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**Cristinel Stoica**: *on* 12/6/11 at 8:13am UTC, wrote Hi, I have some news about the methods of singular semi-Riemannian...

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**Cristi Stoica**: *on* 3/13/11 at 7:23am UTC, wrote Dear Peter, thank you for reading my essay and for your kind observations....

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FQXi FORUM

August 17, 2017

CATEGORY:
Is Reality Digital or Analog? Essay Contest (2010-2011)
[back]

TOPIC: Infinite Resolution by Cristinel Stoica [refresh]

TOPIC: Infinite Resolution by Cristinel Stoica [refresh]

It was reported that General Relativity predicts its own breakdown, because of singularities. We will see that the mathematics of General Relativity can be naturally extended to work fine when the metric becomes degenerate. Then is proposed an extended version of Einstein's equation which remains valid at singularities. The time evolution is expressed by equations which allow passing beyond the singularities. Consequently, the problems of singularities, including Hawking's information paradox, vanish. The core principle used to extend the mathematics and physics of General Relativity beyond the singularities provides a surprising answer to the question: "Is there a deep, foundational reason why reality must be purely analog, or why it must be digital?"

Cristi Stoica is caught between two worlds: the digital world of computational geometry (he works as a computer programmer for a leading provider of cad/cam components), and the continuous world of differential geometry (where he is doing his PhD). The present essay, entitled "Infinite Resolution", is based on his research for his PhD thesis.

Hi Cristi,

Welcome to the game!

I enjoyed your essay. I also addressed the problems of "Infinity" and its inverse, but I used the properties of Scales. Lawrence Crowell also addressed the properties of Black Holes, but his analysis (with qubit entangled strings and the Holographic Principle) admits more discrete structure than does yours.

Good Luck & Have Fun!

report post as inappropriate

Welcome to the game!

I enjoyed your essay. I also addressed the problems of "Infinity" and its inverse, but I used the properties of Scales. Lawrence Crowell also addressed the properties of Black Holes, but his analysis (with qubit entangled strings and the Holographic Principle) admits more discrete structure than does yours.

Good Luck & Have Fun!

report post as inappropriate

Hi Ray!

Thank you for your welcome. I am glad that these problems interest other people too. My approach to the infinities presented here is based solely on the geometry of spacetime in General Relativity - the most difficult part being to extend it in a natural and consistent way beyond the singularities. I look forward to read your essay.

Good luck!

Thank you for your welcome. I am glad that these problems interest other people too. My approach to the infinities presented here is based solely on the geometry of spacetime in General Relativity - the most difficult part being to extend it in a natural and consistent way beyond the singularities. I look forward to read your essay.

Good luck!

Hi Cristi,

I reread your essay. We both agree that infinity is a problem. Your perspective is that information gets overwritten when field lines become degenerate, mine is that infinity cannot exist within a finite Observable Universe (13.7 billion light-years is huge, but finite). As I understand, you essentially are saying that the fields remain continuous ad infinitum, and that infinitesimally small terms are introduced that prevent the true singularity. It seems that these small terms could be introduced via properties of intrinsic and extrinsic curvature.

My perspective is that infinity cannot exist in our reality, and therefore the concept of field lines that are continuous ad infinitum is a slippery slope - what is the definition of "ad infinitum" if infinity doesn't really exist?

You approached this problem from the perspective of General Relativity and handled a tough subject very well in my opinion. I partially addressed the problem of infinity from the perspective of Scales, Supersymmetry and Solid State Physics models. I think that the concept of infinity requires different scales - some larger (Multiverse?) and some smaller (Quantum?) than our classical scale. Perhaps your intrinsic and extrinsic curvatures are related to my ideas of Scales, and are related to each other via a concept similar to Supersymmetry (or perhaps the Haag–Lopuszanski–Sohnius theorem).

In my forum, I have been discussing the idea that a static Black Hole "singularity" is encased by a Buckyball lattice of "quantum spacetime" or "quantum gravity", so that infinity is never truly reached. Because most (perhaps all?) Black Holes rotate, we should expect torsion to morp a nested pair of buckyballs into their homotopic cousin, a lattice-like torus. Although this torus may have lattice properties at a scale of 10^-31 cm, these lattice points are the ends of strings that appear continuous (and seemingly continuous ad infinitum) at scales greater than 10^-31 cm.

The contest period is drawing near the end of Community votes. I would appreciate your feedback on my essay if you have an opportunity.

Good Luck and Have Fun!

Dr. Cosmic Ray

report post as inappropriate

I reread your essay. We both agree that infinity is a problem. Your perspective is that information gets overwritten when field lines become degenerate, mine is that infinity cannot exist within a finite Observable Universe (13.7 billion light-years is huge, but finite). As I understand, you essentially are saying that the fields remain continuous ad infinitum, and that infinitesimally small terms are introduced that prevent the true singularity. It seems that these small terms could be introduced via properties of intrinsic and extrinsic curvature.

My perspective is that infinity cannot exist in our reality, and therefore the concept of field lines that are continuous ad infinitum is a slippery slope - what is the definition of "ad infinitum" if infinity doesn't really exist?

You approached this problem from the perspective of General Relativity and handled a tough subject very well in my opinion. I partially addressed the problem of infinity from the perspective of Scales, Supersymmetry and Solid State Physics models. I think that the concept of infinity requires different scales - some larger (Multiverse?) and some smaller (Quantum?) than our classical scale. Perhaps your intrinsic and extrinsic curvatures are related to my ideas of Scales, and are related to each other via a concept similar to Supersymmetry (or perhaps the Haag–Lopuszanski–Sohnius theorem).

In my forum, I have been discussing the idea that a static Black Hole "singularity" is encased by a Buckyball lattice of "quantum spacetime" or "quantum gravity", so that infinity is never truly reached. Because most (perhaps all?) Black Holes rotate, we should expect torsion to morp a nested pair of buckyballs into their homotopic cousin, a lattice-like torus. Although this torus may have lattice properties at a scale of 10^-31 cm, these lattice points are the ends of strings that appear continuous (and seemingly continuous ad infinitum) at scales greater than 10^-31 cm.

The contest period is drawing near the end of Community votes. I would appreciate your feedback on my essay if you have an opportunity.

Good Luck and Have Fun!

Dr. Cosmic Ray

report post as inappropriate

Dear Cristi,

Your essay is most impressive and you appear to have solved the problem you set out to solve, by making appropriate postulates and interpretations. I have no arguments with your math, but I would like to ask about something that you've obviously thought about a lot.

I got somewhat lost when you gave up orthonormality and when you insist that identical points can exist with no distance between them and still retain their identity.

For example, in Feynman diagrams two identical particles can enter into an interaction and two identical particles can exit the interaction, and it is impossible to track the identity through the interaction--they may, or may not, have switched places. I don't believe there is even the need for assuming zero distance between them. That is, we apparently don't need a singularity to lose track of identity.

I find the idea that black holes can evaporate and all the 'information inside' be reconstructed ridiculous, but I know that others do not do so, so you are addressing a 'legitimate' problem of current physics.

Why would one insist that such is the case? You seem to imply that both classical and quantum time evolution laws are violated if info is lost.

But if, as many fqxi'ers seem to believe, the real nature of time is essentially NOW, and Einstein's block time is an illusion, or at least a mathematical extrapolation that goes beyond reality, then what seems to be necessary is a physics that accurately describes interactions taking place NOW.

And here is my main question:

Can we have gotten to NOW by two (or more) different paths, based on different initial and/or boundary conditions? A sort of generalization of the Feynman example above.

To your knowledge, has anyone proved the 'uniqueness' of the history leading up to NOW?

Elsewhere Jason Wolfe points out that when photons are red-shifted, they lose information. This seems to me to be true (with a caveat that I'm working on.)

I also have the opinion that, as Feynman said of QM, no one understands information. For example, some big names treat information as if it is a particle. Information is not a particle. In this sense I am not sure what is even meant when one speaks of 'information at a point of space', whether or not there is a zero or finite distance from another point.

I look forward to any response you might make, but I am most interested in whether our current physical state of existence NOW is not reachable (in theory) by two or more different histories. It seems to me that only a probabilistic answer is possible, and when probability enters the picture, information becomes even more complicated.

I repeat, to the best of my ability to judge, it does seem that you proved what you set out to prove.

Edwin Eugene Klingman

report post as inappropriate

report post as inappropriate

Your essay is most impressive and you appear to have solved the problem you set out to solve, by making appropriate postulates and interpretations. I have no arguments with your math, but I would like to ask about something that you've obviously thought about a lot.

I got somewhat lost when you gave up orthonormality and when you insist that identical points can exist with no distance between them and still retain their identity.

For example, in Feynman diagrams two identical particles can enter into an interaction and two identical particles can exit the interaction, and it is impossible to track the identity through the interaction--they may, or may not, have switched places. I don't believe there is even the need for assuming zero distance between them. That is, we apparently don't need a singularity to lose track of identity.

I find the idea that black holes can evaporate and all the 'information inside' be reconstructed ridiculous, but I know that others do not do so, so you are addressing a 'legitimate' problem of current physics.

Why would one insist that such is the case? You seem to imply that both classical and quantum time evolution laws are violated if info is lost.

But if, as many fqxi'ers seem to believe, the real nature of time is essentially NOW, and Einstein's block time is an illusion, or at least a mathematical extrapolation that goes beyond reality, then what seems to be necessary is a physics that accurately describes interactions taking place NOW.

And here is my main question:

Can we have gotten to NOW by two (or more) different paths, based on different initial and/or boundary conditions? A sort of generalization of the Feynman example above.

To your knowledge, has anyone proved the 'uniqueness' of the history leading up to NOW?

Elsewhere Jason Wolfe points out that when photons are red-shifted, they lose information. This seems to me to be true (with a caveat that I'm working on.)

I also have the opinion that, as Feynman said of QM, no one understands information. For example, some big names treat information as if it is a particle. Information is not a particle. In this sense I am not sure what is even meant when one speaks of 'information at a point of space', whether or not there is a zero or finite distance from another point.

I look forward to any response you might make, but I am most interested in whether our current physical state of existence NOW is not reachable (in theory) by two or more different histories. It seems to me that only a probabilistic answer is possible, and when probability enters the picture, information becomes even more complicated.

I repeat, to the best of my ability to judge, it does seem that you proved what you set out to prove.

Edwin Eugene Klingman

report post as inappropriate

Dear Eugene,

Thank you for your careful reading and consideration of the implications. I think I can answer you why, on the one hand, you find my solution correct, on the other hand it seems to contradict some other views. My solution takes place within General Relativity, and it makes use of the mathematics of GR - extended to work with degenerate metrics. I did not add other postulates, I just removed one assumption or two, which are made implicit. The things that seem to be contradicted by my findings may be either part of other theories, or consequences of some other assumptions. I will try to address them individually.

> "I got somewhat lost when you gave up orthonormality ..."

In GR orthonormality cannot exist in coordinate systems in general, only in particular cases. It can exist in local frames though. But if the metric becomes degenerate, the length of some vector fields becomes zero, even if they are not zero. Trying to normalize such a vector fields leads to infinities. But GR works fine with non-orthonormal and even non-orthogonal frames.

> "... and when you insist that identical points can exist with no distance between them and still retain their identity."

Geometrically, the simplest example is a 2-dimensional vector space having the inner product given by g=(1,0), not g=(1,1) as in Euclidean geometry, not g=(1,-1) as in Lorentzian geometry.

Having my solution confined to General Relativity (with the small fix I proposed) doesn't exclude QFT, as many work was done in QFT in curved spacetime, which is in my opinion compatible with GR and may very well be enough.

> "For example, in Feynman diagrams..."

Yes, the particles of the same type are identical in QFT. The quantum particles get mixed up, but there is no need to track them back. The evolution is unitary, and here there is no problem with the information, even if we lose track of their identities. The input in a Feynman diagram determines the output - the converse is valid as well.

> "I find the idea that black holes can evaporate and all the 'information inside' be reconstructed ridiculous, but I know that others do not do so, so you are addressing a 'legitimate' problem of current physics."

Of course it is ridiculous, it is like reconstructing the "Total Baseball, The Ultimate Baseball Encyclopedia" from its own ashes. This is not possible in practice, because we don't know the complete information, and even if we do, it may be impossible to calculate from it the original information. The point is that somehow the universe knows all this info, and computes it to find the next state. The laws we know work as well backwards, so they can in principle help reconstructing the past.

> "You seem to imply that both classical and quantum time evolution laws are violated if info is lost."

Maybe in the real world there is no information conservation. But in Quantum Theory the time evolution is unitary, hence the information is preserved. The classical time evolution is deterministic and reversible (once we know the complete configuration), so the information is also preserved. Hawking's paradox states a contradiction between the information conservation and evaporating singularities, and this is what I address.

> "But if, as many fqxi'ers seem to believe, the real nature of time is essentially NOW, and Einstein's block time is an illusion, or at least a mathematical extrapolation that goes beyond reality, then what seems to be necessary is a physics that accurately describes interactions taking place NOW."

Maybe Einstein's block is an illusion, maybe the NOW is an illusion (I think that both are illusions, but it would take many pages to explain how). But General Relativity can be formulated in terms of NOW as well, and you can see in my essay that I addressed the ADM formalism, which does exactly this. So, my solution does not contradict the presentism.

> "Elsewhere Jason Wolfe points out that when photons are red-shifted, they lose information. This seems to me to be true (with a caveat that I'm working on.)"

I don't know, but I think that in a discrete world this should be true.

> "I also have the opinion that, as Feynman said of QM, no one understands information. For example, some big names treat information as if it is a particle. Information is not a particle. In this sense I am not sure what is even meant when one speaks of 'information at a point of space', whether or not there is a zero or finite distance from another point."

In my essay and the articles I referred, I am using the word "information" implicitly as a placeholder for "the complete description of the topology of space, and of the fields defined on the space" - for example the initial data. The fields have definite values at each point, and by this I understand 'information at a point of space'. And by "information loss" I meant the loss of the initial data at a given time.

> "To your knowledge, has anyone proved the 'uniqueness' of the history leading up to NOW?"

No. The 'only' proof is that the laws that seem to us to work best have this property. But I would not exclude the possibility of violation of this 'uniqueness'. The most notable counterexample was the black hole information paradox. And perhaps the state vector reduction in Quantum Mechanics.

--- continued in the following comment ---

Thank you for your careful reading and consideration of the implications. I think I can answer you why, on the one hand, you find my solution correct, on the other hand it seems to contradict some other views. My solution takes place within General Relativity, and it makes use of the mathematics of GR - extended to work with degenerate metrics. I did not add other postulates, I just removed one assumption or two, which are made implicit. The things that seem to be contradicted by my findings may be either part of other theories, or consequences of some other assumptions. I will try to address them individually.

> "I got somewhat lost when you gave up orthonormality ..."

In GR orthonormality cannot exist in coordinate systems in general, only in particular cases. It can exist in local frames though. But if the metric becomes degenerate, the length of some vector fields becomes zero, even if they are not zero. Trying to normalize such a vector fields leads to infinities. But GR works fine with non-orthonormal and even non-orthogonal frames.

> "... and when you insist that identical points can exist with no distance between them and still retain their identity."

Geometrically, the simplest example is a 2-dimensional vector space having the inner product given by g=(1,0), not g=(1,1) as in Euclidean geometry, not g=(1,-1) as in Lorentzian geometry.

Having my solution confined to General Relativity (with the small fix I proposed) doesn't exclude QFT, as many work was done in QFT in curved spacetime, which is in my opinion compatible with GR and may very well be enough.

> "For example, in Feynman diagrams..."

Yes, the particles of the same type are identical in QFT. The quantum particles get mixed up, but there is no need to track them back. The evolution is unitary, and here there is no problem with the information, even if we lose track of their identities. The input in a Feynman diagram determines the output - the converse is valid as well.

> "I find the idea that black holes can evaporate and all the 'information inside' be reconstructed ridiculous, but I know that others do not do so, so you are addressing a 'legitimate' problem of current physics."

Of course it is ridiculous, it is like reconstructing the "Total Baseball, The Ultimate Baseball Encyclopedia" from its own ashes. This is not possible in practice, because we don't know the complete information, and even if we do, it may be impossible to calculate from it the original information. The point is that somehow the universe knows all this info, and computes it to find the next state. The laws we know work as well backwards, so they can in principle help reconstructing the past.

> "You seem to imply that both classical and quantum time evolution laws are violated if info is lost."

Maybe in the real world there is no information conservation. But in Quantum Theory the time evolution is unitary, hence the information is preserved. The classical time evolution is deterministic and reversible (once we know the complete configuration), so the information is also preserved. Hawking's paradox states a contradiction between the information conservation and evaporating singularities, and this is what I address.

> "But if, as many fqxi'ers seem to believe, the real nature of time is essentially NOW, and Einstein's block time is an illusion, or at least a mathematical extrapolation that goes beyond reality, then what seems to be necessary is a physics that accurately describes interactions taking place NOW."

Maybe Einstein's block is an illusion, maybe the NOW is an illusion (I think that both are illusions, but it would take many pages to explain how). But General Relativity can be formulated in terms of NOW as well, and you can see in my essay that I addressed the ADM formalism, which does exactly this. So, my solution does not contradict the presentism.

> "Elsewhere Jason Wolfe points out that when photons are red-shifted, they lose information. This seems to me to be true (with a caveat that I'm working on.)"

I don't know, but I think that in a discrete world this should be true.

> "I also have the opinion that, as Feynman said of QM, no one understands information. For example, some big names treat information as if it is a particle. Information is not a particle. In this sense I am not sure what is even meant when one speaks of 'information at a point of space', whether or not there is a zero or finite distance from another point."

In my essay and the articles I referred, I am using the word "information" implicitly as a placeholder for "the complete description of the topology of space, and of the fields defined on the space" - for example the initial data. The fields have definite values at each point, and by this I understand 'information at a point of space'. And by "information loss" I meant the loss of the initial data at a given time.

> "To your knowledge, has anyone proved the 'uniqueness' of the history leading up to NOW?"

No. The 'only' proof is that the laws that seem to us to work best have this property. But I would not exclude the possibility of violation of this 'uniqueness'. The most notable counterexample was the black hole information paradox. And perhaps the state vector reduction in Quantum Mechanics.

--- continued in the following comment ---

--- continued from the previous comment ---

> "And here is my main question: Can we have gotten to NOW by two (or more) different paths, based on different initial and/or boundary conditions? A sort of generalization of the Feynman example above."

I think that you may be interested in my view on the state vector reduction in Quantum Mechanics. I used this view in my essay about time, and that about possible and impossible (which you know). In Quantum Mechanics, at a given NOW, we can choose what to observe, and the outcome depends of our choice. But we can measure things that happened long time ago, as Wheeler pointed out - do they depend on our present choice? I show there that, if we admit that the initial conditions are not fixed completely in the past, but part of them are delayed at various NOW's when we choose what to observe, the time evolution still can be unitary (hence deterministic and without loss of information). This is a way to get to the same NOW from two different histories, provided that the NOW doesn't have specified the complete data. And Quantum Mechanics seems to show that this is the case. Please note that the information I refer is that required for the time evolution, not what we can record. Here lies the answer: what we can access/observe/record is incomplete, hence it involves indeed probabilities.

I appreciate the time and thoughtful consideration you gave to my essay. I look forward to read yours.

Best regards,

Cristi

> "And here is my main question: Can we have gotten to NOW by two (or more) different paths, based on different initial and/or boundary conditions? A sort of generalization of the Feynman example above."

I think that you may be interested in my view on the state vector reduction in Quantum Mechanics. I used this view in my essay about time, and that about possible and impossible (which you know). In Quantum Mechanics, at a given NOW, we can choose what to observe, and the outcome depends of our choice. But we can measure things that happened long time ago, as Wheeler pointed out - do they depend on our present choice? I show there that, if we admit that the initial conditions are not fixed completely in the past, but part of them are delayed at various NOW's when we choose what to observe, the time evolution still can be unitary (hence deterministic and without loss of information). This is a way to get to the same NOW from two different histories, provided that the NOW doesn't have specified the complete data. And Quantum Mechanics seems to show that this is the case. Please note that the information I refer is that required for the time evolution, not what we can record. Here lies the answer: what we can access/observe/record is incomplete, hence it involves indeed probabilities.

I appreciate the time and thoughtful consideration you gave to my essay. I look forward to read yours.

Best regards,

Cristi

Cristi,

I find your answers enlightening. Thanks for the explanations. You do have an excellent grasp of the issues. Without going over them point by point I will say that I agree with your comments almost wholly, and I now understand your paper even better. I will read your previous essay on time about state vector reduction in QM as I only vaguely remember what you said. I am even...

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I find your answers enlightening. Thanks for the explanations. You do have an excellent grasp of the issues. Without going over them point by point I will say that I agree with your comments almost wholly, and I now understand your paper even better. I will read your previous essay on time about state vector reduction in QM as I only vaguely remember what you said. I am even...

view entire post

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Dear Cristi,

I am glad that you take part in this contest and my first impression is that your essay is very interesting. I wish you success.

All the best, Felix.

report post as inappropriate

I am glad that you take part in this contest and my first impression is that your essay is very interesting. I wish you success.

All the best, Felix.

report post as inappropriate

Dear Felix,

thank you for your appreciation.

Best wishes,

Cristi

thank you for your appreciation.

Best wishes,

Cristi

Hi Dear Christi,

Happy to see you on this contest.

I liked your essay,the singularities and the uniqueness show the road of our real infinities.

The Universe is finite, the spaces infinite, but all that evolves. ....

I wish you all the best in the contest.

Regards

Steve

report post as inappropriate

Happy to see you on this contest.

I liked your essay,the singularities and the uniqueness show the road of our real infinities.

The Universe is finite, the spaces infinite, but all that evolves. ....

I wish you all the best in the contest.

Regards

Steve

report post as inappropriate

Dear Steve,

thank you for your appreciation and kind words. I'm happy to see you too.

Best Regards,

Cristi

thank you for your appreciation and kind words. I'm happy to see you too.

Best Regards,

Cristi

Dear Cristi,

Glad to see you in the contest. I read the essay, it is interesting. I have a question regarding your notes from the references. Why don't you put them on the arxiv? You could get lots of potentially interesting suggestions and/or healthy critical remarks.

Concerning semi-riemannian metrics, do you think is there any way to set up an experiment which could allow to check this hypothesis?

Best,

Marius

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Glad to see you in the contest. I read the essay, it is interesting. I have a question regarding your notes from the references. Why don't you put them on the arxiv? You could get lots of potentially interesting suggestions and/or healthy critical remarks.

Concerning semi-riemannian metrics, do you think is there any way to set up an experiment which could allow to check this hypothesis?

Best,

Marius

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Dear Marius,

thank you for reading and appreciating my essay, and for the suggestions. I agree with you and I plan to arxiv them soon.

About semi-riemannian metric. When it is non-degenerate, it is the core of Relativity, both special and general. All the tests of Relativity also test the semi-riemannian metric. The special relativistic effects show that the Poincare symmetry is valid. The predictions of General Relativity, such as deflection of light, perihelion Advance of Mercury, etc, show that this metric exists but it is curved. One of the problems is the prediction of singularities: under general conditions GR saids that they exist. And these conditions are satisfied in our world - for example the existence of black holes. What the singularity theorems show is that at some extreme places, such as the "core" of a black hole, the metric becomes degenerate. In this case, the important invariants of semi-riemannian geometry become divergent.

So, I would say that I do not propose new hypotheses or laws of nature. I just modify the formalism, to base it on other invariants. For example, when the metric becomes degenerate in a way I named semi-regular, the curvature R^{a}_{bcd} becomes divergent, but its covariant version R_{abcd} remains smooth. Normally, they have the same information, but not when the metric is degenerate. In my approach, it turns out that the fundamental invariant is the covariant version of the Riemann tensor, which is smooth.

As an analogy, think at curves in plane. If we decide to characterize the way they turn by the curvature radius, the straight lines - having infinite curvature radius - will turn out to be singular. So, the best choice was to use the curvature, which is 1/R. In this case, the straight lines have curvature 0, which is more manageable. Basically, this is similar to what I did - I chose other fundamental invariants than it is customary. They contain the same information for non-degenerate metric, but they are not divergent, and continuous, when the metric becomes degenerate.

How can we test directly that the singular semi-riemannian metric occurs somewhere in the universe? It is hard, because the places where it is predicted are at the big bang, and inside the black holes. But maybe we can send some information in a black hole, and check that it survived the evaporation. I think that this is very difficult to do, perhaps impossible - we need to know completely the values of the fields, and to compute the outcome (the information survives, but it is mixed). Another way would be if micro black holes (predicted by Hawking) really exist. In this case, they should occur and evaporate quickly all the time, from quantum fluctuations. If they violate the conservation of information, then we can detect violations of unitarity and of the entanglement, by quantum experiments. Personally, I don't think that these experiments are practical. I just wanted to show that it is premature to reject General Relativity because it predicts singularities, by showing that these singularities are not "malign". I could not find a simple way to show this, so I had to develop some mathematics for the singular semi-riemannian manifolds.

Thank you for your observations, and good luck with the contest,

Cristi

thank you for reading and appreciating my essay, and for the suggestions. I agree with you and I plan to arxiv them soon.

About semi-riemannian metric. When it is non-degenerate, it is the core of Relativity, both special and general. All the tests of Relativity also test the semi-riemannian metric. The special relativistic effects show that the Poincare symmetry is valid. The predictions of General Relativity, such as deflection of light, perihelion Advance of Mercury, etc, show that this metric exists but it is curved. One of the problems is the prediction of singularities: under general conditions GR saids that they exist. And these conditions are satisfied in our world - for example the existence of black holes. What the singularity theorems show is that at some extreme places, such as the "core" of a black hole, the metric becomes degenerate. In this case, the important invariants of semi-riemannian geometry become divergent.

So, I would say that I do not propose new hypotheses or laws of nature. I just modify the formalism, to base it on other invariants. For example, when the metric becomes degenerate in a way I named semi-regular, the curvature R

As an analogy, think at curves in plane. If we decide to characterize the way they turn by the curvature radius, the straight lines - having infinite curvature radius - will turn out to be singular. So, the best choice was to use the curvature, which is 1/R. In this case, the straight lines have curvature 0, which is more manageable. Basically, this is similar to what I did - I chose other fundamental invariants than it is customary. They contain the same information for non-degenerate metric, but they are not divergent, and continuous, when the metric becomes degenerate.

How can we test directly that the singular semi-riemannian metric occurs somewhere in the universe? It is hard, because the places where it is predicted are at the big bang, and inside the black holes. But maybe we can send some information in a black hole, and check that it survived the evaporation. I think that this is very difficult to do, perhaps impossible - we need to know completely the values of the fields, and to compute the outcome (the information survives, but it is mixed). Another way would be if micro black holes (predicted by Hawking) really exist. In this case, they should occur and evaporate quickly all the time, from quantum fluctuations. If they violate the conservation of information, then we can detect violations of unitarity and of the entanglement, by quantum experiments. Personally, I don't think that these experiments are practical. I just wanted to show that it is premature to reject General Relativity because it predicts singularities, by showing that these singularities are not "malign". I could not find a simple way to show this, so I had to develop some mathematics for the singular semi-riemannian manifolds.

Thank you for your observations, and good luck with the contest,

Cristi

Dear Cristi,

Here is a concrete mathematical question. It may be trivial, or not.

Given a semi-riemannian metric g with its covariant version curvature

tensor field R, is there (locally, around a point, or generically, like almost everywhere) another, non degenerate metric g', with the same curvature field R?

Marius

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Here is a concrete mathematical question. It may be trivial, or not.

Given a semi-riemannian metric g with its covariant version curvature

tensor field R, is there (locally, around a point, or generically, like almost everywhere) another, non degenerate metric g', with the same curvature field R?

Marius

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Dear Marius,

sorry, I just saw your post. Nice question, and far from trivial. And I think important, because we may need to know how to obtain the metric from the stress-energy tensor. I don't know how to answer it. Searching the net I found this article, this and this, which seem to show that in general the solution is unique, up to one arbitrary scalar. I don't know if it applies to all cases, and if they apply to the singular case as well.

Best regards,

Cristi

sorry, I just saw your post. Nice question, and far from trivial. And I think important, because we may need to know how to obtain the metric from the stress-energy tensor. I don't know how to answer it. Searching the net I found this article, this and this, which seem to show that in general the solution is unique, up to one arbitrary scalar. I don't know if it applies to all cases, and if they apply to the singular case as well.

Best regards,

Cristi

AM I IN THE ANALOG OR DISCRETE PARTY?

Dear visitor,

I think that this contest stimulated in all of us the curiosity about the orientations of each of the participants: "is he or she on the discrete side, or on the continuous side?".

So I will try to clarify my position, which is somehow ambivalent.

At the present time, I put my hopes in a continuous spacetime, on which continuous fields "grow". This may be obvious from my essay, in which I stated that the solution I propose to the singularities requires a continuous spacetime. (Of course, I may be wrong and the spacetime be discrete.)

On the other hand, I do not exclude the possibility that, even in the conditions of continuous spacetime and fields, the world still may be discrete. I illustrate this with the example of vector graphics format in computer graphics. This type of format allows infinite resolution, but in the same time it is digital. Digital does not necessarily means pixelated, so discrete information still may describe a continuous spacetime.

Similarly, continuous spacetimes endowed with continuous fields may very well be describable by digital information. After all, all books on continuous mathematics and physics can be scanned into a computer.

I argued for the continuity of spacetime and fields, but I do not exclude the possibility that all the information contained in these fields can be compressed in a digital format. I wish I could write about this too in my essay, but I need to do more research in this direction.

Best regards,

Cristi

Dear visitor,

I think that this contest stimulated in all of us the curiosity about the orientations of each of the participants: "is he or she on the discrete side, or on the continuous side?".

So I will try to clarify my position, which is somehow ambivalent.

At the present time, I put my hopes in a continuous spacetime, on which continuous fields "grow". This may be obvious from my essay, in which I stated that the solution I propose to the singularities requires a continuous spacetime. (Of course, I may be wrong and the spacetime be discrete.)

On the other hand, I do not exclude the possibility that, even in the conditions of continuous spacetime and fields, the world still may be discrete. I illustrate this with the example of vector graphics format in computer graphics. This type of format allows infinite resolution, but in the same time it is digital. Digital does not necessarily means pixelated, so discrete information still may describe a continuous spacetime.

Similarly, continuous spacetimes endowed with continuous fields may very well be describable by digital information. After all, all books on continuous mathematics and physics can be scanned into a computer.

I argued for the continuity of spacetime and fields, but I do not exclude the possibility that all the information contained in these fields can be compressed in a digital format. I wish I could write about this too in my essay, but I need to do more research in this direction.

Best regards,

Cristi

Dear Cristi,

Yes, this is a nice question! Choose your side!

My side is that reality is continuous, but it looks discrete sometimes because it has a differential structure which is scale-dependent. Then, making an experiment amounts to draw a map of the part of reality at one scale, to another part of reality, at another scale. There is a mathematical limit of the precision any such map could have.

Physicists somehow are lost in another dream. Despite many claims that physics so inspired modern mathematics, in fact since some decades this is only very limited so, while the most dynamic mathematical fields have little to do with physics, but more with computer science or (less now but more in the future) biology. Or just pure curiosity.

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Yes, this is a nice question! Choose your side!

My side is that reality is continuous, but it looks discrete sometimes because it has a differential structure which is scale-dependent. Then, making an experiment amounts to draw a map of the part of reality at one scale, to another part of reality, at another scale. There is a mathematical limit of the precision any such map could have.

Physicists somehow are lost in another dream. Despite many claims that physics so inspired modern mathematics, in fact since some decades this is only very limited so, while the most dynamic mathematical fields have little to do with physics, but more with computer science or (less now but more in the future) biology. Or just pure curiosity.

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Dear Marius,

indeed, it is hard to choose a side. I agree with you that there is a limit in the precision, this is why I think it is hard to decide :)

My personal view is that the continuum is a differentiable manifold, and many of the discrete aspects occur from its topological properties.

Best regards,

Cristi

indeed, it is hard to choose a side. I agree with you that there is a limit in the precision, this is why I think it is hard to decide :)

My personal view is that the continuum is a differentiable manifold, and many of the discrete aspects occur from its topological properties.

Best regards,

Cristi

AM I IN THE ANALOG OR DISCRETE PARTY?

Part 2

Here is a link to some older work I did, named World Theory. It is a mathematical framework (based on sheaf theory) - a general mathematical structure which speaks about any possible world consisting in space, time, matter and laws of nature. It can be particularized to obtain many of our current theories in the foundational physics. In other words, to make abstraction of the particular solution we adopt, and to say the most general things we can say about the world. The intention was to write the laws in the most general possible form, so that we can compare them, and see which principles really contradict each other and which can be reconciliated on a higher level of generality. It was not a unified theory, just a unified framework.

The mathematical structure defined there can be particularized to most of the continuous, and of the discrete theories which are currently researched. In other words, two theories about space, time and matter, one which is discrete and another which is continuous, are both particular instantiation of the mathematical structure I named there "world". The matter is, in all cases, a section in a sheaf over space and time, and this notion works with both continuous and discrete spacetimes, as the examples I gave there show.

Although it would have been appropriate for the theme, because it brings under the same umbrella discrete and continuous, in the present essay I did not pursue this idea. The reason is that I spoke about the World Theory in my FQXi essay about time, Flowing with a Frozen River. There I used World Theory to discuss time, determinism and causality.

Best regards,

Cristi

Part 2

Here is a link to some older work I did, named World Theory. It is a mathematical framework (based on sheaf theory) - a general mathematical structure which speaks about any possible world consisting in space, time, matter and laws of nature. It can be particularized to obtain many of our current theories in the foundational physics. In other words, to make abstraction of the particular solution we adopt, and to say the most general things we can say about the world. The intention was to write the laws in the most general possible form, so that we can compare them, and see which principles really contradict each other and which can be reconciliated on a higher level of generality. It was not a unified theory, just a unified framework.

The mathematical structure defined there can be particularized to most of the continuous, and of the discrete theories which are currently researched. In other words, two theories about space, time and matter, one which is discrete and another which is continuous, are both particular instantiation of the mathematical structure I named there "world". The matter is, in all cases, a section in a sheaf over space and time, and this notion works with both continuous and discrete spacetimes, as the examples I gave there show.

Although it would have been appropriate for the theme, because it brings under the same umbrella discrete and continuous, in the present essay I did not pursue this idea. The reason is that I spoke about the World Theory in my FQXi essay about time, Flowing with a Frozen River. There I used World Theory to discuss time, determinism and causality.

Best regards,

Cristi

Dear Cristi,

Thank you for clarifying your position. When I read your essay, I was under the impression that you were saying that space and fields are divisible ad infinitum, and thus continuous.

Some of your older work implies that Nature is simultaneously continuous and discrete, and that was also my essay's point.

It seems we have many similar ideas, and I need to read your older work.

Have Fun!

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Thank you for clarifying your position. When I read your essay, I was under the impression that you were saying that space and fields are divisible ad infinitum, and thus continuous.

Some of your older work implies that Nature is simultaneously continuous and discrete, and that was also my essay's point.

It seems we have many similar ideas, and I need to read your older work.

Have Fun!

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Dear Cristi,

In connection with your just stated position, can you comment on the following opinion of Schrödinger (which I quote on p.1 of my essay:

"If you envisage the development of physics in the last half-century, you get the impression that the discontinuous aspect of nature has been forced upon us very much against our will. We seemed to feel quite happy with the continuum. Max Planck was seriously frightened by the idea of a discontinuous exchange of energy ... Twenty-five years later the inventors of wave mechanics indulged for some time in the fond hope that they have paved the way of return to a classical continuous description, but again the hope was deceptive. Nature herself seemed to reject continuous description...

The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of continuous description in space and time. ... So the facts of observation are irreconcilable with a continuous description in space and time."

Did we learned anything fundamentally new which might have changed his opinion?

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In connection with your just stated position, can you comment on the following opinion of Schrödinger (which I quote on p.1 of my essay:

"If you envisage the development of physics in the last half-century, you get the impression that the discontinuous aspect of nature has been forced upon us very much against our will. We seemed to feel quite happy with the continuum. Max Planck was seriously frightened by the idea of a discontinuous exchange of energy ... Twenty-five years later the inventors of wave mechanics indulged for some time in the fond hope that they have paved the way of return to a classical continuous description, but again the hope was deceptive. Nature herself seemed to reject continuous description...

The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of continuous description in space and time. ... So the facts of observation are irreconcilable with a continuous description in space and time."

Did we learned anything fundamentally new which might have changed his opinion?

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Dear Ray,

thank you for the feedback. You understood well what I said in my essay: my solution to the singularities in general relativity requires spacetime and fields to be divisible ad infinitum.

In World Theory I define a mathematical structure named "world", which describe possible matter fields over possible spacetimes, subject to possible laws, "possible" in the mathematical sense. Known theories in physics, discrete or continuous, are find to be particular cases of this structure, in the same way as the definition of group applies to both discrete and continuous groups. But World Theory does not make any implications about the real world, it is just a metatheoretical framework.

But there is enough room for discrete structures in continuous theories, and I will return to this subject soon.

Best regards,

Cristi

thank you for the feedback. You understood well what I said in my essay: my solution to the singularities in general relativity requires spacetime and fields to be divisible ad infinitum.

In World Theory I define a mathematical structure named "world", which describe possible matter fields over possible spacetimes, subject to possible laws, "possible" in the mathematical sense. Known theories in physics, discrete or continuous, are find to be particular cases of this structure, in the same way as the definition of group applies to both discrete and continuous groups. But World Theory does not make any implications about the real world, it is just a metatheoretical framework.

But there is enough room for discrete structures in continuous theories, and I will return to this subject soon.

Best regards,

Cristi

AM I IN THE CONTINUOUS OR DISCRETE PARTY?

Does it matter?

My personal opinion is not that important. I tried to prove something, and only the arguments should be important. My personal opinion can be a complementary information, which helps to put the things in a larger picture, but what matters is what I can support with arguments, what I can prove.

In this line, I would like to say that I am glad to see at this contest such a wide diversity of opinions. The Nature is one, indeed, but we are far from knowing how she really is. So we try to guess the principles guiding her. So far, they are incomplete, and although they complement one another, they are in contradiction. I have no intention to take a side and claim that this is the truth, because I don't know the truth. I am just happy to see each effort, and each progress made by us in various directions. Am I proposing a continuous theory? Sure, but this shouldn't blind me against the discrete approaches. So, if you see me liking your discrete explanation, you should not conclude that I should not like a continuous explanation as well. Important is to progress, to add a new viewpoint, to solve a problem. Hence, we should encourage all good ideas which solve, or have the potential to solve a problem, to enrich our understanding, to widen our vision.

Good luck to all in this contest

Cristi

Does it matter?

My personal opinion is not that important. I tried to prove something, and only the arguments should be important. My personal opinion can be a complementary information, which helps to put the things in a larger picture, but what matters is what I can support with arguments, what I can prove.

In this line, I would like to say that I am glad to see at this contest such a wide diversity of opinions. The Nature is one, indeed, but we are far from knowing how she really is. So we try to guess the principles guiding her. So far, they are incomplete, and although they complement one another, they are in contradiction. I have no intention to take a side and claim that this is the truth, because I don't know the truth. I am just happy to see each effort, and each progress made by us in various directions. Am I proposing a continuous theory? Sure, but this shouldn't blind me against the discrete approaches. So, if you see me liking your discrete explanation, you should not conclude that I should not like a continuous explanation as well. Important is to progress, to add a new viewpoint, to solve a problem. Hence, we should encourage all good ideas which solve, or have the potential to solve a problem, to enrich our understanding, to widen our vision.

Good luck to all in this contest

Cristi

Dear Cristi,

Thanks for the good wishes! My best wishes to you too.

Just one little note. You know, of course, that at least 99.99% of researchers prefer to work in the old kitchen: it is so much more comfortable, but ... our scientific intuition, as was Schrödinger's, should not be biased by such comfort. ;-)

Besides, all the comfort may turn out to be illusory, and so in the long run, the research life might turn out to be wasted (which wouldn't be so comfortable after all). ;-) The latter possibility has always been my greatest fear.

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Thanks for the good wishes! My best wishes to you too.

Just one little note. You know, of course, that at least 99.99% of researchers prefer to work in the old kitchen: it is so much more comfortable, but ... our scientific intuition, as was Schrödinger's, should not be biased by such comfort. ;-)

Besides, all the comfort may turn out to be illusory, and so in the long run, the research life might turn out to be wasted (which wouldn't be so comfortable after all). ;-) The latter possibility has always been my greatest fear.

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Dear Lev,

I invite you to visit my "kitchen" and taste my recipe. Please don't judge me only for using honey instead of sugar for this recipe; taste instead the cookie. Aren't the 99.99% you mention said that this meal cannot be cooked? How many of them do you see in my kitchen, cooking the same recipe?

I invite you to visit my "kitchen" and taste my recipe. Please don't judge me only for using honey instead of sugar for this recipe; taste instead the cookie. Aren't the 99.99% you mention said that this meal cannot be cooked? How many of them do you see in my kitchen, cooking the same recipe?

Dear Dr. Stoica,

I enjoyed reading your essay before you lost me in mathematical technicalities. Nevertheless, before you launch into your thesis, you ask the following 3 questions, which I hope to answer:

1) Is reality discrete or continuous? At least ‘matter’ and ‘radiation’ are discrete.

2) (Was) it possible to find the answer by experiment? With respect to ‘matter,’ no, because we did not know what to look for.

3) Or at least from theoretical arguments? Yes.

You also point out that the reason we cannot decide between the continuous or discrete is that: “(T)he theories we know so far don't seem to make use in an essential, irreducible way of the discrete or continuous nature of the reality they propose.” And further: “In addition, this theory should be mathematically and logically consistent, very well corroborated by experiments, and as simple as possible.”

You then proceed to make your case for your ‘version of General Relativity.’

While I am unable to mathematically argue the merits of your thesis, in terms of foundations it would be pointless to do so, as I show in my essay. Furthermore, the derived foundations in my essay are “mathematically and logically consistent, very well corroborated by experiments, and as simple as possible.”

I merely wish to bring this to your attention, for unless I am fooling myself the derived foundations leave no room for debate.

Kind regards,

Robert

P.S – If I am fooling myself, could you please leave me a post to let me know.

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I enjoyed reading your essay before you lost me in mathematical technicalities. Nevertheless, before you launch into your thesis, you ask the following 3 questions, which I hope to answer:

1) Is reality discrete or continuous? At least ‘matter’ and ‘radiation’ are discrete.

2) (Was) it possible to find the answer by experiment? With respect to ‘matter,’ no, because we did not know what to look for.

3) Or at least from theoretical arguments? Yes.

You also point out that the reason we cannot decide between the continuous or discrete is that: “(T)he theories we know so far don't seem to make use in an essential, irreducible way of the discrete or continuous nature of the reality they propose.” And further: “In addition, this theory should be mathematically and logically consistent, very well corroborated by experiments, and as simple as possible.”

You then proceed to make your case for your ‘version of General Relativity.’

While I am unable to mathematically argue the merits of your thesis, in terms of foundations it would be pointless to do so, as I show in my essay. Furthermore, the derived foundations in my essay are “mathematically and logically consistent, very well corroborated by experiments, and as simple as possible.”

I merely wish to bring this to your attention, for unless I am fooling myself the derived foundations leave no room for debate.

Kind regards,

Robert

P.S – If I am fooling myself, could you please leave me a post to let me know.

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Cristi: I'm even happier that you like my essay after reading yours. I appreciated your resolution of the singularity and information paradox -- I found myself wondering why nobody had thought of your degenerate metric idea. I want to look further into your smooth quantum mechanics by way of your paper. Anyway, I am glad to find new and innovative arguments for (fundamentally) continuous nature that supplements the work by Zeh et al., so thank you for that, and good luck!

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Dear Karl,

Thank you for your appreciation. Yes, when I initially considered to apply degenerate metrics to singularities I though it will be simple. It turned out that it was not that simple, because of the divergences and other problems which occur by the normal methods. The hardest part was to find a way to avoid them.

Good luck to you too,

Cristi

Thank you for your appreciation. Yes, when I initially considered to apply degenerate metrics to singularities I though it will be simple. It turned out that it was not that simple, because of the divergences and other problems which occur by the normal methods. The hardest part was to find a way to avoid them.

Good luck to you too,

Cristi

"AM I IN THE CONTINUOUS OR DISCRETE PARTY?"

There is obviously both the continuous and the discrete in reality. The empirical evidence for this is obviously that we are all able to see bits and pieces of reality somehow and some say they can divide the bits and pieces to infinitely ever-increasing entropy.

It does not really take peculiar minds to see the obvious. But understanding the obvious takes peculiar minds, because pointing out the obvious for peculiar minds to understand is quite difficult.

The difficulty comes because of the unclear picture of the foundational ideas. Once the foundational ideas are clarified, eveyone could readily adapt their interpretations and most everyone will see that we have been describing the same thing, albeit unclearly.

The answer to "what of reality is continuous and what is discrete" can readily be clarified if we take the analysis down to the fundamentals.

Perhaps you will agree with me regarding what is continuous and what is discrete in reality if you read my essay... http://www.fqxi.org/community/forum/topic/835

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There is obviously both the continuous and the discrete in reality. The empirical evidence for this is obviously that we are all able to see bits and pieces of reality somehow and some say they can divide the bits and pieces to infinitely ever-increasing entropy.

It does not really take peculiar minds to see the obvious. But understanding the obvious takes peculiar minds, because pointing out the obvious for peculiar minds to understand is quite difficult.

The difficulty comes because of the unclear picture of the foundational ideas. Once the foundational ideas are clarified, eveyone could readily adapt their interpretations and most everyone will see that we have been describing the same thing, albeit unclearly.

The answer to "what of reality is continuous and what is discrete" can readily be clarified if we take the analysis down to the fundamentals.

Perhaps you will agree with me regarding what is continuous and what is discrete in reality if you read my essay... http://www.fqxi.org/community/forum/topic/835

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Dear Castel,

I think that most of us would agree with you that "There is obviously both the continuous and the discrete in reality". Probably the divergences occur when discussing what of the two features is fundamental, and what is derived. If "some say they can divide the bits and pieces to infinitely ever-increasing entropy", my essay doesn't support their claim.

I think that most of us would agree with you that "There is obviously both the continuous and the discrete in reality". Probably the divergences occur when discussing what of the two features is fundamental, and what is derived. If "some say they can divide the bits and pieces to infinitely ever-increasing entropy", my essay doesn't support their claim.

Dear Christi,

I don't exactly know how to put this. But what makes the "degenerate metrics" degenerate - what is the cause of the process? Does the degeneration bring about "particles" or "voids"?

Just a thought...

Castel

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I don't exactly know how to put this. But what makes the "degenerate metrics" degenerate - what is the cause of the process? Does the degeneration bring about "particles" or "voids"?

Just a thought...

Castel

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Dear Castel,

the metric defines the distance, and it can be represented (by choosing a coordinate chart) as a matrix whose entries depend on the position in space and on time. As a matrix, it has a determinant, which also depends on the position in space and time. This matrix determines (curvature, who determines) gravity, and it is in turn determined by the fields of matter (particles). There is no law able to prevent its determinant to become 0. It can become 0 because it changes, and the change is governed by Einstein's equations, or if you prefer to emphasize the time evolution, by the ADM equations. The particles have their role, because matter determines (curvature, who determines) metric, and if matter is dense enough, it evolves in a black hole - leading to singularities. That's pure General Relativity.

Cristi

the metric defines the distance, and it can be represented (by choosing a coordinate chart) as a matrix whose entries depend on the position in space and on time. As a matrix, it has a determinant, which also depends on the position in space and time. This matrix determines (curvature, who determines) gravity, and it is in turn determined by the fields of matter (particles). There is no law able to prevent its determinant to become 0. It can become 0 because it changes, and the change is governed by Einstein's equations, or if you prefer to emphasize the time evolution, by the ADM equations. The particles have their role, because matter determines (curvature, who determines) metric, and if matter is dense enough, it evolves in a black hole - leading to singularities. That's pure General Relativity.

Cristi

Dear Christi,

Regarding "change," - you say "change is governed by Einstein's equations or, if emphasis is on time evolution, by the ADM equations."

By the phrase "prefer emphasis on time evolution," there seems to be the clear implication that there are other possible preferences that can be emphasized on. The idea looks like it is in accordance with Einstein's "arbitratry...

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Regarding "change," - you say "change is governed by Einstein's equations or, if emphasis is on time evolution, by the ADM equations."

By the phrase "prefer emphasis on time evolution," there seems to be the clear implication that there are other possible preferences that can be emphasized on. The idea looks like it is in accordance with Einstein's "arbitratry...

view entire post

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Dear Castel,

If it was unclear for you what I said when discussing Einstein's equation vs. the ADM formalism, and if you are genuinely interested in the subject, please refer to one of the many GR books presenting it, for example the classic "Gravitation" by Misner, Thorne and Wheeler, or "Gauge Fields, Knots and Gravity" by John Baez and Javier Muniain. For many, the book "Gravitation" is an "authority", but please do not take these readings this way. I give you these references to read them critically, to check their calculations, their arguments relating the theory with experiment, and not to blindly accept them. Many accept them blindly, and many reject them blindly just because they think that if they don't understand the math, it is because the math has no "good sense in the interpretation". I would suggest to those an easier reading, such as La Fontaine's "Le Renard et les Raisins".

Cristi

If it was unclear for you what I said when discussing Einstein's equation vs. the ADM formalism, and if you are genuinely interested in the subject, please refer to one of the many GR books presenting it, for example the classic "Gravitation" by Misner, Thorne and Wheeler, or "Gauge Fields, Knots and Gravity" by John Baez and Javier Muniain. For many, the book "Gravitation" is an "authority", but please do not take these readings this way. I give you these references to read them critically, to check their calculations, their arguments relating the theory with experiment, and not to blindly accept them. Many accept them blindly, and many reject them blindly just because they think that if they don't understand the math, it is because the math has no "good sense in the interpretation". I would suggest to those an easier reading, such as La Fontaine's "Le Renard et les Raisins".

Cristi

Dear Christi,

You haven't answered my questions. Instead you have given me advice. Nevertheless, thank you for the advice. :)

Castel

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You haven't answered my questions. Instead you have given me advice. Nevertheless, thank you for the advice. :)

Castel

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Dear Castel,

Presuming that you are genuinely interested in answers: why don't you ask one clear question, and then let me answer, and then continue with another clear question and so on?

Cristi

Presuming that you are genuinely interested in answers: why don't you ask one clear question, and then let me answer, and then continue with another clear question and so on?

Cristi

Dear Cristi,

You say: 'I think that most of us would agree with you that "There is obviously both the continuous and the discrete in reality".'

Even though it is a popular view, nature cannot be both discrete and continuous, simply because "discrete" means non-continuous.

Regarding semi-metrics, I started my research work with them for somewhat similar reasons you went after them. But from the beginning I went after semi-metrics because one wants to allow some *structural* data to be at a zero distance from each other while still being distinct. However, under the conventional, "point", representation, the concept of semi-metric does look odd, mainly because two *points* at zero distance cannot be distinct. So don't be surprised, Cristi, that it might be questioned. ;-)

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You say: 'I think that most of us would agree with you that "There is obviously both the continuous and the discrete in reality".'

Even though it is a popular view, nature cannot be both discrete and continuous, simply because "discrete" means non-continuous.

Regarding semi-metrics, I started my research work with them for somewhat similar reasons you went after them. But from the beginning I went after semi-metrics because one wants to allow some *structural* data to be at a zero distance from each other while still being distinct. However, under the conventional, "point", representation, the concept of semi-metric does look odd, mainly because two *points* at zero distance cannot be distinct. So don't be surprised, Cristi, that it might be questioned. ;-)

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Dear Lev,

I said indeed 'I think that most of us would agree with you that "There is obviously both the continuous and the discrete in reality".'

But I continued with "Probably the divergences occur when discussing what of the two features is fundamental, and what is derived."

So the contradiction you are trying to show in what I said does not hold, and I will give you a few well known examples.

1. A string on a guitar can oscillate, and the way in which it can do this is discrete. This has nothing to do with the fact that the string is made of atoms.

2. Quantum Mechanics. It describes the states as vectors in a Hilbert space (here comes the continuum). When we try to measure a certain observable, the result is an element of the spectrum of the operator describing that observable. The spectrum may very well have discrete parts. For example, the states in which an electron can be in an atom are labeled bu the discrete part of the spectrum of the energy. When it is outside the atom, the spectrum is continuous. Hence, at least in the standard version of Quantum Mechanics, the continuum is discrete.

3. A topological manifold - that is, a space with n dimensions, is continuous. But many of its topological properties are discrete. For example, the possible closed surfaces are classified by discrete numbers. The topology of a 4-dimensional manifold (like spacetime) has more and complex discrete properties. These are used in the attempts to describe particles as instantons or as topological charges.

It seems to me that these well known examples show clearly how genuine discrete properties can arise from fundamentally continuous structures.

There is nothing wrong in asking questions ;-)

I said indeed 'I think that most of us would agree with you that "There is obviously both the continuous and the discrete in reality".'

But I continued with "Probably the divergences occur when discussing what of the two features is fundamental, and what is derived."

So the contradiction you are trying to show in what I said does not hold, and I will give you a few well known examples.

1. A string on a guitar can oscillate, and the way in which it can do this is discrete. This has nothing to do with the fact that the string is made of atoms.

2. Quantum Mechanics. It describes the states as vectors in a Hilbert space (here comes the continuum). When we try to measure a certain observable, the result is an element of the spectrum of the operator describing that observable. The spectrum may very well have discrete parts. For example, the states in which an electron can be in an atom are labeled bu the discrete part of the spectrum of the energy. When it is outside the atom, the spectrum is continuous. Hence, at least in the standard version of Quantum Mechanics, the continuum is discrete.

3. A topological manifold - that is, a space with n dimensions, is continuous. But many of its topological properties are discrete. For example, the possible closed surfaces are classified by discrete numbers. The topology of a 4-dimensional manifold (like spacetime) has more and complex discrete properties. These are used in the attempts to describe particles as instantons or as topological charges.

It seems to me that these well known examples show clearly how genuine discrete properties can arise from fundamentally continuous structures.

There is nothing wrong in asking questions ;-)

Dear Lev, part 2,

"the concept of semi-metric does look odd, mainly because two *points* at zero distance cannot be distinct. So don't be surprised, Cristi, that it might be questioned. ;-)"

Of course it is counterintuitive. So I will give you two examples, one of them being also in my essay.

1. In special relativity, let's consider that Alice sends a photon (event A). Let's say that the photon travels undisturbed and undeviated, and one second later Bob receives it (event B). Can you tell the 4-distance (as given my the metric tensor) between the two events, A and B? What if Bob receives the photon one year later?

This 4-distance coincides with the proper time of the photon, and you may take a guess that it is 0. But A and B are distinct. How can the time between A and B be 0 for the photon, and yet A and B be distinct?

2. In Newtonian mechanics, let's say that a body is at the position (0,0,0) in whatever units you want, when the time is 0 (event A). Assume that the body doesn't move, so at the time 1 it is in the same place (event B). If we represent the Newtonian spacetime as space x time, the two events are (0,0,0,0) and (0,0,0,1), so they are distinct. What is the distance between them?

You see, in the first example the metric is a matrix having on its diagonal (1,1,1,-1), the other entries being 0. This allows to have the hypotenuse = 0, even if the catheti are not 0. In the second example, the metric has on its diagonal (1,1,1,0), other entries being 0. This is degenerate. Using it to calculate the distance between A and B gives 0.

"the concept of semi-metric does look odd, mainly because two *points* at zero distance cannot be distinct. So don't be surprised, Cristi, that it might be questioned. ;-)"

Of course it is counterintuitive. So I will give you two examples, one of them being also in my essay.

1. In special relativity, let's consider that Alice sends a photon (event A). Let's say that the photon travels undisturbed and undeviated, and one second later Bob receives it (event B). Can you tell the 4-distance (as given my the metric tensor) between the two events, A and B? What if Bob receives the photon one year later?

This 4-distance coincides with the proper time of the photon, and you may take a guess that it is 0. But A and B are distinct. How can the time between A and B be 0 for the photon, and yet A and B be distinct?

2. In Newtonian mechanics, let's say that a body is at the position (0,0,0) in whatever units you want, when the time is 0 (event A). Assume that the body doesn't move, so at the time 1 it is in the same place (event B). If we represent the Newtonian spacetime as space x time, the two events are (0,0,0,0) and (0,0,0,1), so they are distinct. What is the distance between them?

You see, in the first example the metric is a matrix having on its diagonal (1,1,1,-1), the other entries being 0. This allows to have the hypotenuse = 0, even if the catheti are not 0. In the second example, the metric has on its diagonal (1,1,1,0), other entries being 0. This is degenerate. Using it to calculate the distance between A and B gives 0.

Cristi,

Part I

1. I didn't get your point here, since both of these are discrete.

2. Hilbert space is not part of Nature.

3. Manifolds, including surfaces, are also not part of Nature. ;-)

Part 2

My point here was that a more transparent emergence of semi-metric can be seen IF we postulate a structural, rather than a point, form of representation: zero distance means that in a particular setting, or field, they are indistinguishable though different.

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Part I

1. I didn't get your point here, since both of these are discrete.

2. Hilbert space is not part of Nature.

3. Manifolds, including surfaces, are also not part of Nature. ;-)

Part 2

My point here was that a more transparent emergence of semi-metric can be seen IF we postulate a structural, rather than a point, form of representation: zero distance means that in a particular setting, or field, they are indistinguishable though different.

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If as I believe my thoery is true and it will be accepted by one compentant judge it will be a considerable step in science.Charles Darwin............No that I say Darwinism is right just that I think this quote applies equally to a thoery of ex-nihilo creation.

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If as I believe my thoery is true and it will be accepted by one compentant judge it will be a considerable step in science.Charles Darwin............No that I say Darwinism is right just that I think this quote applies equally to a thoery of ex-nihilo creation.

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If as I believe my thoery is true and it will be accepted by one compentant judge it will be a considerable step in science.Charles Darwin............No that I say Darwinism is right just that I think this quote applies equally to a thoery of ex-nihilo creation.

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If as I believe my thoery is true and it will be accepted by one compentant judge it will be a considerable step in science.Charles Darwin............No that I say Darwinism is right just that I think this quote applies equally to a thoery of ex-nihilo creation.

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Well, i understand little Physics and hardly Mathematics. What i like is Philosophy from where both these disciplines emerged. We have made these disciplines so complex by now that hardly we get out of our ' specializations '. Only a broad enough mind can comprehend the Huge Huge mind of the Creator. The latter is not human. It is 'consciousness ultimate' which had some logic for creation. Mathematically, i may say that it is Infinite Potential Field with Ultimate Intelligence imaginable/ unimaginable!

Your text of the Essay is beyond me and thus my comments are vague and inadequate, my dear Christi

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Your text of the Essay is beyond me and thus my comments are vague and inadequate, my dear Christi

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Dear Narendra,

thank you for going through my essay and provide feedback - I am happy for this.

I tried to split the essay in two parts: a story, and the proof for that story.

The "Story":

I tried to explain the main ideas in the Prologue and the first section, using as simple language as I could. These parts were addressed to the intuition of the reader. My hope was that the reader with less background in the mathematics of General Relativity will understand at least the story.

The Proof:

Starting with the second section I tried to provide the mathematical backup of the story I told in the first pages, for the reader who wants to understand the mathematics behind the story. Nevertheless, I tried to keep the mathematical explanation as much as possible at a conceptual level, and I postponed the equations to the endnotes and to the references.

I appreciate very much Philosophy, and I admire you for this passion. My "story" part contains just the explanation of my ideas, and I cannot expect the reader to consider it philosophy. Philosophy is a well-constituted discipline, and not every story has the qualities which makes it good philosophy. Mine is just an explanation.

Even if the reader resumes to the story and skips the mathematics, there are in the story some conceptual leaps. From the feedback I got, the most counterintuitive is the possibility to have the distance 0 between distinct points of the space. Unfortunately, I could not find support for it in Philosophy. In Mathematics instead, such a distance is a banal fact. Unfortunately, although the starting idea is simple, to provide the mathematical backup for it I had to write 120 pages (they can be found through the References section) - contributing even more to making "these disciplines so complex by now that hardly we get out of our ' specializations '" :-). I hoped that providing a logical and mathematical support for the story I told did not hurt it.

Sorry for giving such a long answer. The "story" part of my present essay has much to do with the feedback you and other readers gave me for a previous essay. You suggested me that I have to explain more, and to present with more patience the concepts. I tried to apply this advice at the form of the present essay.

Best regards,

Cristi

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thank you for going through my essay and provide feedback - I am happy for this.

I tried to split the essay in two parts: a story, and the proof for that story.

The "Story":

I tried to explain the main ideas in the Prologue and the first section, using as simple language as I could. These parts were addressed to the intuition of the reader. My hope was that the reader with less background in the mathematics of General Relativity will understand at least the story.

The Proof:

Starting with the second section I tried to provide the mathematical backup of the story I told in the first pages, for the reader who wants to understand the mathematics behind the story. Nevertheless, I tried to keep the mathematical explanation as much as possible at a conceptual level, and I postponed the equations to the endnotes and to the references.

I appreciate very much Philosophy, and I admire you for this passion. My "story" part contains just the explanation of my ideas, and I cannot expect the reader to consider it philosophy. Philosophy is a well-constituted discipline, and not every story has the qualities which makes it good philosophy. Mine is just an explanation.

Even if the reader resumes to the story and skips the mathematics, there are in the story some conceptual leaps. From the feedback I got, the most counterintuitive is the possibility to have the distance 0 between distinct points of the space. Unfortunately, I could not find support for it in Philosophy. In Mathematics instead, such a distance is a banal fact. Unfortunately, although the starting idea is simple, to provide the mathematical backup for it I had to write 120 pages (they can be found through the References section) - contributing even more to making "these disciplines so complex by now that hardly we get out of our ' specializations '" :-). I hoped that providing a logical and mathematical support for the story I told did not hurt it.

Sorry for giving such a long answer. The "story" part of my present essay has much to do with the feedback you and other readers gave me for a previous essay. You suggested me that I have to explain more, and to present with more patience the concepts. I tried to apply this advice at the form of the present essay.

Best regards,

Cristi

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Dear Cristi,

You very often used the notion singularity, and you wrote:"the most counterintuitive is the possibility to have the distance 0 between distinct points of the space".

I am suggesting to reinstate old notions more precisely .

Regards, Eckard

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You very often used the notion singularity, and you wrote:"the most counterintuitive is the possibility to have the distance 0 between distinct points of the space".

I am suggesting to reinstate old notions more precisely .

Regards, Eckard

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Dear Eckard,

thank you for reading my previous comment. If you are interested, you can read how the distance between distinct points can be 0 in my essay. The distance is given by the metric, and the definition of a degenerate metric is well-known for long time (although spacetimes whose metric can become degenerate were too little studied because the standard methods don't work well in this situation), and it is very precise. For your convenience, I quote from the essay two places where the definition is implicit (but precise): "the metric has an inverse - i.e. it is non-degenerate", and "the metric becomes degenerate - its determinant becomes 0". To read more, please refer to the references.

If I misunderstood your suggestion, I would appreciate if you will restate it more precisely.

Regards,

Cristi

thank you for reading my previous comment. If you are interested, you can read how the distance between distinct points can be 0 in my essay. The distance is given by the metric, and the definition of a degenerate metric is well-known for long time (although spacetimes whose metric can become degenerate were too little studied because the standard methods don't work well in this situation), and it is very precise. For your convenience, I quote from the essay two places where the definition is implicit (but precise): "the metric has an inverse - i.e. it is non-degenerate", and "the metric becomes degenerate - its determinant becomes 0". To read more, please refer to the references.

If I misunderstood your suggestion, I would appreciate if you will restate it more precisely.

Regards,

Cristi

"EXPLANATION" BETWEEN CONCRETE AND ABSTRACT

I realized that an apparently well-understood word, "explanation", may lead to controversies in discussions about the foundations of physics. The foundations are already controversial enough, but this adds even more to the confusion. It gives you a double featured feeling: on the one hand, of being misunderstood, and on the other hand, that you don't understand where the interlocutor is going on.

What is an "explanation"? Probably the most usual meaning is that explanation is to reduce the unknown to the known, the unfamiliar to the familiar. When this happens, we get the sense of understanding.

Even since childhood, we had so many questions, and the grown ups explained them - reduced the unfamiliar to more familiar notions. In school, the teachers continued to provide us explanations, and we appreciated most the teachers who managed to make the unclear things more intuitive for us. When reading about the foundations of physics, we usually start with popular physics books. The most recommended such books are those providing the feeling of understanding, appealing to our intuition. When we try to read something more advanced, even if it is recommended by our favorite pop-sci books, we find ourselves in a totally different situation. Instead of finding the deeper explanations we are looking for, we find ourselves thrown in the turbulent torrents of the abstract mathematics, drifting without an apparent purpose. And what is most annoying, these textbooks and articles full of equations actually claim to explain things!

Why is this happening? I think that they are guided by another meaning of the term "explanation": "to give an explanation to a phenomenon is to deduce the existence of that phenomenon from hypotheses considered more fundamental. For example, when from the principles of General Relativity was deduced the correct value from the perihelion precession of Mercury, it was considered that GR explained this precession. On the other hand, the deflection of light by the Sun was considered a prediction. After the full experimental confirmation, it became an explanation. I consider that "prediction" is just a temporary status of a scientific explanation, and that the fact that many explanations are first predictions is a historical accident.

There seem to be a similarity between principles/phenomena and axioms/theorems. This similarity suggests the reason why mathematics plays such an important role in the explanation of phenomena. To deduce more from less, complicated from simple, diverse from universal, this means to use logic and mathematics. And there is no limit of the difficulty of the needed mathematics, even if the principles are not that difficult.

This notion of explanation, I understand now, it is not shared by all of us. The reason is simple: because "explanation" usually means to reduce the unfamiliar to familiar. When somebody claimed to explain a phenomenon, we expect him to show how this strange phenomenon can be described in more familiar, concrete terms. Instead, we find that he or she starts describing it in more abstract terms. How come that such more and more abstract terms are shamelessly named "more basic principles", "more elementary principles" and so on? Isn't this a lie?

Maybe the explanation by "reducing to concrete things" has pedagogical reasons, and the explanation by "reducing to universal principles" is in fact foundational research. But does this means that the gap between pedagogical and scientific explanation should grow as it does nowadays? Wouldn't be much, much better to have a mechanistic explanation? After all, Maxwell sought for such an explanation of the electromagnetic waves, even though he had the equations! The ether theorists of the XIXth century tried to reduce electromagnetism to vibrations in a medium. This tradition still continues, and we encounter on a daily basis renowned scientists trying to explain things which other renowned scientists consider to be already explained: electromagnetism, wave-particle duality, gravity, entropy, the Unruh effect, spacetime, time, black holes and so on.

Probably it would be better to have a mechanistic explanation of everything. This would definitely help the public outreach of physics, and will help physics to advance faster. This may have a huge impact on technology, and on our lives. But who can bet that God, when created the world, bothered about our need to reduce the things to what we know? Why would the universe care about our limited understanding, when decided what principles to follow? Who are we, why would we be so important? I think that, although it would be desirable to find concrete, familiar universal principles behind this complex and diverse world, we have no guarantee that this will ever happen. "You shall not make for yourself a carved image, or any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth."

The definition of "explanation" as a reduction to universal principles has its own advantages, given that we do not take these principles as ultimate truths, but just as hypotheses. One of these advantages is that it allows us to equally appreciate theories which seem to contradict each other. We can appreciate its explanatory power in the sense stated above: as its efficient encapsulation of a wide variety of phenomena in fewer, simpler, and more general principles. This doesn't mean that we should consider these principles as being "true". It is not about being "true", just about encapsulating as much phenomena as possible in as few principles as possible, even if these principles are more abstract. If we insist to become fans of one theory or another as the ultimate "truth", we may reduce our capacity to grasp other explanations. This would not be a problem, if we could prove our theories beyond any doubt, but the truth is that we cannot, no matter how convincing they may look to us.

Cristi

I realized that an apparently well-understood word, "explanation", may lead to controversies in discussions about the foundations of physics. The foundations are already controversial enough, but this adds even more to the confusion. It gives you a double featured feeling: on the one hand, of being misunderstood, and on the other hand, that you don't understand where the interlocutor is going on.

What is an "explanation"? Probably the most usual meaning is that explanation is to reduce the unknown to the known, the unfamiliar to the familiar. When this happens, we get the sense of understanding.

Even since childhood, we had so many questions, and the grown ups explained them - reduced the unfamiliar to more familiar notions. In school, the teachers continued to provide us explanations, and we appreciated most the teachers who managed to make the unclear things more intuitive for us. When reading about the foundations of physics, we usually start with popular physics books. The most recommended such books are those providing the feeling of understanding, appealing to our intuition. When we try to read something more advanced, even if it is recommended by our favorite pop-sci books, we find ourselves in a totally different situation. Instead of finding the deeper explanations we are looking for, we find ourselves thrown in the turbulent torrents of the abstract mathematics, drifting without an apparent purpose. And what is most annoying, these textbooks and articles full of equations actually claim to explain things!

Why is this happening? I think that they are guided by another meaning of the term "explanation": "to give an explanation to a phenomenon is to deduce the existence of that phenomenon from hypotheses considered more fundamental. For example, when from the principles of General Relativity was deduced the correct value from the perihelion precession of Mercury, it was considered that GR explained this precession. On the other hand, the deflection of light by the Sun was considered a prediction. After the full experimental confirmation, it became an explanation. I consider that "prediction" is just a temporary status of a scientific explanation, and that the fact that many explanations are first predictions is a historical accident.

There seem to be a similarity between principles/phenomena and axioms/theorems. This similarity suggests the reason why mathematics plays such an important role in the explanation of phenomena. To deduce more from less, complicated from simple, diverse from universal, this means to use logic and mathematics. And there is no limit of the difficulty of the needed mathematics, even if the principles are not that difficult.

This notion of explanation, I understand now, it is not shared by all of us. The reason is simple: because "explanation" usually means to reduce the unfamiliar to familiar. When somebody claimed to explain a phenomenon, we expect him to show how this strange phenomenon can be described in more familiar, concrete terms. Instead, we find that he or she starts describing it in more abstract terms. How come that such more and more abstract terms are shamelessly named "more basic principles", "more elementary principles" and so on? Isn't this a lie?

Maybe the explanation by "reducing to concrete things" has pedagogical reasons, and the explanation by "reducing to universal principles" is in fact foundational research. But does this means that the gap between pedagogical and scientific explanation should grow as it does nowadays? Wouldn't be much, much better to have a mechanistic explanation? After all, Maxwell sought for such an explanation of the electromagnetic waves, even though he had the equations! The ether theorists of the XIXth century tried to reduce electromagnetism to vibrations in a medium. This tradition still continues, and we encounter on a daily basis renowned scientists trying to explain things which other renowned scientists consider to be already explained: electromagnetism, wave-particle duality, gravity, entropy, the Unruh effect, spacetime, time, black holes and so on.

Probably it would be better to have a mechanistic explanation of everything. This would definitely help the public outreach of physics, and will help physics to advance faster. This may have a huge impact on technology, and on our lives. But who can bet that God, when created the world, bothered about our need to reduce the things to what we know? Why would the universe care about our limited understanding, when decided what principles to follow? Who are we, why would we be so important? I think that, although it would be desirable to find concrete, familiar universal principles behind this complex and diverse world, we have no guarantee that this will ever happen. "You shall not make for yourself a carved image, or any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth."

The definition of "explanation" as a reduction to universal principles has its own advantages, given that we do not take these principles as ultimate truths, but just as hypotheses. One of these advantages is that it allows us to equally appreciate theories which seem to contradict each other. We can appreciate its explanatory power in the sense stated above: as its efficient encapsulation of a wide variety of phenomena in fewer, simpler, and more general principles. This doesn't mean that we should consider these principles as being "true". It is not about being "true", just about encapsulating as much phenomena as possible in as few principles as possible, even if these principles are more abstract. If we insist to become fans of one theory or another as the ultimate "truth", we may reduce our capacity to grasp other explanations. This would not be a problem, if we could prove our theories beyond any doubt, but the truth is that we cannot, no matter how convincing they may look to us.

Cristi

Dear Cristi,

Instead of verbose explanations I am soliciting comments. You wrote: "who can bet that God, when created the world, ...". Does your belief matter in physics? I see it a hypothesis that cannot be confirmed and also at best rendered harmless and unlikely.

May I ask you to comment on my argument that analog computers did mimic integration rather than differentiation because real processes are obviously ongoing superpositions of influences? Do you have a counterargument?

Regards,

Eckard

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Instead of verbose explanations I am soliciting comments. You wrote: "who can bet that God, when created the world, ...". Does your belief matter in physics? I see it a hypothesis that cannot be confirmed and also at best rendered harmless and unlikely.

May I ask you to comment on my argument that analog computers did mimic integration rather than differentiation because real processes are obviously ongoing superpositions of influences? Do you have a counterargument?

Regards,

Eckard

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maybe it was a metaphor :))

Dear Eckard,

I'll develop my brief answer.

> "Does your belief matter in physics?"

I try to be objective. I need to try hard, because I am just a human being, and human beings tend to be biased. The effort to be objective is present in my essay: I wrote what the starting hypothesis is (general relativity) and I developed the consequences by mathematical proof. Please note that I considered general relativity just a hypothesis, although a hypothesis very well-corroborated by experiment. I tried not to appeal to authority. You half-quoted a phrase where I used the word "God". I learned from the examples of Einstein, Penrose, and Hawking, who used the term "God" as a metaphor for the physical law. I did not gave it other meaning in that comment.

> "May I ask you to comment on my argument ..."

Sure, but I don't have any comments.

Cristi

I'll develop my brief answer.

> "Does your belief matter in physics?"

I try to be objective. I need to try hard, because I am just a human being, and human beings tend to be biased. The effort to be objective is present in my essay: I wrote what the starting hypothesis is (general relativity) and I developed the consequences by mathematical proof. Please note that I considered general relativity just a hypothesis, although a hypothesis very well-corroborated by experiment. I tried not to appeal to authority. You half-quoted a phrase where I used the word "God". I learned from the examples of Einstein, Penrose, and Hawking, who used the term "God" as a metaphor for the physical law. I did not gave it other meaning in that comment.

> "May I ask you to comment on my argument ..."

Sure, but I don't have any comments.

Cristi

Pi taken to infinity approximpation.

22/7*4/3.99999= 3.142865

Pi*3.99999/4=3.1415848

So you can add INFINITY 2+2=4.

And you can add meanigless infinities to get meaningful figures..

Einsteins other equation for mass apporaching the speed of light not E=MC^2.

Is an equation for momentum.And when we know that momentum is determined by state then four states in one produces a new kind of E=MC^2 that is right for the big bang.

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22/7*4/3.99999= 3.142865

Pi*3.99999/4=3.1415848

So you can add INFINITY 2+2=4.

And you can add meanigless infinities to get meaningful figures..

Einsteins other equation for mass apporaching the speed of light not E=MC^2.

Is an equation for momentum.And when we know that momentum is determined by state then four states in one produces a new kind of E=MC^2 that is right for the big bang.

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Cristinel,

Does your theory hold true with a cyclical universe as proposed in "Endless Universe" by Steinhardt and others?

Thanks for the read.\

Jim Hoover

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Does your theory hold true with a cyclical universe as proposed in "Endless Universe" by Steinhardt and others?

Thanks for the read.\

Jim Hoover

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Dear Jim,

If I understand well, in their cyclic model of the universe, Steinhardt and Turok use the M-theory and a fifth dimension to avoid the problem of singularities. My proposal works well with extra dimensions, but it doesn't require them. It allows a wide variety of cyclic cosmologies with four dimensions - so long as the metric is semi-regular at singularities. One example is a FLRW cyclic model obtained as a warped product with warping function that can become 0 at big bang/big crunch. The singularities obtained are semi-regular, so the densitized version of the Einstein equation can be written at big bang/big crunch points too. Another way to obtain a semi-regular metric is by rescaling a nondegenerate one with a conformal factor that is allowed to become 0, and obtain a cyclic cosmology similar to Penrose's.

Best regards,

Cristi

If I understand well, in their cyclic model of the universe, Steinhardt and Turok use the M-theory and a fifth dimension to avoid the problem of singularities. My proposal works well with extra dimensions, but it doesn't require them. It allows a wide variety of cyclic cosmologies with four dimensions - so long as the metric is semi-regular at singularities. One example is a FLRW cyclic model obtained as a warped product with warping function that can become 0 at big bang/big crunch. The singularities obtained are semi-regular, so the densitized version of the Einstein equation can be written at big bang/big crunch points too. Another way to obtain a semi-regular metric is by rescaling a nondegenerate one with a conformal factor that is allowed to become 0, and obtain a cyclic cosmology similar to Penrose's.

Best regards,

Cristi

Dear Cristinel Stoica,

I feel your essay can convince anyone, on General-Relativity and its implications,without doubt but not so when it comes to recociling digital and analog nature of reality.If you want to know how to reconcile digital nature of reality with its analog nature,please read my essay and express your openion on it.

Good luck and best wishes.

Sreenath B N.

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I feel your essay can convince anyone, on General-Relativity and its implications,without doubt but not so when it comes to recociling digital and analog nature of reality.If you want to know how to reconcile digital nature of reality with its analog nature,please read my essay and express your openion on it.

Good luck and best wishes.

Sreenath B N.

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Dear Cristi,

Thank you for your kind remarks. I have enjoyed reading your essay about doing general relativity with degenerate metrics. I had a small curiosity : it was not clear to me whether while discussing gravitational singularities you make reference to geodesic incompleteness - this I thought is conceptually more general than metric degeneracy.

Good luck for the contest,

Tejinder

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Thank you for your kind remarks. I have enjoyed reading your essay about doing general relativity with degenerate metrics. I had a small curiosity : it was not clear to me whether while discussing gravitational singularities you make reference to geodesic incompleteness - this I thought is conceptually more general than metric degeneracy.

Good luck for the contest,

Tejinder

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Dear Dr. Tejinder Singh,

thank you for reading and commenting my essay. Your observation is correct: geodesic incompleteness is more general than degenerate metrics. But maybe we don't need to resolve all possible types of singularities, since not all occur from Einstein's equation. For example, Penrose's Cosmic Censorship hypothesized that naked singularities don't occur. My approach works fine with naked singularities, if they are "semi-regular". Semi-regular singularities behave well at the densitized versions of the Einstein and the ADM equations I proposed.

Good luck for the contest to you too,

Cristi

thank you for reading and commenting my essay. Your observation is correct: geodesic incompleteness is more general than degenerate metrics. But maybe we don't need to resolve all possible types of singularities, since not all occur from Einstein's equation. For example, Penrose's Cosmic Censorship hypothesized that naked singularities don't occur. My approach works fine with naked singularities, if they are "semi-regular". Semi-regular singularities behave well at the densitized versions of the Einstein and the ADM equations I proposed.

Good luck for the contest to you too,

Cristi

You're my kind of scientist, Cristi. Excellent essay!

You might be interested in another model of metric degeneracy, with analytic continuation, in my ICCS 2006 paper ("self organization in real and ocmplex analysis").

And I hope you get a chance to read my essay entry, too.

Thanks for a great read.

Tom

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You might be interested in another model of metric degeneracy, with analytic continuation, in my ICCS 2006 paper ("self organization in real and ocmplex analysis").

And I hope you get a chance to read my essay entry, too.

Thanks for a great read.

Tom

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Thank you Ray, I'm glad you like it. Congratulations for your essay too.

Best regards,

Cristi

Best regards,

Cristi

Dear Cristi,

Fascinating idea to extend general relativity to prevent mathematical singularities. Interesting to make use of ADM formalism to eliminate divergences. I enjoyed your essay very much. It is well-organized and nicely illustrated with the colorful diagrams you used.

Best wishes,

Paul

Paul Halpern, The Discreet Charm of the Discrete

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Fascinating idea to extend general relativity to prevent mathematical singularities. Interesting to make use of ADM formalism to eliminate divergences. I enjoyed your essay very much. It is well-organized and nicely illustrated with the colorful diagrams you used.

Best wishes,

Paul

Paul Halpern, The Discreet Charm of the Discrete

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Dear Paul,

thank you for reading my essay and I am happy you appreciate it. I like your essay and how you use the idea of lowest wavelength fields to explain discrete aspects of the physical phenomena.

Best regards,

Cristi

thank you for reading my essay and I am happy you appreciate it. I like your essay and how you use the idea of lowest wavelength fields to explain discrete aspects of the physical phenomena.

Best regards,

Cristi

Your essay is interesting. There are some questions I do have about some of this, in particular the nature of this caustic smoothng. In particular it seems to imply more degrees of freedom to spacetime.

However, what you wrote is interesting. This is one of the better papers.

Cheers LC

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However, what you wrote is interesting. This is one of the better papers.

Cheers LC

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Dear Lawrence,

thank you for the appreciation and careful reading. I would like to answer your questions. You say: "In particular it seems to imply more degrees of freedom to spacetime". Could you please explain me what degrees of freedom do you refer to? I tried to understand by myself, but I am not sure I did. What I can say is that my approach reduces to the "standard" one on the regions of nondegenerate metric (and have the same degrees of freedom), and about the regions of degenerate metric the standard GR doesn't comment. I am interested in understanding your observation.

Best regards,

Cristi

thank you for the appreciation and careful reading. I would like to answer your questions. You say: "In particular it seems to imply more degrees of freedom to spacetime". Could you please explain me what degrees of freedom do you refer to? I tried to understand by myself, but I am not sure I did. What I can say is that my approach reduces to the "standard" one on the regions of nondegenerate metric (and have the same degrees of freedom), and about the regions of degenerate metric the standard GR doesn't comment. I am interested in understanding your observation.

Best regards,

Cristi

Dear Christi

I've just re-read your essay and congratulate you, though areas were outside my more conceptional and empirical approach. In particular I reconsidered in a broader sense your 'boxed' comment;

"The best evidence for the continuity of reality would be provided by a theory which is based in an essential, irreducible way, on the necessity that spacetime and the values of the fields are divisible ad infinitum."

I realised that this is at face value entirely equivalent to the model of discrete fields which I describe, itself also consistent with Edwin Klingmans Cfield. Your route otherwise is largely finer than my 'overview' approach, and would not be qualified to comment. My discrete fields are consistent with Einstein's descriptions, applicable from a single ion up to the (or each) universe where in relative motion.

I've just posted a logical assessment of where it derives SR may reconnect with it's quantum mechanism, where GR also more naturally emerges. I am not a mathematical physicist but a logician and architect, viewing the matter from a different place, and that job is done. I really hope you're able to read and comprehend the dynamic variable logic in the style written.

I feel it is important, and your views would be appreciated, if you have the time and am not suffering the eye strain I am!

Many thanks

Peter

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I've just re-read your essay and congratulate you, though areas were outside my more conceptional and empirical approach. In particular I reconsidered in a broader sense your 'boxed' comment;

"The best evidence for the continuity of reality would be provided by a theory which is based in an essential, irreducible way, on the necessity that spacetime and the values of the fields are divisible ad infinitum."

I realised that this is at face value entirely equivalent to the model of discrete fields which I describe, itself also consistent with Edwin Klingmans Cfield. Your route otherwise is largely finer than my 'overview' approach, and would not be qualified to comment. My discrete fields are consistent with Einstein's descriptions, applicable from a single ion up to the (or each) universe where in relative motion.

I've just posted a logical assessment of where it derives SR may reconnect with it's quantum mechanism, where GR also more naturally emerges. I am not a mathematical physicist but a logician and architect, viewing the matter from a different place, and that job is done. I really hope you're able to read and comprehend the dynamic variable logic in the style written.

I feel it is important, and your views would be appreciated, if you have the time and am not suffering the eye strain I am!

Many thanks

Peter

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Dear Peter,

thank you for reading my essay and for your kind observations. I am happy for the connections you pointed with what you and Edwin say. As certainly you observed, I am only trying to show that GR is not guilty of a sin of which it is accused by many, and which became widely accepted - that it predicts its own "breakdown". In this essay I did not discuss the connections of this theory with the experiment, or how to find an explanation for its principles, and I limited myself to discuss only the problem of singularities, although there are certainly other problems too. Definitely all other areas are important and should be analyzed thoroughly, but I could concentrate only on one issue. I hope what I say may be helpful even to modified versions of GR, since it concerns the geometric structure of space and time in general.

Best regards,

Cristi

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thank you for reading my essay and for your kind observations. I am happy for the connections you pointed with what you and Edwin say. As certainly you observed, I am only trying to show that GR is not guilty of a sin of which it is accused by many, and which became widely accepted - that it predicts its own "breakdown". In this essay I did not discuss the connections of this theory with the experiment, or how to find an explanation for its principles, and I limited myself to discuss only the problem of singularities, although there are certainly other problems too. Definitely all other areas are important and should be analyzed thoroughly, but I could concentrate only on one issue. I hope what I say may be helpful even to modified versions of GR, since it concerns the geometric structure of space and time in general.

Best regards,

Cristi

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Christi

I agree. Far too many have simply treated this as an opportunity to give us a lecture on physics history as they see it, which is far from the point! And many of those are higher than yours, which I think very wrong. Your point is very important and well made, and I'm glad you saw the important aspect of my own as a way round the present illogicalities in relativity.

You referred to possible predictions varying from our current understanding. I responded - that I've made many, but in front of mobile goalposts they all fall to nought! I've just made some more ref the 'ignorosphere' (NS) we're about to explore. I won't repeat them as they're here.http://fqxi.org/community/forum/topic/803

Many thanks, and best of luck

Peter

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I agree. Far too many have simply treated this as an opportunity to give us a lecture on physics history as they see it, which is far from the point! And many of those are higher than yours, which I think very wrong. Your point is very important and well made, and I'm glad you saw the important aspect of my own as a way round the present illogicalities in relativity.

You referred to possible predictions varying from our current understanding. I responded - that I've made many, but in front of mobile goalposts they all fall to nought! I've just made some more ref the 'ignorosphere' (NS) we're about to explore. I won't repeat them as they're here.http://fqxi.org/community/forum/topic/803

Many thanks, and best of luck

Peter

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Dear Cristinel,

Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,

Alan

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Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,

Alan

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Dear Alan,

thank you for your kind words. I looked at what Edwin answered to you, and I concur with what he said.

When Maxwell wrote his equations, he realized that they are totally different than Newton's mechanics, which was then the accepted paradigm. His equations unified the electric and the magnetic forces, and I think that he was unhappy because they were not unified with Newton's mechanics too. This is why he tried to obtain the electromagnetic field from some gear mechanisms, and from ether mechanical waves. These attempts are based on the assumption that Newton's mechanics is more fundamental, and electromagentism somehow emerges. Physics evolved a bit from those times. Special relativity appeared because Maxwel's electromagnetism and Newton's mechanics not only were not unified, but they could not be unified in principle. General relativity appeared because Special relativity is incompatible with Newton's gravity. Einstein's gravity is pure geometrical, and the gauge theoretical view on the electromagnetism (and the weak and strong forces) are pure geometrical too, and they are compatible. My personal, subjective, biased if you want, opinion is that something like this really happens. This explanation is far from being perfect (mostly because quantum theory seems to be of a totally different nature), but this is what makes sense to me.

You say: "An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't.". For me, the meaning of what you said is that "an Archimedes screw model" makes more sense to you than "the 'fabric' of spacetime" makes sense to you. Im my opinion, your question boils down to what makes more sense to you, vs. what makes more sense to me. I wrote a comment somewhere above: "Explanation" Between Concrete and Abstract. I made there the observations that there are two understandings of the word "explanation", and I think that this may help you realize why what makes sense to you may not make sense to me and conversely.

Best regards,

Cristi

thank you for your kind words. I looked at what Edwin answered to you, and I concur with what he said.

When Maxwell wrote his equations, he realized that they are totally different than Newton's mechanics, which was then the accepted paradigm. His equations unified the electric and the magnetic forces, and I think that he was unhappy because they were not unified with Newton's mechanics too. This is why he tried to obtain the electromagnetic field from some gear mechanisms, and from ether mechanical waves. These attempts are based on the assumption that Newton's mechanics is more fundamental, and electromagentism somehow emerges. Physics evolved a bit from those times. Special relativity appeared because Maxwel's electromagnetism and Newton's mechanics not only were not unified, but they could not be unified in principle. General relativity appeared because Special relativity is incompatible with Newton's gravity. Einstein's gravity is pure geometrical, and the gauge theoretical view on the electromagnetism (and the weak and strong forces) are pure geometrical too, and they are compatible. My personal, subjective, biased if you want, opinion is that something like this really happens. This explanation is far from being perfect (mostly because quantum theory seems to be of a totally different nature), but this is what makes sense to me.

You say: "An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't.". For me, the meaning of what you said is that "an Archimedes screw model" makes more sense to you than "the 'fabric' of spacetime" makes sense to you. Im my opinion, your question boils down to what makes more sense to you, vs. what makes more sense to me. I wrote a comment somewhere above: "Explanation" Between Concrete and Abstract. I made there the observations that there are two understandings of the word "explanation", and I think that this may help you realize why what makes sense to you may not make sense to me and conversely.

Best regards,

Cristi

Dear Cristi,

I much appreciate your explanation of events, it was most useful. We have a difference of opinion on this one then. The layperson will be on my side though I think, given time. Scientific American could make a great cover story out of the Archimedes screw idea, we'll have to wait and see(!).

Best wishes,

Alan

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I much appreciate your explanation of events, it was most useful. We have a difference of opinion on this one then. The layperson will be on my side though I think, given time. Scientific American could make a great cover story out of the Archimedes screw idea, we'll have to wait and see(!).

Best wishes,

Alan

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Hi,

I have some news about the methods of singular semi-Riemannian geometry and singular general relativity described in my essay.

I constructed analytic extensions of the black hole solutions. These extended metrics are smooth at the singularities (and degenerate), and are obtained by coordinate changes which are themselves singular at the metric's singularities (somehow similar to the Eddington-Finkelstein coordinates, but the resulted metric is degenerate). These extensions are detailed, for the standard black hole solutions, here:

Schwarzschild Singularity is Semi-Regularizable

Analytic Reissner-Nordstrom Singularity

Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike Curves

The Kerr-Newman black hole is normally accompanied by closed timelike curves, but my solution allows us to avoid them naturally.

These solutions allow us to find maximal globally hyperbolic spacetimes with black holes, which admit foliations with Cauchy hypersurfaces. This restores the evolution equations, and the information is no longer lost:

The Cauchy Data in Spacetimes With Singularities

I developed the mathematical apparatus of singular semi-Riemannian geometry in the following papers:

On Singular Semi-Riemannian Manifolds

Warped Products of Singular Semi-Riemannian Manifolds

Cartan's Structural Equations for Degenerate Metric

Best regards,

Cristi Stoica

I have some news about the methods of singular semi-Riemannian geometry and singular general relativity described in my essay.

I constructed analytic extensions of the black hole solutions. These extended metrics are smooth at the singularities (and degenerate), and are obtained by coordinate changes which are themselves singular at the metric's singularities (somehow similar to the Eddington-Finkelstein coordinates, but the resulted metric is degenerate). These extensions are detailed, for the standard black hole solutions, here:

Schwarzschild Singularity is Semi-Regularizable

Analytic Reissner-Nordstrom Singularity

Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike Curves

The Kerr-Newman black hole is normally accompanied by closed timelike curves, but my solution allows us to avoid them naturally.

These solutions allow us to find maximal globally hyperbolic spacetimes with black holes, which admit foliations with Cauchy hypersurfaces. This restores the evolution equations, and the information is no longer lost:

The Cauchy Data in Spacetimes With Singularities

I developed the mathematical apparatus of singular semi-Riemannian geometry in the following papers:

On Singular Semi-Riemannian Manifolds

Warped Products of Singular Semi-Riemannian Manifolds

Cartan's Structural Equations for Degenerate Metric

Best regards,

Cristi Stoica

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