Dear Lawrence,

Interesting calculations. But my substitution (r, t) --- > (tau, xi tau^4) is suited for the Schwarzschild coordinates, so I don't expect them to work as well with the Eddington-Finkelstein's or to Kruskal-Szekeres coordinates, which indeed have the 1/r blow up.

As I said, in differential geometry it is known that you can't cover any surface with only one coordinate system. For example, for the sphere you need at least two coordinates, otherwise you have coordinate singularities. If it is impossible to cover the sphere with only one coordinate system, this doesn't mean the sphere is singular. It just means that it has different topology than R^2. In general relativity the things are more complicated, for two reasons: first, we deal with 4 dimensions. Second, the metric is indefinite, and it imposes constraints, in addition to those imposed by topology. But, if we can find, for a given spacetime, two or more coordinates which cover it and generate an atlas, and in which the singularity problems are resolved, the problem is solved. So, for r=2m there are the Eddington-Finkelstein, as well ass Kruskal-Szekeres, and Gullstrand-Painlevé coordinates, which preceded all. They work only for the event horizon, not for r=0. Mine just repair the metric at r=0, in the sense that they make it benign. They apply for r

the text is broken... I'll try to rewrite the continuation:

They [my coordinates] apply for r less than 2m. For r=2m we can apply Eddington-Finkelstein's. Together they cover the Schwarzschild spacetime, extended at r=0 by my coordinates. This is enough, we have a covering of the spacetime, we don't need a covering by just one coordinate patch, it is fine to have it by two or more patches. Having only one global coordinate system would be just a bonus, but it is not necessary.

There's no reason to apply my coordinate transformation directly to Eddington-Finkelstein's. Eddington-Finkelstein's solution is obtained from Schwarzschild's by a coordinate transformation, call it T1. Let's call mine, (r, t) --- > (tau, xi tau^4), T2. To go from Eddington-Finkelstein's coordinates to mine, one should apply the transformation T2*T1^{-1}, and not T2 directly. And even if we do this, we would not find global coordinates in which both the r=0 and the r=2m singularities are nice.

I know you know all these, but I wanted to make the things clear. Thank you for the analysis.

Best regards,

Cristi

  • [deleted]

  • Post #52

I will try in the next few days to do as you indicate. In thinking about the Kruskal-Szekeres coordinates I see potentially a similar issue. The Kruskal-Szekeres coordinates in (t, r, θ, φ) coordinates are

U = sqrt{1 - r/2m}e^{r/4m}cosh(t/4m)

V = sqrt{1 - r/2m}e^{r/4m}sinh(t/4m)

in the exterior region. If I let (t, r) --- > (τξ^a, ξ^b) thse become

U = sqrt{1 - ξ^b/2m}e^{ξ^b/4m}cosh(τξ^a/4m)

V = sqrt{1 - ξ^b/2m}e^{ξ^b/4m}sinh(τξ^a/4m)

and the differential of these are

dU = [(1 - ξ^b/2m)^{1/2}bξ^{b-1}/4m} -

(1 - ξ^b /2m)^{-1/2}]e^{ξ^b/4m}cosh(τξ^a/4m) +

{1 - ξ^b/2m}e^{ξ^b/4m}e^{ξ^b/4m}sinh(τξ^a/4m)(ξ^adτ + aξ^{a-1}dξ)/4m

dV = [(1 - ξ^b/2m)^{1/2}bξ^{b-1}/4m} -

(1 - ξ^b/2m)^{-1/2}]e^{ξ^b/4m}cosh(τξ^a/4m) +

{1 - ξ^b/2m}e^{ξ^b/4m}e^{ξ^b/4m}sinh(τξ^a/4m)(ξ^adτ + aξ^{a-1}dξ)/4m

which are rather complicated. The whole metric is

ds^2 = 32m^3e^{-ξ^b/2m)/(ξ^b)(dU^2 - dV^2) + r^2dΩ^2

with the dU and dV substituted from above.

In looking at this without working out the exact metric coefficients there are some things I can see. The ξ^{-b} counters the ξ^{2b-2} into ξ^{b-2}, the ξ^{2a} into ξ^{2a-2b} and the ξ^{2a-2} into ξ^{2a-2b -1}. These do not diverge if a = 1 and b >= a +1/2. However, there is a term (1 - ξ^b/2m)^{-1/2}, which diverges for ξ^b = 2m. As a result we are left with the horizon blow up.

I'd have to work this out more explicitly (rather than typing in the analysis as I think it) to make sure this is on the mark. We seem to be left with a singular condition at the horizon still.

Cheers LC

  • [deleted]

  • Post #53

Hello,

I didn't know these equations,(Kerr,...)

It seems very relevant considering my theory of spherization. I see that the Schwarzschild metric is relevant also about the spherical symmetry.Apparently it is a good road to understand the einstein field equations. In my model, the BH spheres turn.the rotation is weak but it turns.

The coordonates of Kruskal-Szekeres seems intersting for a kind of taxonomy of events. The diagram of Minkowski is in the same rational logic it seems to me in 1 dimension.

The extension is made by Penrose in 2 diemnsions after the extension of spacetime by Kruskal-Szekeres.

I see that we can extend still in 3D spherical spcetime with a closed evolutive system and a time constant of evolution.See that the diagrams of Penrose are very relevant for the superimposings of spacetime.I ask me what shall be the results if the spherical coordonates are inserted with the pure thermodynamics and the volumes. See that the serie of Uniqueness is essential also. It permits to extend to the central universal BH, the most important.In my model,this BH does not turn, so its mass is relevant .The extention towards the singularities and the singularity can be relevant if the series are rational.Finite even for the serie of uniqueness. The extention must in fact showing the road towards this central BH of our Universal sphere. We can even class all the otehr BH.In fact the singularities are fascinating when we consider the uniqueness !

But OF COURSE THE INFINITY AND THE FINITE GROUPS MUST BE ANALYZED WITH THE BIGGEST RATIONALISM .If not the serie is not a finite groups with a central sphere !!!

Regards

5 days later

  • [deleted]

  • Post #54

Dear Cristi Stoica,

Although due to my lack of education I was unable to fully understand some of your mathematical commentary about physics abstract assumptions, I still found your scrupulously argued exceptionally well written essay quite absorbing. I was considerably helped by the exquisitely constructed graphics. As best as I could tell, you meticulously adhered to the contest's rules by clearly identifying those physics foundational assumptions you thought were erroneous, and you gave a valid explanation for why you thought the way you did. May I please make a comment? In my essay Sequence Consequence, I thoughtfully point out that all scientific studies of snowflakes have determined that of the trillions that have fallen, no snowflakes have ever been found to be identical. Physical laws are supposed to be consistent throughout the Universe, so it seems reasonable to assume that each star is different in composition and size than every other star in the Universe is, and more importantly, each star is set at a differing intervening distance than every other star is. In your graphics, you use seemingly identical circles and triangles and squares. You use identical numbers and identical symbols in your mathematical equations. As the only realist at this site, I have to reluctantly tell you that I think one real indivisible Universe can only be occurring eternally once in one dimension.

    Dear Joe Fisher,

    Thank you very much for the attention given to my essay, and for the kind words. I also thank you for the interesting comments. Indeed, every star is unique. Maybe when they become black holes, they lose "hair" and become identical (asymptotically), modulo few parameters like mass, angular momentum and electric charge, as some theorems seem to show. But until then, they are unique. Unfortunately, I had to cut the original manuscript, containing the mathematical descriptions of every unique star, due to the regulations of the contest (there is a limitation in the number of pages and characters) :). By the way, I don't know if you have noticed, your final remark, as well as your essay, is written in "seemingly identical" words, despite the fact that "no snowflakes have ever been found to be identical" :)

    Best regards,

    Cristi

    • [deleted]

    • Post #56

    Dear Christinel Stoica,

    "By the way, I don't know if you have noticed, your final remark, as well as your essay, is written in "seemingly identical" words, despite the fact that "no snowflakes have ever been found to be identical" :)" Touché, although I did notice that all of the physical representations of all of the real characters that made up the words I used, and all of the real spaces separating the letters and punctuation marks you mentioned are unique. I have also noticed real holes and every one of them I have seen can only best be described as being black. For instance every person totes nine unique major holes around with them wherever they go. They also have innumerable real unique tiny black holes all over their skins. Now I do not know what real methodology you could use to measure any one of these real unique black holes that would scientifically establish that they originally came either just before or immediately after there was a singular state of nothing. Nor can I see how any real measurement of any of these real black holes could determine their modern uses and functions and their actual relationship to other holes. So while I do not mind in the least anyone speculating that an imaginary old time black hole behaved differently from a real one, I do wish these dreamers would take the advice my daughters often give to me and "Get real."

    Dear Joe Fisher,

    "I did notice that all of the physical representations of all of the real characters that made up the words I used, and all of the real spaces separating the letters and punctuation marks you mentioned are unique."

    Same for images and equations. Can one say with only words more than with words and pictures and equations?

    "I have also noticed real holes and every one of them I have seen can only best be described as being black. For instance every person totes nine unique major holes around with them wherever they go."

    I have to agree with your daughters :)

    Returning to:

    "I thoughtfully point out that all scientific studies of snowflakes have determined that of the trillions that have fallen, no snowflakes have ever been found to be identical."

    I don't contradict you. On the other hand, you are missing something. Among these trillions which you claim were compared, is there any snowflake which violates the known mathematics and physics of ice crystals? The moral is that scientists try to find the general laws, the universal equations. It would be unrealistic to try to write down all possible solutions of these equations. What is realistic is to propose the equations, and then check that the new instance found in Nature respects the equations or violates them. So I have to say that you have the wrong picture of what science is, and it is that wrong picture you are fighting against.

    Best regards,

    Cristi

    • [deleted]

    • Post #58

    Dear Christinel Stoica,

    "It would be unrealistic to try to write down all possible solutions of these equations. What is realistic is to propose the equations, and then check that the new instance found in Nature respects the equations or violates them." Thank you so much for providing pragmatic proof that my assessment of reality is correct. All mathematical equations are unrealistic because all equations are abstractions. One real Universe can only be eternally occurring in one dimension once. The only accurate mathematical equation that could persist is 1=U where 1 is really equal to the one real Universe. While abstract definitions of abstract Nature may provide suitable instances for dreamy speculation about the abstract unification of abstract separated undefined elements such as the abstract ability for abstract total energy to always abstractly equal abstract total amounts of abstract mass times abstract total amounts of light squared as in e=mc², real energy actually only equals one real Universe, or 1=U. I think I do have a wrong picture of science. I do not picture reality, I live in it.

    Dear Cristi:

    Enjoyed reading your essay and agree with the conclusions of the paper that God did not divide by zero.

    The singularities can be shown to arise from the missing physics in GR as described in my paper - -" From Absurd to Elegant Universe" that integrates the missing physics of spontaneous decay into a simplified form of general relativity that includes specific relativity and gravitational potential. The results of the model show that the relativistic gravitational effects at quantum scale can be successfully predicted without any singularities experienced by GR. This also eliminates the need or relevance of the so far unsuccessful efforts of unifying the gravity and other fundamental forces of the standard model. The model also resolves many other paradoxes and inconsistencies of modern physics and explains relativistic understanding of the inner workings of QM.

    I would greatly welcome your thoughts on the above and comments on my paper.

    Best Regards

    Avtar Singh

      Dear Avtar Singh,

      Thank you for the kind comment. I look froward to reading your essay. Good luck with the contest!

      Best regards,

      Cristi

      Cristi,

      Congrats on a thought-provoking and splendidly illustrated essay. I need to read some of your other papers to fully understand your program, but it seems evident to me that the ideas are good whether or not they ultimately describe physics.

      My background is algebraic geometry, and your ideas brought to mind a couple of concepts from that field that might be interesting to compare to what you are doing. The first is resolution of singularities (Hironaka); particularly blowups. When you blow up at a singularity, every point in the singular locus corresponds to an entire subspace in the exceptional fiber, so in some sense you could consider points in the blowup "zero distance apart" if they coincide in the original singular variety. The second is the fact that algebraic tangent spaces change dimension at singularities. Perhaps some analogue of these concepts in the analytic category is relevant?

      My own point of view is much different; I tend to regard manifolds as too good to be true, and prefer to regard them as emerging from a more primitive structure, as I briefly describe in my essay:

      On the Foundational Assumptions of Modern Physics

      In any case, I enjoyed your essay and hope to learn more about your approach. Take care,

      Ben Dribus

        Dear Ben,

        Thank you for the interest in my essay and for the comments. I am answering you with a delay because I was on a short vacation. Indeed, we can relate what I did with Hironaka's resolutions of singularities in Algebraic Geometry, and I did mention this connection in this paper. I've made this parallel only to show a situation in which a singularity due to the way a space (the variety) sits in a higher dimensional space is singular, but the singularity can be "blown-up". In Singular General Relativity, my point is that the blow-up is not needed in reality, but only to repair some solutions obtained under the assumption that one should identify points which are topologically distinct, but the distance between them (as measured by the metric tensor) is zero.

        I look forward to reading your essay.

        Good luck with the contest!

        Cristi

        Dear Cristi,

        Very interesting essay. In particular, it resolves a conundrum of myself. I was suspect about the problem why no one used different coordinate patches to solve problems in GR. Every problem I studied uses a global coordinate system but (as you also mentioned) for most manifolds there is no one. I always thought that I'm to stupid and there is an easy argument which I miss.

        But as you showed, this kind of thinking is correct: singularities can be resolved by using a covering with more than one coordinate patch.

        I will never forget this lesson.

        Best

        Tosrten

          Dear Torsten,

          Thank you for the nice comments. About the little usage of different coordinate patches in GR, one possible explanation is that geometers and physicists think differently. Geometers insist so much on using a manifestly invariant language, and tend to consider physicists who work in coordinates careless. But Einstein's summation convention is a powerful tool, which reveals the common grounds of the metric as a scalar product, contraction, the lowering and raising isomorphisms, the relation between various apparently distinct tensors, which turn out to be the same by the musical isomorphisms. I was always wondering why geometers avoid working with indices (but I suspected that they use them in private, and in the papers rewrite everything in the coordinate-free notation :) ). Especially since there is a manifestly invariant version, Penrose's abstract index notation. Another possible explanation for working in one coordinate, as if it is global, may be due to the fact that in general we tend to see time as flowing. There are several hints which support the idea of a global time: quantum mechanics and QFT, the condition of global hyperbolicity, the necessity to have a global spin structure. The existence of a global time may suggest that the coordinates should be global too, although this is not necessarily true.

          Best regards,

          Cristi

          9 days later

          • [deleted]

          • Post #66

          I gave your paper a high score which popped it up a bit. I am a bit dismayed at some of the higher ranked papers, some of which look like pure fluff and nonsense. Papers submitted a month ago seem to be collectively sinking like a stone.

          Cheers LC

            • [deleted]

            • Post #67

            Thank you. I'd like to see your essay and other good essays receiving more attention.

            Best regards,

            Cristi

            • [deleted]

            • Post #68

            Are you bargaining, Lawrence Crowell and Cristinel Stoica? Is that fair?

            Pentcho Valev

            Pentcho Valev, first, thanks for checking my page, despite your rude comment. I'll answer for myself. I am not bargaining (and I don't see where did you see this). I wouldn't consider it fair. If you have any doubts, you are free to report us.