There is an even simpler thought experiment you can do to reach this conclusion: consider what the result of measuring anything to an infinite precision could possibly look like. It would require somehow recording an infinite amount of information. How would you do that, particularly when you take into account that everything you can interact with to make an information storage device is subject to the Heisenberg uncertainty principle?
> consider what the result of measuring anything to an infinite precision could possibly look like. It would require somehow recording an infinite amount of information
This is Zeno's dichotomy paradox [1]. Finitely-defined infinitely-complex systems (e.g. fractals and anything chaos theory) are the escape.
> a much simpler escape: That space is ultimately discrete (i.e. that there's an elementary length) rather than infinitely continuous
Sure. The point is the gedankenexperiment proves nothing. We don't need to "[record] an infinite amount of information" to encapsulate the infinity between any pair of real numbers.
Division? Our productorial sect believes the universe does not have such a useless operation. Multiplication by fractions is the true implementation of the NaN one!
More expensive, harder to maintain and keep current, and a tangent to the core matter; Starlink satellites leak radiation and could be shielded, Starlink satellites could be {switched off | turned low} off over Quiet Zones but are not.
Statements were made that shielding would improve after Ver 1.0 .. it got worse. Statements were made that sats would go low power over quiet zones, they do not.
Returning to your erudite point "and stuff"
The NASA Cosmic Background Explorer (COBE) satellite orbited Earth in 1989–1996 ...
Inspired by the COBE results, a series of ground and balloon-based experiments measured cosmic microwave background anisotropies on smaller angular scales ...
The sensitivity of the new experiments improved dramatically, with a reduction in internal noise by three orders of magnitude.
If it were not so you could encode an arbitrary amount of information into the specific length of a one dimensional object. It would be like a physical Taylor series, but since you can go arbitrarily small you can encode arbitrary coefficients. In fact, if you had a physical disc you could encode everything at every point along its circumference. Which is, like, everything squared or something.
There are a large number of continuous physical quantities, not only length (though all continuous quantities are dependent in one way or another on space or time, which are the primitive continuous quantities), and the reason why you cannot encode an arbitrary amount of information into a specific value of such a quantity is because it is impossible to make an object for which such a quantity would have a perfectly constant value. All the values of such quantities are affected by noise-like variations so you could store information only in the average value of such a quantity, computed over some time and any such average would still be affected by uncertainties that limit the amount of information that can be stored.
One of the most constant lengths that have ever characterized an artificial object has been the length of the international prototype meter kept in France and used to define the meter until 1960. To minimize the length variations, that meter bar was made of platinum-iridium alloy and it was measured at a temperature as constant as possible.
Despite the precautions, which included gentle removing of the dust and handling with soft grippers, the length of that meter bar fluctuated continuously. Even if it was attempted to keep a constant temperature, very small fluctuations in temperature still caused thermal expansions and contractions. Every time the bar was touched, a few metal atoms were removed from it, but other atoms from the environment remained stuck to its surface, changing the length.
All these continuous variations have nothing to do with the possibility of the space being discrete, but they limit the amount of information that can be stored in any such value.
For now there exists absolutely no evidence about the space or time being discrete and not continuous. There have been attempts to make theories based on the discreteness of the space and/or time, but until now they have not provided any useful result.
Sorry, my mistake, I was distracted when I wrote that reply. Yes, I did write that, but it's not actually essential to the point I was trying to make, which was: what could the result of measuring anything to an infinite precision possibly look like?
> what could the result of measuring anything to an infinite precision possibly look like?
Depends on what you're measuring. To illustrate why that isn't a facetious response, consider the difference between 'measuring' pi, 'measuring' a meter and 'measuring' the mass of a proton. (Or, for that matter, the relative mass of three of something to one of it.)
By repeatedly throwing a needle on a striped pattern: [1]. Obviously, you will need an infinite number of throws for an infinitely precise measurement of pi.
You can calculate or 'measure' an arbitrary approximation of that ratio by various methods, but calculating all of it takes infinite time, which I don't have and thus can't do it.
Considering that we don't know the value of pi (not that we could write it out nor read it), I'm not sure your definition of "measure" is the same as mine or most people's.
The article made the wrong statement. The thought experiment isn't that you can't measure length with infinite precision. It's that you can't measure length with precision better than the Planck length. No infinities are involved here.
How does this not break the foundations of quantum theory? For example the Heisenberg uncertainty principle itself implies that the conjugate of a discrete variable must have a continuous spectrum. Thus if there are no continuous variables, there can be no discrete ones either. Either this or we need to throw out one of the variables and call it non-physical/observable -- and yet it very much seems like both position and momentum are things.
The Pontryagin dual of a discrete (locally compact abelian) group is a compact group, and the Pontryagin dual of a compact abelian group is a discrete (locally compact abelian) group…
Hm.
Momentum space being compact does seem weird..
Of course, if rather than a discrete group for space, you just have a discrete uh, co-compact(? Unsure of term. Meaning, there is a finite radius such that the balls of that radius at each of the sites, covers the entire space [edit: “Delone set” is the term I wanted.]), uh,
if you take a Fourier transform of that lattice…
Err… wait, but if the lattice is a subgroup, how does the Fourier transform relate to…
I think the Fourier transform of a Dirac comb is also a Dirac comb (with the spacings being inversely proportional)
If you multiply the Dirac comb by something first…
Well, if you multiply it pointwise by e^(i x p_0 /hbar) , then the Fourier transform will have whole thing shifted by p_0 , and this is periodic in (width of the spacing of the comb in momentum space)
So, if you consider all the pointwise multiples of a Dirac comb in position space (multiplying it by arbitrary functions), then I guess the image of that space under the Fourier transform, is going to in some way correspond to functions on S^1, I guess it would be functions periodic in the width of the comb in momentum space.
So, if instead of a regular comb, you jostle each of the Dirac deltas in the position space comb by a bit first (a different random amount for each)… I’m not sure quite what one would get…
> it very much seems like both position and momentum are things.
The operative word being "seems". Position and momentum (and indeed real numbers in general) are mathematical models that predict observations. But the observations themselves are the results of physical interactions that transfer energy, and those can only ever be discrete because energy is quantized.
Energy levels in simple finite systems are indeed quantized, but this does not mean we can not make the energy quanta be continuously parameterized. For instance, if your system is two mirrors facing each other and you are using the quantum description of the light trapped between these mirrors, you can pick any real value for the energy separation between levels of this system simply by continuously varying the distance between the mirrors.
Maybe one can make the argument that position itself is quantized (thus the position of the mirrors can not be varied continuously), but we do not have experimental reasons to believe space is discrete (and quantum mechanics does not require it to be discrete). And while it is pleasing to imagine it discrete (it is more "mathematically elegant"), we do not have any significant rigorous reasons to believe it is.
Edit: Moreover, if you want to describe (in quantum mechanics) the interaction between a finite system and the open environment around it, the only way to get a mathematical description that matches real-world experiments is to have continously parameterized energy levels for the systems making up the open environment. If you assume that only discrete values are possible, you will simply get the wrong result. Most quantum optics textbooks have reasonably good discussion of this. E.g.:
Quantum Optics by Walls and Milburn
Quantum Optics by Scully and Zubairy
Methods in Theoretical Quantum Optics by Barnett and Radmore
The Planck Length is a practical limit to the precision you can possibly attain in space.
The electron might be smaller. Its diameter is known to be smaller than 10^-22m, but could be much smaller than that.
Further below the Planck Length, there are strong indications that the universe isn't continuous -- it's discrete. That there's an absolute limit to precision, something really quite analogous to a pixel. This elementary length could be somewhere around 10^-93m.
The attainable precision is limited to much lower values by much simpler causes.
The theory that the Planck length has any significance is just a speculation.
Nobody knows how interactions would behave at distances so small and there are no known methods that could compress anything into volumes so small. There is no basis to believe that extrapolating the behavior from normal distances and sizes to the scale of the Planck length is valid.
There are pure speculations that are interesting, but in my opinion any speculation about the Planck length is not interesting, because nobody has been able to formulate any prediction based on such a speculation that can be verified in any way.
Most speculations about the Planck length are made by people who obviously know very little about the meaning of the so-called fundamental constants or of about the significance of the useful natural units for physical quantities, to which the Planck length does not belong.
The Planck length is just one way to express the intensity of the gravitational interaction, i.e. an alternative to Newton's constant of gravitation. Its numeric value does not say anything about any other physical phenomena.
The numeric smallness of Planck's length is just an expression of how weak the gravitational interaction is in comparison with the other interactions. It does not have any other significance.
Im my murky conception of reality, the existence of Planck limits indicates that reality is discrete, and that therefore quantum uncertainty must exist.
For example, I pound the picnic table. Presumably this is somehow transmitted thru the entirety of the Earth, or at least thru a tiny portion of it. But is there a cutoff ? Where is the cutoff ? Where is the effect simply too small to "register" in any conception of reality ?
> It would require somehow recording an infinite amount of information
You're assuming spacetime behaves like the set of reals (something with cardinal ℵ1, if you accept the continuity hypothesis), an object that even if you stay confined within the bounds of pure mathematics, behaves in very, very weird ways.
It may be that spacetime at small scales maps better to a different kind of mathematical object and not even a grid-like one.
The observational limits described here remind me very much (albeit that I read it 40 years ago) of Blood Music by Greg Bear. The ending (as I remember it) has nano-scale intelligences observing the universe so closely that the fabric of spacetime starts buckling under the strain.
I still don't understand why a black holes needs an inside at all. If they are equivalent to their surface then why not dispense with having an interior and just be a surface?
Why isn't the surface smaller then? Probably something inside is pushing out? It's full? Also on the way to a black hole bodies clearly have insides. Do they somehow evaporate the moment a black hole forms?
Edit: My understanding is that all bodies are the size that they are because the inner/outer pressure equalizes, and this has many equilibriums based on the makeup of the body. Black holes are the ultimate degenerate last-stand where the make up is basically raw "information" which cannot be compressed any further while allowing said information to be recovered, which seems to be a fact of our universe. And it just so happens that the amount of information is proportional to the surface area of the black hole rather than its volume, which is probably a statement about how efficiently information can be compressed in our universe. One dimension is redundant?
"Pressure" as a concept doesn't apply to black holes. They are the size they are because of their mass. The bigger the mass, the larger area where their gravity is so great light can't escape. Scientists model black holes as only have a mass and a spin on the inside because that's all the external universe cares about. Information being inscribed on the exterior is an artifact of tike dilating as an object approaches a black hole, iirc.
> Why isn't the surface smaller then? Probably something inside is pushing out?
The surface of space doesn't require something in a higher dimension pushing it out. That such an object may appear to have internal volume from our perspective doesn't need to be any more real than the apparent depth behind a mirror.
Agreed. Couldn’t black holes warp spacetime to the extent that there is no such place as “inside”? Time dilation is infinite at the event horizon, after all.
As you approach the event horizon, your frame of reference slows asymptotically to match that of the black hole while the universe around you fast-forwards toward heat death. I’d expect the hawking radiation coming out at you to blue shift the closer you got until it was so bright as to be indistinguishable from a white hole. You’d never cross the event horizon; you’d be disintegrated and blasted outward into the distant future as part of that hawking radiation.
The time dilation at the event horizon is infinite for an external observer. It appears that the person falling into the black hole slows down and never passes the event horizon. They redshift until you can't see them anymore.
For the unfortunate person falling into the black hole, there is nothing special about the event horizon. The spacetime they experience is rotated (with respect to the external observer) in such a way that their "future" points toward the black hole.
In a very real sense, for external observers there isn't really an interior of the black hole. That "inside" spacetime is warped so much that it exists more in "the future" than the present.
Professor Brian Cox also says that from a string theory perspective there isn't really an inside of a black hole, it's just missing spacetime. I tried to find a reference for this but I couldn't find one. Perhaps in his book about black holes.
I'm no physicist so happy to be corrected on any of the above!
> For the unfortunate person falling into the black hole, there is nothing special about the event horizon.
This is from a simplified model using black holes with infinite lifetime, which is non-physical. Almost all textbook Penrose diagrams use this invalid assumption and shouldn't be relied upon..
Fundamentally, external observers and infalling observers can't disagree on "what happens", just the timing of events. If external observers never see someone falling in, then they didn't fall in.
Most that I know would say that it was disapointingly too big and too general to make specific predictions tied to this specific universe we occupy, although it had early promise.
Brian Cox didn't even make the wikipedia page so its difficult to claim he had any major role in perpertaring it as a large scale fraud.
Laughing at someone who says “hey an idea that isn’t falsifiable isn’t a good theory and certainly not something that any other ideas or theories should be constructed upon is I think more serious than not.
There's a theological doctrine called https://en.wikipedia.org/wiki/Eschatological_verification that claims that statements like "there's God" or "there's afterlife" are perfectly fine verifiable scientific statements, except that their verification come after death or Second Coming.
Your comment reminds me of it.
A surface implies an interior, otherwise it's a just a point. A surface is a boundary, by definition there is another side, something that is being partitioned.
The interior contains a singularity, which may as well be the entirety of the interior. Maybe it has a "degenerate interior", which is very different than a region of space.
I'm not a topological expert, but I'm pretty sure you can have a surface without an interior. A unit sphere would be a good example of a surface without an interior.
I remember as a kid asking how many possible speeds there are between 60 mph and 0 mph. Infinite right? So how does the car get from 60 to stopped when mom hits the brakes?
I’ve said this so many times now. Contemporary physics would greatly benefit from reading Kant. The extent of his influence on contemporary physics, especially with regard to space and time, is so great and the knowledge of his work so little in the scientific community of today. Almost all the great physicists of the 20th century were familiar with Kantian philosophy and were heavily informed by it.
I assume you mean the Critique of Pure Reason? Kant's Oeuvre is quite vast, though it wouldn't be unreasonable to have read the 3 critiques twice.
>What would Kant add to this discussion that the physicists in the article haven't considered?
Kant himself, I'm not sure, but its his model of the cosmos that we employ today, and spatio-temporality is a development out of his critical philosophy, especially his aesthetics. If you want to break space and time out of spatiotemporality it helps if you are familiar with the metaphysical undergirding of contemporary physics, since we have not treated them as separate since Einstein, even though Kant originally kept them as completely separate intuitions and did not seek to unify them but only to try and see what happens when they are set in relation. That is to say that spatiotemporality is, if we are being good Kantians, an entirely negative, transcendental view of space and time since it does not appear at the level of the senses but rather as an abstraction from them. But if we treat the second-level abstraction as real then we are bound to make errors about the empirical world, as returning to the critical project would, I believe, greatly help in re-evaluating our empirical methods.
To be clear, you're suggesting that physisticsts reject general relativity's unification of spacetime because Kant, who obviously had no knowledge of GR nor the empiricism that supports it, did not unify them? Kant's Prize essay also pre-dates e.g. Gödel. That doesn't mean every modern mathemetician must first consider a Kantian slant to their work before rejecting it for well-established and obvious reasons. (Nor that Gödel disproves Kant.)
Deducing that Kant would want us to reject modern science because it's not based on our senses ignores entirely his work as a mathemetician. Kant was, in his own time, a modernist. Not a proto flat earther.
The (what very much feels like an) assertion that "If a collision concentrates enough energy in a small enough region, the particles form a black hole" seems very much rabbit out of a hat.
That supposes in particular that general relativity is still a valid theory at these minuscule scales, something that I believe has never been experimentally verified.
If general relativity's equations do not work at the planck scale, we know strictly nothing about black hole formation.
Please stop trying to present information in this style of webpage, I am begging you. Besides being an abysmal way to present scientific information, every time someone posts one of these it happens to be a topic I am extraordinarily interested in, but due to disabilities I have, I cannot read it even if I wasn't tremendously annoyed by it.
I agree, this is abysmal ux. Doesn't even remotely work on my phone. All those stupid animations literally removing information from my screen before I've had a chance to read it. Drastically reducing the "reading surface" of the actual information attempting to be conveyed. Animations are cool and useful but they too could just be placed on a static page.
The whole thing seems like some over excited marketing person enshittifying the literal idea of static pages of informative just to make something "new".
If you're a reader of Quanta, there's nothing new on this page. It seems like a repackaging of some simple concepts in a pretty web format to attract a less sophisticated audience.
I was disappointed to see reader mode wasn't available, and tried to find some other plugin to just show me the text. The couple plugins I tried could not parse it, nor could the text-only browser I downloaded called ViolonCello. (It only renders header and footer text.) It's very frustrating.
I liked the presentation and it worked great on my phone.
I'm sorry you have issues but I'm glad the world doesn't cater to a single individual's issue.
I can't swim because of a whole in my ear drum from when the Nun at the free clinic my poor mother took me to popped that bad boy with a enthusiastic squeeze from an ear syringe and my tinnitus rings like a son-of-a-bitch when I wear ear plugs but I don't demand they fill in every swimming pool with concrete. I just walk by on those hot summer days wistfully jealous of the guy doing a cannonball and the lady doing the hand stand thing where your feet are in dry air but your head is 2 feet below the water level.
The analogy would be more like a library forcing everyone to swim through a pool in order to read a particular book. And if you complain you have a disability, someone says that the world shouldn't cater to an individual.
The analogy is ridiculous, yes. As it is ridiculous to build such a website that disabled people cannot possibly read. You don't have to make it perfect for them, just don't make it impossible.
Fortunately or not, largest artificial accelerators are still not powerful enough to produce artificial black hole.
Sometimes from deep Universe, appear particles with much more energy than achieved at labs, and some theories say, their energy enough to create BH, but have not confirmation with decades observations, may be because scientific method need tens or better hundreds appearances in one place to confirm, but have less than half dozen.
But physicists already made spacetime redundant by dividing it back to space and time. This was started by Dirac who restated general relativity with Hamilton formalism. The slicing of spacetime was completed in the sixties with ADM formulation. Also we know spacetime does not exist in practice because when we say "universe is expanding" we mean "space is expanding." It makes no sense to say spacetime is expanding.
Good lord that was awful to read. Text popped into existence half-off the screen, disappeared just as it was entering the screen, etc. I'm operating on a 15" MacBook with a reasonable-sized window, and yet I had to scroll carefully back and forth to catch the text. And no reader mode with just the text.
"Please don't complain about tangential annoyances—e.g. article or website formats, name collisions, or back-button breakage. They're too common to be interesting."
Besides the silly, but inevitable HN complaints about the format of the webpage presentation, (great presentation btw)
The fundamental challenges these experiments (and others) surface is a deep challenge to the traditional narratives of Materialism or 'Physicalism' as our understanding of what existence is. In essence science and human knowledge has lept forward technologigcally over the past 400 and esp the past 100 years because we started assuming the world was physical in nature, material and metaphysically, ie that it reduced to fundamentally physical things we could quantify and measure.
Yet, the older I get the more inclined I am to believe in some form of Idealism.. Not only in Idealism but I'm leaning towards the belief that some kind of fundamental universal Consciousness is the only fundamental property or baseline to the universe or to existence.
Time and Space is not fundamental. Locality isnt true.
There is an even simpler thought experiment you can do to reach this conclusion: consider what the result of measuring anything to an infinite precision could possibly look like. It would require somehow recording an infinite amount of information. How would you do that, particularly when you take into account that everything you can interact with to make an information storage device is subject to the Heisenberg uncertainty principle?
> consider what the result of measuring anything to an infinite precision could possibly look like. It would require somehow recording an infinite amount of information
This is Zeno's dichotomy paradox [1]. Finitely-defined infinitely-complex systems (e.g. fractals and anything chaos theory) are the escape.
[1] https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Dichotomy_p...
There's a much simpler escape: That space is ultimately discrete (i.e. that there's an elementary length) rather than infinitely continuous.
> a much simpler escape: That space is ultimately discrete (i.e. that there's an elementary length) rather than infinitely continuous
Sure. The point is the gedankenexperiment proves nothing. We don't need to "[record] an infinite amount of information" to encapsulate the infinity between any pair of real numbers.
The universe runs on IEEE floats. The missing matter and energy are due to roundoff error. We pray to the great NaN in the sky.
I'd subscribe to your religion but for the division.
Division? Our productorial sect believes the universe does not have such a useless operation. Multiplication by fractions is the true implementation of the NaN one!
That sounds a bit shifty.
If space is discretized, we should see "upscaling artifacts" in the CMB, but we don't.
Maybe we would .. but Musk has to turn off his leaky Starlink over the Murchison (not so) Quiet (anymore) Zone first.
We got space telescopes and stuff.
More expensive, harder to maintain and keep current, and a tangent to the core matter; Starlink satellites leak radiation and could be shielded, Starlink satellites could be {switched off | turned low} off over Quiet Zones but are not.
Statements were made that shielding would improve after Ver 1.0 .. it got worse. Statements were made that sats would go low power over quiet zones, they do not.
Returning to your erudite point "and stuff"
~ https://en.wikipedia.org/wiki/Cosmic_microwave_backgroundHmmm, it appears the ground based results were a dramatic improvement over the sat based data.
Is there? Planck length isn't a pixel.[0]
[0] https://www.physicsforums.com/insights/hand-wavy-discussion-...
If it were not so you could encode an arbitrary amount of information into the specific length of a one dimensional object. It would be like a physical Taylor series, but since you can go arbitrarily small you can encode arbitrary coefficients. In fact, if you had a physical disc you could encode everything at every point along its circumference. Which is, like, everything squared or something.
No, this is a wrong argument.
There are a large number of continuous physical quantities, not only length (though all continuous quantities are dependent in one way or another on space or time, which are the primitive continuous quantities), and the reason why you cannot encode an arbitrary amount of information into a specific value of such a quantity is because it is impossible to make an object for which such a quantity would have a perfectly constant value. All the values of such quantities are affected by noise-like variations so you could store information only in the average value of such a quantity, computed over some time and any such average would still be affected by uncertainties that limit the amount of information that can be stored.
One of the most constant lengths that have ever characterized an artificial object has been the length of the international prototype meter kept in France and used to define the meter until 1960. To minimize the length variations, that meter bar was made of platinum-iridium alloy and it was measured at a temperature as constant as possible.
Despite the precautions, which included gentle removing of the dust and handling with soft grippers, the length of that meter bar fluctuated continuously. Even if it was attempted to keep a constant temperature, very small fluctuations in temperature still caused thermal expansions and contractions. Every time the bar was touched, a few metal atoms were removed from it, but other atoms from the environment remained stuck to its surface, changing the length.
All these continuous variations have nothing to do with the possibility of the space being discrete, but they limit the amount of information that can be stored in any such value.
For now there exists absolutely no evidence about the space or time being discrete and not continuous. There have been attempts to make theories based on the discreteness of the space and/or time, but until now they have not provided any useful result.
Just because you can't record something doesn't mean it doesn't exist.
Sure but i can invent infinitely many unfalsifiable claims that mean nothing
Who said anything about recording? What would the subjective experience of measuring something with infinite precision possibly be like?
> would require somehow recording an infinite amount of information...
>> Just because you can't record something...
>>> Who said anything about recording?
Sorry, my mistake, I was distracted when I wrote that reply. Yes, I did write that, but it's not actually essential to the point I was trying to make, which was: what could the result of measuring anything to an infinite precision possibly look like?
> what could the result of measuring anything to an infinite precision possibly look like?
Depends on what you're measuring. To illustrate why that isn't a facetious response, consider the difference between 'measuring' pi, 'measuring' a meter and 'measuring' the mass of a proton. (Or, for that matter, the relative mass of three of something to one of it.)
You'd need to somehow record refinements endlessly? I don't get what you're getting at.
How do you measure pi?
By repeatedly throwing a needle on a striped pattern: [1]. Obviously, you will need an infinite number of throws for an infinitely precise measurement of pi.
[1] https://en.wikipedia.org/wiki/Buffon%27s_needle_problem
> How do you measure pi?
Pick your method. It’s the ratio of a circle’s circumference to its diameter.
You can calculate or 'measure' an arbitrary approximation of that ratio by various methods, but calculating all of it takes infinite time, which I don't have and thus can't do it.
Considering that we don't know the value of pi (not that we could write it out nor read it), I'm not sure your definition of "measure" is the same as mine or most people's.
I think your definition of "know" is unreasonably strict. Especially because we can write out pieces of algebra that are exactly pi.
I think it's reasonable to say we can't truly measure pi, though.
And you can neither know nor measure a random real.
The article made the wrong statement. The thought experiment isn't that you can't measure length with infinite precision. It's that you can't measure length with precision better than the Planck length. No infinities are involved here.
https://en.wikipedia.org/wiki/On_Exactitude_in_Science
Jorge Borges' way of telling a story as analogy is beautiful and simple.
It takes the resources of the universe to simulate the universe.
How does this not break the foundations of quantum theory? For example the Heisenberg uncertainty principle itself implies that the conjugate of a discrete variable must have a continuous spectrum. Thus if there are no continuous variables, there can be no discrete ones either. Either this or we need to throw out one of the variables and call it non-physical/observable -- and yet it very much seems like both position and momentum are things.
The Pontryagin dual of a discrete (locally compact abelian) group is a compact group, and the Pontryagin dual of a compact abelian group is a discrete (locally compact abelian) group…
Hm.
Momentum space being compact does seem weird..
Of course, if rather than a discrete group for space, you just have a discrete uh, co-compact(? Unsure of term. Meaning, there is a finite radius such that the balls of that radius at each of the sites, covers the entire space [edit: “Delone set” is the term I wanted.]), uh, if you take a Fourier transform of that lattice…
Err… wait, but if the lattice is a subgroup, how does the Fourier transform relate to…
I think the Fourier transform of a Dirac comb is also a Dirac comb (with the spacings being inversely proportional) If you multiply the Dirac comb by something first… Well, if you multiply it pointwise by e^(i x p_0 /hbar) , then the Fourier transform will have whole thing shifted by p_0 , and this is periodic in (width of the spacing of the comb in momentum space)
So, if you consider all the pointwise multiples of a Dirac comb in position space (multiplying it by arbitrary functions), then I guess the image of that space under the Fourier transform, is going to in some way correspond to functions on S^1, I guess it would be functions periodic in the width of the comb in momentum space.
So, if instead of a regular comb, you jostle each of the Dirac deltas in the position space comb by a bit first (a different random amount for each)… I’m not sure quite what one would get…
> it very much seems like both position and momentum are things.
The operative word being "seems". Position and momentum (and indeed real numbers in general) are mathematical models that predict observations. But the observations themselves are the results of physical interactions that transfer energy, and those can only ever be discrete because energy is quantized.
Energy levels in simple finite systems are indeed quantized, but this does not mean we can not make the energy quanta be continuously parameterized. For instance, if your system is two mirrors facing each other and you are using the quantum description of the light trapped between these mirrors, you can pick any real value for the energy separation between levels of this system simply by continuously varying the distance between the mirrors.
Maybe one can make the argument that position itself is quantized (thus the position of the mirrors can not be varied continuously), but we do not have experimental reasons to believe space is discrete (and quantum mechanics does not require it to be discrete). And while it is pleasing to imagine it discrete (it is more "mathematically elegant"), we do not have any significant rigorous reasons to believe it is.
Edit: Moreover, if you want to describe (in quantum mechanics) the interaction between a finite system and the open environment around it, the only way to get a mathematical description that matches real-world experiments is to have continously parameterized energy levels for the systems making up the open environment. If you assume that only discrete values are possible, you will simply get the wrong result. Most quantum optics textbooks have reasonably good discussion of this. E.g.:
It would simply be written down on an infinitely long strip of paper.
You could use a Turing machine tape for the job. I'm told they're stored in the parentheses mines beneath MIT, somewhere near the point masses.
The shelving down there is frictionless, so they're often not so nearby.
Möbius strip for double surface area....I can practically see the cheese TV comercial
The Planck Length is a practical limit to the precision you can possibly attain in space.
The electron might be smaller. Its diameter is known to be smaller than 10^-22m, but could be much smaller than that.
Further below the Planck Length, there are strong indications that the universe isn't continuous -- it's discrete. That there's an absolute limit to precision, something really quite analogous to a pixel. This elementary length could be somewhere around 10^-93m.
The attainable precision is limited to much lower values by much simpler causes.
The theory that the Planck length has any significance is just a speculation.
Nobody knows how interactions would behave at distances so small and there are no known methods that could compress anything into volumes so small. There is no basis to believe that extrapolating the behavior from normal distances and sizes to the scale of the Planck length is valid.
There are pure speculations that are interesting, but in my opinion any speculation about the Planck length is not interesting, because nobody has been able to formulate any prediction based on such a speculation that can be verified in any way.
Most speculations about the Planck length are made by people who obviously know very little about the meaning of the so-called fundamental constants or of about the significance of the useful natural units for physical quantities, to which the Planck length does not belong.
The Planck length is just one way to express the intensity of the gravitational interaction, i.e. an alternative to Newton's constant of gravitation. Its numeric value does not say anything about any other physical phenomena.
The numeric smallness of Planck's length is just an expression of how weak the gravitational interaction is in comparison with the other interactions. It does not have any other significance.
Im my murky conception of reality, the existence of Planck limits indicates that reality is discrete, and that therefore quantum uncertainty must exist.
For example, I pound the picnic table. Presumably this is somehow transmitted thru the entirety of the Earth, or at least thru a tiny portion of it. But is there a cutoff ? Where is the cutoff ? Where is the effect simply too small to "register" in any conception of reality ?
> there are strong indications that the universe isn't continuous -- it's discrete
There are indications discrete space is plausible. It's actively debated.
There are also strong indications space is continous, e.g. Lorentz symmetry. (This was recently the death knell for a branch of LQG.)
I have similar reasons for not believing that the world that we experience is a computer simulation.
> It would require somehow recording an infinite amount of information
You're assuming spacetime behaves like the set of reals (something with cardinal ℵ1, if you accept the continuity hypothesis), an object that even if you stay confined within the bounds of pure mathematics, behaves in very, very weird ways.
It may be that spacetime at small scales maps better to a different kind of mathematical object and not even a grid-like one.
The observational limits described here remind me very much (albeit that I read it 40 years ago) of Blood Music by Greg Bear. The ending (as I remember it) has nano-scale intelligences observing the universe so closely that the fabric of spacetime starts buckling under the strain.
I still don't understand why a black holes needs an inside at all. If they are equivalent to their surface then why not dispense with having an interior and just be a surface?
Why isn't the surface smaller then? Probably something inside is pushing out? It's full? Also on the way to a black hole bodies clearly have insides. Do they somehow evaporate the moment a black hole forms?
Edit: My understanding is that all bodies are the size that they are because the inner/outer pressure equalizes, and this has many equilibriums based on the makeup of the body. Black holes are the ultimate degenerate last-stand where the make up is basically raw "information" which cannot be compressed any further while allowing said information to be recovered, which seems to be a fact of our universe. And it just so happens that the amount of information is proportional to the surface area of the black hole rather than its volume, which is probably a statement about how efficiently information can be compressed in our universe. One dimension is redundant?
"Pressure" as a concept doesn't apply to black holes. They are the size they are because of their mass. The bigger the mass, the larger area where their gravity is so great light can't escape. Scientists model black holes as only have a mass and a spin on the inside because that's all the external universe cares about. Information being inscribed on the exterior is an artifact of tike dilating as an object approaches a black hole, iirc.
Black holes also have a charge!
Yes, that's true, thanks.
> Why isn't the surface smaller then? Probably something inside is pushing out?
The surface of space doesn't require something in a higher dimension pushing it out. That such an object may appear to have internal volume from our perspective doesn't need to be any more real than the apparent depth behind a mirror.
Since when is the surface of the universe that of a hypersphere?
Agreed. Couldn’t black holes warp spacetime to the extent that there is no such place as “inside”? Time dilation is infinite at the event horizon, after all.
As you approach the event horizon, your frame of reference slows asymptotically to match that of the black hole while the universe around you fast-forwards toward heat death. I’d expect the hawking radiation coming out at you to blue shift the closer you got until it was so bright as to be indistinguishable from a white hole. You’d never cross the event horizon; you’d be disintegrated and blasted outward into the distant future as part of that hawking radiation.
The time dilation at the event horizon is infinite for an external observer. It appears that the person falling into the black hole slows down and never passes the event horizon. They redshift until you can't see them anymore.
For the unfortunate person falling into the black hole, there is nothing special about the event horizon. The spacetime they experience is rotated (with respect to the external observer) in such a way that their "future" points toward the black hole.
In a very real sense, for external observers there isn't really an interior of the black hole. That "inside" spacetime is warped so much that it exists more in "the future" than the present.
Professor Brian Cox also says that from a string theory perspective there isn't really an inside of a black hole, it's just missing spacetime. I tried to find a reference for this but I couldn't find one. Perhaps in his book about black holes.
I'm no physicist so happy to be corrected on any of the above!
> For the unfortunate person falling into the black hole, there is nothing special about the event horizon.
This is from a simplified model using black holes with infinite lifetime, which is non-physical. Almost all textbook Penrose diagrams use this invalid assumption and shouldn't be relied upon..
Fundamentally, external observers and infalling observers can't disagree on "what happens", just the timing of events. If external observers never see someone falling in, then they didn't fall in.
You know string theory is now generally considered a large scale fraud perpetrated by cox to maintain funding right?
Got a source for either part of that?
* generally considered a large scale fraud,
* perpetrated by (UK's Professor Brian) cox
Most that I know would say that it was disapointingly too big and too general to make specific predictions tied to this specific universe we occupy, although it had early promise.
Brian Cox didn't even make the wikipedia page so its difficult to claim he had any major role in perpertaring it as a large scale fraud.
https://en.wikipedia.org/wiki/String_theory
Found Sabine Hossenfelder's HN account!
I am, of course, joking but she posts this sort of easy and empty clickbait.
Yup, her content is not very appealing these days.
Laughing at someone who says “hey an idea that isn’t falsifiable isn’t a good theory and certainly not something that any other ideas or theories should be constructed upon is I think more serious than not.
Because if you free-fall into a black hole you can go past the event horizon.
Can we? Is there a way to test this assumption? If not, then it's not science, right?
If you free fall into a black hole you are testing it.
I don’t think that not being able to communicate your results makes it not scientific.
There's a theological doctrine called https://en.wikipedia.org/wiki/Eschatological_verification that claims that statements like "there's God" or "there's afterlife" are perfectly fine verifiable scientific statements, except that their verification come after death or Second Coming. Your comment reminds me of it.
I think the test involves communicating your results. It's the same thing with the afterlife
Publish or perish made manifest.
> if you free-fall into a black hole you can go past the event horizon
Falling "through" a hologram on the surface would be physically indistinguishable to the person falling from falling into a volume.
A surface implies an interior, otherwise it's a just a point. A surface is a boundary, by definition there is another side, something that is being partitioned.
The interior contains a singularity, which may as well be the entirety of the interior. Maybe it has a "degenerate interior", which is very different than a region of space.
> surface implies an interior, otherwise it's a just a point
Space-time is not Euclidean geometry under GR.
But it is continuous, at least until you hit or pass through the singularity.
> it is continuous
We don’t know this. It has been as far as we’ve measured. But there are compelling reasons to at least consider discrete spacetime.
I'm not a topological expert, but I'm pretty sure you can have a surface without an interior. A unit sphere would be a good example of a surface without an interior.
? It by definition has a radius of 1.
unit sphere != unit ball
The former is the boundary, the latter is the interior + boundary. One of the great arbitrary naming conventions of math.
Minor nitpick, the ball might be closed or open, depending on whether the boundary is included or not, respectively.
Black hole insides is another Penrose universe.
I remember as a kid asking how many possible speeds there are between 60 mph and 0 mph. Infinite right? So how does the car get from 60 to stopped when mom hits the brakes?
> how does the car get from 60 to stopped when mom hits the brakes?
The sum of an infinite series can be finite [1].
[1] https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-conver...
That you can split any finite length into an infinite amount does not equate to having to traverse infinite numbers.
Not infinite if we take into the account that we have a physical car. Speed many but not infinite steps.
Everything is infinite if we think this way.
Zeno’s paradox.
I’ve said this so many times now. Contemporary physics would greatly benefit from reading Kant. The extent of his influence on contemporary physics, especially with regard to space and time, is so great and the knowledge of his work so little in the scientific community of today. Almost all the great physicists of the 20th century were familiar with Kantian philosophy and were heavily informed by it.
What would Kant add to this discussion that the physicists in the article haven't considered?
(Saying this as someone who's read Kant twice and agrees with most of what he claimed. Outside his taste in music.)
>Saying this as someone who's read Kant twice
I assume you mean the Critique of Pure Reason? Kant's Oeuvre is quite vast, though it wouldn't be unreasonable to have read the 3 critiques twice.
>What would Kant add to this discussion that the physicists in the article haven't considered?
Kant himself, I'm not sure, but its his model of the cosmos that we employ today, and spatio-temporality is a development out of his critical philosophy, especially his aesthetics. If you want to break space and time out of spatiotemporality it helps if you are familiar with the metaphysical undergirding of contemporary physics, since we have not treated them as separate since Einstein, even though Kant originally kept them as completely separate intuitions and did not seek to unify them but only to try and see what happens when they are set in relation. That is to say that spatiotemporality is, if we are being good Kantians, an entirely negative, transcendental view of space and time since it does not appear at the level of the senses but rather as an abstraction from them. But if we treat the second-level abstraction as real then we are bound to make errors about the empirical world, as returning to the critical project would, I believe, greatly help in re-evaluating our empirical methods.
English and German, hence twice.
To be clear, you're suggesting that physisticsts reject general relativity's unification of spacetime because Kant, who obviously had no knowledge of GR nor the empiricism that supports it, did not unify them? Kant's Prize essay also pre-dates e.g. Gödel. That doesn't mean every modern mathemetician must first consider a Kantian slant to their work before rejecting it for well-established and obvious reasons. (Nor that Gödel disproves Kant.)
Deducing that Kant would want us to reject modern science because it's not based on our senses ignores entirely his work as a mathemetician. Kant was, in his own time, a modernist. Not a proto flat earther.
I too am wanting to know the answer to this question, sounds like an intriguing concept!
The (what very much feels like an) assertion that "If a collision concentrates enough energy in a small enough region, the particles form a black hole" seems very much rabbit out of a hat.
That supposes in particular that general relativity is still a valid theory at these minuscule scales, something that I believe has never been experimentally verified.
If general relativity's equations do not work at the planck scale, we know strictly nothing about black hole formation.
Please stop trying to present information in this style of webpage, I am begging you. Besides being an abysmal way to present scientific information, every time someone posts one of these it happens to be a topic I am extraordinarily interested in, but due to disabilities I have, I cannot read it even if I wasn't tremendously annoyed by it.
It's really bad even for me on my phone. The text content fades away before I can scroll it completely into view. No reader mode available.
Same here. I gave up on reading it once the diagram animations made it near impossible
I agree, this is abysmal ux. Doesn't even remotely work on my phone. All those stupid animations literally removing information from my screen before I've had a chance to read it. Drastically reducing the "reading surface" of the actual information attempting to be conveyed. Animations are cool and useful but they too could just be placed on a static page.
The whole thing seems like some over excited marketing person enshittifying the literal idea of static pages of informative just to make something "new".
If you're a reader of Quanta, there's nothing new on this page. It seems like a repackaging of some simple concepts in a pretty web format to attract a less sophisticated audience.
As a less sophisticated audienc emember, can someone PDF the thing for me my vision wetware can't grok it
How could it be made more accessible to you?
They could be regular web pages without the silly scroll animations, just like in the good old days.
Just make it a web page, not a powerpoint presentation.
reader mode supported. sometimes they are sometimes no.
I was disappointed to see reader mode wasn't available, and tried to find some other plugin to just show me the text. The couple plugins I tried could not parse it, nor could the text-only browser I downloaded called ViolonCello. (It only renders header and footer text.) It's very frustrating.
I liked the presentation and it worked great on my phone.
I'm sorry you have issues but I'm glad the world doesn't cater to a single individual's issue.
I can't swim because of a whole in my ear drum from when the Nun at the free clinic my poor mother took me to popped that bad boy with a enthusiastic squeeze from an ear syringe and my tinnitus rings like a son-of-a-bitch when I wear ear plugs but I don't demand they fill in every swimming pool with concrete. I just walk by on those hot summer days wistfully jealous of the guy doing a cannonball and the lady doing the hand stand thing where your feet are in dry air but your head is 2 feet below the water level.
The analogy would be more like a library forcing everyone to swim through a pool in order to read a particular book. And if you complain you have a disability, someone says that the world shouldn't cater to an individual.
The analogy is ridiculous, yes. As it is ridiculous to build such a website that disabled people cannot possibly read. You don't have to make it perfect for them, just don't make it impossible.
You're a moron if you believe these two things compare even remotely
>If a collision concentrates enough energy in a small enough region, the particles form a black hole and never reach a detector.
Is this actually experimentally confirmed?
Fortunately or not, largest artificial accelerators are still not powerful enough to produce artificial black hole.
Sometimes from deep Universe, appear particles with much more energy than achieved at labs, and some theories say, their energy enough to create BH, but have not confirmation with decades observations, may be because scientific method need tens or better hundreds appearances in one place to confirm, but have less than half dozen.
But physicists already made spacetime redundant by dividing it back to space and time. This was started by Dirac who restated general relativity with Hamilton formalism. The slicing of spacetime was completed in the sixties with ADM formulation. Also we know spacetime does not exist in practice because when we say "universe is expanding" we mean "space is expanding." It makes no sense to say spacetime is expanding.
Ah yes a debate between scientists with theories which are so far removed from Science that it can’t be called anything other than fairy tails
For those who aren’t in the know physics is in a crisis where huge portions of theoretical physics are turning out to be complete nonsense.
Good lord that was awful to read. Text popped into existence half-off the screen, disappeared just as it was entering the screen, etc. I'm operating on a 15" MacBook with a reasonable-sized window, and yet I had to scroll carefully back and forth to catch the text. And no reader mode with just the text.
"Please don't complain about tangential annoyances—e.g. article or website formats, name collisions, or back-button breakage. They're too common to be interesting."
https://news.ycombinator.com/newsguidelines.html
Besides the silly, but inevitable HN complaints about the format of the webpage presentation, (great presentation btw)
The fundamental challenges these experiments (and others) surface is a deep challenge to the traditional narratives of Materialism or 'Physicalism' as our understanding of what existence is. In essence science and human knowledge has lept forward technologigcally over the past 400 and esp the past 100 years because we started assuming the world was physical in nature, material and metaphysically, ie that it reduced to fundamentally physical things we could quantify and measure.
Yet, the older I get the more inclined I am to believe in some form of Idealism.. Not only in Idealism but I'm leaning towards the belief that some kind of fundamental universal Consciousness is the only fundamental property or baseline to the universe or to existence.
Time and Space is not fundamental. Locality isnt true.