# B 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?

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#### jtbell

Mentor
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

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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

Mentor
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

Mentor
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

Mentor
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

Mentor
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

Mentor
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

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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.

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#### 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

Mentor
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.

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#### 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

Mentor
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

Mentor
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

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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.

"Relativistic tips of a propeller"

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