Mathematicians Disprove Conjecture Made to Save Black Holes
quantamagazine.org
quantamagazine.org
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.
Today's society values drama above everything else. It's a shame (either that they do, or that I don't fit in ;-) ).
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.
All those articles can go burn in a hot, hot fire.
I wish somebody out there could cover social science research and politics with this kind of attitude. This is really good science writing.
What on earth would discontinuous spacetime involve? It sounds like a sort of shattered chaos of torn-up bits of space.
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).
Plenty of strange things can happen with derivatives, e.g. https://en.m.wikipedia.org/wiki/Cantor_function
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.
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.
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.
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.")
(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.)
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.
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
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)
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)
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.)
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.
I think Feynmann would disagree with you.
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 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.
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.
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.
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.
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 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.
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.
So you agree that planetary orbits are computed by numerical simulation as I claim. Then why do you object at what I'm saying?
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"
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.
However, "Norton's dome" is a good example of a situation in which Newtonian mechanics might arguably be non-deterministic: