I did not know that problem, even so, to a theory that would provide a signature ++++, rotatable 4D vectors and invariance of c, would not affect it, would not it?

Dear Ed,

Hamilton Quaternions solve, as you said, the gimbal lock problem introducing an additional degree of freedom.

This would suggest us that nature is expressible with at least 4 degrees of freedom instead of 3. Although I really don't see how the 4th can possibly be related to space-time since time in this case does not enter the game as the additional degree of freedom in the quaternion "problem"...

This additional degree of freedom digs in the "mystery" of the angular momentum conservation and the pendulum plane independence (Foucault pendulum).

Yes but the movement is not given by time, time is just how we perceive the evolution of phenomena through sequences of events, in agreement with what are called "spontaneous transformations"....

time has to be an external variable, as Carlo Rovelli supports in his theory of loop quantum gravity...time does not have an ontology... although we usually give it to it....

The gimbal lock is a technical problem which has nothing to do at all with SR in spacetime interpretation vs. the Lorentz ether. If you have a device which has a gimbal lock, you will have it in the 3D + time description as well as in the 4D description. And if you solve the problem with, say, a forth gimbal, you solve it in the 3D + time description as well as in the 4D description.

All the other points you mention have also nothing to do with spacetime vs. Lorentz ether desciption. So, the mathematical expression of 4D curvature used in GR does not need any embedding in some higher (5, 6 or 7)-dimensional space but only the 4 dimensions. The information which is locally available at a given event is also available in 3D plus time.

Your (4) is really funny. I read this as "I'm right, search yourself for the proof".

The situation remains unchanged, you have no empirical evidence against the Lorentz ether, thus, no empirical evidence in favor of 4D spacetime really existing.

I need no empirical evidence, because the gimbal lock happens for similar constructions in the 4D spacetime too.

I have not claimed empirical evidence for 3D reality really existing, 4D remains a theoretical possibility. But you have no empirical evidence against 3D only reality. So, your claim that there exists such evidence is wrong.

So what? It is a technological problem with devices which manage rotations. The group of rotations SO(3) is, by accident, also 3-dimensional, and this is the only connection with 3D space. And, btw, the quaternionic solution does not need 4D, because all it needs is the unit sphere of quaternions, which is a three-dimensional sphere.

Anyway, the only mystery is why you think this gimbal lock has anything to do with the 4D spacetime interpreation of relativity vs. the Lorentz ether.

It is S^3, and is a three-dimensional manifold. It is the unit sphere in R^4. The sphere in our three-dimensional space is two-dimensional.

The importance of Quaternions in quantum mechanics is well known. The Dirac matrices are quaternions, they describe the spin of the charges for example...

Any speed extrapolates outside the experimental data range. The proposition may be proven for low speeds, but not necessarily for extreme high speeds. Gimbals may not be able to function at extreme speed, in which situation a different technology would be needed.

Any number of gimbals can experience a locked pair position. When four or more gimbals are used then a locked pair in a system of three angular control functions can be arbitrarily unlocked.

At extreme speeds time may pass differently for each gimbal, assuming the gimbal is able tp operate at such speed.

I believe you have proved the point, but don't understand why a proof is needed, considering that nearly every researcher accepts the 4D+. Maybe a physical proof is preferred, but Minkowski was convincing enough.

Jerry, can you please explain me where you see a "proof", given that the Lorentz ether interpretation, which works also for the Einstein equations of GR (see http://ilja-schmelzer.de/ether ) shows (and essentially proves) that one can nicely live with classical 3D space filled with matter changing in time, with time only as a parameter?

Note that this question is quite different from the question if the Lorentz ether is a preferable interpretation. For the point that there is no proof for a necessity of 4D it is sufficient that an interpretation without 4D exists and is viable.

Dear Ilja,

since a complete Lorentz electrodynamics theory is missing, while SR is the argument of most of the text books, it would be the case that you write something about it and clarify some aspects and differences, since there are clear differences....

Ed, your arguments I have seen and discussed here and in several other discussions. I don't think you have presented anything that would show that the Lorentz ether has a problem. I have seen that there are two more answers in the quoted discussion [16], they give nothing new. [17] is on vixra, and looks like it is not on arxiv for good reasons. So, these is no reason for accepting their claims.

I have always said that there are solutions of GR where the Lorentz ether can be introduced only locally, like wormholes, closed causal loops and other science fiction solutions of GR. We have not observed them yet, thus, this does not endanger the viability of the Lorentz ether. And as long as the Lorentz ether is viable, 3D is sufficient, and 4D an unnecessary addition.

Stefano, I think all you need about the generalization of the Lorentz ether to gravity you can find on my homepage http://ilja-schmelzer.de/ If you have questions, please ask them.

The most important difference to the ether of Lorentz time is that it is no longer electromagnetic only, but universal: All fields which have a c in their equations, that means, all fields we know, describe properties of the ether, and there is nothing but the ether.

Again, I think I have answered all arguments of [16] resp. [17], at least all what I have been able to identify as an argument.

Why you think the Mahsroom article is in anyway relevant for this discussion is beyond me. It is about something completely different. To quote: "the hypothesis of locality posits the local equivalence of an accelerated observer with a comoving inertial observer". Compare with your "any physics model that uses only one inertial frame, or even an infinity of inertial frames [16], can only be valid locally". You obviously seem to confuse observers (which is what the article is about) with frames. So again: Any system of coordinates (frame) describes completely the whole physics, including all observers. The only thing which connects observers with frames is that in an inertial frame the inertial observer moves along an especially simple trajectory, a straight line. That's all.

Dear Ilja Schmelzer

since you are one of the few experts in the world about the LET, I invite you to reply to this question:

https://www.researchgate.net/post/The_trajectory_of_light_and_the_time_dilation_in_Lorentz_ether_theory_and_in_Special_Relativity

thank you

A new preprint for this question, with a shorter title, is at:

Preprint On Proof, At Any Speed, of an Underlying 4D Spacetime

Perhaps relevant, see also:

Preprint The Einstein Phenomenon and "fake news"

Although almost all those points, and others not listed, are in my RG publications, I tend to see things through QM.

Reality is observer-dependent, in QM. Starting with the Heisenberg principle, observer and experiment cannot be dissociated. There is no objectivity in QM -- but there is a coherent abstract view one can pursue in QM.

Now, from that common, coherent, abstract view in QM, one can pursue instances of a subjective, intersubjective, and objective natures, and mixed, ordering them into a scale, even contradictory to themselves. It is a cornucopia of choices! All validated by the same common, coherent, abstract view that has the "seed thought" of the subject. It is like a manifold, connecting different domains to one another, although no connection is seen directly between them.

QM and SR are necessary for EM, so its easy to have RQM -- it is a "match made in heaven". Gravity is modeled by extrinsic curvature of spacetime in GR, when QM is utterly ignored, incompatibly, but more than that, it is of a different type, in HoTT and TCS terminology.

In this abstract view, one is concerned with testable relationships, one is focusing on the essence of a hypothesis in the Scientific Method. A hypothesis is not a guess, a hypothesis is a testable relationship, no more and no less.

Accordingly, SR is not a guess, but a testable relationship.

In that abstract sense, SR passed the QM test and the EM test, and yet it MUST be observer-dependent, which seems to contradict several old "principles", but actually extends them, it does NOT deny them. When I summarize, for example:

"Comoving observers NEVER measure time dilation or length contraction, whereas non-comoving ALWAYS do. It does not matter the acceleration, nor the velocity."

then goodbye to "relativistic speed" or "inertial observer", they are limited to a platonic model, a human-made domain, not Nature.

In Nature, an "inertial observer" can see different realities, whether comoving, or non-comoving; reality can be the same also, no matter the acceleration, nor the velocity.

What really distinguishes observers, is whether they are comoving or not. Here, as normal in Physics, comoving is a technical word, and means "same position and velocity", not necessarily identical objects; in cosmology comoving observers see the same wavefront, so it is coherent with other related fields.

Depending on the SR formulation as the stance that one uses, even no SR, more or less "paradoxes" will result. The less "paradoxes" the better. The same with EM, thus we want to eliminate what is not necessary. Can the Maxwell equations be excluded in EM? Other questions follow.

Ilja Schmelzer

The 4-D of Minkowski gives a mathematical method of making predictions that are confirmed in experiments and observations.

Physics tells us that our 4-D does not give the complete description of everything.

https://www.researchgate.net/post/Can_one_at_the_same_time_accept_special_relativity_SR_by_Einstein_and_non-SR