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
Thank you Ray, I'm glad you like it. Congratulations for your essay too.
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
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
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
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
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 in finitum."
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
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
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
thank you, Peter, and best of luck to you too!
Cristi
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
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
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
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
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