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Here's my question;

If we have a line of connected cogs, arranged in such a way that for any one cog to move each other cog would necessarily have to move also, and that line of cogs were of an arbitrarily large distance, would we be able to induce motion in the first cog at one end of the line immediately, thus acting upon the farthest cog faster than $c$, or would the line of cogs become an immovable object for a period of time due to constraints regarding the speed at which information can propagate?

To help with answering the question assume these things to be true;

  • The cogs can not be made to go out of alignment.
  • The energy needed to move every cog is arbitrarily small.
  • The cogs can maintain their physical integrity even when subject to arbitrarily large amounts of force.

Thank you all for any time you take to answer my question.

The speed of sound i think may not be a problem here, because the moment a force is applied on one side of the first cog and reaches the first point of the second cog, the balance of the second cog would change. Since the second cog is put off balance, gravity would begin to act on that second cog inducing motion on its farthest point at the speed of sound, but faster than the information would travel from the closest point of the first cog to the farthest point of the second. Its the instantaneous action upon the farthest most points of each cog created by gravity or another constant force that happens at the moment that force is applied to the closest most points of those cogs, that overcomes the speed of sound barrier, as the information telling each cog to move in a certain way is communicated to every point in the cog either by the force itself or by the nearest most points which contain that information, those points being nearer to the center most points of the cogs than the closests most points are. The distance that any informatjon has to travel is less than the distance which light has to travel, and therefore as the length of the line of cogs increases, the distance information has to travel to move each cog within the line, relative to the distance light has to travel to cross the same distance, approaches zero.

This is the sort of fun thing we could get into, if people didnt just assume this question is the same as the rigid pole question.

This question is substantially different from the pole question for numerous reasons, and the differences make the appropriate discussion about this different from that had concerning the rigid pole.

For instance, each cog is individually bound by the EM force, where as the pole is a single object bound by the EM force, this makes a couple things different but one simple thing is that while pushing a long pole would compromise the integrity of the pole itself, stopping one cog does not compromise the integrity of the others because they are individually bound and therefore the EM force which binds the farthest cog together is not compromised when a force is applied to the first cog, that difference is hugely substantial. The motion of each cog is dependent on the cogs next to it, whereas the motion of one end of the rigid pole is dependent on action at the other end, this difference is also substantial. in these ways and others the cog question is substantially different from the pole question, even though rigidity is a factor the way rigidity is discussed would neccessarily be substantially different, and even though the EM force should be mentioned it does not play the same role as it does with the pole. This question and the pole question are fundementally different and they should be treated as such, even though the difference may be subtle, they are fundemental non-the-less.

I used postulates specifically so i wouldnt get answers that are mostly irrelevant to the given context, and they were largely ignored for their intended purpose, and it seems mainly considered only so much as it allowed for my question to be answered as something else.

The cogs cant go out of allignment, is like saying the pole cant bend, because were talking about cogs not a pole and the physics are different.

The cogs can maintain their physical integrity, is like saying the pole cant break, if each cog on its own can withstand a substantial amount of force without breaking than infinitesimal differences between the motion of one cog will not cause the structure of the surrondings cogs to be functionally compromised

I can not stress enough, that im grateful for any effort that people have given, and will give, im just a little bummed out is all.

Could somebody delete this question.

Lgnttsrm
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    Possible duplicate of Is it possible for information to be transmitted faster than light by using a rigid pole? – sammy gerbil Mar 16 '17 at 09:43
  • Im aware that they seem to be duplicates. The difference is non arbitrary. – Lgnttsrm Mar 16 '17 at 09:55
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    @Louie: The difference is the set up, but the fundamental question is the same so the answer is going to be the same as well. – Kyle Kanos Mar 16 '17 at 10:05
  • What is the difference? There does not seem to be a fundamental difference. As John Rennie says, the same answer applies. There are many variations on the same idea, but the same principle is involved : information being passed at infinite speed through a rigid body. – sammy gerbil Mar 16 '17 at 10:08
  • The pertinent portion of the cogs question can be rephrased as "If a line of cogs of an arbitrarily large length were already rotating, and the cogs were alligned so that if any one cog stopped rotating the others would be unable to rotate..." that difference is non arbitrary. Im going to edit the question, so as to go into that a little mote clearly than i may be able to in a comment. – Lgnttsrm Mar 16 '17 at 10:10
  • @Kyle, the difference in the set up is obvious, the difference in the fundemental question is less obvious but there,. – Lgnttsrm Mar 16 '17 at 10:30

4 Answers4

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Your postulates violate special relativity. Specifically,

"The cogs can maintain their physical integrity even when subject to arbitrarily large amounts of force."

and

"The cogs can not be made to go out of allignment"

are both incompatible with special relativity, because they imply an arbitrarily high speed of sound. Since sound can transmit a signal, this is in conflict with the universal signal speed limit $c$. In practice, the cogs would deform elastically, and the effect would be transmitted at the speed of sound through the system, at a speed considerably below $c$.

Selene Routley
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This is effectively a duplicate of Is it possible for information to be transmitted faster than light by using a rigid pole? and the same answer applies.

In relativity there is no such thing as a perfectly rigid object because the impulse that moves the material from which the object is made cannot move faster than light. So when you rotate the first cog it, and every subsequent cog, must deform slightly no matter how rigid it is. The result is that the motion cannot reach the terminal cog faster than the speed of light.

Community
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John Rennie
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  • It only seems like that at first, there are similarities, but theres substantial enough difference in the physics of things. – Lgnttsrm Mar 16 '17 at 09:56
  • @Louiearmstrong: that's why I didn't close your question as a duplicate. However the underlying principle is the same in both cases. No motion can propagate through a material faster than the speed of sound, and the speed of sound cannot be faster than the speed of light. So regardless of whether the motion is rotation (as in your case) or linear (as in the pole) it cannot propagate faster than light. – John Rennie Mar 16 '17 at 10:00
  • What is the "substantial difference in the physics"? – sammy gerbil Mar 16 '17 at 10:05
  • The substantial difference is that the same things which make inducing movement or stopping movement of the rigid pole an impossibility, make the continued movement of the cogs a physical impossibility were the question rephrased so as to concern stopping the cogs instead of moving the cogs. – Lgnttsrm Mar 16 '17 at 10:51
  • You can look at the rigid pole question, and on way that its substantially different should become immediately obvious. Since cogs are seperate objects, the role of the EM force in binding the pole together doesnt apply to cogs, the rigidity of any one cog and each cog in the system is not compromised by the length of the system. There are other substantial difference, but thats one of them. – Lgnttsrm Mar 16 '17 at 11:05
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It turns out that your condition:
"The cogs can maintain their physical integrity even when subject to arbitrarily large amounts of force."
Is actually not compatible with special relativity.

And even simpler setup like your idea, and relies on mechanical forces, is a long rod. Push one end, and the other end moves. But in special relativity, the information that you pushed one end cannot travel to the other end faster than light, so the end can't know to move. This means in special relativity, the concept of a rigid rod or cog is just not possible.

PPenguin
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  • Its a distinct question. The cogs question be rephrased as follows; If a line of cogs of an arbitrarily large length are rotating at a constant speed, and are arranged in such a way that if any one cog stops moving each other cog will be unable to move.....t – Lgnttsrm Mar 16 '17 at 10:01
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    Its the same problem. If a cog stops turning, how does this information travel infinitely fast to all the other cogs to tell them to stop? It can't. Rigid object mechanics like this is not possible in SR. One is forced to consider objects as having finite elastic properties. – PPenguin Mar 16 '17 at 10:04
  • Its not the same question, i dont mean to be rude but your just not getting it yet, im sure your very intelligent, i considered all these things being brought up before i asked. What allows the cogs on the other end to countinue moving, if continued movement becomes a physical impossiblity? – Lgnttsrm Mar 16 '17 at 10:41
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    @Louiearmstrong What allows the cogs at the other end to continue moving is that the cogs in between are not, and can not be rigid. –  Mar 16 '17 at 11:24
  • @Louiearmstrong Try imagining the cogs with nonzero moments of inertia $I$ and with with finite elastic constants (say torsion modulusses $G$). When you jam a cog at one end, the force slowing the cogs propagates along (actually it will reflect back and forth) the line at a finite speed as an acoustic wave. You could probably set this problem up as a linear system of coupled, torsional pendulums. Your postulated situation is simply this one with $I\to0$ and $G\to\infty$. Let this happen. A you increase $G$ / decrease $I$, the acoustic speed approaches $c$. At this point, you can either .... – Selene Routley Mar 16 '17 at 11:41
  • Rigidity, is not the issue, its my understanding that the deformation of objects is proportinate to the sum of forces acting on those objects. The limits of deformation relative to the sum of force applied is the more material matter. – Lgnttsrm Mar 16 '17 at 11:45
  • conclude that you model is wrong, because relativity will somehow forbid this speed to exceed $c$ (even though there is no obvious mechanism) or you can look at the dynamics carefully: as the acoustic speeds become relativistic, different parts of the system become linked by nontrivial Lorentz transformations, and this changes the dynamical equations to prevent faster than $c$ acoustic propagation. – Selene Routley Mar 16 '17 at 11:46
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Regarding the correct answers citing your "rigid rod" error, the usual textbook end-of-chapter exercise concerning this misconception is the paradox described as follows. Suppose a car is driving over a large pothole that's exactly the size of the car. But the car's moving very fast. So, from the car's frame of reference, length contraction makes the pothole seem shorter, and the car doesn't fall in (but probably needs a wheel alignment). Contrariwise, from the pothole's frame of reference, the car seems shorter and falls in. So what actually happens? And why? I won't tell you, but hint: the answer involves (you guessed it) rigidity.

Another typical misconception that fools people into thinking they can design some faster-than-light apparatus is confusion between phase velocity https://en.wikipedia.org/wiki/Phase_velocity (potentially greater than $c$) and group velocity https://en.wikipedia.org/wiki/Group_velocity (which is what can carry information from point to point and can't be greater than $c$, but see comments below). For example, suppose you're at the center of a very, very big shell, say a light-year in radius, with a very,very bright flashlight in your hand. You point the light at the shell and sweep out a $180^o$ arc in a second. A year later, the first spot of light hits the shell where you first pointed. And a year-plus-a-second later the last spot of light hits the shell at its diametrically opposite point. So, circumference-wise, the spot travelled $\pi$ light-years in one second. But that's phase velocity, not group velocity. And there's no way the "travelling spot" could've carried information from the first point to the last. I'll leave you, if motivated, to do the reading and think about it.

Edit re-reading my answer, I noticed that I neglected to emphasize the major point I'd wanted to make with the car/pothole paradox exercise. Reading some preceding answers, it seemed they might leave the (mis-)impression that non-rigid means the objects "physically deform" subject to the usual forces that typically deform objects, like when you (or maybe Superman) bend(s) a metal rod in your hands. But it's really a purely geometrical effect due to relativistic spacetime relations. In particular, the car/pothole makes that clearer because the car does fall into the pothole: it "bends" through it by looking like a "${}^{---}\backslash{}_{---}$" shape rather than its usual straight "$----$" shape. And it's usually pretty clear that's not "physical bending". It's just a geometrical effect. (Note: I'm vaguely recalling from this exercise, long, long ago, that there's also some interesting conformal stuff to talk about, regarding all those bending angles, but I'm not recalling the details nor immediately seeing how to derive them. If anybody can fill that in, please add another Edit below discussing it, especially if it's actually an interesting discussion. Thanks...)

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    Actually group velocity can exceed $c$ for some weird materials: it's only an approximation to the signal speed, the latter always being less than or equal to $c$. – Selene Routley Mar 17 '17 at 07:57
  • @WetSavannaAnimalakaRodVance You mean $>c$ in a Cherenkov sense? Or, if not, then how would that work? (I couldn't get google to answer that for me in a $\pm$minute, and didn't have the extra time to try harder) –  Mar 17 '17 at 09:29
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    No, I mean in the "supraluminal" sense: it's simply that the group velocity isn't always an accurate representation of the signal velocity. Mostly the two are very close, but sometimes ...... The end of the group velocity Wiki page gives some references that describe experiments showing faster than $c$ group velocities whilst the true signal velocity of course stays below $c$. Dissipative or high gain materials tend to show this behavior. – Selene Routley Mar 17 '17 at 09:36
  • @WetSavannaAnimalakaRodVance Okay, and thanks for the reference. I'll check that out a bit later and (hopefully) fix my misunderstanding. I'd have thought just your mention of it would jog some long-ago lecture and/or reading memory in my head, but I'm coming up blank. Must have been asleep that day. –  Mar 17 '17 at 09:52