3 ms·

A New Geometry for Einstein's Theory of Relativity

135 points by jandrewrogers 1y ago

deleted 1y ago
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deleted 1y ago
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WCSTombs 1y ago
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...
- xelxebar 1y ago
  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.
- pkoird 1y ago
  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.
- bmacho 1y ago
  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.
  - youcandoittoo 1y ago
    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 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 http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf Taken from the site of Yann Ollivier http://www.yann-ollivier.org/rech/index http://www.yann-ollivier.org/rech/index
Sniffnoy 1y ago
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!
- btilly 1y ago
  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-singularities-extreme-physics-special/ 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.
ggdG 1y ago
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.
- layer8 1y ago
  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.
cluckindan 1y ago
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.
- kittikitti 1y ago
  Do people not know the standard for theories in physics? Is the brain rot this bad?
  - cluckindan 1y ago
    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.
    - 1y ago
kittikitti 1y ago
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.
michaelsbradley 1y ago
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 https://www.euclideanrelativity.com

135 points by jandrewrogers 1y ago

deleted 1y ago

[deleted]

deleted 1y ago

[deleted]

WCSTombs 1y ago

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...

xelxebar 1y ago
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.
pkoird 1y ago
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.
bmacho 1y ago
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.
- youcandoittoo 1y ago
  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 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 http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf Taken from the site of Yann Ollivier http://www.yann-ollivier.org/rech/index http://www.yann-ollivier.org/rech/index

Sniffnoy 1y ago

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!

btilly 1y ago
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-singularities-extreme-physics-special/ 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.

ggdG 1y ago

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.

layer8 1y ago
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.

cluckindan 1y ago

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.

kittikitti 1y ago
Do people not know the standard for theories in physics? Is the brain rot this bad?
- cluckindan 1y ago
  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.
  - 1y ago

kittikitti 1y ago

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.

michaelsbradley 1y ago

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 https://www.euclideanrelativity.com