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The deep conflict between Theory of Relativity and Planck Constant


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Planck constant is routinely described as "representing the smallest quantity that exists in reality", or even worse, as "the smallest quantity imaginable".

Certainly it is not the smallest "imaginable" quantity. Take Planck length - 1.61622837 × 10^-35 meters. Now imagine a smaller number. There, you did it.

But is it fair to think of it as "the smallest quantity that exists in reality". In physics, people often adopt a philosophy where things that cannot be measured also "do not exist". From that perspective they might say that "nothing smaller than Planck length exists" - since it would be fundamentally outside of our ability to experience it; thus there's no reason to assume it exists.

Unfortunately, by saying that one is muddying the philosophical waters much more than by simply saying "it represents our observation limit within the framework of modern physics". In general the problem of assuming that things beyond an observation limit literally do not exist is that you may well end up placing unnecessary mysticism on other parts of your world model without realizing it - for example relativistic simultaneity makes reality static if you presume that observation limit represents also the limit of "what exists".

As far as I'm concerned, the only reason there are so many exotic interpretations of Quantum Mechanics is this same exact philosophical trap. The observation that matter absorbs enerqy in discrete quantities does not prove that energy always exists in discrete quantities. Obviously it's possible that only the absorption event is quantized. Since we can only produce observations with matter, it is easy to fall into the trap of thinking that we can always think of EM energy itself as quantized packets (photons). But it's equally true that quantized energy absorption mechanism of matter simply places an observation limit to our experiments. That idea yields a very simple local realist interpretation of the entire quantum theory, including a full local realist explanation to Bell Experiments. The topic I'm about to get to is connected to that local realist interpretation but I thought it deserves a thread of its own. So let's get to it!

CMBR thought experiment

I've commented more than once in various threads around here about the fact that the cosmic microwave background radiation (CMBR) is emanating from a single reference frame - commonly thought to originate from the moment in past when the universe first became transparent (i.e. when universe cooled down sufficiently from energy levels where everything would have been opaque plasma). Meaning, we can very trivially measure the motion of earth against CMBR, because we see blue-shifted frequency in one direction, and red-shifted in the opposite direction.

image.png.672390287b5a891119ffef533ee6369a.png

https://en.wikipedia.org/wiki/Cosmic_background_radiation#/media/File:Cobe-cosmic-background-radiation.gif

This should raise a lot of interesting questions in your mind, one of which is the very obvious point that CMBR appears to represents a rather pragmatic universal reference frame for us. Certainly relativistic calculations can still be done in any frame we choose - Lorentz transformation is still symmetrical between frames. But from philosophical standpoint it's a bit of a bother to place the entirety of known universe and CMBR into a space where it might not be at rest (Sort of anti-Mach's principle). Especially when CMBR is viewed as a remnant of a Big Bang event in which all of space was created "simultaneously" - if so it would stand to reason that simultaneity still exists

But that's not the fundamentally problematic part of this. Here's the problematic part; Since the energy of electromagnetic radiation depends on its frequency (as per the photoelectric effect), it means that when we accelerate into any direction in the universe, we would be running against more and more energetic CMBR radiation. Following the definitions of modern physics, at some point the energy of CMBR in front of us would hit the Planck limit. That limit is essentially the moment when the wavelength of a photon becomes smaller than its Schwarzschild radius - meaning it becomes a black hole.

(Yes, before we'd get that far we would hit many other obstacles - for one the increasing energy of CMBR would eat away the kinetic energy of the ship - but for the purpose of this thought experiments we might as well discuss the black hole limit)

Now, in terms of relativity principle, we have already broken this picture into a somewhat incoherent mess. We can't say that an object out there turns into a black hole because we are accelerating. We are not changing reality out there - we are merely changing our own representation of reality. (And this is why it always bothers me when people claim that length contraction is "real"...)

The object looks like a black hole only in our reference frame, but not in others. The problem with this is that Planck limit implies a very observable change in the behavior of an element (a black hole does not let information pass through it). An object either really is a black hole, or it is not, and we can find that out. We can find out if an apparent black hole is really a black hole, by observing whether information manages to pass through it. If the answer is "yes", we know we are merely looking at the situation from a wrong reference frame (and thus also length contracting Planck length becomes completely unproblematic - we just need to know the factor of error implied by our "wrong" reference frame.. ...or maybe choose a more convenient reference frame)

With this thought experiment, we seem to have somewhat broken the relativity part of Theory of Relativity. Not so much with CMBR, but rather with Planck constant. CMBR was just a convenient vehicle to discuss the thought experiment, but we could always construct the same experiment without relying on CMBR at all. If we have any EM wave, we can always blueshift it into an apparent blackhole by changing our reference frame. Put this together with the fact that EM radiation propagates coherently (the waves from different objects don't overtake one another), and we have means to differentiate between "correct" and "wrong" reference frames;

There's a lot of evidence suggesting that it is in principle possible to push enough energy into a small amount of space, and thus produce an actual black hole, not just an apparent one. The difference with this method of producing a black hole would be that it acts like a black hole when observed from any reference frame. By finding out the limits of producing an actual black hole, we would find out the actual rest frame of C (at least locally). Thus, we would have implicitly found one-way speed of C, and we would have established simultaneity in our local environment. Lorentz-transformed frames would be correct up to the Planck limit only, meaning a reasonable interpretation of Special Relativity is, well, becoming very close to Lorentz' aether theory.

Again, not too surprising being that Quantum Field Theory is basically conceptually what I'm describing, with the interesting difference that in Quantum Field Theory, the fields themselves are assumed to be quantized. I'm slightly puzzled why that assumption exists, being that we already think that all EM absorption events are quantized by themselves. That already means that our observation limit would make the fields look quantized, even if they are not (this goes back to the same philosophical trap that I discussed in the first couple of paragraphs).

The role of Planck Constant

I think at this point is best to take a few steps back, and discuss where did we get Planck constant in the first place. It is quite simply connected to the idea that atoms (or electrons) fundamentally store energy as harmonic waves.

This produces a wave-matter picture where we have rather obvious reasons to some fundamental observation limits. We can only probe particles or waves of reality by how they interact with other particles or waves of reality, and since the frequency-energy dependency of those objects implies a direct coupling between "more accurate measurement" and "more energy being pushed into the system by the observation mechanism itself", it means we can only get limited accuracy out of our experiments.

But an observation limit doesn't automatically give us the limits what can actually exist in reality! When we assume it does, we are starting to see the incompatibility between Quantum Mechanics and General Relativity raising its ugly head. As discussed, Special Relativity (and General Relativity) trivially lead into "apparent singularities" when describing objects from wrong reference frames. This is one of the problems. Quantizing time relationships is another one. Based on the thought experiment I described above, we are looking at a very plausible possibility that singularities in Relativity theory can be simply a result of describing a situation from a wrong reference frame (note that when describing an accelerating frame, there is always an event horizon at some distance behind you - a clear indication of using a wrong reference frame. Likewise, the event horizon of a gravity well can only be defined from a specific reference frame).

Some hypothetical possibilities for Quantum Gravity

So, if the problem is that we are applying the principle of Relativity in places where it doesn't work, what can we do instead? Hopefully I have at this point managed to open your mind into multitudes of valid possibilities still open for us. From this point on, I'd wish to discuss somewhat hypothetical musings that I've been thinking about personally, but not developed very far. Maybe they spark some ideas in your minds for further developments.

The most obvious step that comes to my mind is to first apply a healthy dose of Mach's principle into this situation. EM waves / photons propagate coherently (they can't pass each other), so we should be modeling a uniform structure to support this propagation. I think it's naive to assume that reality is made out of decoupled "matter" and "space" (it's naive to consider these as two separated things). We can't really even define what is "matter", beyond it being a form of electromagnetic behavior itself. At this point already there is no clear discrete boundary between matter and space. Matter clearly can be in motion. If matter and space are coupled, think about what that means for the "rest frame of space"...?

The success of Quantum Field Theory already suggests that it's perfectly valid to model matter as merely excitations of space. This implies it is equally valid to model space as excitations of matter - think of matter as a harmonic wave center, and space as simply the extended "veil" of that "wave center", where the veils of all matter combine into single  coherent structure. You can think of this veil as the means for the harmonic wave centers to communicate (which they constantly do - we call it thermal radiation)

Thus we'd have a model where the existence of matter defines the rest frame for space (locally). This model gives you - potentially - a trivial explanation to otherwise strange phenomenon like frame dragging and anomalous galaxy curves (the matter of the galaxy yields a partially rotating rest frame for the space that is local to itself, thus requiring larger orbit speeds to the outer stars). But also it gives you the potential to combine quantum theory and gravity.

I realize that most people might not be well versed with Relativity enough to realize how relativistic time relationships fall into this picture, so let's discuss that briefly. First of all, Special Relativity is trivial, it's just a result of all natural observers and devices being electromagnetic systems; Basically exactly the argument given by Lorentz when when he developed Lorentz transformation. Mathematically identical to SR, philosophically just means you'd probably prefer one frame for your calculations (but up to the Planck limit you don't really have to).

For General Relativity, a full proof of General Relativity running with universal reference frame is little more involved, but described by Richard Stafford (who frequented this forum) in his book here; https://drive.google.com/file/d/0B-26v8xdJjFrNEZWMml5aUJCd1E (page 82 onwards for the critical bits).

Probably shouldn't be too surprising result though, as otherwise GR would require relativistic simultaneity -> static universe.

And here we finally get back to the connection to the Quantum Absorption interpretation - where I propose that the basis of the (apparent) quantum indeterminism is that all the excess energies that were not absorbed by electrons simply continue to propagate to the universe.

In Richard's book, from page 97 onwards, he discusses an interesting idea of gravity as refraction between pieces of matter. In principle any mechanism that produces jitter during energy exchange between microscopic objects, with magnitude that is inversely proportional the distance squared, would yield exactly what we call gravity (to understand exact details of why, you may have to go through the whole analysis, as it boils down to relativistic time relationships for objects with crooked paths). It would not be a force in itself, it's just that the jittered paths would yield refraction across the volume of the objects, that would bend their paths towards one another.

It is rather natural to consider the possibility that such jitter mechanism might be related to EM propagation, being that gravity shares some properties with EM radiation. Couple of interesting possibilities come to mind.

One is effectively a version Richard mentions. Neutrally charged macroscopic objects are only net neutrally charged - their microscopic elements are still charged, with interaction strength being inversely proportional to the distance squared. If the charge interaction events are quantized, then charged elements would be giving jolts to one another, with strength of the jolt becoming weaker by distance - this would refract the paths of the objects, and act like gravity.

Another possibility is that the ordinary black body radiation, after being absorbed by the surface of an object, will continue to travel under absorption threshold through the object. This wave in itself could possibly yield jitter to microscopic elements (that can't absorb it). Or potentially it could constructively interfere with other low-level energies and create virtual particles inside macroscopic objects (energy levels that do get absorbed). In both cases, the absolute (first case) or statistical (latter case) magnitude of the implied jitter would be inversely proportional to the distance squared.

All of these ideas are pure speculation, but I think they are rather good examples of somewhat plausible possibilities of marrying gravity with a quantum mechanical picture. The point is, there is no reason to presume that gravity has to be explained with a dedicated interaction carrier. There's no reason to muddy up the picture with Plank time problems. And there's no reason to constrain oneself to the traditional relativistic picture of time as a distance metric in itself (and thus vulnerable to Planck limits). These are all examples of getting rid of a purely philosophical trap that most of physics tends to get tangled up, simple because of insisting that unobservable things cannot exist. I think that idea is holding us back from finding simpler and better models, and I think I have now given few pretty good examples why, and how gravity could fall in line with Quantum Theory.

---------

Another paper about the topic

One last thing for anyone who made it this far. While looking for references to this post, I ran into a paper that brought up many of the same issues I'm bringing here, and I think it's really worth reading. His conclusions might be little different than mine, but the problem is exactly the same.

Enjoy it here;
https://arxiv.org/pdf/gr-qc/9610066.pdf

Does matter differ from vacuum?
Christoph Schiller

Abstract:
A structured collection of thought provoking conclusions about space and time is given. Using only the Compton wavelength \lambda = \hbar/mc and the Schwarzschild radius r_s = 2Gm/c^2, it is argued that neither the continuity of space-time nor the concepts of space-point, instant, or point particle have experimental backing at high energies. It is then deduced that Lorentz, gauge, and discrete symmetries are not precisely fulfilled in nature. In the same way, using a new and simple Gedankenexperiment, it is found that at Planck energies, vacuum is fundamentally indistinguishable from radiation and from matter. Some consequences for supersymmetry, duality, and unification are presented

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12 hours ago, AnssiH said:

These are all examples of getting rid of a purely philosophical trap that most of physics tends to get tangled up, simple because of insisting that unobservable things cannot exist. I think that idea is holding us back from finding simpler and better models, and I think I have now given few pretty good examples why, and how gravity could fall in line with Quantum Theory.

Hi AnssiH,

This reminds me of a thread on another forum (13,863 posts) that was basically about the difference between mind dependent reality (MDR) and mind independent reality (MIR) particularly with regards to scientific consensus.

In essence MDR is a 'now' concept at the core of the scientific method that appears to be short sighted and have no real understanding of its own evolution to the extent that it discounts the 'reality' of that which it doesn't understand (MIR).

In a similar way the requirement that our 'universe' expands into nothing leads our physics into the dim realms of dark energy and dark matter because we cannot currently measure anything outside of our light limited visible universe.

Lets just hope that some scientists start looking beyond the obvious limitations of the MDR.

https://forum.cosmoquest.org/showthread.php?153299-The-last-and-final-argument-about-reality

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10 hours ago, AnssiH said:

Now that's a pretty epic thread 😅

And yeah, ignoring all "stuff" outside of our observation sphere seems to be indeed a good example of the same philosophical trap...

-Anssi

When things are either too big to measure or too small to measure the end result is the same if you ignore what you can't measure, yet! 😉

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Thing is though. Ignoring all the mind stuff. There is no conflict between hbar and relativity, because not only can we utilise it in GR equations as we often have to without any choice. Take for instance the dispersion relation which is a facet of GR. Another example is the Einstein slope. There's so many reasons why we cannot invite the unification of quantum theory without inviting a sense of relativity.

Edited by Dubbelosix
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Posted (edited)

Hi Dubbelosix

Hmm, I'm not quite sure why do you mention these examples... By dispersion relation are you referring to the use of Planck constant in energy <-> frequency relationship?

And by "Einstein slope", do you refer to photoelectric effect?

If you had something interesting in mind about these specific examples, I'm interested of hearing some elaboration.

But in the meantime I'll just comment that of course Planck Constant appears in relativistic contexts often. But being able to use it in some circumstances is different from being able to use it in all circumstances.

For example, a relativistic circumstance involving photo-electric effect would be a case of a spaceship accelerating into high enough speed to start experiencing photo-electric effect on its own surface, induced by CMBR. Since the effect is described as an interaction between two elements, we get consistent expectations regardless of what reference frame we describe that situation from. Intuitively it's quite easy to understand too - the surface of the spaceship is really experiencing high energy radiation, no matter where any "true" frame might or might not exist.

Problems really only arise when we describe something where high energies would imply absolute effects to their environment (such as the formation of a black hole) - at that point we can hardly claim to reach those energies simply by choosing an appropriate reference frame. At that point those energy levels must be real only in the frame of their actual environment (which may well be local of course).

Btw, another little detail that I find curious. People often claim that classical physics lead into the so-called ultraviolet catastrophe, where the energy output of black body radiation approaches infinity towards higher frequencies. What that actually means is that someone is bluntly assuming there's  no maximum limit to thermal radiation frequency. That is already clearly a logical error, if we at the same time see thermal radiation as originating from the kinetic energy of the atoms making up some material - literally from jiggling motion of the elements yielding electromagnetic oscillations. That idea already places an upper limit to the frequency output of a body - the higher the frequency, the faster those elements must jiggle. Faster oscillations imply harder acceleration on each cycle, and that implies also higher energy requirement to drive the material into that point. At some point you would be pushing each element with more force than they can take without breaking off from the material - clearly there cannot exist an arbitrarily high energy potential in those jiggling elements, even if they acted as continuous energy storage.

The point is, there never was any reason to expect ultraviolet catastrophe, other than staring too much at the math and not thinking what kind of mechanism might be behind it. The same mistake is happening in many areas of modern physics today.

That is not to say that Planck's Law is unnecessary - it is clearly an improvement. The only problem is in its common interpretation. It seems far more likely that the discrete energy steps implied by Planck's law are related to a limit that exists in the energy absorption mechanism of matter (which is also true in traditional view of physics - electrons are viewed as some kind of harmonic wave system), but not a limit that exists in "all things in reality" (this is included as redundant assumption in modern physics).

And while it's so popular to claim that "nothing beyond Planck limit can exist", at the same time our physical models are full of other things that cannot be directly observed, and do not necessarily exist (like photons). Which just goes to show how strongly physical theories still are, and always have been, social constructions.

-Anssi

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On 4/25/2021 at 8:47 PM, AnssiH said:

The most obvious step that comes to my mind is to first apply a healthy dose of Mach's principle into this situation. EM waves / photons propagate coherently (they can't pass each other), so we should be modeling a uniform structure to support this propagation. I think it's naive to assume that reality is made out of decoupled "matter" and "space" (it's naive to consider these as two separated things). We can't really even define what is "matter", beyond it being a form of electromagnetic behavior itself. At this point already there is no clear discrete boundary between matter and space. Matter clearly can be in motion. If matter and space are coupled, think about what that means for the "rest frame of space"...?

The success of Quantum Field Theory already suggests that it's perfectly valid to model matter as merely excitations of space. This implies it is equally valid to model space as excitations of matter - think of matter as a harmonic wave center, and space as simply the extended "veil" of that "wave center", where the veils of all matter combine into single  coherent structure. You can think of this veil as the means for the harmonic wave centers to communicate (which they constantly do - we call it thermal radiation)

Thus we'd have a model where the existence of matter defines the rest frame for space (locally). This model gives you - potentially - a trivial explanation to otherwise strange phenomenon like frame dragging and anomalous galaxy curves (the matter of the galaxy yields a partially rotating rest frame for the space that is local to itself, thus requiring larger orbit speeds to the outer stars). But also it gives you the potential to combine quantum theory and gravity.

Related to the idea above, there's a nice little rundown of some of the problems of a Dark Matter hypothesis here (starting from 5:11):

Think about those problems from the perspective that what we call space is just an extension of matter - basically defining the (local) propagation speed of EM radiation.

The anomalies mentioned here seem to be fairly easily explainable by that idea, as the rotation of a large galaxy cluster would influence the propagation speeds of EM radiation in its vicinity (and thus how the influence of gravity itself is felt). The question is, is there a simple rule for describing how extended parts of matter "amalgamate" (as in, what is the contribution of the rotation galaxy itself vs the contribution of distant galaxies), that would give us exactly the results we observe, straight out of visible matter distribution. (I find the direct correlation of deviations in the galaxy rotation curve, and visible matter distribution rather interesting...)

Seems like this line of reasoning opens a possibility to develop a serious contender for MOND and Dark Matter hypothesis, but I've never heard anyone try... And since I've never heard anyone thinking about it, I've also never heard anyone say anything to rule it out. 😄

-Anssi

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 Don't get me started with Sabine. She's a cowboy physicist who gets things wrong a lot. I also discovered back in 2019 when I hypothesised dark matter was in fact a drag effect of gravity, she had gone and published a paper the same year saying it was a gravitational drag of Baryons. I don't like her because she did something similar to a physicist friend of mine where she literally took ideas from conversations they had. She doesn't give credit where credit is due and she only spouts what she learns like a parrot what she learns. But doesn't seem to add anything of significance to physics. Any lecture I've watched by her is incredibly boring. I also asked her a question a while time back, and she demanded money from me, and by then I knew what only she was interested in.

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Yeah well, I don't really know anything about her so can't say much there, but I do share the "well trained parrot" sentiment about pretty much all Youtube physics channels I'm aware of. I mean, I don't expect pure gold from everywhere and I appreciate the difficulty of translating complex concepts into simple explanations, and maybe it's often a case of poor communication, but most of the time anything I watch makes me feel like they really have very shallow idea of what they are talking about, and probably end up spreading more misinformation than actual information.

Anyway, the reason I was referring to that video was the nice rundown it contains about of some problems of Dark Matter hypothesis. I have no idea why her conclusion is to think of both MOND and Dark Matter as valid ideas.... How about considering neither as valid?

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On 4/26/2021 at 8:11 AM, LaurieAG said:

Hi AnssiH,

This reminds me of a thread on another forum (13,863 posts) that was basically about the difference between mind dependent reality (MDR) and mind independent reality (MIR) particularly with regards to scientific consensus.

In essence MDR is a 'now' concept at the core of the scientific method that appears to be short sighted and have no real understanding of its own evolution to the extent that it discounts the 'reality' of that which it doesn't understand (MIR).

In a similar way the requirement that our 'universe' expands into nothing leads our physics into the dim realms of dark energy and dark matter because we cannot currently measure anything outside of our light limited visible universe.

Lets just hope that some scientists start looking beyond the obvious limitations of the MDR.

https://forum.cosmoquest.org/showthread.php?153299-The-last-and-final-argument-about-reality

Today, we don't really say it expands into nothing, simply that there is no outside to the universe. It is self contained we say, and because of that, it expands in itself, instead of in nothing.

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But who knows right? It's not as if anyone can be outside from the inside looking in? There is another idea that our universe has a boundary floating in higher dimensions, some say it completes several problems by inviting higher dimensions. Others think of you have only four and you can explain reality with those four dimensions, then that should suffice.

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The seven or ten. To just our usual three or four is one of the hottest debates in physics. I did learn independently that if bivector gravity was the correct model, it can only be applied I'm either three or seven dimensions. It was such a stringent result that it almost convinces me it was either, on the fence so to speak.  This because the cross product in bivector gravity will only have solutions for either three or seven dimensions and no more.

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This is somewhat off-topic, but regarding the dimensionality question, it seems pretty clear to me that dimensionality is fully open parameter of our world model. To explore that thought more deeply, you might be interested of checking out the that analysis I referred to in the opening post
 

I realize it's not read and understood in a jiffy, so to give a hand-wavy rundown of the relevant bits here; it's an analysis for general data mapping, exploring some general constraints that come out purely from self-consistency requirements of a prediction function. From that point you can get effectively all of modern physics simply by choosing your representation form out of some arbitrary choices (meaning the other choices would be just as valid, just semantically different).

When it comes to dimensionality, purely from data plotting point of view it's trivial to see that dimensionality is a fully open parameter for your data plotting.  (See p. 124) That's why there exists different models with different dimensionality. The identities of "objects" will become somewhat different, and the rules of behavior (or at least their approximations) will look different, but any data that can be mapped into any dimensionality picture.

Less than three spatial dimensions seems to yield very complex rules, but it can certainly be done.

More than three spatial dimensions can give you elegance in some rules, but the identifiable objects become much more complex.

In fact from p. 140 onward there's a description of data mapping scheme where you map all the data (n data points) from the entire universe into a single n-dimensional object. In this case, all the possible dynamics can be represented by rotations of that object. Which may or may not be useful for representing some situations. But the point is, it is equally valid as a traditional 3 dimensional picture.

It might be that for our intuitive world model handling, the 3 dimensional picture is simply a nice compromise between two types of complexities.

Also, when it comes to orthogonality of space, that can be also viewed as an arbitrary choice of a world model. Orthogonal coordinates just means you are plotting data parameters that are independent from one another. Other types of coordinate systems are representations of dependencies between data points.

General relativity is exactly that type of model. From p. 82 onwards there's an orthogonal space version of all the dynamics that GR represents. Einstein's proof of requiring a four dimensional space-time construction doesn't hold if you flatten out one dimension in the final result (which is already flattened out for multitude of other reasons that are someone what out of the scope of this post... just let it be said that positions are not a knowable / relevant property of that dimension in his picture, by the very definition of the information it holds)

Sooo, when people so hotly debate about what dimensionality choice is "real", they are really only debating about their beliefs. That's not very scientific. Our intuitive world models have already defined what are the identities of some objects and how they behave. Getting the dimensionality of that picture is just reverse-engineering of "choices that were already made". Different choices would have yielded different looking picture - the look of that picture would not prove the dimensionality choice was "the only possible one".

-Anssi

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20 hours ago, AnssiH said:

In fact from p. 140 onward there's a description of data mapping scheme where you map all the data (n data points) from the entire universe into a single n-dimensional object. In this case, all the possible dynamics can be represented by rotations of that object. Which may or may not be useful for representing some situations. But the point is, it is equally valid as a traditional 3 dimensional picture.

It might be that for our intuitive world model handling, the 3 dimensional picture is simply a nice compromise between two types of complexities.

Also, when it comes to orthogonality of space, that can be also viewed as an arbitrary choice of a world model. Orthogonal coordinates just means you are plotting data parameters that are independent from one another. Other types of coordinate systems are representations of dependencies between data points.

That reminds me of the matter ratio's in the Lambda CDM model results, albeit not GR, when you consider a 'flat' universe modeled on the surface of a sphere. In both the PLANCK and WMAP data the ratio of total calculated universal matter percentages (dm + vm) divided by vm equals 2*pi +/- 1.1% (i.e. the relative density of all matter to all visible matter equals 2*pi).

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