the infalling observer, from any finite radius r to r = 0, the center of the hole, is finite.


What I am saying is meant in the context of the non-steady-state collapsing sphere, not the steady-state Schwarzschild black hole. I never doubted the validity of these equations, I just doubted them to accurately describe reality, as they neglect an explanation why the collapsing star should ever reach a steady state. If you have such an explanation, please tell me, I'm interested.


> What I am saying is meant in the context of the non-steady-state collapsing sphere

That doesn't change my comment. The spacetime exterior to the collapsing sphere is still Schwarzschild, by Birkhoff's Theorem, and therefore has the same properties as the spacetime around a steady-state black hole.

> they neglect an explanation why the collapsing star should ever reach a steady state

Are you familiar with the "no hair" theorems for black holes? They are the explanation.


It's been a while since I was that deep in this, so I can't counter on a technical level. I won't have time to look the details up, at least in the near future.


> I can recommend this paper [0] that talks about simulations that show a black hole just implodes in a weird way and kind of does not stop imploding, due to space-time getting stretched with a speed above that of light, as measured within a frozen moment in time, and summed over some line from inside to outside

I would not put too much credence in this paper. First, it seems to be contradicting the well-known singularity theorems proven in the late 1960s and early 1970s, and that doesn't give me very high confidence in its results. Second, just reading section 1 of the paper I'm seeing what look like obvious misunderstandings of the simple collapsing dust model; for example, this:

"Consider the case of a particular set of initial data or conspiring alien civilization that seeks to extract mass from this infalling cloud by moving much larger and denser bodies near it so that some of the particles leave the collection and follow and, possibly, join the larger mass. The aliens have infinite time to pursue this project..."

This is wrong: the aliens do not have infinite time to extract mass from the collapsing cloud. The fact that light signals emitted outward by the cloud will continue arriving at the aliens' ship, far away, indefinitely (in practice how long will depend on how low frequency radiation they can detect) does not mean that the aliens can continue going down to the cloud, getting material from it, and bringing it back up indefinitely. In fact it's simple to show (and is a common textbook problem in GR) that, if we call the time by the aliens' clock at which the collapse starts t = 0, there is a finite time t > 0 after which even a light signal emitted by the aliens towards the cloud will not reach the cloud until after it has collapsed beneath an event horizon and formed a black hole. (And for a black hole of mass a few times that of the Sun, this time is not very long: for example, if the hole has 10 times the Sun's mass and is collapsing from a starting radius of a billion kilometers, which is about 100 million times the horizon radius for that mass, the time t is about a million seconds, or about 12 days.)

In other words, the paper is making a simple mistake: it's treating processes going inward towards the collapsing cloud, as though they worked the same as processes coming outward from the cloud. But the two are not the same, because of the asymmetry of the cloud's gravity: it pulls things inward. And when I see a paper making this kind of simple mistake in the beginning, my credence in the rest of what it says goes way down.


As I recall, Gen. Rel. continues to explain all directly observable phenomena within limits of resolution. That's damn powerful.

As for 'black holes', well ... believe what you choose. A prof. once told me that Einstein 'wasted 30 years' looking for unified theory. By that standard, so did Hawking I guess.


> In classical physics, the universe is predictable: If you know the laws that govern a physical system and you know initial state, you should be able to track its evolution indefinitely far into the future. ... [According to physicist Demetrios Christodoulou:] "This is the basic principle of all classical physics going back to Newtonian mechanics. You can determine evolution from initial data."

This is a famous mantra repeated by physicists but it is not correct. Newtonian physics cannot even predict the future positions of three body from their initial positions. And Newton knew and stated that his doctrines could not predict planetary orbits in long term and he invoked the very scientific and physical notion (or maybe footballers term) of Hand of God. Thus, Newton claimed his doctrine could not make accurate prediction not because they were wrong but because God erred to create the universe according to Newton’s doctrines. Consequently, according to Newton, God once in a while nudged the orbits to make them move correctly according to Newtonian doctrines. According to Newton himself initial states cannot predict long term behavior.

So how come NASA can predict so accurately planetary motions by using the so-called Newtonian Mechanics? The answer is easy: by not using Newtonian mechanics. NASA uses sophisticated mathematical methods or numerical integration to calculate orbits. But since they use as a unit conversion factor the strategically named Newton's Constant of Gravitation as one of their mathematical terms they feel they are justified to declare that they use Newtonian mechanics to compute orbits.

So what happened is that at some point, maybe in the 18th century this philosophical -not physical- assumption entered the physics literature and gained the status of truth after centuries’ of repetition. But if we question the mantra we see that the so-called classical theories do not claim that they can predict future states by the initial state. Phycists do.


I am quite stupefied by this statement. NASA and other organisations most certainly uses Classical mechanics (whether in its Newtonian, Lagrangian, or Hamiltonian formulations does not impinge upon the central issue) when simulating celestial motions and calculating trajectories. That there exists an acknowledged ”Three Body Problem” (i.e., that the trajectories followed by more than two bodies interacting gravitationally are in general not algebraic) matters not one iota for numerical simulations such as those performed by aforementioned organisations for aforementioned purposes. Some kind of relativistic correction might be made for motions that occur within the orbit of Mercury, but again, those are numerical in nature. And as far as I know there are no exact solutions known for general-relativistic interactions between two gravitational objects, placing it at an even greater ’disadvantage’ compared to the ’Newtonian’ mechanics you erroneously deplore.


> matters not one iota for numerical simulations such as those performed by aforementioned organisations for aforementioned purposes.

So you agree that planetary orbits are computed by numerical simulation as I claim. Then why do you object at what I'm saying?


The general relativistic two body problem is hard mostly because the topological maths get really tough real quick symbolically. In 2D space, this problem is solved, but not in 4D. The topology involved makes string theory look easy...


> Newtonian physics cannot even predict the future positions of three body from their initial positions.

You're confusing chaotic systems with random systems. Chaotic systems are still deterministic. [1]

> according to Newton, God once in a while nudged the orbits to make them move correctly according to Newtonian doctrines

Newton thought that because he didn't know perturbation theory. Perturbation theory doesn't change the laws, it reformulates them as "simple thing + small tweaks + more complicated even smaller tweaks + ...". This allows you to quantify how stable the solar system is.

1: https://en.wikipedia.org/wiki/Chaos_theory "these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved"


> Newtonian physics cannot even predict the future positions of three body from their initial positions.

Uh... yes it can. We don't have a closed-from solution [1] for the three-body problem, but we do have existence proofs that, for any initial condition that does not result in a collision, there is a well-defined, unique solution for the motion of bodies given that initial condition. (This kind of uniqueness/existence proof is quite hard in differential equations, hence why you win $1 million if you can prove the same thing for Navier-Stokes equations).

[1] Actually, reading on Wikipedia, we do have a series expansion for the 3-body problem. It's just a really, really, really slow converging series.


What on earth? Not having a closed form for something is decidedly not the same as not being able to use it to predict anything. That’s like saying so-called computer programmers can’t actually predict anything about their programs will do.


Any (non-pathological) terminating computer program (including a non-terminating program that includes a step to forcibly stop at a certain clock time) has a closed form, it's just phenomenally complex. :)


As others have pointed out there is an important distinction to be made between determinism and predictability.

However, "Norton's dome" is a good example of a situation in which Newtonian mechanics might arguably be non-deterministic:

https://www.pitt.edu/~jdnorton/papers/DomePSA2006_final.pdf

https://en.wikipedia.org/wiki/Norton%27s_dome




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