Did God Divide by Zero? by Cristinel Stoica

HI Cristi

I do believe this "the physical world space is not composed of an infinite number of points between the endpoints of a segment" because if we assume that the physical space can be infinitely divided we will arrive at the dichotomy paradox, experience tells us however that we can always reach the opposite endpoint which evidently suggests the assumption that a physical distance is not composed of infinitely many points but only of segments greater than zero. I agree with you that there is so far, to the best of my knowledge, no unambiguous mathematical proof and that the notion of the continuum and discreteness are not well understood yet. I also agree that theories based on the continuum have been quite successful. But I think that when a line of reasoning leads to a paradox implies that some of our ideal assumptions (which were aimed at representing experience) is wrong.

On the other hand, one should keep in mind that a singularity usually leads to an infinity, but so far, no one understands the physical meaning of infinity. When we say "let us consider a physical object at an infinite distance", what do we physically mean by that? Mathematically "makes sense", physically not. I will give another physical example to make clear that there is some sort of a disparity between the physical world and the abstract one. With this I hope I could persuade you that singularities might not exist in the physical world. Suppose you have an apple and two people. Then you want to share the apple between the number of people that you have. So, we ask the question: How much apple each person will obtain? The answer is clear, 1/2. Now, if we have only one person and we ask the same question, the answer will be 1. Imagine now that there is nobody to share the apple and we ask the same question. In such case we can reply with two equivalent arguments: (1) The question no longer applies or (2) simply, the information (i.e. 1 apple and 0 persons) that I have is not the appropriate one to answer the question. It is clear that the question is demanding physical quantities different from 0 to be answered. This example teaches us that when we face a singularity is because we insist in asking the same question when the physical elements of a particular situation no longer comply with the situation when there is no singularity. As long as we have 1 apple, 0 persons and we keep asking the same question where the question is no longer legitimate, we will never get an answer. This is the case with singularities and infinities. The expression "1/x=" is a question and when we arrive at x=0 it is said that the expression is singular because it is not defined at that point, i.e., at that point the question no longer applies. We should recall that in mathematics the zero was introduced by mathematical convenience though the symbol 0 per se was not considered as a number like 1 or 2 since its meaning represents nothingness, vacuum, void, etc.. As time went by people realized its usefulness and included it in the list of numbers like any other, though we all know that the zero have some privileges in mathematics. This is one reason why one should be well aware of the distinction between the physical reality and its abstract representation.

Israel

Hi Cristi

Thanks for you comments. I agree with what you said in your first paragraph. With respect to the second, I'm giving the same arguments that Aristoteles gave to explain the "fallacy". But formally, to the best of my knowledge, Zeno' paradoxes haven't been solved yet. In the literature this is still an active topic of research, there is not any general consensus among specialists. Like I said before, the notion of continuum is not well understood on epistemological terms. As to the twin paradox I agree that the paradox does not invalidate the theory but certainly it is an indicative that there is something strange going on; we all know that all theories have weaknesses. My work have something to say about the origin of the paradox. I hope you will read and comment about it.

Best regards

Israel

I did the following calculation. I start with the Eddington-Finkelstein coordinates

ds^2 = (1 - 2m/r)dt'^2 +/- (4m/r)dt'dr + (1 + 2m/r)dr^2 + r^2dΩ^2

where t' = t +/- (r* - r) and r* = r + 2m ln|r/2m - 1| or

dr*/dr = (1 - 2m/r)^{-1}

I then use your substitution (r, t) --- > (τ, ξτ^4). I compute the metric components

g_{ττ} = 8(ξτ^2 sqrt{τ^2 - 2m} +/- 2m/sqrt{τ^2 - 2m})

g_{ξξ} = (τ^2 - 2m)τ^6

g_{ξτ} = 8τ^3(ξτ^3(τ^2 - 2m) +/- 2m)

g_{ξr} = 4mτ^2 dξdτ

g_{τr} = 16τ(ξ +/- m/(τ^2(τ^2 - 2m)).

g_{rr} = 1 - 2m/τ^2

The metric components g_{ττ} and g_{τr} blow up at τ^2 = 2m. Also the g_{rr} component blows up at τ^2 = 0.

I suspect this is still a coordinate singularity and not a real singularity. The KruskalSzekeres coordinates with metric

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

for

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

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

It appears we still have a blow up problem with the 1/r ~ 1/τ^2.

Cheers LC

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

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

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 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

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 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

I responded to your comments on my essay blog.

Cheers LC