Infinite Resolution by Cristinel Stoica

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

Dear Lev,

thank you for your explanations and your patience. I am grateful for having this conversation, because now I feel that I understand you better.

Dear Eugene,

thank you for your intervention, which helped us in this conversation.

Best regards,

Cristi

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

"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

maybe it was a metaphor :))

@admin: since you have removed re castel's comment, my previous reply (#29648 on Jan. 30, 2011 @ 09:37 GMT) makes no longer sense. could you please remove it as well?

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

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

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