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.
The Schrödinger wave-function is expressed in a unit which is the square root of an inverse cubic meter. This fact alone makes clear that the wave-function is an abstraction, forever hidden from our view. Nobody will ever measure directly the square root of an inverse cubic meter.
Freeman Dyson, Why is Maxwell’s Theory so hard to understand?
I think the Pythagoreans had the same silly crisis about square root of two that you have about the real-valued measurements. And the solution, I believe, would be the same: there simply are two physical things with lengths that relate to each other exactly as sqrt(2) to 1.
So, how would result of measuring e.g. length of something to an infinite precision look like? It would look like two particles that are kept at rest relative to each other; the distance between them is the measured distance. Whether this distance has to be commeasurable with the Planck scale or not is an interesting question but it really can go either way.
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.
> 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 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.
Integers have perfect precision, but finite storage. For example, pi is infinite information and digits in float, but finite when represented as a single symbol
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 ?
I am not sure what the thought experiment is here — it is more like two facts, one about the Planck scale (and things break there) and the other about a black hole (its information is proportional to its surface).
However, there are deeper things around. Seth Lloyd suggested that we use information density to derive general relativity from quantum theory:
https://arxiv.org/abs/1206.6559
Not only are these not thought experiments by most definitions, this is also not really an article by most definitions. It's as if the author had a few interesting yet unrelated thoughts, scribbled them down and covered up for the lack of depth with fancy illustrations and transitions.
They're not unrelated thoughts. The author is describing current mysteries in physics related to the edges of what we could theoretically measure.
The article actually seems clear and straightforward to me. I'd only add that I wish there were links at the end regarding what scientists are proposing right now for resolving those mysteries.
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.
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.
> 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.
thats interesting, but im curious what the basis for that thought comes from?
Did quanta get bought by Forbes or something? I seem to recall they had a lot more informative articles than whatever this was. Further, there is no indication on how the first two postulates/statements force one to conclude the latter. Also? The whole format of the thing screams "LOOK AT ME". This is a very weird... thing (I hesitate to call it "article") coming from a site that previously had some interesting content (in the actual sense of the word, not the current colloquial sense)
Is the problem the author can't let go of not understanding? That they need everything to be, for lack of a better term, quantifiable? That there must always be no boundary to our ability to measure? Do they demand an answer to why there is a limit to what we can see at the end of the universe (beginning/surface)?
Is this something AI shat out for clicks? Did they fire actual writers at quanta? Did they smoke a bunch of DMT? Are you ok, quantamagazine? Do you need us to call for help? I'm a bit annoyed that I had to read that, thinking there would be some point, that the top thread was exaggerating, but they weren't.
One of my favorites is a different interpretation of the events happening inside a black hole being inscribed in its surface: that there is no inside and the events happen at the surface, which seems totally normal because spacetime is so extremely stretched there things don't realize they look like a 2D surface from outside observers.
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?
Hoberman spheres expand and contract via forces that act only along the structures that make up the surface, and this is a simple classical object. I don't see why a more exotic physical object like a black hole couldn't only have properties defined by its surface.
> 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!
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.
I like to picture poking your finger into a loosely knit jumper so that the weave bunches up densely around the outside of a large hole. If you think of an ant walking around on the threads, it would realise that there's an area of increased density. It would also notice that there's a boundary it can't get past, but if you asked it what the topology of the threads are on the "other side" of the boundary it wouldn't be able to give you an answer.
In my mind that is what a black hole is, a spherical hole in the fabric of spacetime with matter bunched up around it in a very thin shell. That's why their area is proportional to their mass instead of their volume, because there is no volume.
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.
Readit News
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...
The Schrödinger wave-function is expressed in a unit which is the square root of an inverse cubic meter. This fact alone makes clear that the wave-function is an abstraction, forever hidden from our view. Nobody will ever measure directly the square root of an inverse cubic meter.
Freeman Dyson, Why is Maxwell’s Theory so hard to understand?
https://www.clerkmaxwellfoundation.org/DysonFreemanArticle.p...
So, how would result of measuring e.g. length of something to an infinite precision look like? It would look like two particles that are kept at rest relative to each other; the distance between them is the measured distance. Whether this distance has to be commeasurable with the Planck scale or not is an interesting question but it really can go either way.
And how do you do that in the face of Heisenberg uncertainty?
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…
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.
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.
Jorge Borges' way of telling a story as analogy is beautiful and simple.
It takes the resources of the universe to simulate the universe.
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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 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.
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.)
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 ?
However, there are deeper things around. Seth Lloyd suggested that we use information density to derive general relativity from quantum theory: https://arxiv.org/abs/1206.6559
The article actually seems clear and straightforward to me. I'd only add that I wish there were links at the end regarding what scientists are proposing right now for resolving those mysteries.
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.
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.
Is the problem the author can't let go of not understanding? That they need everything to be, for lack of a better term, quantifiable? That there must always be no boundary to our ability to measure? Do they demand an answer to why there is a limit to what we can see at the end of the universe (beginning/surface)?
Is this something AI shat out for clicks? Did they fire actual writers at quanta? Did they smoke a bunch of DMT? Are you ok, quantamagazine? Do you need us to call for help? I'm a bit annoyed that I had to read that, thinking there would be some point, that the top thread was exaggerating, but they weren't.
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?
Hoberman spheres expand and contract via forces that act only along the structures that make up the surface, and this is a simple classical object. I don't see why a more exotic physical object like a black hole couldn't only have properties defined by its surface.
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.
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.
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!
Deleted Comment
Falling "through" a hologram on the surface would be physically indistinguishable to the person falling from falling into a volume.
Deleted Comment
In my mind that is what a black hole is, a spherical hole in the fabric of spacetime with matter bunched up around it in a very thin shell. That's why their area is proportional to their mass instead of their volume, because there is no volume.
Space-time is not Euclidean geometry under GR.