B Relativistic tips of a propeller

[Ibix]

If you spin up the propellor arbitrarily slowly (i.e. $d\omega/dt \rightarrow 0$) then you can treat the propellor as rigid, I think, in the usual sense of "transients die away quickly". Alternatively one could attach rockets along the length of the blades and adjust their thrust to maintain uniform (in some inertial frame) rotational motion. However, there is an upper limit to the angular velocity you can achieve like this - namely $\omega<c/r$ if r is the radius of the propellor. But there is nothing stopping you spinning the boss up to a greater angular speed than this - which is what is baking the OP's noodle, I think.

At that point, you can't treat the material as rigid, infinitely or otherwise. To maintain rigidity would require the propellor tips to force the boss to stay below the critical angular velocity, and that is problematic. What must happen is that the blades flex so that the tips (and indeed the entire blades) are contained in a radius $r'$, $r'<r$, so that $\omega r' <c$. Either that or they disintegrate - which is much more plausible.

 

[PeterDonis]

If you spin up the propellor arbitrarily slowly (i.e. $d\omega/dt \rightarrow 0$) then you can treat the propellor as rigid, I think, in the usual sense of "transients die away quickly". Alternatively one could attach rockets along the length of the blades and adjust their thrust to maintain uniform (in some inertial frame) rotational motion.

Actually, even these limited claims aren't quite true. The Herglotz-Noether theorem says, in part, that it is impossible to have a Born rigid timelike congruence with an angular velocity that varies with time. That means it is literally impossible, according to SR, to "spin up" any object from non-rotating to rotating without distorting it in some way--i.e., without the proper distances between some of the parts changing. Even if you attach a rocket to every little part, and program them to deliver precisely timed accelerations, as you would to make an object accelerate in a straight line in a Born rigid manner, you can't do the same with rotation. The reason this is worth mentioning is that the proof of the theorem does not depend, at least not in any obvious way, on the restriction that internal forces in the object can only propagate at the speed of light. It's a theorem about the possible geometries of timelike congruences of worldlines.

 

[Ibix]

I was going to come back and say that I was beginning to be doubtful that one could get away with "waiting for the transients to die away" because the time needed in the inertial rest frame of the propellor (i.e. its center of mass frame) seems likely to scale with $\gamma$, which grows without bound as the tips approach c. That you can't even do it in a "brute force" way with rockets is surprising to me. I'll investigate Herglotz-Noether.

 

[pervect]

My $.02. Rotating disks (or in this case, propellors) are just not a good way to understand why one can't exceed the speed of light, which was supposedly the OP's motivation according to his original post. It'd be both a lot easier and more productive to study the case of continuing accelertion, the relativistic rocket, http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html than to worry about relativistic disks/propellors.

Note though that the article on the relativistic rocket I quoted just gives the results, it doesn't derive them. So it's not as useful as it could be in explaining the "why". But it's probably best to leave that to another post.

After one has grasped enough relativity to understand why one doesn't reach the speed of light by accelerating, then it might be possible to dig into the Ehrenfest and the relativistic disk. But I'd say it's pretty hopeless for someone who hasn't mastered the basics of relativity to tackle the rotating disk straight off.

 

[A.T.]

1. In the "barn door/ladder paradox", to make this thought experiment physically real requires closing the barn doors instantly and a signal receipt, processing and response system that acts instantly. That violates Newton's laws of motion and SR.
2. In the twins paradox, accelerations are assumed instantaneous, also in violation of Newton's laws and SR.

In both your examples we are not interested in the torques/forces needed to accelerate the doors/rocket. But here you trying to make conclusions about the torque needed at the hub to accelerate a perfectly rigid propeller.

In both your examples introducing perfect rigidity of extended objects would lead to contradictions, because it allows instantaneous signals and contradicts relativity of simultaneity.

 

[Quickbobo]

because it allows instantaneous signals and contradicts relativity of simultaneity.

I'll probably get head slapped for this, but if Bohr's assumptions were correct, violation of Bell inequalities, what then? This still has not been decided/ proven, after 70 some odd years.

 

[Nugatory]

I'll probably get head slapped for this, but if Bohr's assumptions were correct, violation of Bell inequalities, what then? This still has not been decided/ proven, after 70 some odd years.

Violations of Bell's inequality do not allow instantaneous signalling and do not pose problems for relativity of simultaneity. There are many threads over in the quantum physics section.

 

[russ_watters]

In both your examples we are not interested in the torques/forces needed to accelerate the doors/rocket. But here you trying to make conclusions about the torque needed at the hub to accelerate a perfectly rigid propeller.

Correct. You asked for additional [different] examples of where simplifying assumptions are used that contradict the very theory being investigated. I provided them.

 

[xpell]

Wow, so many answers! Thank you all very much!

But there is nothing stopping you spinning the boss up to a greater angular speed than this - which is what is baking the OP's noodle, I think.

Rotating disks (or in this case, propellors) are just not a good way to understand why one can't exceed the speed of light, which was supposedly the OP's motivation according to his original post. It'd be both a lot easier and more productive...

Actually I wanted to learn more about how this can't happen, and this simple rotating propeller (or disk) felt like more "technologically feasible" (!) than an "infinitely" fueled ultra-massive rocket. I of course knew that at some point I'd need infinite energy (and torque, and resistance to stress, and everything) to make the tips/border graze real c, and that's impossible.

My motivation was more about learning what would happen (specially to the "unobtainium" propeller tips or disk border) if we stubbornly attempted to spin it up and up until we definitely ran out of energy, as in what kind of relativistic effects could be expected, how that would interact with the inner parts of the disk which are vastly sub-c, the difference between doing in in a vacuum or in an atmosphere, etc. It sounded to me a way more interesting question than "yeah, I know I can't accelerate a rocket infinitely up to c, and further just by crashing against the interplanetary medium at moderate sub-relativistic speeds it's going to be destroyed, so what?"

No sci-fi intended, just trying to better understand the effects of relativity.

 

[pervect]

Wow, so many answers! Thank you all very much! Actually I wanted to learn more about how this can't happen, and this simple rotating propeller (or disk) felt like more "technologically feasible" (!) than an "infinitely" fueled ultra-massive rocket. I of course knew that at some point I'd need infinite energy (and torque, and resistance to stress, and everything) to make the tips/border graze real c, and that's impossible.

Interesting, I'm not sure why it feels that way. I'd say that things are the other way around, it's much easier to deal with an infinitely advanced rocket - or even a fleet of infinitely advanced cars, each car in the fleet going 1 meter/second faster than the previous car (as measured by the previous car in the fleet) - than it is to deal with the rotating disk.

The most basic lesson that can be learned from the rotating disk is that perfectly rigid objects just don't exist in relativity (or in reality). One way that I like to look at it (I won't offer the full justification here) is that you can judge the rigidity of an object by the speed of sound in that material. More rigid objects have a higher speed of sound. Thus, an object with the speed of sound equal to the speed of light would be the most rigid possible object. So when we switch our length standards from platinum bars (the old SI standard) to "the distance light travels in a certain fraction of a second" (the new SI standard), all we're really doing is defining a more rigid ruler. Which doesn't have much effect, especially as we usually try to set up experiments so that we don't have to deal with how non-rigid rulers distort, we maintain the ideal of a rigid ruler by arranging the experiment, as much as possible, so that the rulers don't experience any forces that would deform them.

Going back to your propeller- you can ask how your non-rigid propelller distorts as it approaches the speed of light - trying to imagine what a non-existent perfectly rigid propeller only results in the conclusion that such a thing can't exist.

My motivation was more about learning what would happen (specially to the "unobtainium" propeller tips or disk border) if we stubbornly attempted to spin it up and up until we definitely ran out of energy, as in what kind of relativistic effects could be expected, how that would interact with the inner parts of the disk which are vastly sub-c, the difference between doing in in a vacuum or in an atmosphere, etc. It sounded to me a way more interesting question than "yeah, I know I can't accelerate a rocket infinitely up to c, and further just by crashing against the interplanetary medium at moderate sub-relativistic speeds it's going to be destroyed, so what?"

No sci-fi intended, just trying to better understand the effects of relativity.

The question needs to be modified some before it can be analyzed, as the analysis of the original question just gives the answer "it can't happen", which isn't very helpful. One possible modification of the question is Greg Egan's "hyperelastic" materials, discussed on his webpage under the title "Rotating Elastic Rings, Disks, and Hoops" Unfortunately, his paper isn't peer reviewed, and if you dig it up and look at it, you'll also see it's fairly advanced and not suitable for an introductory level question. Even though it's not peer reviewed, it's probably the best treatment of the subject I've seen.

 

[xpell]

Interesting, I'm not sure why it feels that way. I'd say that things are the other way around, it's much easier to deal with an infinitely advanced rocket - or even a fleet of infinitely advanced cars, each car in the fleet going 1 meter/second faster than the previous car (as measured by the previous car in the fleet) - than it is to deal with the rotating disk.

Honestly I don't know. That's how it felt to me, most possibly because I've seen way more mundane machines (like basic turbochargers) rotating at high speeds than ultra-advanced rockets. Anyone can easily see (or imagine) a turbocharger spinning at around 250,000 rpm and intuitively (and wrongly, I know) think: "OK, just enlarge the blades of this thing, reinforce it, use a more powerful source of power if needed, and their tips will achieve an arbitrarily high linear speed." Further, I know of some pieces of way heavier industrial machinery spinning at similar rotational speeds, so the thought is (wrongly) straightforward: "Just make this thing bigger. Or faster."

Actually I wonder why the "centrifugal cannon" hasn't been more thoroughly explored (partially or totally) to shoot hypersonic projectiles, instead of railguns and the like. According to my calculations, the border of a disk or the tip of a blade barely 2 meters in diameter spinning in a vacuum at those same turbocharger-style 250,000 rpm has a linear/tangential velocity of over 26 km/s (sure a ship or even some large aircrafts could handle that!) It would probably have to be a sturdy machine but it would be simpler than railguns etc., and I can't see anything intrinsecally impossible. Just spin the thing up and release the projectile at the appropriate instant. A system of locks or the like would be able to transfer it from vacuum to standard atmosphere. Add a magnetic drive for extra precision or advanced guidance onboard if you'd like. There sure must be a reason because this hasn't been done for ages, but I don't really grasp it. Anyway, I guess this is a question for the "General Engineering" forum, not this one. ;)

The most basic lesson that can be learned from the rotating disk is that perfectly rigid objects just don't exist in relativity (or in reality). One way that I like to look at it (I won't offer the full justification here) is that you can judge the rigidity of an object by the speed of sound in that material. More rigid objects have a higher speed of sound. Thus, an object with the speed of sound equal to the speed of light would be the most rigid possible object. So when we switch our length standards from platinum bars (the old SI standard) to "the distance light travels in a certain fraction of a second" (the new SI standard), all we're really doing is defining a more rigid ruler. Which doesn't have much effect, especially as we usually try to set up experiments so that we don't have to deal with how non-rigid rulers distort, we maintain the ideal of a rigid ruler by arranging the experiment, as much as possible, so that the rulers don't experience any forces that would deform them.

This is something I absolutely didn't know at all and I thank you very much for teaching it to me!

Going back to your propeller- you can ask how your non-rigid propelller distorts as it approaches the speed of light - trying to imagine what a non-existent perfectly rigid propeller only results in the conclusion that such a thing can't exist.

The question needs to be modified some before it can be analyzed, as the analysis of the original question just gives the answer "it can't happen", which isn't very helpful. One possible modification of the question is Greg Egan's "hyperelastic" materials, discussed on his webpage under the title "Rotating Elastic Rings, Disks, and Hoops" Unfortunately, his paper isn't peer reviewed, and if you dig it up and look at it, you'll also see it's fairly advanced and not suitable for an introductory level question. Even though it's not peer reviewed, it's probably the best treatment of the subject I've seen.

I'd sure take a look at it anyway and see if I can grasp anything. Thanks a lot again!