a-cosmology-group/acg#99

This is a calculation based on [how a clock’s velocity affects the clock’s ticking]. Same calculation as in the Hafele-Keating experiment’s “special relativity” component.

My goal is to first establish how much or the maximum a clock drifts by walking it 3km (bounded here to << 1ns, right?), and then send a beam each way and time each way. Sending a laser one-way and timing it, back and forth, and even better, all x y z dimensions.

I suspect https://en.wikipedia.org/wiki/Einstein_synchronisation is a problem. It prevents measuring a change. Similar calibration error to how Big Bang cosmologists miscalibrate supernovae light curves. They distort the data and then claim that proves space is expanding faster than the speed of light.

It’s not clear to me if you want to do an experiment or repeat a calculation…

Transporting slowly an accurate atomic clock will disturb it and make it lose accuracy. Transporting it quickly will simply repeat experiments that have already been done.

If it’s a calculation you want to make, then I could only echo something that was said some time ago: “tes calculs sont corrects, mais ta physique est abominable…” 🤣

Why are you shortchanging this idea?

This is about getting a sense of how much transporting a clock affects it. Which I think we agree is << 1 ns in this case. And then from here, it is possible to measure the one-way time of flight without the distortive Einstein synchronization calibration.

It’s not possible to measure the one-way speed of light this way.

Firstly, it’s circular: you need to have some idea of what it means for two clocks to be in sync - you need to pick a synchronisation convention to talk about this. Every synchronisation convention is equivalent to a description of the one-way speed of light - for example if you use Einstein synchronisation then the one-way speed of light is c. (To derive this, imagine sending light at a mirror one light-second away. You receive the reflected signal 2s after you send the initial signal. By Einstein synchronisation convention, the light hit the mirror half way through that, I.e. After 1s. So, it took 1s to travel 1 light-second in both directions, hence it travelled at c is both directions.) So, by establishing what you mean by synchronized, you’re already assuming the one-way speed of light.

You have an experiment that I think goes something like: have two clocks in sync at the same location, slowly transport one clock to a distance, measure the one-way speed of light across that distance, using the reading of the two clocks. If we know how out of sync the moved clock is due to time dilation, then we have a measurement of the one-way speed of light.

However, your time dilation factor of sqrt(1 - v²/c²) when used this way already assumes Einstein synchronisation convention. If you look at how that time dilation factor is derived it assumes that light travels the same speed across each direction of the zig-zag.

Basically it’s impossible to calculate how out of sync your remote clock has become without a synchronisation convention, and using that convention and then trying to measure the one way speed of light will always give you the one way speed of light mandated by that synchronisation convention. Your calculation is using Einstein synchronisation, which assumes the one-way speed of light is C, and that’s exactly what you’d get if you measured it.

As a toy example, suppose we had the convention where light travels at c/2 East, and instantly West. (Special Relativity allows for this.) We conduct your experiment. For this case, the time dilation factor in the East direction is actually sqrt(1 + 2v/c). (You can calculate this using the same light clock example from the Wikipedia page I linked.) So if I move one clock distance d away to the East at a speed of v let’s say, then it will be out of sync by d/v × (sqrt(1 + 2v/c) - 1). As v goes to 0, this tends to d/c, so even if you slowly transport the clock, your clock will be out of sync by at least d/c. If you slowly move the clock 1 light-second away, then it will be behind by around 1 second. Now do your measurement of the one-way speed of light. You’ll find that the difference between the time received (according to the remote clock) and the time sent (according to the local clock) is 1 second. This is consistent with our calculations: We assume that light travels at c/2 eastwards, so it takes 2 seconds, but the clock is behind by 1, and 2-1=1 as required. Similarly, if you transported the clock West, it would actually get time dilated the other way, and would be ahead by 1s. We assume that light travels instantly West, so it takes 0s, but the clock is ahead 1s, and 0+1=1 as required.

It turns out that no matter what experiment you do, it will be consistent with the calculations for any synchronisation convention you choose, as long as you do the calculations carefully.

@chriscoomber I like the way you explain that the one-way speed ultimately depends on your definition of simultaneity. Indeed, this is the crux of relativity. However, I do not agree with:

As a toy example, suppose we had the convention where light travels at c/2 East, and instantly West. (Special Relativity allows for this.)

In my understanding, relativity tacitly assumes space homogeneity and isotropy in its only required postulate: “Physics is the same in all inertial frames of reference.”

@ftessier Yep, you’re right. I had it in my head that this convention was still left undeclared in SR, and that SR only concerned itself with what you can measure and experience (which is indeed the same in every inertial frame, and for which you can make any choice about synchronization and get the same answer). But actually SR is a model that heavily uses Einstein synchronization convention - it’s needed for SR’s only postulate and all of the math that follows. (Is that right? I’m not a physicist.)

Nevertheless, I think my comment holds. In @sahil5d attempting to use the time dilation factor of sqrt(1 - v²/c²) to calculte how synchronized their clocks are, they’ve already assumed Einstein synchronization, and therefore that the one-way speed of light is c. They’re not going to be able to devise an experiment that shows the one-way speed of light to be anything other than c. (Unless SR is wrong or they’re hitting up against the limitations of SR.)

Einstein synchronization rests on the second postulate, that the speed of light is constant in all frames. Einstein allegedly recognized that this postulate was superfluous, but decided to keep it for the sake of simplicity in his derivation. And there was already a physical hint behind this ansatz, given that from Maxwell’s equation a wave velocity $\large c = 1/\sqrt{\epsilon_0 \mu_0}$ is derived, without having specified a reference frame.

However, it is possible to derive the Lorentz transformations from the first postulate, without the synchronization convention. You only need to assume homogeneity and isotropy (implied in first postulate) and require that the transformation to and fro the boosted frame is an identity. You obtain the Lorentz transformation with some maximum velocity, the value of which has to be determined experimentally.

Note: the derivation is simple in 1 dimension, but very challenging in 3d (I have yet to see a convincing proof). Perhaps that is why Einstein opted for the second postulate?

I rederived for myself (pdf attached) the article Nothing but relativity by Palash B. Pal published in the European Journal of Physics in April 2003. Mermin also famously derived this form the addition of velocities in his 1984 paper Relativity without light. Pal’s derivation is much simpler.

Reading it again, I note my comment that there is another assumption underlying the derivation: that the speed of frame $S$ in $S’$ is equal to $-v$, the inverse of the speed of $S’$ in $S$. This is fully consistent with the principle of relativity, but strictly speaking it remains an assumption.

relativity-wihtout-light.pdf

(There are a few typos in my version, notably some equation references are mixed up. Left as an exercise to the reader to untangle 😉)

I’m not using or depending on any sort of Einstein synchronization, which I think is a miscalibration. @chriscoomber

The calculation which I asked for feedback on

So Clock2 is slower by ~ 8.3 * 10^-14 seconds << 1 nanosecond. Is this correct?

only depends on an idea of [how a clock’s velocity affects the clock’s ticking]. No calibration, no space stretching or squishing, none of that. What I think is going on (and I’m not the first to propose this) is that the internalcirculation of an atom (or any particle) slows when the atom speeds. Both

• the [muon moving at .98c through the atmosphere and having a longer lifetime]
• the [Hafele-Keating experiment’s clock drift component from moving at the airplane’s velocity]

can be consistent with this idea.

For example, when two gamma photons collide and transform into an electron and positron, I think it’s not unreasonable to imagine that each photon is somehow going in circles, and that’s what a particle is. That’s what I’m referring to with ‘internalcirculation’.

The calculation which I asked for feedback on

So Clock2 is slower by ~ 8.3 * 10^-14 seconds << 1 nanosecond. Is this correct?

only depends on an idea of [how a clock’s velocity affects the clock’s ticking]. No calibration, no space stretching or squishing, none of that.

Clock2 is slower in what sense? What does it mean for two clocks that aren’t at the same frame of reference to be synchronized? That’s where you’re using Einstein synchronisation. As an example I gave an alternative synchronisation which would give you a totally different number of how much slower Clock2 is.

Also, when you use the formula sqrt(1 - (5m/s / c)^2) you’re using the time dilation formula from Special Relativity, which depends on either Einstein Synchronization or isotropy (see discussion above), both of which imply that the one-way speed of light is equal to the two-way speed.

But your calculation was correct I think, with all of those things assumed. And if you use your calculation to try to measure the one-way speed of light you’ll get C. But we already knew it would be because that already followed from the assumptions you used for that calculation.

Have you ever used 3D modeling software, like Blender? https://docs.blender.org/manual/en/latest/editors/3dview/startup_scene.html Consider a Blender scene where objects are in space and have properties.

Yes, I use Blender in my work.

Let the Blender scene be the absolute stillness frame, and the scene’s grid be absolute length, and the scene’s timeline be absolute time. Objects in the scene have an internal ticking. If an object has 0 velocity, it ticks at the same rate as absolute time. If an object has velocity, it ticks slower. Also, if an object is close to other objects, it ticks slower.

Also, I realize I made a mistake in the calculation at the start of this discussion.

If the park has an absolute velocity v_abs and is in the same dir as the path (most perceptible change), then the calculation is

Clock1 will have ticked for 3km / 5m/s * sqrt(1 - (v_abs / c)^2) Clock2 will have ticked for 3km / 5m/s * sqrt(1 - ((v_abs + 5m/s) / c)^2)

And the difference is not ~ 8.3 * 10^-14 seconds but instead

~ 3km / 5m/s * 1/2 * 1/c^2 * ((v_abs + 5m/s)^2 - v_abs^2) ~ 3km / 5m/s * 1/2 * 1/c^2 * (v_abs^2 + (5m/s)^2 + 2*v_abs*5m/s - v_abs^2) ~ 3km / 5m/s * 1/2 * 1/c^2 * ((5m/s)^2 + 2*v_abs*5m/s) ~ 3km / 5m/s * 1/2 * 1/c^2 * (2*v_abs*5m/s) ~ 3km * v_abs / c^2

Notably, v_abs shows up in the result. Clock2 is 3km * v_abs / c^2 ticks behind Clock1. This quantity also is the extra time of travel expected for a signal from Clock1 to Clock2. This is why v_abs can’t be derived from this approach. This doesn’t prove there is no v_abs. This only proves v_abs can’t be derived from this kind of experiment.

This doesn’t prove there is no v_abs. This only proves v_abs can’t be derived from this kind of experiment.

Yep, not from this kind of experiment nor from any kind of experiment designed so far. Glad to see we agree 😉

I agree with regard to this kind of experiment not being able to measure v_abs. I disagree with the additional claim that [because this kind of experiment can’t measure a v_abs that there is no v_abs at all, and therefore one can say a muon moving at .98c into the atmosphere is actually a muon moving at 0 m/s and rather all 5.5 quadrillion tons of atmosphere are moving at .98c]. That I disagree with.

I am now researching coupled shutter experiments. They’re essentially a Fizeau experiment with two wheels, right?

I think they can be refined and have a high likelihood of measuring v_abs.

The part that requires care is making sure the axle is rigid (or that there is no differential torsion, which is something that can be minimized to 0).

I disagree with [because this kind of experiment can’t measure a v_abs that there is no v_abs at all, and therefore one can say a muon moving at .98c into the atmosphere is actually a muon moving at 0 m/s and rather all 5.5 quadrillion tons of atmosphere are moving at .98c]

Your wording assumes an absolute frame of reference. Galilean relativity says

1. the muon is moving at 0.98c relative to the atmosphere,
2. the atmosphere is moving at 0.98c relative to the muon,

are indistinguishable.

These statements do not say [a muon moving at 0 m/s] nor [tons of atmosphere moving at 0.98c] because a velocity is always given with respect to something else. You seem to erroneously believe in the “Blender absolute reference frame”, thinking that it expresses physical properties…

1. the muon is moving at 0.98c relative to the atmosphere,
2. the atmosphere is moving at 0.98c relative to the muon,

are indistinguishable.

I think that is false. They are distinguishable.

Firstly, there are cosmic ray muons going thru the atmosphere, all sides of the sphere of the atmosphere, and the atmosphere is not zig zagging in all spherical directions at .98c to please each new muon’s ego.

Secondly, a photon (consider from a typical cellphone signal) can go +3*10^8m/s and -3*10^8m/s wrt to the atmosphere, but not so for each cosmic ray muon. A photon can go +.06*10^8m/s (+ meaning same direction as) and -5.94*10^8m/s wrt to the muon.

@ftessier this is indeed a fine piece of mathematics by yourselves and Palash B. Pal! Another disclaimer: I am not a Special Relativity dissident. Even though I might not agree with the premises, the mathematical outcome is valid, and also I have learned not to disagree with Einstein :). My own work on aether supports SR.

I will comment on this: “We do not suppose anything about the speed of light, and rely solely on the assumptions of spacetime homogeneity, space isotropy, and the principle of relativity, which states that the laws of physics are the same in all inertial frames.”

Although Palash’s derivation didn’t require the speed of light, the assumptions above are very similar to “speed of light is same in all directions”, since a homogenous and isotropic spacetime is achieved by exactly that.

This is a hard case to argue as many have tried before and failed. I will therefore not try too hard here.

To only elaborate on my stance: Bob has a machine that converts coal to velocity. It takes 1 lump of coal (very precise coals these are) to achieve a velocity v. Bob goes ahead and burns 1 coal. He feels himself accelerating and concludes he is now at velocity v. Alice observed all this, and while ‘standing still’ and knowing for certain that they are not in a gravitational well, nor did they accelerate themselves, agrees that Bob is now at velocity v.

Bob does it again, and by their own maths concludes that they are now speeding along at 2v. Alice is a little perplexed and only sees Bob’s velocity increase by 1/gamma: $1/γ * v $

Thus far all can be explained by SR and even by Palash’s very clever derivation.

However, in real life, at some point Bob realises that their acceleration is not as impressive as before. This must be noticable when γ becomes large, and the increase in v is very small. Bob checks the coal and the machine, but all checks out ok.

Even much further, Bob realises that the lump of coal no longer gives any acceleration. (ok, they had LOTS of coal) Alice agrees. Bob’s velocity is no longer increasing, or maybe it is only increasing by such a small amount that Alice cannot measure anymore.

I guess this means, if I had to reply to some of your posts above and below, that I personally believe that “the laws of physics are the same in all inertial frames”, is an incorrect assumption. That Maxwell’s equations rule, and space itself is the ultimate limiter.

(I only rewrote Pal’s derivation for clarity, I did not participate in any derivation of special relativity transformation without the second postulate.)

I find the scenario your penned down interesting, as I never heard it expressed in that way before! But it involves acceleration, so I refrain from commenting: I am just an enthusiastic special relativity hobbyist 😄.

@Francois-Zinserling

However, in real life, at some point Bob realises that their acceleration is not as impressive as before. This must be noticable when γ becomes large, and the increase in v is very small. Bob checks the coal and the machine, but all checks out ok.

This is incorrect. In real life, Bob realises that their change-of-velocity is always the same $\Delta\vec{v}_B$ for each piece of coal. Bob checks the coal and the machine and all checks out ok (this applies to Bob who is moving at $\vec{v}$ relative to Alice).

However, Alice (in an inertial frame of reference) realises that Bob’s acceleration $\Delta\vec{v}_A = \Delta\vec{v}_B/\gamma$ is not as impressive, since when $\gamma = (1-v^2/c^2)^{-1/2}$ becomes large, $\Delta\vec{v}_A$ becomes small. To Alice, Bob is not accelerating as much, as if their inertial mass had increased by a factor $\gamma$. $\ $

Bob does it again, and by their own maths concludes that they are now speeding along at 2v.

This is incorrect. The mistake is to think velocity is additive. After Bob has burned $n>1$ pieces of coal, $\vec{v} \ne n \ \Delta\vec{v}_B\ $! When Bob does it again, their velocity increases by $\Delta\vec{v}_B$ relative to their previous frame of reference, but not relative to Alice’s frame of reference. $\ $

“the laws of physics are the same in all inertial frames”, is an incorrect assumption.

Yes, but if written more accurately,

the laws of physics are the same in any inertial frame, to an observer in that inertial frame

is a correct assumption.

@RedshiftDrift I agree that is what mainstream science teaches us.

However, we will always be Alice and never Bob, because we actually don’t have that many pices of coal. We are trundling through space at our own v<<c and are able to observe ‘things’ travelling at v–>c, but we cannot experience that for ourselves, so the ‘correct assumption’ is not a proven one.

Then you would also know about the countless paradoxes that cannot be explained without circular reasoning. And as @ftessier pointed out somewhere in this discussion, the twin that ages slower is the one that travelled. <- This does not even make SR sense because it loses it’s special relativity.

It makes sense with an aether because moving at v slows your own clock. Even gravity slows down clocks with a very peculiar t’ relative to the exact escape velocity where you find yourself. Thus if clocks are velocity dependent, then space is not ‘nothing’ and v_abs=0 must exist.

What I pointed out is that the twin paradox DOES make SR sense if you take into account that there are 3 reference frames, not two. It is an apparent paradox, which is resolved in SR.

What do you mean that we travel in space at a speed v much less than c?

the laws of physics are the same in any inertial frame, to an observer in that inertial frame

Except that there are no true inertial frames in our experience of physical reality, only approximately inertial conditions where SR considerations are adequate for some calculations. Even the weak gravitational field of the Earth requires GR corrections for GPS satellites.

In the case of the highly accelerated Bob in the current discussion, he has no stable inertial frame available - he can continue to accelerate (to diminishing effect} or he can decelerate. It is similar to someone bicycling up a steep hillside, they can peddle up or coast back down but the bike is not stable if they try to pedal sideways, parallel to the base of the slope. Bob can also be considered to be in an analogous situation to someone in a steep gravitational well - SR conditions do not apply, not even approximately,

GR conditions have asymmetric consequences unlike SR, where each observer sees the others clock as running slow but in fact, neither clock is locally altered. As with the case of the traveling twin, there are real physical consequences to having undergone acceleration to velocities approaching light speed with reference to the original “home” frame Bob shared with Alice. Both Alice and Bob perceive Bob’s velocity to increase less per unit of fuel, and his clock to be running slower than Alice’s - that’s just standard GR,

@Francois-Zinserling I’m not sure what “countless paradoxes” you are referring to but in my experience misunderstandings usually arise from failing to mind the distinction between SR and GR and/or attempting to cram the square peg of a “universal” framework into the round hole of Relativity Theory.

@ftessier I am referring to our local speed/velocity, which is an addition of earth’s rotation speed, orbit speed around the sun, sun’s speed around the Milky Way, and Milky Way’s apparent speed toward ‘The Great Attractor’. If I recall all together ~600km/s with some cyclic ups and downs. That’s about $c/500$.

@budrap00 yes, I was being overly dramatic. The day SR can tell me which person is moving and which person’s clock will really slow down, I will be at peace. Right now I live in the fear that someone will come racing past earth at v–>c and all our clocks will mysteriously slow down to –> 0. (no, I’m just kidding)

@budrap00 you agree that the GPS clocks conditions are not ‘observations’ only? The clocks really do slow down due to SR(velocity) and they do run faster due to GR(gravity). In the case of gravity, we could say the nearby mass slows the clocks, or we could by extension say (from GR) that mass curves space, and curved space slows the clock. This is a real effect, no paradox. GPS clocks are slowed less than our own here on earth surface (GR component).

What I pointed out was that if you are standing on the earth, your clock will be slowed (GR component) by the exact same amount as if you were travelling through space (SR related) at exactly the escape velocity from earth surface. Coinicidence? I think NOT!

So clocks are REALLY slowed down when there is a real velocity. <- I need someone to confirm or contradict this please.

According to SR, clocks do not slow down for someone in the inertial frame of the clock (no relative movement with respect to the clock). What is the velocity of Great Attractor?

Again, I point to the Lorentz transformations, which derive directly from the four given axioms; which one do you want to abandon? It is always fun to change axioms and explore the logical consequences.

@Francois-Zinserling

What I pointed out was that if you are standing on the earth, your clock will be slowed (GR component) by the exact same amount as if you were travelling through space (SR related) at exactly the escape velocity from earth surface. Coinicidence? I think NOT! So clocks are REALLY slowed down when there is a real velocity.

You are on the right track, but this is incorrect because you still assume that an “absolute velocity” is detectable. (Nobody has answered the challenge to design an experiment that would tell us what our absolute velocity is…)

However, your first point that “a clock is slowed down down near a mass” is correct, as long as it is clear that the clock is compared with a clock at a large distance from the mass. Also, you are correct in that the slowing down corresponds to an escape velocity (relative to the mass). But instead of thinking in terms of velocity, think in terms of the change of velocity (acceleration) that makes the clock tick more slowly. Also, in pursuing this idea, realize that an acceleration in direction $\vec{D}$ will cancel an acceleration in direction $-\vec{D}$ and the clock will be back ticking at the same rate once it has been “accelerated and equally decelerated”. By keeping track of the vectorial nature of acceleration (which is absolute) you can explain special relativity with clocks ticking more slowly (and rulers changing lenghts). This effect is the same as placing the clock near a mass where acceleration, gravitational this time, has the same effect on ticking rate.

This is a different interpretation of relativity that doesn’t change its equations. It doesn’t change any of @ftessier’s axioms, and (to me at least) is more acceptable than space-time distortions (and the “black hole” that is a consequence of those distortions). But even with that interpretation, there is no way to detect absolute motion.

Nobody has answered the challenge to design an experiment that would tell us what our absolute velocity is…

No, that is not so. There are experiment designs here, https://github.com/orgs/a-cosmology-group/discussions/99#discussioncomment-6533895. A rigid axle with the same torque applied at each end can be rotating in phase on each end. This is clock sync achieved mechanically. Then, a difference in time of flight can be detected using a spinning gear on each end. Like the light-chopping Fizeau experiment.

Several issues in the ideas you and @ftessier bring up are still pending. https://github.com/orgs/a-cosmology-group/discussions/99#discussioncomment-6535763

Also, I think there must be a distinction between the solipsistic aspects of relativity and the particulate aspects of relativity. @Francois-Zinserling this is connected to your question.

The solipsistic aspects are

• [the stretching and shrinking of lengths]
• [a photon can move +c or -c relative to anything with any velocity]
• [no absolute velocity exists]
• [no absolute space exists]
• [no absolute time exists]

The particulate aspects are

• [a particle’s velocity affects the particle’s ticking]
• [a particle’s gravity index affects the particle’s ticking]
• [a particle at near-c velocity, when energy is applied to it, has that energy go not into its velocity but into itself]

In my opinion, relativity is undisentanglable from its solipsistic aspects, and it should be dropped. Meanwhile, the particulate aspects are very much verified every day with satellites, with muons, with particles in cyclotrons, and so I’d like to suggest they be allowed to be described independently and referred to as cyclotronic phenomena or phenomena_cyclotronic.

There are a few other aspects of relativity, [gravitational lensing] and [gravitational redshift and blueshift]. I suggest they be extracted too. [gravitational lensing] and [gravitational redshift and blueshift] can be gravity index phenomena or phenomena_gravityindex.

There has been research on modeling gravity as a refractive index, which I think is promising because photons do slow down and bend more in thicker gravity.

@Francois-Zinserling The day SR can tell me which person is moving and which person's clock will really slow down, I will be at peace.

The answer to that is straightforward - it is the person who has changed velocity. Changing velocity involves in most cases an acceleration which is a non-inertial, non-symmetric event in SR terms. In cases like the twins, one twin remains on Earth and one accelerates away. The accelerated clock is effectively slowed by undergoing a non-inertial event.

You are trying to understand a highly constrained textbook example of two inertial observers moving at some constant velocity with respect to each other. Each observer considers themselves at rest and the other to be moving absent any other reference frame. This is not a realistic scenario.

Realistically there are always other local frames that can be used for reference. This is often taken to mean that SR assumes the absence of a universal frame but it is not so much an assumption as it is a feature of the model. It is, by any means, a lesser assumption than the counter-assumption that such a universal frame does exist despite the absence of any empirical evidence for its existence.

The clocks really do slow down due to SR(velocity) and they do run faster due to GR(gravity).

You’re just confusing a GR system with reasoning based on SR theory in which there are two inertial reference frames. There is no SR velocity between two inertial frames involved. The GPS satellite was launched into orbit from the rest frame of the Earth - it accelerated to its orbital velocity. An observer on the earth observes the satellite to have some orbital velocity - observers on a satellite observe themselves to be traveling with the same orbital velocity with respect to the Earth. There is no symmetric SR velocity between inertial reference frames because there are no inertial frames involved. As a general rule I would say that it is impossible to understand relativity theory by focusing on SR and trying to apply it to real world considerations where it does not apply. That’s where the “paradoxes” arise.

@everyone just to put it out there that I am not an SR nor GR dissident. I am quite familiar with the maths, also happy to accept the maths. My arguments are thus not against SR or GR, but at our own interpretations there-of. And of course, I like to push against boundaries .. I have been thinking more about @ftessier paper that he had linked, with the SR (and Lorentz) derivations without bringing the speed of light into the derivation. This is an even bigger confirmation that SR effects are due to properties of space, and not ‘the speed of light is the same in all reference frames’.

@budrap00 I always value your responses but today you are cheating :)

“The answer to that is straightforward - it is the person who has changed velocity.” This is is not from SR. There is no SR component that deals with acceleration or change in velocity. You cannot borrow from GR to explain SR. Cheating! :) Look at it this way, from a practical point of view: 2 reference frames with a relative velocity between them. Pure SR. A sees B’s clock as slow, and B sees A’s clock as slow. The SR theory does not hint at which slowing effect is ‘real’. From experiments we know at least one side is real. But the maths say they both see each other’s clock as slow. <- here’s another problem. Let’s say I am A and B sees my clocks as slowed. And let’s say by whetever our reasoning is, my clock is the one that is really slowed. (I will be the twin that ages less). That means my clocks actually slowed (even though I cannot determine it for myself). This is perfectly allowed in SR (and my interpretation) because I am the one moving through space. Here comes the next problem. If my clock is really slowed, (all my atoms are ticking slower) then all other (unchanged) clocks must immediately appear to run fast to me. For me to even ‘see’ B’s clock as slowed is a further contradiction.

“You’re just confusing a GR system with reasoning based on SR theory in which there are two inertial reference frames” It took me a very long time to come to terms with SR, and I don’t even profess to know it extremely well. However, I don’t think I am confusing GR and SR components. Here’s a brilliant GPS analysis. Neil Ashby of U Colorado This one focuses only on SR and GR adjustments to GPS Ohio

@Francois-Zinserling, I wasn’t cheating. I was just pointing out that you can’t get the results you desire (a “really” slowed clock) without introducing GR conditions.;)

This is indeed the problem:

2 reference frames with a relative velocity between them. Pure SR. A sees B's clock as slow, and B sees A's clock as slow. The SR theory does not hint at which slowing effect is 'real'.

The answer to the question of which clock is “really” slow is neither. The slowed clocks only appear slow to the remote observers. Neither one of them is in fact running slower that the other. The apparent slowness is a point-of-view phenomenon. It is analogous to the following situation:

You are standing at the edge of a large field and at the far edge you see your friend who you know to be the same height as you. Yet from your point of view your friend appears smaller than that. To your friend, you are the one who appears smaller. Your question in this context is which one is really smaller and the correct answer (as you well know) is neither. It is exactly the same type of situation under SR conditions, the clocks only appear slow to the remote observers. The difference in the SR case is, of course, that you cannot bring the clocks back together for comparison without violating the SR conditions (introducing GR effects).

From experiments we know at least one side is real.

I don’t know of any such experiments. Please cite one or more.

Let's say I am A and B sees my clocks as slowed. And let's say by whetever our reasoning is, my clock is the one that is really slowed. (I will be the twin that ages less). That means my clocks actually slowed (even though I cannot determine it for myself). This is perfectly allowed in SR (and my interpretation) because I am the one moving through space.

I’m afraid that is just circular reasoning - your conclusion is in the premise. You are also invoking, to support your argument, the ad hoc hypothesis of a causally interactive “space”, when there is no empirical evidence for its existence. The case of the traveling twin is not an SR situation where each sees the others clock running slower. There is no way to get SR to produce a result where the stay at home twin ages more slowly than the traveler (home clock runs slower that the traveling clock.) It is the act of leaving their shared frame by accelerating to some relativistic velocity that produces the “really” slower clock of the traveler.

The point again is that there isn’t a “really” slower clock under SR. To get a really slower clock you have to incur a non-inertial (non-SR) effect of some sort. That’s why, as Louis pointed out, the twins paradox isn’t a paradox - as long as you consistently apply relativity theory.

I really like your simile with a person’s apparent height as a concrete and tangible example of the relational or perspectival aspect of physics. I will certainly keep this one in mind when explaining the next time I explain SR. Thank you!

(Updated thought)

I don’t think that step (call it the “walk-back test” ?) will help test whether [Clock1 Clock2 diff] << 1 nanosecond.

Here are some cases. Now I’ll be specific about the direction of the possible absolute velocity v_abs and the direction of the 3km park path.

• If
  • v_abs >> 0 (possibly in the 100s of km/s based on the evidence of the CMB dipole and Distant Quasars dipole),
  • and the 3km path is aligned with v_abs,
  • then
    • bringing Clock2 back would undo the drift,
  • thus
    • the test isn’t helpful.
• If
  • v_abs >> 0
  • and the 3km path is perpendicular to v_abs,
  • then
    • the estimated drift from slow transport would be based on a velocity diff of not 5m/s but something << .01m/s (sqrt((100km/s)^2 + (5m/s)^2) - 100km/s),
  • thus
    • the clock drift one way would be even smaller than the ~ 8.3 * 10^-14 seconds estimated before, and even doubling that tiny drift would be <<< 1 ns, so the test isn’t helpful.
• If
  • v_abs = 0,
  • then
    • bringing Clock2 back would add ~ 8.3 * 10^-14 seconds to the estimated drift (moving at an absolute -5m/s has the same effect as moving at an absolute 5m/s),
  • thus
    • maybe it helps test, but still ~ 16.6 * 10^-14 seconds << 1 ns.

So, the walk-back test does not help test for [Clock1 Clock2 diff] << 1ns.

I think we can trust the calculation with [how a clock’s velocity affects the clock’s ticking]. That’s reliable.

In all three cases above, bringing Clock2 back to Clock1 results in a time difference of 166 fs between the initially synchronized clocks (the effect cannot be “undone”).

Instead of a walk in the park, clock transport is continuously tested at higher velocity with GPS clocks: Clock 1 from A to B, Clock 2 from B to C, and Clock 3 from C to A.

With Clock 1 and Clock 2 synchronized at B, and Clock 2 and Clock 3 synchronized at C, Clock 3 returns to A no longer synchronized with Clock 1. The time difference (hundreds of ns) agrees with relativity and is the same for any direction of motion of the Earth w.r.t. the CMB dipole or Distant Quasars dipole (and by the way, these dipoles disagree in magnitude and direction by $5\sigma$.)

The relativistic calculation only uses relative velocities of the satellites w.r.t. each other, and works for any “triangle” formed by the orbits of the GPS constellation.

I think that is not so, for case 1 and 2, because your description assumes no v_absolute possible. I think that’s making a similar mistake to the one you accurately criticize here, https://github.com/orgs/a-cosmology-group/discussions/96#discussioncomment-6409516.

In the GPS example, are those clocks Einstein synchronized? https://en.wikipedia.org/wiki/Einstein_synchronisation.

CMB dipole or Distant Quasars dipole (and by the way, these dipoles disagree in magnitude and direction

Yes, by 2x in magnitude (~ 400km/s and ~ 800km/s), and by 28 degrees. They’re still both interesting candidates for a rest frame, and they’re both in the 100s of km/s.

What I explain there is that a fourth physical phenomenon is responsible for the cosmological redshift. Here, I claim that general relativity works well for all experiments involving velocities below 1000 km/s.

The GPS example above is synchronized with clock transport around the triangle ABC, no Einstein synchronization is needed.

What I explain there

I referenced that because the essence was the same. In response to a question about {what potential flaws there were that biased the results…}, you wrote {Every paper based on that false model is flawed because their interpretation of x is limited to x false explanations…}.

On the GPS example, do you have a link to a more detailed description? How does the comparison from Clock3 to Clock1 happen? Have the satellites made an orbit? Also, what sort of satellites move east to west?

false explanation…

I meant “the explanation x does not solve the problem” (even if x might be a correct theory).

On the GPS example, do you have a link to a more detailed description? Y? Z? …?

Read the manuals and technical papers, the GPS system is too complex to be described in one place.

• NIST documentation,
• papers found by Google Scholar about GPS synchronization,
• more specifically, search “clock transport synchronization” GPS on google Scholar. It lists this paper that seems interesting (but I haven’t read it), written by Gérard Petit (data from all atomic clocks were sent to him as part of an international collaboration), and this one also seems interesting,
• many GPS experts attend the IEEE Frequency Control Symposium, which offers courses at their meetings.

You might object to the techniques that are used for clock synchronization, but in the end the GPS works without any reference to an “assumed motion of Earth w.r.t. some absolute rest frame”.

Can you expand on why you support it? Feel free to comment on Pal’s derivation relativity-wihtout-light.pdf and let me know where you disagree or what is unclear. Granted it is a principled approach (based on the relativity principle), but a tight one in my humble opinion. I am keenly interested in your comments, to help me clarify or buttress the derivation.

Maybe your idea will work if it leverages the other idea here https://github.com/orgs/a-cosmology-group/discussions/99#discussioncomment-6461277. @Francois-Zinserling

The [rigid axle to establish simultaneousness].

The key idea (which I am trying to figure out how to execute, and which I hope others do) is that a spinning rigid axle can establish simultaneousness at two points some distance apart. The axle revs can be counted as ticks.

I think the concern of [What if there’s a twist in the axle?] (a differential torsion, a part of the axle lagging the other, an axle segment out of phase) is solved by applying torque with two motors, one at each end. Even if one motor starts later than the other, a nanosecond later, or a minute later, or an hour later, with both motors running, the discomfort in the axle will diffuse. If the two motors are applying the exact same torque, any twist in the axle will soon go to 0. Right?

Rigidity is also limited by a maximum speed, the simultaneity problem remains in full force with “rigid” objects. This issue is fundamental, it cannot be resolved by clever engineering, I reckon. Again, review Pal’s derivation, which IS fundamental, and let’s identify the point where you stop agreeing with it!

The issue you bring up is [What if there’s a twist in the axle?], right? I think that is solved with two motors, one on each end, applying the same torque.

• If the motors start at the same instant (unlikely), then there will be no twist.
• If they don’t start at the same instant, when motor 1 starts, there will be a twist (because no axle is infinitely rigid), and then when motor 2 starts, any twist will diffuse.

The reasoning behind the 2nd point is from considering the cases, the twist increasing, staying the same, or decreasing. The twist definitely won’t increase, because no torque is applied in the opposite dir. So the twist might stay the same or decrease. But it might not stay the same, because there is no longer an asymmetric torque, so the molecular bonds restorative force will win out, and any twist will diffuse and decrease. So both ends of the axle will be in phase.

No, this is not what I bring up. What I am trying to point out is that there is a fundamental reason for the Lorentz transformations once the principle of relativity is postulated. This reason is simply demonstrated (and verified everyday, for over 100 years). A more convoluted example is not convincing to me, because it is complicating the problem while not addressing the simpler, more fundamental reasoning.

The thing verified everyday is that [an atom’s velocity affects the atom’s ticking] and [an atom’s gravity index affects the atom’s ticking].

The stretching or shrinking of lengths is false, and the idea that a photon can move +c or -c relative to anything with any velocity is also false https://github.com/orgs/a-cosmology-group/discussions/99#discussioncomment-6463012 and is reminiscent of the solipsism of the geocentric model of the planets.

Ok, then let us identify what is erroneous in Pal’s derivation attached above. That should be straightforward, since it is a very simple argument. Let me know what you think, I am keenly interested in finding the misstep.

@ftessier I will read the pdf and comment. I liked your comments regarding the 1-direction of light tests.

As for my support toward an existence of V_abs

1. Michelson Morley did not disprove the existence of aether. Their failure to detect it does not mean it does not exist. Even by your arguments above, proving that we cannot detect it, still does not disprove its existence.
2. In my pursuit of push-gravity, it emerges that Fatio’s Flux must exist, not in the form he anticipated, but that it must be the elusive aether. For me personally this was even a bigger ‘aha’ moment than figuring out how gravity works. This is personal though, until I can give more strength to my arguments.

An aether paper is in the works.

What’s erroneous is pages 1 and 2, which repeat the same false assumptions that [a photon can move +c or -c relative to anything with any velocity] and that [one can put 0 velocity at a muon hurtling at .98c and expect all physics to make sense].

There is no claim about the speed of light in the argument. So we disagree on the principle of relativity itself, of course; this is where we started. According to the derivation, without this principle, we have to drop at least one of the following:

1. homogeneity of space and time;

2. isotropy of space;

3. that transforming coordinates to a different inertial frame and back to the original frame yields coordinates different from the original ones.

4. the relative speed between the two frames is different in the two frames.

Dropping any of these is completely reasonable, since they are postulated. Which one(s) would you say we shall drop. Let’s work out the consequences!

(Disclosure: my personal inclination is that space and time are not fundamental, so in that sense I view relativity as correct but incomplete: it arises from our insistence on having an x and a t, both complex phenomena emerging from simpler considerations.)

There is no claim about the speed of light in the argument.

Really? Then what is the meaning of “rely solely on… laws of physics are the same in all inertial frames.”? The “laws of physics are the same in all inertial frames”. Is that a vapid expression or does it stand for something?

There is nothing relating to the speed of light in that statement, and nothing about a maximum speed at this starting point. Only the concept of relativity is invoked. Again, let us work out the consequences of dropping assumption(s) from the 4 mentioned above (or any other one if I missed it).

Do you think that assumption (#3 or #4 however you define it) is not inextricably associated with [a photon can move +c or -c relative to anything with any velocity]? If it is not, then what the heck is the meaning of “rely on laws of physics are the same in all inertial frames”?

Drop all the assumptions.

Apologies, I mixed up positives and negatives in my list earlier (!) The menu of assumptions that can we can consider dropping are:

1. homogeneity of space and homogeneity of time;

2. isotropy of space;

3. the relative speed between the two frames is the same in both frames.

4. transforming coordinates to a different inertial frame and back to the original one yields the original coordinates.

These four assumptions lead directly to very narrow constraints on the possible coordinate transformations. This is why I am asking: which one should we investigate dropping? Dropping any one of these is completely reasonable at the outset.

None of the four assumptions allude to the speed of light or photons, but in my haste I may have missed it. Please exlpain how any of these statements is inextricably associated with photons.

“Rely on laws of physics are the same in all inertial frames” precisely means: “we are looking for physical equations that we don’t have to rewrite every time our relative speed changes with respect to something else.”

Whether Nature behaves that way, or we ourselves choose to consider only equations that are serving our lazy condition that we don’t have to rewrite physics all the time, is an epistemological question.

It has nothing to do with light, by any means (until someone measures something 😉).

The following two articles - 1 and 2 - have been sent to me by a creationist friend. Special Relativity is rejected, or anyway very much restricted in the first one. Perhaps this opens a new perspective for the discussion as it is taking place here.

You’re now advocating the exact benefits of absolute space. Like the Blender absolute stillness frame.

What I am advocating for is: let’s make (or remove) some assumptions, and verify the implications. Let’s propose genuine experiments to falsify our theory!

@HanDeBruijn, the first article leads with: 1) relativity rests on the speed of light being constant, which it does not, as discussed above; 2) interestingly, the orientation-dependent mass, which Einstein himself uses in his original paper on special relativity, however the concept of “relativistic mass” was abandoned early on (including by Einstein himself);

The arguments for stating that there is a problem with special relativity are: 1) there is a serious contradiction (twin paradox), which is false in this context; 2) some conclusions are unrealistic, which is not a meaningful argument in science (and moreover the theory does not conclude anything, it makes predictions to be tested experimentally; 3) it fails to provide answers to some problems, but again the theory is only making a testable prediction, not aiming to solve problems; 4) has been develop from wrong interpretation of experiments, but that is the bread and butter of science: observe contradictory evidence, make a theory, test its predictions.

On these grounds, I don’t find this article rejects the predictions of special relativity, or that it makes a convincing argument that it ought to be.

the twin paradox, which is not within the purview of special relativity because the frames are not inertial (we can still insist and analyze it considering two pairs of inertial frames, in which case — spoiler! — both twins agree that the travelling one is younger upon reuniting).

@ftessier, the twin paradox can be reproduced without accelerations by using triplets! A, B and C are all inertial frames. When B meets A at $X$, they synchronize their clocks, then B moves away from A. At $Y$, C synchronizes its clock with B. When C reaches A at $Z$, clock A is ahead of clock C.

   B A             C
     X           C
     A B       C
     A   B   C
     A     Y
     A   C   B
     A C       B
     Z           B

Fig. 1 Time is vertical, space is horizontal, as seen in A’s frame of reference.

Indeed, but when drawing regular light rays emitted at regular intervals by all participants (as a form of clock), they all agree in the end that the travelling twin(s) are younger.

I wonder if it works if lthey are not real twins 😂

Hi Louis,

That layout does not seem to capture the twin paradox which of necessity involves a change of inertial frames. In your example all observers remain in their initial inertial frames. It is the change of inertial frames in the twins paradox that results in the traveling twin having aged more slowly upon returning to the “home” frame,

Without a change of frame each inertial observer according to SR (moving with some constant velocity relative to each other) perceives the others clock as running more slowly while in fact neither clock is running slowly in the local frame. The two cannot be brought together for comparison without one or both changing inertial frames, the result of which is that the observer who has has changed frames returns with a clock that has run slower relative to the clock that has not changed its inertial frame.

Normally changing inertial frames involves an acceleration. What happens in your example is that the observers have not changed frame but the reference clock has via the synchronization of B & C clocks. This demonstrates that acceleration per se doesn’t produce the twin paradox, it is the change from one inertial frame of reference to another that does.

As soon as you change inertial frames you are no longer dealing with the symmetrical SR condition where each inertial observer sees the other’s clock as slowed. That’s why you don’t need to calculate the acceleration to get the aged twin. The change of frame is connoted effectively by the velocity differential (which is initially zero in the twin paradox.) Regards,

Bud

:+1: and that’s why the description called “twin paradox” is not a paradox.

e.g., there are no particles travelling faster than light, evidence of which immediately brings relativity into question

Well, Newtonian gravity is travelling faster than light. Gravitons (if they exist) are travelling faster than light. Now guess what Einstein did. With General Relativity he reduced gravity to 4-D geometry and herewith ingeniously “solved” that problem.

This (first) statement is highlighting my point, exactly.

To be precise, the coordinate transformation constraints derived with the relativity postulate (without light) admits two solutions (if we restrict coordinates to be real numbers): Galilean relativity ($\large \kappa = 0$), and special relativity ($\large \kappa > 0)$. Which one best describes our observations is a matter of experimental determination.

One possible avenue is testing the prediction of a maximum speed, which is very distinct between Galilean (there is no maximum speed) and Special (there is a maximum speed) relativities. In the Galilean case, the burden of falsification is to show experimentally that we cannot observe any particle travelling faster than a limiting speed. In special relativity, the burden of falsification is to observe a single one that does. In that sense, special relativity is a more robust theoretical stance at the outset (before experiments begin, to settle the question): it can be falsified by one observation that something travels faster than the proposed maximum speed.

Beyond that, the confidence in special relativity rests on the large amount of corroborating evidence accumulated up to now, as already mentioned. If I don’t use relativistic equations of motions for electrons in our accelerator, I get cross sections that don’t match experiments, by a large margin.

SR is the limiting case for GR.

Nope. GR and SR are completely different theories. What’s true is that SR spaces are locally tangent to GR space (all in theory).

a fully relativistic theory that first encompassed the Maxwellian treatment of electromagnetic radiation and was then extended to include gravity/acceleration.

Nope again. Electromagnetism and Gravity have never been successfully unified, as can been read in Classical Unified Theories (Wikipedia).

the problem that remains is, the geometry of what?

Are you going to try throwing doubt on classical Euclidean geometry, Bud? I hope not.

Han,

Nope indeed. Once again we are at the math is not physics divide which distinction you like to blur with either non sequiturs dressed as straw man arguments or straw man arguments in the guise of non sequiturs. I’m never sure which. In either case you are ignoring what I actually said, which concerned physics, and wandering off into mathland.

GR and SR are completely different theories.

That is ahistorical nonsense. GR and SR use different math but math, once again, is not physics. They were developed conceptually from the Galilean principle of relativity, extending it first to the behavior of electromagnetism in inertial frames (SR) and then to non-inertial frames (GR).

Electromagnetism and Gravity have never been successfully unified...

I did not say or imply anything about EM or GR being unified or anything about GR being a unified field theory which is just an ill-considered mathematicist project. GR treats both electromagnetism and and gravity successfully (at least in the solar system).

Are you going to try throwing doubt on classical Euclidean geometry, Bud? I hope not.

Do you actually think before you write this stuff? My question regarded what the geometry of GR referred to - what is it the geometry of? And that’s your answer?

@budrap00 :

but you can’t make ’em

Bud you can’t make ’em check dimensions:

$$ G = \sqrt{\frac{4\times 10^{-12}}{3c}} $$

I don’t have a reference better than the reasoning that The axle twist would go to 0 when the exact same torque is applied at both ends, because in such a case there wouldn’t be an asymmetric torque, and if there happened to be a twist in the axle, then the molecular restorative force would win out and any twist would decrease. The discomfort in the axle would diffuse.

The [rigid axle with 2 motors to achieve synchronization] is possible whether the mechanical force between the atoms happens at the speed of electricity or the speed of sound in the metal or whether it is faster or slower down one way of the axle, as long as the motors on each end are applying the same torque (in other words, as long as the torque diff is 0).

Optical sync is, informally as I understand it, sending one pulse of light so one motor can talk to the other motor. But mechanical sync is, informally as I understand it, a lattice of atoms constantly talking to each other back and forth. Whereas the lone pulse of light does its job and then that’s the end of that, the atomic lattice continues to exist between the motors and has an equilibrium state reachable.

How to you synchronize the torques?

Buy identical motors (https://www.google.com/search?q=buy+electric+motors+100000+rpm) and turn the dial (or whatever the controller is) on each motor to the same torque.

All right. Assuming that the two motors remain synchronized assumes (tacitly):

• Homogeneity of space and time
• Isotropy of space
• the speed of motor 2 with respect to motor 1 is negative that of motor 1 with respect to motor 2

We are half way there! :wink:

@sahil5d If you buy 2 equivalent motors, one will always work slightly harder. <- which might be ok in some cases but I don’t think here. In industry there are controllers available that do exactly that. Motor 1 is set to a fixed speed, and motor 2 will sense differences in torque and adjust.

Cases are e.g. when the load is too high for 1 motor, or 2 motors needed for safety (e.g. elevator), or a long axle is driven that might skew under load. However, these can synchronise the torque at best to a few decimal positions. Also, it is reactive, so if it senses a difference in torque, one motor (always only one) will react and adjust up or down. This results in continuous (very small adjustments) and when you zoom in on the kW curve there will always be some ups and downs.

Is this good enough for what you want to achieve? Also, how many decimal places of control is required? I am thinking if you are trying to do tests with light speed, then 2 or 3 decimal places will not be good enough?

@Francois-Zinserling Maybe don’t use a reactive motor. Find a motor that can be set to a fixed torque or a fixed speed. Or monitor both motors with a tachometer and make sure they’re spinning at the same speed.

On decimal places… The expectation is v_abs is somewhere in the 100s of km/s (~ c/1000), based on the CMB dipole and the Distant Quasars Dipole. So let’s try an axle length of 10 meters, a geartooth gap of 100 microns, and a gear radius of 30cm. These seem feasible? They can be modified.

In the case the photon is going in the same dir as v_abs, the time of travel of the photon
  = 10m / (c - v_abs)
  ~= 10m / .999c
  ~= 33.39ns  https://www.wolframalpha.com/input?i=10m+%2F+.999c

In the case the photon is going in the opp dir as v_abs, the time of travel of the photon
  = 10m / (c + v_abs)
  ~= 10m / 1.001c
  ~= 33.32ns  https://www.wolframalpha.com/input?i=10m+%2F+1.001c

How fast and precise does a gear have to spin to detect the cases above?

The time of travel of the geartooth
  = 100microns/v_gear
  = 100microns/(r_gear * ω_gear)
  ~= 100microns/(30cm * ω_gear)

The idea of the light-chopping experiment is to spin the gear fast enough so the time of travel of the photon == time of travel of the geartooth. So set them equal and solve.

For case [photon in same dir as v_abs]
  33.39ns = 100microns/(30cm * ω_gear)
  ω_gear ~= 9983 rad/s
         ~= 9983 rad/s * 1rev/(2pi rad) * 60s/1min
         ~= 95330 rpm https://www.wolframalpha.com/input?i=100microns%2F%2830cm+*+%2810m+%2F+.999c%29%29

For case [photon in opp dir as v_abs]
  33.32ns = 100microns/(30cm * ω_gear)
  ω_gear ~= 10003 rad/s
         ~= 10003 rad/s * 1rev/(2pi rad) * 60s/1min
         ~= 95522 rpm https://www.wolframalpha.com/input?i=100microns%2F%2830cm+*+%2810m+%2F+1.001c%29%29

So in this example, the required precision is increments between 95,330 rpm and 95,522 rpm. Seems possible.

Perhaps you can consider a single motor at the centre of the axle? By symmetry, both ends would be synchronized, i.e. it might prove easier to build a near-ideal axle than two nearly identical motors. regardless, radial acceleration is involved, so you would have to worry about that if you are trying to test SR, which does not admit accelerations.

Maybe. The torque propagation thru one axle half might take less time than the torque propagation thru the other axle half. I’m not sure.

Also, I think it’s worth proving the axle’s maximum twist, to know if the maximum twist angle <<? d_geartoothgap / r_gear. The formula for angular deflection of a shaft is theta = torque * length / (second polar moment of area * modulus of rigidity) https://en.wikipedia.org/wiki/Second_polar_moment_of_area#Application. Can it be bound to << 0.1 degrees, even in the least favorable setup with only one motor at one end of the axle?

Also, if the setup is frictionless, once the axle is up to a desired rpm, no further torque is needed, and then the twist angle, the angular deflection of the shaft, would be 0. Synchronization achieved.

Also, I’m fairly sure that two off-the-shelf motors can achieve single-digit precision in the 10,000-100,000 rpm range.

It’s still there on 2023-9-6 14:25 UTC…