Did God Divide by Zero? by Cristinel Stoica

I am unsure why Hawking radiation is a problem. Physics is independent of the particular coordinate system we impose.

I was a bit glib with the language, comparing the benign singularity at det(g) = 0 as "removed," when this is just a "converse" of the malign singularity with g_{ij} -- > ∞.I originally did this with the idea of working with a black hole that sent the singularity off to infinity. In that way the analytic functions across the horizon of a Rindler wedge could be compared to a meremorphic function: analytic everywhere but at infinity.

There seems to be something odd going on here. Your equation 8 is singular at τ^2 = 2m. This appears to exchange the malign singularity at r = 0 with a singularity of some type on the horizon. As I communicated with you a few months ago this seems to have something to do with a dualism between quantum state interior to a black hole, or on the singularity, and the holographic states of a black hole as seen from the exterior.

Cheers LC

A quick corroboration of my worries about the worlds that singularity connects by this manipulation is the respond by L. B. Crowell :"... this seems to have something to do with a dualism between quantum state interior to a black hole, or on the singularity, and the holographic states of a black hole as seen from the exterior."

P.S. Virtual part: interior to a black hole ; Real part: exterior view of BH. (A look to my essays may be helpful).

Hello to both of you,

I agree also with Christi, I prefer when the serie is finite. After all , a BH is just a sphere with an enormous mass, it is logic that we do not see the light. If a correct picture of horizons and events must be analyzed.So the foundamentals must be respected. It is like the einstein effect in fact with finite groups and with a spherical volume more important.It is just the same road with limits.So indeed why this infinity in the equation. Now if we want to see what is a BH really, the road is more comple about the add of series of uniquness. The superimposings so are under several universal laws. The number of spheres even in the galaxy is specific, perhaps that the serie of quantum uniquenss is also correlated with its own volumes ! Now it is sure that the serie possesses volumes more important more we go towards the main central sphere, so the main singularity and its codes.So the codes of comportments inside the 3D sphere and its intrinsic cosmological spheres and quantum spheres can be seen in a pure finite road.

I must tell you that I am also intrigued by these BH , central to Galaxies. I know that they have a rule of mass for the universal rotations around the central sphere inside the universal sphere.But what are they if we see them really at the present and at a kind of locality. The volulmes of spheres become universal keys. The recycling of spheres become intriguing.When we insert the informations in this line of reasoning and the volumes of spheres of light....see that it becomes very relevan,t about our main codes inside our main central spheres.The central sphere is fascinating at all scale.The Universal singularity and the singularities in evolution.....

All this reasoning implies that our quantum uniquenss is like a relativistic foto of our universal sphere and its cosmological spheres. So we have an interesting link between the central universal sphere and the central quantum spheres. What is the result is we make a simple /, Volume of the central universal sphere/ vol.of the quantum central sphere.Now it is intriguing about the volumes of singularities. We can make the same with the mass of these spheres and their rotations spinal and orbital.The / is relevant , you can even make a / of these results .The constants must appear.

The BH have a pure complementary rule.The real universal secret is the optimization, so the spherization :)if they exist these spheres, they are reasons.They embark of course their stars and planets and moons and all quantum spheres. but I am persuaded that they have still a lot of unknown properties. The light is perhaps only for our stars and planets.Perhaps that we can see BH particules with new properties above the relativity and c.c is a probelm for the universal travel, more far, above the galaxies.Perhaps that after all, these BH have a lot of secrets for the travel between galaxies.But of course, we have already difficult to travel inside our milky way, so you imagine between galaxies. If we can go more quickly than c for a mass turning in the other sense than c, so we can extrapolate that it exists perhaps other angles of rotation for BH for example, implying so a possible travel at very interesting velocity above c. c is relevant for the teleportation, so we change m in c and after c in mass with codes.But for the universal travel, it is not sufficient. Probably that it exists several secrets due to rotations of spherical volumes.See thatt he serie of uniquenss is very relevant for the correlated link between the volumes of singularities.

The equations , derivations and integrations can be superimposed in the two 3D scales.(quant. and cosmol.)

First, I would like to apologize till I am an amateur (in your field) and not even a physicist, so my ability to express ideas by physics' (or even worse by mathematics') language is rather limited. Your essay was the motive to have a helpful Internet tour to the subject of Black Holes.

According to my essays I suggest that our Universe is consisted by two subUniverses (the "real" and the "virtual") that are presented as two hyperspheres (for 3 plus 1 D) that are connected with a singularity at the BB (start, Big Bang at the one pole of hyperspheres). So, there is a singularity in between the real and virtual subUniverse. The Universe expands and half the way of its life the reality of its inhabitants switches from real to virtual (I call this edge:horizon; not unrelated to Black Hole's). This is the "time" that Universe starts to shrink until it concludes to new singularity(ies) (Big Bounce(s) at the other pole of the two hyperspheres). This new singularities transform the old Universe (pre-Universe) to the new one (post-Universe) which has bigger dimensionality (part because we start now from two singularities). This sequence of transformations are followed up to present.

All this is referred to the major Universe (the Whole) which is the Universe we are living in. Into this major Universe numerous other universes (Black Holes, galaxies, stars,...) appear and disappear but their dimensionality can not exceed the dimensionality of the major Universe.

Hence, there are two types of examining singularities: the procedure that connects the two subUniverses (real-virtual of the same dimensionality) and the procedure that transforms a Universe to its post-Universe (higher dimensionality) or pre-Universe (less dimensionality).

I suppose: tau=0 is the BB singularity, tau^2=2m is Big Bounce (a singularity too), and tau^2=m is the event horizon that it does not seem a singularity to me.

This discussion was very helpful for me but I am not going to bother you any more (for the time being), Ioannis

It seems plausible that the τ = sqrt{2m} singularity is a pure coordinate singularity. If this is removed by going to Eddington-Finkelstein coordinates, then it would be interesting to work your problem to start in those coordinates. I might this weekend try to work with this some to see what happens.

As for Hawking radiation, you are talking about the final "POW!" when the black hole is completely evaporated. That part of the problem is not well known in any coordinates. Your objection does have a certain classical logic to it. However, by the time the black hole is down to its last 10^4 or 10^3 Planck mass units the black hole itself is probably quantum mechanical. In my coordinates (assuming they are unique to me, which is not likely) the singularity at infinity may not have to "move" from infinity. There may be some nonlocal physics which causes its disappearance without having to move at all.

I think your work might provide some machinery for looking at nonlocal quantum physics of black hole and possibly some duality between quantum fields (information) on a stretched horizon has some duality to quantum field interior to the black hole. It would be interesting if this could be demonstrated without violating the quantum Xerox principle (no-cloning). The appearance of the τ = sqrt{2m} singularity, even if it is a coordinate singularity, gives me pause to question whether information concerning the interior appears at the horizon.

Hi Cristinel

Congratulations for your nice and well written essay. It is clear and easy-to-read. I would like to leave my opinion about your question. I think that one has to differentiate the physical world and the mathematical representation of the world. Many physicists think that for every element of reality there most be an element in a mathematical structure. This could be the case, but sometimes the abstraction of reality becomes paradoxical. Recall that one of the fallacies of Zeno's paradoxes is to assume that space and time are continuous. This implies that between two real numbers we will find an infinite set of real numbers. If we follow this line of reasoning then the persistent Achilles will never catch up with the tortoise. This experiment immediately tell us that in the physical world space is not composed of an infinite number of points between the endpoints of a segment. If we accept this, we also have to accept that singularities only exist in mathematics but not in the physical world. The function 1/x is said to have a singularity at x=0, this is because at that point the function is not defined. But in the physical world a distance cannot be equal to zero, if a distance were equal to zero it would not be a distance.

I'd like to invite to read my essay, I would be happy if you could leave some comments on my essay entitled THE PREFERRED SYSTEM OF REFERENCE RELOADED.

Best wishes

Israel

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.