Is Spacetime Countable? by Sean Gryb

Dear Sean

You wrote "...so that there can be no minimum length"

I agree but on other reason

http://vixra.org/abs/1301.0191

Good luck

Yuri

Nice essay, well-written and clearly explaining its topic. It does seem less deeply connected to the question of "It from bit" than some of the other essays I've read, though. I guess countability could be related to the notion of an exact description via a sequence of bits... but if one can imagine eliciting an infinite such sequence, the space of possible sequences could be uncountable...

I"ll be interested to see how this stereographic/de Sitter duality is or isn't useful in future physics... curious on how you view its relation to the conformal/anti-de Sitter duality that is talked about a lot by string theorists?

Hello Sean

I agree with Howard. A lovely essay. I wonder how your concept of there being no minimum length would play out in the context of a proof I created in a Master's thesis (see attached).

Please read my essay on what I call the Armchair Universe which provides a foundation for spacetime, which arose initially after considering scaling bothers. I would be very interested to see how your work might be connected to mine.

Best wishes

Stephen AnastasiAttachment #1: 1_A_problem_for_geometry_1.pdf

Very interesting. I also enjoyed your previous paper I found on the arXiV (Right about Time). I had not previously made the connection between countability (one to one relation with the integers) and the continuum problem. Of course, you are right, its all about manifesting discreteness.

However, I would appreciate your opinion other aspects of countability. Consider, for example, the distinction between identity and indistinguishability. If we can identify particles as being discrete, but they are indistinguishable (for example in a Bose-Einstein condensate), then are they countable? More generally, do we need "identity" before the notion of countability is valid?

Also, what about recurrence? Imagine a pendulum (or other simple harmonic oscillator). If there is no physical mechanism to count beyond a modulo (in this case 2), then there could be an infinite number of recurrences (cycles of oscillation) before the event we observe, and after the event we observe. Can we refer to them as countable?

More importantly, if you are asleep, and someone wakes you, and you see the oscillator make a few cycles (you can even count them), and then you sleep for a while, but you don't know how long you have slept and someone wakes you again, can you still say that the number of cycles is countable -- if you have absolutely no external mechanism whatsoever to count the cycles that went by when you were asleep -- can you still say that the cycles are countable?

These may be easy questions for you to answer, nonetheless, they are interesting questions to me regarding "Is Spacetime Countable". For example, we could just as easily say "are cycles countable in the quantum domain"?

Kind regards, Paul

Hi Sean,

Thanks for your helpful response. I don't want to make too big an issue of this -- and given your busy schedule, please don't feel a need to respond -- but when you say (on p. 2 of your essay) that scale-invariance implies that a theory stops changing as we make our measurements more precise, this raises the question of whether there are any limits on how precise our measurements can be. On the face of it, the Heisenberg Uncertainty Principle suggests that there are such limits. However, your reference to the "infinitely small" on p. 8, and the surrounding discussion there, suggests that you're denying any such limits. But in that case, it appears that Planck's constant h must get "scaled down" to the scale of the infinitely small; otherwise, observers at the infinitely small scale would, in attempting to perform measurements there, be completely "engulfed" by quantum uncertainty and consequently unable to make definite measurements at all. But if h gets scaled down, then presumably related constants such as G and c would also get scaled down; in the case of c, this would mean abandoning relativity altogether, and in the case of G we would end up with some sort of "degravitation" -- and these major theoretical changes, as we go down in scale, clearly violate scale-invariance.

On the other hand, if there are limits on the precision of measurements, then it would seem that we have to settle for some sort of minimal scale. Of course, there's still a kind of scale-invariance here, since increasing the energy scale that we use to probe this minimal scale doesn't lead to new results or change our theories. But this energy-scale-invariance goes hand in hand with a minimal (Planckian) length, as the Spallucci/Ansoldi paper shows. So, I'm questioning whether scale-invariance should be viewed as an actual alternative to a minimal length, and to discreteness. In saying this, of course, I'm not rejecting your idea of scale-invariance; I'm simply denying that scale-invariance and discreteness are inherently opposed.

One additional, minor point: in using the word "fundamental" in my earlier comment, I was simply following your own usage on p. 2, where you say that there's no way to prove that a purportedly fundamental theory is truly fundamental.

Best Wishes,

Willard

In re-reading the comment I just made, I see that, in the final sentence of my second paragraph, the clause following the semicolon should be deleted as irrelevant, since you're not claiming that there's an inherent opposition between scale-invariance and discreteness. I apologize for my carelessness.

-Willard

Dear Sean,

Very good essay that, in my opinion, that tackles the ambitious task of unifying QM and GR in the right way.

My topic has to do with "dessins d'enfants" and contextuality. In the wikipedia paper you have a picture of the triangulated hyperbolic plane.

http://en.wikipedia.org/wiki/Dessin_d'enfant

In his book "The icosahedron and the solutions of equations of the fifth degree" (Dover, NY, 1956), Klein's antcipates the Belyi functions for platonic solids by using the stereographic projection.

My essay is here

http://www.fqxi.org/community/forum/topic/1789

I don't look at scale invariance, I don't look at the large scale universe but strangely there are closely related mathematical objects at the starting point.

Good luck,

Michel

Hello Sean

Richard Feynman in his Nobel Acceptance Speech (http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html)

said: "It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don't know why that is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature."

I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

The belief that "Nature is simple" is however being expressed differently in my essay "Analogical Engine" linked to http://fqxi.org/community/forum/topic/1865 .

Specifically though, I said "Planck constant is the Mother of All Dualities" and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism ... and so on.

Taken two at a time, it can be read as "what quantum is to classical" is similar to (~) "what wave is to particle." You can choose any two from among the multitudes that can be found in our discourses.

I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

Since "Nature is Analogical", we are free to probe nature in so many different ways. And you have touched some corners of it.

Best

Than Tin

Dear Dr. Gryb,

interesting essay. I like the idea of shape dynamics. One point is especially interesting. You wrote about scale invariance and I think this point of view has something to do with my approach using exotic smoothness. In my essay, I also discuss the question about the countability of the spacetime. I came to the same conclusion, in particular I do not put the smooth manifold away. There is another possibillity to kill the infinities (I believe). All relevant structures in 3D and 4D are very rigid, for instance one has Mostow rigidity for hyperbolic 3-manifolds etc.

I certainly have to read your other papers.

Good luck for the contest

Best

Torsten

Dear S. Gryb,

The reduction of dimensions like the use of meter defined in S. R. simultaneity and time as nothing inherently more than the ticks of hands pointing to positions on a dial indicate a departure form dimension of some if not all quantities by a type of foreshadowing.

Thankful for the good read,

Amos.

Dear Sean,

Please pardon my straying from subject matter. As I am not a professional physicist and just for me to be clear and learn from the experts, especially those with a relationist leaning: Is it being implied by the relational view of space and as suggested by Mach's or other relational principle that what decides whether a centrifugal force would act between two bodies in *constant relation*, would not be the bodies themselves, since they are at fixed distance to each other, nor the space in which they are located since it is a nothing, but by a distant sub-atomic particle light-years away in one of the fixed stars in whose reference frame the *constantly related* bodies are in circular motion? Or in which other relational reference frame can such circular motion be described?

You can reply me here or on my blog. Much appreciated. And please pardon my naive view of physics.

Accept my best regards,

Akinbo

Dear Sean,

You propose to trade the scale-invariance of special relativity to explain the real universe in a way that its expansion is not influenced by any form of matter and energy. In fact, there already exists one such theory - the Milne's model which is a kinematic theory based on the cosmological principle.

It has recently been shown [Phys. Scripta, 5, 055901, (2013)] that this theory explains all the cosmological observations without even requiring the speculative dark matter, dark energy and inflation. Moreover, this theory evades the long-standing problems of the standard cosmology. It would not be correct to say, as is generally (mis)believed, that the Milne model represents an empty universe.

As the Milne's model appears as a homogeneous, isotropic solution of equations R^{ik}=0 (see my essay in the present contest), and as equations R^{ik}=0 present an scale-invariant theory, it appears that your holy grail is the theory I have presented! I would appreciate your comments on my essay.

Best Regards.

___Ram

Dear Sean,

One single principle leads the Universe.

Every thing, every object, every phenomenon

is under the influence of this principle.

Nothing can exist if it is not born in the form of opposites.

I simply invite you to discover this in a few words,

but the main part is coming soon.

Thank you, and good luck!

I rated your essay accordingly to my appreciation.

Please visit My essay.

Dear Dr. Gryb,

A clear and nicely illustrated essay (the pictures help). If I understand correctly you are proposing scale invariance as a guiding principle to give a small scale structure for space-time which is smooth rather than foamy or discrete. This is done in the context of dS space-time rather than Minkowski space-time. This is a nice idea and may have some connection with an article by Aldrovandi and Pereira "Is Physics Asking for a New Kinematics?" Int.J.Mod.Phys.D17:2485-2493,2009. The abstract reads "It is discussed whether some of the consistency problems of present-day physics could be solved by replacing special relativity, whose underlying kinematics is ruled by the Poincare' group, by de Sitter relativity, with underlying kinematics ruled by the de Sitter group. In contrast to ordinary special relativity, which seems to fail at the Planck scale, this new relativity is "universal" in the sense that it holds at all energy scales." Anyway the two features of using dS and being valid at all energy scales was what made me think there might be a connection.

Also there may be *experimental* support for a smooth space-time structure at the smallest scales. About a year ago the Fermi satellite timed three gamma ray photons which arrived essentially at the same time after traveling a great distance from a gamma ray burst. Since the photons had slightly different energies one would expect *if* space-time were discrete, foamy etc. that the different energy photons would disperse due to interactions with the space-time fine structure. Since they didn't, within limits this implies that space-time is smooth within some limits. The original paper is "Bounds on Spectral Dispersion from Fermi-detected Gamma Ray Bursts", R. J. Nemiroff, et al. arXiv:1109.5191 [astro-ph.CO]

Finally if your fundamental theory is scale invariant at some point to connect with GR and the SM at low energy you need to introduce a scale somehow. Any ideas along these lines? Or in fact the cosmological constant does have a scale to it although I;m not sure if this will do.

Anyway a clever idea.

BEst,

Doug

Hi Sean,

Within so many essays it is very hard to find something so interesting as your essay.

We do not agree in 100% but we are very close. My key concept for the unification in physics is scale invariant metric. I have even proposed a simple spin experiment to find out if that metric exists.

Despite the differences between our views (we could discuss them if you read my concept) your essay deserves the highest rating.

Best regards