As a one-time mathematician, this was a really fascinating article. The similarities seem to be entirely coincidental, but what would have been my doctoral dissertation was also about generalizing some concepts from smooth manifolds to a "non-smooth" setting, and the crux of my work also hinged on optimal transport.
Actually I feel optimal transport is a pretty underrated concept in both pure and applied math, and I would have loved to explore it had I continued in academia. But oh well, one must make choices in life...
Small world. My graduate research was precisely on this topic as well. I was going in a more algebraic direction, though. My master's thesis was essentially about different discrete analogues of curvature using cooked-up cohomological constructions.
I really wish academia consistently provided as much security as industry. Would have loved to continue this line of research.
I can't simply help but think that optimal transport is intricately linked to the principle of least action (and as we know POLA is everywhere in nature). At the end, natural interactions seem to be one big optimization problem.
Do I understand correctly that with sectional curvature/triangle comparison methods you can do differential geometry on non-smooth manifolds (e.g. on a cube)? If so, I've completely missed this fact before.
This is all really cool, but is getting new singularity theorems really a positive sign? Like, my understanding was that it was generally hoped that an improved, quantum theory of gravity would eliminate such singularities -- that such singularities were generally considered to be non-physical artifacts that occur in GR due to its deficiencies at the most extreme scales (where quantum gravity would be relevant), not that they are in fact real and physical. So I'd consider it a better sign if these predicted black holes, which we see, but without singularities!
It is a positive sign, and here are three reasons why.
First, even if space is smooth, it is sometimes well-approximated by a singularity. In which case understanding that approximation has value for real universe predictions. As https://www.scientificamerican.com/article/naked-singulariti... points out, models strongly suggest that it is possible for naked singularities to form in GR. If we understand better how GR with singularities behaves, we may be able to make testable predictions about what astronomers should look for to verify them.
Second, it may be that the right quantum theory of gravity, contains singularities after all. QM is filled with smooth fields that are quantized particles. For example smooth electromagnetic waves give rise to discrete photons. Shouldn't we expect that a graviton, in the right quantized particle, also looks like a discrete particle? In that case, shouldn't it be some kind of singularity? If so, then a better understanding of singularities in GR may help us find a unified theory.
And third, extending from a smooth model to one with singularities, may result in developing better mathematical tools. For a historical example, consider the development of distributions such as the Dirac delta as an extension of theories built using Calculus on smooth functions. There is a chance that history will repeat. But we won't know until we try to develop these new tools.
> [Einstein's General Relativity] tells us that the universe is expanding
Does GR really tell us that though?
The way I understood it, GR's differential equations will produce solutions for many different constraints and initial conditions you throw at them. Including the constraints & conditions informed by astronomical observation.
GR tells us that a homogeneous and isotropic universe either expands or contracts. Besides being the default assumption, our universe also looks quite homogeneous and isotropic at large scales, and definitely looks like it fits the expanding option rather than the contracting one.
I have another theory. And yes, I know it’s not rigorous.
Fundamentally, all that exists is oscillations (think vibrating strings).
Those oscillations are not three-dimensional, but as they oscillate, their sum goes through states which correspond with fundamental interactions, one by one, in series, giving rise to fundamental ”particles”.
As this state proceeds to evolve, consonance/dissonance between interactions gives rise to higher order oscillations, which yield even higher order configurations of oscillation.
These oscillatory configurations eventually start to resist change, yielding mass. They become logically separated from other oscillation that is not coherent with their structure, yielding multidimensional causal structures in their interactions.
We are observers inside this system, ourselves made of innumerable such fundamental structures. We cannot experience or sense the non-locality directly, for all our sensing-structures are made of higher order oscillatory structures which have mass and locality.
To us and to our instruments of perception, existence appears dimensionally separated, even though everything is dancing in a conga line to the exact same tune.
So, perhaps in a singularity, these structures become so tightly packed that the innermost one breaks down, no longer has mass or locality, and leaves an inherent ”quantum vacuum” in its place, which is then immediately filled with the next structure, which also gets broken down, etc. What remains is pieces of configuration which, no longer supporting the notions of ”mass” or ”locality”, are free of the gravitational pull of a black hole and capable of radiating out.
I was using the word ”theory” in the colloquial sense, not in the academic sense. The above is barely even a hypothesis.
It does touch on string theory, pilot wave theory as well as causal set theory and would explain Hawking radiation, so the idea is clearly not entirely without merit.
1: No idea, it is just a model. String theory also does not answer this question. It is a bit like asking ”which instrument does the universe play?”
2: See string theory, which kind of works similarly, assuming 1-dimensional strings.
3: Nobody has detected Hawking radiation as it is presumably very faint. It would explain the mechanics of its origin and its ability to radiate out of black holes (which no other form of radiation can do).
This differs from Hawking’s explanation of spontaneous matter/antimatter pair generation at the event horizon in that here the radiation would genuinely originate from inside the event horizon. It is a more simple explanation: spontaneous pair generation is not required to explain the same phenomenon.
4: All oscillation resists change. If you place multiple metronomes on a desk, they will synchronize, but not instantly. If you play two strings on an instrument that are tuned almost the same, they both affect each other, each trying to resonate on its own frequency while resisting the resonance imparted by the other string - unless the strings are tuned in unison or in a harmonic.
The two extreme examples are dissonance (resisting change) and consonance (accepting change, i.e. destructive and constructive interference.
These principles are found everywhere from nature to the cosmos and to quantum mechanics.
Why that would ”create mass” (as in, make it possible for a waveform to interact with itself/other waveforms in a way that looks like ”mass” to us) is a good question! As far as I understand, it has something to do with conservation of angular momentum and waveform curvature.
If you have more questions, I will try to answer them!
Hawking explained the origin of Hawking radiation. That’s kind of why it’s called Hawking radiation. His explanation includes why it comes out of a black hole.
The rest of it is, respectfully, “not even wrong”, so it wouldn’t really benefit either of us for me to try to engage further. Thanks for your response though. If you actually want to develop these ideas I would strongly encourage you to do some actual work to understand first classical mechanics and then the standard model. For example, if you read Kolenkow and Kleppner and do the exercises, you will realise the flaws in what you have written about oscillation and interference. Oscillation and the phenomenon of resonance really doesn’t work anything like you have written.
> Hawking explained the origin of Hawking radiation. That’s kind of why it’s called Hawking radiation. His explanation includes why it comes out of a black hole.
Not only that; it's never been physically proven to exist, so there's absolutely no reason to explain it if his theories are wrong.
Well, I’m certainly not qualified to argue against Hawking. Still, the theory is very difficult to test empirically, so I would argue the bets are still open!
I would assume my model is wrong in many ways. The reason I’m sharing my thoughts is not to claim they are the way things are, but to hopefully inspire someone to try building a theory that could explain the empirical, top-down-oriented Standard Model using a novel, minimalistic bottom-up approach.
I could also be getting some terms wrong, English is not my first language and certainly was not the language of natural sciences instruction in my university.
No, the bets are definitely not still open. Hawking described a model of particle behaviour on the event horizon of a black hole. His model predicts radiation arising from the mechanism described in the model. That may not be what happens in reality because as you pointed out, the radiation is predicted to be extremely faint and so it has not yet been detected, and therefore it is currently not empirically verified.
You claim that your proposal predicts radiation from black holes (without explaining any mechanism other than it’s different from Hawking’s model). But here’s the thing: If it doesn’t happen the way Hawking’s model says, then it isn’t Hawking radiation. It could be that cluckindan radiation becomes a thing, but that’s not Hawking radiation.
You’re not getting terms wrong, you’re making empirically unjustifiable statements that do not connect with each other or intersect with the observations we have made of the universe in any meaningful way. That’s what I mean by “not even wrong”.
Quantum models tend to require mathematical interpretation before they are in any way empirically testable, as in verifiable against observations.
In that sense, one could call most models in quantum mechanics ”empirically unjustifiable”.
Would you say models like QCD and QFT are ”not even wrong”, too, even though the Standard Model is based directly on QFT, which lacks a formal, generally agreed-upon mathematical foundation, and has multiple competing interpretations?
Textbook-thumping zealotry has no place outside a church.
It's not the same "new geometry" as OP, but I was fascinated to learn (20 years ago, wow, time flies) that it's viable to formulate GR with the Euclidean metric instead of the traditional Minkowski metric:
This is some really fancy marketing for "yet to be discovered" much less peer reviewed. I guess it fills the Einstein fandom clickbait quota. I hope these mathematicians can glaze Einstein's Theory of Relativity (not a theory in physics anymore, they can only legally say it's Einstein's theory) so it actually works with modern physics.
As a one-time mathematician, this was a really fascinating article. The similarities seem to be entirely coincidental, but what would have been my doctoral dissertation was also about generalizing some concepts from smooth manifolds to a "non-smooth" setting, and the crux of my work also hinged on optimal transport.
Actually I feel optimal transport is a pretty underrated concept in both pure and applied math, and I would have loved to explore it had I continued in academia. But oh well, one must make choices in life...
Small world. My graduate research was precisely on this topic as well. I was going in a more algebraic direction, though. My master's thesis was essentially about different discrete analogues of curvature using cooked-up cohomological constructions.
I really wish academia consistently provided as much security as industry. Would have loved to continue this line of research.
I can't simply help but think that optimal transport is intricately linked to the principle of least action (and as we know POLA is everywhere in nature). At the end, natural interactions seem to be one big optimization problem.
Do I understand correctly that with sectional curvature/triangle comparison methods you can do differential geometry on non-smooth manifolds (e.g. on a cube)? If so, I've completely missed this fact before.
Sure, see (2010) A curved Brunn-Minkowski inequality on the discrete hypercube, Or: What is the Ricci curvature of the discrete hypercube? http://www.yann-ollivier.org/rech/publs/cube.pdf
Or this: (2011) A visual introduction to Riemannian curvatures and some discrete generalizations http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf
Taken from the site of Yann Ollivier http://www.yann-ollivier.org/rech/index
This is all really cool, but is getting new singularity theorems really a positive sign? Like, my understanding was that it was generally hoped that an improved, quantum theory of gravity would eliminate such singularities -- that such singularities were generally considered to be non-physical artifacts that occur in GR due to its deficiencies at the most extreme scales (where quantum gravity would be relevant), not that they are in fact real and physical. So I'd consider it a better sign if these predicted black holes, which we see, but without singularities!
It is a positive sign, and here are three reasons why.
First, even if space is smooth, it is sometimes well-approximated by a singularity. In which case understanding that approximation has value for real universe predictions. As https://www.scientificamerican.com/article/naked-singulariti... points out, models strongly suggest that it is possible for naked singularities to form in GR. If we understand better how GR with singularities behaves, we may be able to make testable predictions about what astronomers should look for to verify them.
Second, it may be that the right quantum theory of gravity, contains singularities after all. QM is filled with smooth fields that are quantized particles. For example smooth electromagnetic waves give rise to discrete photons. Shouldn't we expect that a graviton, in the right quantized particle, also looks like a discrete particle? In that case, shouldn't it be some kind of singularity? If so, then a better understanding of singularities in GR may help us find a unified theory.
And third, extending from a smooth model to one with singularities, may result in developing better mathematical tools. For a historical example, consider the development of distributions such as the Dirac delta as an extension of theories built using Calculus on smooth functions. There is a chance that history will repeat. But we won't know until we try to develop these new tools.
FTA:
> [Einstein's General Relativity] tells us that the universe is expanding
Does GR really tell us that though?
The way I understood it, GR's differential equations will produce solutions for many different constraints and initial conditions you throw at them. Including the constraints & conditions informed by astronomical observation.
GR tells us that a homogeneous and isotropic universe either expands or contracts. Besides being the default assumption, our universe also looks quite homogeneous and isotropic at large scales, and definitely looks like it fits the expanding option rather than the contracting one.
I have another theory. And yes, I know it’s not rigorous.
Fundamentally, all that exists is oscillations (think vibrating strings).
Those oscillations are not three-dimensional, but as they oscillate, their sum goes through states which correspond with fundamental interactions, one by one, in series, giving rise to fundamental ”particles”.
As this state proceeds to evolve, consonance/dissonance between interactions gives rise to higher order oscillations, which yield even higher order configurations of oscillation.
These oscillatory configurations eventually start to resist change, yielding mass. They become logically separated from other oscillation that is not coherent with their structure, yielding multidimensional causal structures in their interactions.
We are observers inside this system, ourselves made of innumerable such fundamental structures. We cannot experience or sense the non-locality directly, for all our sensing-structures are made of higher order oscillatory structures which have mass and locality.
To us and to our instruments of perception, existence appears dimensionally separated, even though everything is dancing in a conga line to the exact same tune.
So, perhaps in a singularity, these structures become so tightly packed that the innermost one breaks down, no longer has mass or locality, and leaves an inherent ”quantum vacuum” in its place, which is then immediately filled with the next structure, which also gets broken down, etc. What remains is pieces of configuration which, no longer supporting the notions of ”mass” or ”locality”, are free of the gravitational pull of a black hole and capable of radiating out.
Do people not know the standard for theories in physics? Is the brain rot this bad?
Humor me, please.
I was using the word ”theory” in the colloquial sense, not in the academic sense. The above is barely even a hypothesis.
It does touch on string theory, pilot wave theory as well as causal set theory and would explain Hawking radiation, so the idea is clearly not entirely without merit.
Well to start with, in your theory
1) If all that exists is vibrations, what is vibrating?
2) You say the vibrations are not three-dimensional, but you don’t say what they are. What are they?
3) If it would explain Hawking radiation, how would it explain it, and why do you consider Hawking not to have explained Hawking radiation?
4) What causes the oscillation to resist change? What does that mean and why does that create mass?
1: No idea, it is just a model. String theory also does not answer this question. It is a bit like asking ”which instrument does the universe play?”
2: See string theory, which kind of works similarly, assuming 1-dimensional strings.
3: Nobody has detected Hawking radiation as it is presumably very faint. It would explain the mechanics of its origin and its ability to radiate out of black holes (which no other form of radiation can do).
This differs from Hawking’s explanation of spontaneous matter/antimatter pair generation at the event horizon in that here the radiation would genuinely originate from inside the event horizon. It is a more simple explanation: spontaneous pair generation is not required to explain the same phenomenon.
4: All oscillation resists change. If you place multiple metronomes on a desk, they will synchronize, but not instantly. If you play two strings on an instrument that are tuned almost the same, they both affect each other, each trying to resonate on its own frequency while resisting the resonance imparted by the other string - unless the strings are tuned in unison or in a harmonic.
The two extreme examples are dissonance (resisting change) and consonance (accepting change, i.e. destructive and constructive interference.
These principles are found everywhere from nature to the cosmos and to quantum mechanics.
Why that would ”create mass” (as in, make it possible for a waveform to interact with itself/other waveforms in a way that looks like ”mass” to us) is a good question! As far as I understand, it has something to do with conservation of angular momentum and waveform curvature.
If you have more questions, I will try to answer them!
Hawking explained the origin of Hawking radiation. That’s kind of why it’s called Hawking radiation. His explanation includes why it comes out of a black hole.
The rest of it is, respectfully, “not even wrong”, so it wouldn’t really benefit either of us for me to try to engage further. Thanks for your response though. If you actually want to develop these ideas I would strongly encourage you to do some actual work to understand first classical mechanics and then the standard model. For example, if you read Kolenkow and Kleppner and do the exercises, you will realise the flaws in what you have written about oscillation and interference. Oscillation and the phenomenon of resonance really doesn’t work anything like you have written.
> Hawking explained the origin of Hawking radiation. That’s kind of why it’s called Hawking radiation. His explanation includes why it comes out of a black hole.
Not only that; it's never been physically proven to exist, so there's absolutely no reason to explain it if his theories are wrong.
Well, I’m certainly not qualified to argue against Hawking. Still, the theory is very difficult to test empirically, so I would argue the bets are still open!
I would assume my model is wrong in many ways. The reason I’m sharing my thoughts is not to claim they are the way things are, but to hopefully inspire someone to try building a theory that could explain the empirical, top-down-oriented Standard Model using a novel, minimalistic bottom-up approach.
I could also be getting some terms wrong, English is not my first language and certainly was not the language of natural sciences instruction in my university.
No, the bets are definitely not still open. Hawking described a model of particle behaviour on the event horizon of a black hole. His model predicts radiation arising from the mechanism described in the model. That may not be what happens in reality because as you pointed out, the radiation is predicted to be extremely faint and so it has not yet been detected, and therefore it is currently not empirically verified.
You claim that your proposal predicts radiation from black holes (without explaining any mechanism other than it’s different from Hawking’s model). But here’s the thing: If it doesn’t happen the way Hawking’s model says, then it isn’t Hawking radiation. It could be that cluckindan radiation becomes a thing, but that’s not Hawking radiation.
You’re not getting terms wrong, you’re making empirically unjustifiable statements that do not connect with each other or intersect with the observations we have made of the universe in any meaningful way. That’s what I mean by “not even wrong”.
Quantum models tend to require mathematical interpretation before they are in any way empirically testable, as in verifiable against observations.
In that sense, one could call most models in quantum mechanics ”empirically unjustifiable”.
Would you say models like QCD and QFT are ”not even wrong”, too, even though the Standard Model is based directly on QFT, which lacks a formal, generally agreed-upon mathematical foundation, and has multiple competing interpretations?
Textbook-thumping zealotry has no place outside a church.
How would you respond to the anti-science folks who respond "it's just a theory" at anything they can find?
Please refrain from strawmen and ad hominem arguments.
I am not anti-science and what I wrote above is not something I found.
It's not the same "new geometry" as OP, but I was fascinated to learn (20 years ago, wow, time flies) that it's viable to formulate GR with the Euclidean metric instead of the traditional Minkowski metric:
https://www.euclideanrelativity.com
This is some really fancy marketing for "yet to be discovered" much less peer reviewed. I guess it fills the Einstein fandom clickbait quota. I hope these mathematicians can glaze Einstein's Theory of Relativity (not a theory in physics anymore, they can only legally say it's Einstein's theory) so it actually works with modern physics.