Relativistic tips of a propeller

[xpell]

Hi! Yes, I know that faster-than-light travel is impossible. But please stay with me for a while to help me understand this. Let's imagine we take some unobtainium and build a 12-km-radius propeller, attached to an engine able to accelerate it up to 250,000 rpm (like a turbocharger, or not a few large industrial motors and turbines.) Then we plug this engine to the nearest sun or whatever and smoothly accelerate the thing.

According to my (classical) calculations, the tips of the propeller would reach a linear velocity of c slightly under 239,000 rpm, and we'd still have over 11,000 remaining rpm's to (classically) accelerate them above c. To keep the thing stable, we maybe could add another counter-rotating propeller, as in Kamov-style helicopters.

I know Relativity totally forbids this. But... what would happen as we approach the 239,000 rpm mark to make it impossible, please?

 

[jtbell]

It happened before you even started spinning up the thing, when you said "unobtanium." No "obtanium" is perfectly rigid. In fact, "perfect rigidity" is impossible in relativistic physics. Your propeller will inevitably start to deform/bend/break before it reaches a relativistically significant speed.

 

[xpell]

It happened before you even started spinning up the thing, when you said "unobtanium." No "obtanium" is perfectly rigid. In fact, "perfect rigidity" is impossible in relativistic physics. Your propeller will inevitably start to deform/bend/break before it reaches a relativistically significant speed.

Yes I know. But it was intended as a "thought experiment", precisally to understand what would happen (relativistically) and learn not only that it is not possible (which I already know), but how it is not possible. That's what I chose that unobtainium. :) And the sun plug too, not to mention the engine.

 

[HallsofIvy]

I presume you know that as something's speed gets greater and greater, its total energy gets greater and greater (I'm trying to avoid talking about "mass" as getting greater). That means that more and more force would be required to increase the speed at all. That is essentially the same reason an object, with an unlimited fuel supply still cannot move faster than the speed of light.

 

[Orodruin]

To put it differently: Any material is being held together by interactions which propagate (at most) at the speed of light. As you approach the speed of light, the forces would not propagate fast enough to withstand the (humongous) stresses involved. Of course, any real material is going to break long before then.

 

[HallsofIvy]

Yes I know. But it was intended as a "thought experiment", precisally to understand what would happen (relativistically) and learn not only that it is not possible (which I already know), but how it is not possible. That's what I chose that unobtainium. :) And the sun plug too, not to mention the engine.

But even in "thought experiments" about relativity, you cannot assume things that violate relativity!

 

[jim mcnamara]

These kinds of what if questions which are not physically possible because of the energies involved and all kinds of nasty mass-energy issues are discussed in a comic strip writer's column - here is a link to the 'relativstic baseball pitch'. Bear in mind that the mass of a baseball is far less than that of a propeller tip. Much much less than entire propeller.

https://what-if.xkcd.com/1/

I think this is an appropriate answer in a lot of ways.

 

[xpell]

But even in "thought experiments" about relativity, you cannot assume things that violate relativity!

Agreed, but I wanted to know why/how they violate relativity!

 

[Dale]

Agreed, but I wanted to know why/how they violate relativity!

1. It would require an infinite amount of energy
2. It would generate an infinite amount of stress
3. Angular acceleration cannot be rigid even with finite energies and stresses

 

[pixel]

...3. Angular acceleration cannot be rigid even with finite energies and stresses

What does it mean for angular acceleration to be "rigid?"

 

[russ_watters]

Yes I know. But it was intended as a "thought experiment", precisally to understand what would happen (relativistically) and learn not only that it is not possible (which I already know), but how it is not possible. That's what I chose that unobtainium. :) And the sun plug too, not to mention the engine.

Even if the material were infinitely rigid, you'd still run into the problem of needing an infinite amount of torque to accelerate it past the speed of light. Essentially, from your point of view, you'd apply an infinite amount of torque and it wouldn't go any faster.

 

[JVNY]

Here is another try at asking expell's question. Consider a batting machine and a pitching machine in a vacuum. There are no atmospheric particles to strike anything as described in the comic.

Someone expends nearly but not quite infinite energy to accelerate the batting machine to 0.99c relative to the pitching machine. The two machines are now inertial.

According to the principle of relativity there is no way for an observer to tell which machine is moving. Any limit based on on energy, mass or the like must account for the fact that the pitching machine can no more cause the ball to travel at c relative to the batting machine than can the batting machine cause the ball to move at c relative to the pitching machine. This must be true even though no one applied any force to the pitching machine to cause it to be in relativistic motion with respect to the batting machine. So what stops the pitching machine from expending all of its stored energy for the first time and throwing the ball to reach c relative to the batting machine?

 

[PeterDonis]

So what stops the pitching machine from expending all of its stored energy for the first time and throwing the ball to reach c relative to the batting machine?

The fact that the pitching machine only contains a finite amount of energy, and it would take an infinite amount of energy to throw the ball at c relative to the batting machine (or anything else). Do the math and see.

 

[PAllen]

Here is another try at asking expell's question. Consider a batting machine and a pitching machine in a vacuum. There are no atmospheric particles to strike anything as described in the comic.

Someone expends nearly but not quite infinite energy to accelerate the batting machine to 0.99c relative to the pitching machine. The two machines are now inertial.

According to the principle of relativity there is no way for an observer to tell which machine is moving. Any limit based on on energy, mass or the like must account for the fact that the pitching machine can no more cause the ball to travel at c relative to the batting machine than can the batting machine cause the ball to move at c relative to the pitching machine. This must be true even though no one applied any force to the pitching machine to cause it to be in relativistic motion with respect to the batting machine. So what stops the pitching machine from expending all of its stored energy for the first time and throwing the ball to reach c relative to the batting machine?

The algebra of velocity addition. If the pitching machine is moving at .9c relative to the batting machine, and pitches the ball .9c towards the batting machine relative to itself, the ball is then moving at .9945 c relative to the batting machine not 1.8c.

To apply this fundamental line of reasoning to the propeller case, note that if there is any observer relative to whom the propeller tip is moving < c, then it is true for all observers by the algebra of velocity composition. Thus, for the propeller tip to exceed c, it must magically exceed c for all observers (because if < c for any possible observer, it is <c for all).

I think the clearest conceptual solution to such problems is not to focus on the (true) fact energy and stress becoming infinite, but on the fact that Newtonian vector addition is wrong. It does not apply to our universe except approximately for low speeds.

However, I want to add to Dale's point about infinite stress, which I think hasn't been addressed so much in earlier posts. To force the propeller tip to move in a circle you must apply a force on it proportional to <inertia>v²/r. This force approaches infinite as v approaches c. Thus it is not possible, in principle, to bend the tip to a circle when v=c. This would require 'more than infinite' force.

 

[Dale]

What does it mean for angular acceleration to be "rigid?"

Rigid means that the proper distances between different parts do not change. If you accelerate an object linearly then it can remain rigid, but not in rotational acceleration.

 

[PeroK]

Hi! Yes, I know that faster-than-light travel is impossible. But please stay with me for a while to help me understand this. Let's imagine we take some unobtainium and build a 12-km-radius propeller, attached to an engine able to accelerate it up to 250,000 rpm (like a turbocharger, or not a few large industrial motors and turbines.) Then we plug this engine to the nearest sun or whatever and smoothly accelerate the thing.

According to my (classical) calculations, the tips of the propeller would reach a linear velocity of c slightly under 239,000 rpm, and we'd still have over 11,000 remaining rpm's to (classically) accelerate them above c. To keep the thing stable, we maybe could add another counter-rotating propeller, as in Kamov-style helicopters.

I know Relativity totally forbids this. But... what would happen as we approach the 239,000 rpm mark to make it impossible, please?

You can simplify this to see what happens. Imagine your propeller is a light connecting rod joining the engine to a small mass at its tip. A very classical set-up!

Now, as the system rotates, the small mass gains speed. Classically, of course, for a given torque, the mass accelerates indefinitely - ignoring any resisting forces.

But, as the mass reaches relativistic speeds, the acceleration reduces - as it would for a linear acceleration.

It doesn't matter that you assume the materials can withstand the forces or that the system remains rigid. The speed of the mass can only asymptotically approach $c$.

In short, once you clear away all the extraneous details such as rigidity, you simply hit the same constraint as a linear particle accelerator.

 

[A.T.]

Even if the material were infinitely rigid, you'd still run into the problem of needing an infinite amount of torque to accelerate it past the speed of light.

If the material was infinitely rigid, relativity would be wrong. So under these assumptions, what are you basing that infinite amount of torque on?

 

[russ_watters]

If the material was infinitely rigid, relativity would be wrong. So under these assumptions, what are you basing that infinite amount of torque on?

No. It is possible essential to good problem solving skills to analyze one piece of a problem at a time to look for separate errors. We make physically wrong simplifying assumptions about problems all the time - there is no reason why it can't be done here.

The "If relativity were wrong, what would XXX say about relativity" retort used here is practically a meme around here and it is a wrong approach to problem solving. But since it has become a punch-line people are no longer putting any thought into it. Scientists and engineers make physically wrong assumptions in problem solving all the time (and the "infinitely strong" or rigid one is a very, very common one). That's a critical skill in the art of problem solving.

Edit:
Indeed, I wold say this is the most common simplifying assumption used. It is used dozens of times a day on PF.

 

[A.T.]

The "If relativity were wrong, what would XXX say about relativity" retort used here ...

Your argument rather boils down to: "Even if relativity was wrong, you still couldn't do X, because relativity says X needs infinite torque"

 

[russ_watters]

Your argument rather boils down to: "If relativity was wrong, you still couldn't do X, because relativity says X needs infinite torque"

Repeating the punch-line will not help you analyze and understand the "joke". Please read the rest of the post and put some thought into what I said.

 

[JVNY]

I think the clearest conceptual solution to such problems is not to focus on the (true) fact energy and stress becoming infinite, but on the fact that Newtonian vector addition is wrong. It does not apply to our universe except approximately for low speeds.

I agree entirely. The cartoon accepts that the pitcher can throw the ball at 0.9c without using infinite energy. But even so, 0.9c plus 0.99c does not result in the ball traveling at c or greater relative to the batting machine as the ball passes it -- even without worrying about what would happen if the batting machine hit the ball. xpell might be looking for something else, but the answer is just that Newtonian vector addition is wrong.

 

[russ_watters]

I think the clearest conceptual solution to such problems is not to focus on the (true) fact energy and stress becoming infinite, but on the fact that Newtonian vector addition is wrong. It does not apply to our universe except approximately for low speeds.

While I agree that that's true, I think the energy implication of this is a useful way to view it as well. Using common simplifying assumptions, (a point mass at the end of an infinitely rigid and massless rod) yields a device that is basically the same as (can be analyzed the same as) a particle accelerator. They are commonly described in terms of energy.

 

[PeroK]

Your argument rather boils down to: "Even if relativity was wrong, you still couldn't do X, because relativity says X needs infinite torque"

It's not entirely clear that the loss of rigidity scuppers the whole experiment.

What would actually be, the maximum possible speed of the tip of a propeller? Is it 0.1c? Or, 0.2c. Or,perhaps, 0.99c?

Or, perhaps it's difficult to set a definite limit on what is possible. The only true relativistic limit is c. In the sense that you can get, in theory, arbitrarily close.

Anything lower depends on specific engineering limitations, as it does for particle accelerators.

 

[A.T.]

Please read the rest of the post and put some thought into what I said.

OK

We make physically wrong simplifying assumptions about problems all the time

There is a difference between:
a) Making predictions, while ignoring aspects that have negligible quantitative effect on the result.
b) Trying to prove something, based on a set of mutually contradictory assumptions

 

[A.T.]

The only true relativistic limit is c.

Which also rules out perfect rigidity. If you assume perfect rigidity then you cannot also assume that limit, to prove anything.

 

[russ_watters]

OK

There is a difference between:
a) Making predictions, while ignoring aspects that have negligible quantitative effect on the result.
b) Trying to prove something, based on a set of mutually contradictory assumptions

Your "b" is what you think I did, and it isn't correct. The purpose of simplifying assumptions is always the same and does not hinge on the effect being negligible. As the name suggests, the purpose is to make a problem easier to solve (isolate an effect you want to examine) by ignoring other effects you don't care about. Again, this is done all the time, particularly in Relativity threads, where impossible scenarios are proposed a dozen times a day.

Edit:
I'll be specific:
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.

 

[PeroK]

Which also rules out perfect rigidity. If you assume perfect rigidity then you cannot also assume that limit, to prove anything.

There is no need to assume perfect rigidity, only that the blades remain approximately rigid. Which is a common approach to physics problems.

And I really don't see how that's any different from classical physics where everyone who studies rigid body problems knows that all bodies deform to some extent.

If we had a rule that all questions posted on PF make no impossible assumptions like rigid body, massless string, perfect sphere, uniform gravitational field, frictionless surface, no air resistance etc. then there wouldn't be much left outside the maths forum!

 

[A.T.]

If we had a rule that all questions posted on PF make no impossible assumptions...

It's not about making impossible assumptions, but combining mutually contradictory assumptions.

 

[PeroK]

It's not about making impossible assumptions, but combining mutually contradictory assumptions.

What are the mutually contradictory assumptions in this case?

 

[russ_watters]

There is no need to assume perfect rigidity, only that the blades remain approximately rigid. Which is a common approach to physics problems.

I agree, but still wish to emphasize that while it is possible to calculate how almost completely rigid it needs to be to demonstrate the effect we care about, if we don't do that calculation, we are not assusing it is approximately rigid, we are assuming it is exactly rigid...er, to at least our level of significant digits (which are also often assumed exact).

 

[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.