Dear Pentcho,

Thanks for the feedback. Causality is a rather "old" concept, at least according to Aristotle. In any case, to paraphrase an "old" proverb, "mathematics is a good servant but a bad master." Regarding the "bad master" part of this phrase, you might have noticed that I've repeatedly expressed the opinion that physical theories should not substitute convenient or "elegant" mathematical constructs (such as manifolds over the continuum) for clear, motivating physical principles. Hence, my own ideas are based on the simplest physical principles you can imagine: order, cause, and effect. However, regarding the "good servant" part, one cannot afford to do without the mathematical tools necessary to implement these simple principles. The physics comes first; the mathematics must be whatever is required to get the job done. Nature demands no less. Take care,

Ben

Jonathan,

As per my entry, that is my subject.

To clarify the point regarding the initial twin thought experiment, if time is a vector from past to future, one would assume the twin in the faster frame would travel into the future quicker, but the opposite is true, she moves into the past quicker. When her twin returns from the trip, in the slowed frame, she is dead and with every passing day, her live recedes further into the past. That is because she ages quicker and that is comparable to a faster metabolic rate, which was where we first diverged in our view of the discussion. It was you who transferred the topic to two observers passing each other.

Dear Benjamin

I see that we both intuitively believe that space-time is background free. You also understand my conclusion about my derivation of special relativity, a referee only wrote that Minkowski spacetime is so defined that time flows also in empty space. :) He also claimed many time repeated words that "derivation with relativistic mass does not give anything new". (I admit that this version needs some corrections about Duff's claims.) I hope that someone will generalize my derivation into general relativity, that influence of diffeomorphism will be easier explained.

I agree with Machian rule, which is in one version advocated by Barbour. I think that causal relations are based on Machian rule. If we said that spin of universe is zero, then Newton's bucket do not contradict with Mach. Do you agree?

You write that you hope that your theory will give SR and GR at large scales. It seems to me that such theories need be so clear, that we will see this without long calculations. But, who knows. Thus, for instance, I more like Weizsacker explanation of 3D than with triangulation of Loll. Maybe the theory still needs clarification about foundations, and then quantum graphity will be easier.

I you can see I avoided causal metrics to give masses to Planckian black holes, but it will be necessary some day.

Probably we disagree about consciousness. It seems to me that the money which enables Higgs boson, will enable also physical explanation of consciousness. Then we will see, how it is important for physics.

Those are some my disagreements, but because of your right intuition (I hope) and because of your big mathematical knowledge I give you 10.

Best regards Janko Kokosar

p.s.

I wrote the wrong book of Feynman. Feynman has a Lecture book on gravity available. Here I found his explanation why gravity of anti-particles is not negative. I do not remember it precisely, but it is worth to read it.

Any theory dealing with (or rejecting) spacetime should be able to provide an EXPLICIT DERIVATION of the constancy/inconstancy of the speed of light, or at least advance either the constancy or the inconstancy as an assumption and then derive conclusions from it. I am afraid your confession:

"Regarding the constancy of the speed of light, my guess would be that a concept like this only makes sense at sufficiently large scales. (...) You'll have to remember that my background is mostly mathematical, and therefore I'm inclined to consider the possibility of things that most physicists "know" are wrong. This might be useful in some cases; in others it only reflects my own ignorance."

...shows that your approach is incommensurable with the speed of light problem. So is Julian Barbour's Shape Dynamics.

Pentcho Valev

Benjamin,

If "The central new principle I propose is the causal metric hypothesis, which states that the metric properties of classical spacetime, up to overall scale, arise from a binary relation, which I will call a causal relation, on a set, which I will call a universe... Mathematical tools necessary to implement these ideas include a synthesis of multicategory theory and categorification in abstract algebra, involving interchangeability of objects, morphisms, elements, and relations; a version of graph 11 dynamics more sophisticated than versions involving uniform probabilities;and the theory of noncommutative algebras over sets with partially defined operations"

Then your attempt to introduce Some sort of Set-theoretical /Category-theoretical Evolution of Minkowsky spacetime Without Complex Numbers can faced with unavoidable necessary of destruction of some advanced areas of established mathematics and quantum physics... I know you understand it, but pure mathematical doubts on Minkowsky-Einstein complex( number) theory of time can suggest more elegant way to prove some your intuitions...?

    Dear Pentcho,

    Well, you will have to give me time (no pun intended). I have never represented the ideas I discuss in my essay as a complete "theory." The causal-metric hypothesis is just that, a hypothesis, and the ideas so far I term an "approach." A physical theory must eventually make predictions and connect with known physics, or it is not worth the paper it is written on. However, this takes time. I have been thinking about this for three years. Your criticism might be perfectly valid if 1000 people had been thinking about it for 30.

    You seem delighted with my admission of possible ignorance, so I have given you an even better quote above: "A physical theory must eventually make predictions and connect with known physics, or it is not worth the paper it is written on." Feel free to repost it as many times as you wish. No one knows very much about physics compared to what there is to know.

    The problem we were asked to address was "which of our basic physical assumptions are wrong?" It was not, "analyze the implications of the Michelson-Morley experiment." I will opt to interpret your interest in the foundations of relativity in a positive light, as a reflection of your zeal for the integrity of natural philosophy. But please bear in mind that it is counterproductive for everyone to be forced to work on the same problem. Cause and effect is more fundamental than the speed of light, or any speed for that matter, yet I don't insist that you make any definitive statement about causality. Feel free to work on any problem you wish, and please suffer me to do the same. Take care,

    Ben

    Dear Janko,

    Thanks for the feedback. I will see if I can find what Feynman said about this... I always appreciate his point of view.

    Regarding Machian ideas and shape dynamics, a number of us (me, Lawrence Crowell, Sean Gryb, Flavio Mercati, Daniel Alves) have been discussing the possibility of a duality or complementarity between causal theory and shape dynamics. Shape dynamics derives an arrow of time essentially from an asymmetry in configuration space, whereas I take it to be fundamental, but these two ideas need not necessarily contradict each other. Part of that discussion is on my thread, and part of it is scattered about on the other threads.

    Regarding the recovery of relativity, many the ideas specific to relativity (as opposed to Newtonian mechanics) are clear in causal theory; in fact, causal theory is more a generalization of relativity than anything else, since it is in relativity that the primacy of the causal structure and the ideal of background independence are clearly embodied. The technical difficulty comes in recovering manifold structure, and the point of view is that this is really unnecessary to all the important physical ideas of relativity. This will take work, because these ideas are currently expressed in geometric language. If Riemann had been around to advise Einstein rather than Minkowski and his friends, he might have showed him that there are many possible formalisms for expressing his physical ideas that don't involve continuum manifolds.

    I don't know enough about consciousness to agree or disagree with anyone about it, though I think it can't be deterministic. I like to read about it, though! Also, I appreciate the rating! Take care,

    Ben

    Dear Michael,

    Thanks for the feedback! The complex numbers are undoubtedly the workhorse of much of modern physics, and avoiding them, if it's even possible to do so, will certainly require a great deal of work to achieve the same explanatory power, let alone surpass it. You might ask, "why, then, abstain from the complex numbers at all, if they are so useful?" The important word here is "useful." Many of the properties that characterize the real number continuum, and by extension the complex numbers, are mathematically "useful" without having any clear physical meaning. Even if there were not physical reasons to doubt that spacetime is infinitely divisible (of which there are many), properties like the least upper bound property of the real numbers would never be directly relevant to physics. Even worse, paradoxes like the Banach-Tarski paradox (you can pull apart a solid sphere into nonmeasurable sets and reassemble them into two spheres identical to the original sphere) show that the real and complex numbers have properties that are not only physically irrelevant, but also physically absurd.

    But I am not sure if this answers your question? (I suspect it doesn't!) In any case, I see that you have an essay here about an interesting subject, so I will read what you wrote before making any further remarks. Take care,

    Ben

    Benjamin,

    Since you want Minkowski spacetime to emerge from your "causal relations", you will have to assume, additionally, constancy of the speed of light:

    John Norton: "That the speed of light is a constant is one of the most important facts about space and time in special relativity. That fact gets expressed geometrically in spacetime geometry through the existence of light cones, or, as it is sometimes said, the "light cone structure" of spacetime. (...) So if we mean a spacetime that also behaves the way special relativity demands, then we have a Minkowski spacetime."

    Then it may turn out that your "causal relations" are superfluous - Minkowski spacetime emerges from the assumption that the speed of light is constant and that's it.

    Pentcho Valev

      Dear Pentcho,

      I have heard of a certain company that offers to name distant celestial objects after its clients in exchange for a fee. Calling the causal relations of the universe "mine" would be an even more absurd presumption of ownership.

      I'm afraid that suggesting that causal relations are superfluous because spacetime is a large-scale manifestation of them is like suggesting that the pixels on your screen are superfluous because they form a picture. Try doing without them!

      Regarding the constancy of the speed of light, "speed" means "change in distance per unit time." Distance and time are both metric concepts. When one begins with something other than a metric, "speed" must be viewed as a secondary, rather than a primary, concept. No assumption must be made about it whatsoever. Take care,

      Ben

      Dear Benjamin,

      I found your essay to be one of the most all-encompassing in the contest. The paper paints a fairly complete picture of the current state of things and yet it is not lacking in details. It surely merits the high rating it has received.

      One comment I can pick out that especially triggered a response is:

      "Causality is often formalized at the classical level as an irreflexive, acyclic, transitive binary relation on the set of spacetime events."

      I agree. Though the approach I wrote about is classical or semi-classical, and therefore embedded solely in U(1) topology, the rigorous consideration of rotations really requires formulation in SU(2) or higher topologies because of the difference in rotational characteristics of the electric and magnetic fields, path dependencies and the cyclic nature of waves. Then the algebra accommodates reflections and cyclic relations. And there the non-commutative nature of the algebra naturally leads to quantization.

      It's hard to picture just what you mean when you make this intriguing statement:

      "I will call the relation induced by these transitions the universal relation, and each of its subrelations a kinematic scheme. Under suitable assumptions, a pleasing fractal picture emerges, in which kinematic schemes share most of their abstract properties with the causal relations of their constituent universes, and quantization becomes an iteration of structure, at least locally"

      Perhaps that is related to the point made in the essay at note 1. The statement conjured up reflections on an interesting but perplexing book "Fourier Analysis on Finite Groups and Applications" by Audrey Terras. I've not yet fully fathomed its nuances but the author demonstrates how Fourier Analysis and associated sets can be used in a surprisingly general series of applications where the domain can be interpreted as having some type of cyclic basis. She doesn't mention Quantum Mechanics, as I recall, but maybe there is a natural fit there.

      Congratulations on a fine essay,

      Steve

        Dear Ben,

        Earlier today I made a short comment (hard for you to to find) related to your conversation with Daniel Burnstein. Forget about it. What I wanted to say was that I'm hopefully beginning to begin to grasp your very interesting ideas. As you already know, I'm neither a physicist nor a mathematician. But re-reading your essay today, and following the above conversation, has been most instructive. It also gave me an idea, which I will come back to later, after more reading and thinking. I very much look forward to follow this conversation further. I have so much to learn!

        Best regards,

        Inger

          Dear Steve,

          Thanks for bringing this to my attention. I had checked your thread a couple of times after I posted there originally, but there are so many to look at that I missed seeing this. I don't know if you saw the discussion about antimatter antigravitation in my thread above or my brief mention of the possibility in my essay, but this is something I find very interesting. Take care,

          Ben