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We know that the faster a propeller rotates, the faster the attached vehicle (airplane, submarine, etc.) moves. (Assume that the propeller is normal to the motion direction.)

On the other hand, special relativity asserts that in high-speed linear motions, the clocks slow down. If we consider the propeller as a mechanical clock, it is anticipated that $-$ due to the direct effect of time dilation $-$ it runs very slowly when the airplane travels at a considerable fraction of light speed. I just cannot understand how is it possible to see an airplane moving through the air at, say, $0.9999999999999999c$ (digit 9 is repeated 16 times), while its propellers spin extremely slowly.

Remember that in the airplane rest frame the propeller can rotate at a maximum angular velocity of $\omega=c/r$, where $r$ is the propeller's radius. If we assume that this maximum angular velocity is capable of setting the airplane in motion at $0.9999999999999999c$ as seen by the lab observer at rest with respect to the air, this observer sees that the angular speed of the propeller reduces to $\omega^\prime=\sqrt{1-0.9999999999999999^2}\omega$ $\approx 10^{-8}\omega$. Now if $r=10\space m$, we have $\omega \approx 3×10^{7}\space rad/s$ and $\omega^\prime=0.3\space rad/s$.

I wonder how it can be rational for the lab observer to see an aircraft moving very close to the speed of light, while its propeller rotates very slowly at just $\omega^\prime=0.3\space rad/s$? Does not it violate this sentence that the faster a propeller rotates, the faster the attached vehicle moves?

Mohammad Javanshiry
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  • Propellers stop providing thrust when the tip reaches the speed of sound. A relativistic propeller is nonsense. Now, you could modify it to be a subsonic propeller in a relativistic tailwind. That would be a sensible question. – Dale Nov 14 '19 at 11:04
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    @Dale Propeller blades work the same way aircraft wings work. If it was not possible for propeller blades to work at supersonic speeds, then it would not be possible to build airplanes that fly at supersonic speeds. Supersonic propellers have been built and flown. http://fighterwriter101.blogspot.com/2015/08/the-xf-88b-voodoo.html . – Solomon Slow Nov 14 '19 at 13:33
  • @Solomon Slow I think Dale meant the speed of sound in the material of which the propeller has been made, rather than that in the air. – Mohammad Javanshiry Nov 14 '19 at 13:42
  • @SolomonSlow good point. In any case, I don't know of a relativistic propeller theory, so we would need some "toy" theory to use for this question. – Dale Nov 14 '19 at 16:08
  • the propeller would be seen moving slower, but this is unrelated to the instantaneous speed of the airplane. it is not the instantaneous speed of the propeller which gives the airplane its instantaneous speed, the speed was "built" in the times before you observe it –  Nov 15 '19 at 00:49

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That's a lovely conundrum, and can be generalised to other means of propulsion (eg rotating wheels, rotating turbines, the rotating crankshaft of a petrol engine etc). It can also be made more abstract to talk about the energy consumption of the propelling device being reduced as a consequence of time dilation.

In the case of a wheeled vehicle, the paradoxical nature of the situation is compounded by the fact that the part of the wheel in contact with the ground is momentarily stationary in the ground frame of reference, while the top of the wheel is moving at twice the speed of the vehicle.

There are various explanations of these effects on the internet. I have not seen one specifically relating to propellors before, but I suspect the principles are the same. My physics instinct tells me that a possible resolution of the paradox is that while the propellor will be seen to be rotating more slowly to the observer, each rotation will sweep a correspondingly larger volume of air because the pitch of the propellor relative to the air will also appear changed, with the result that the propulsive effect will remain unchanged.

As you will see from the related articles on the internet, the actual calculations of these effects can be quite involved and laborious, so if you want a quantitative answer rather than a conceptual one, I will leave that to other posters with more enthusiasm for mathematics.

Marco Ocram
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It completely makes sense.

From aircraft pilot's perspective, as propeller rotates faster, the air itself due to the Lorentz contraction becomes more dense. And because of that, pilot will claim that air can not arrive at speed of light, because at that speed, density of air becomes infinity (That's, pilot's velocity w.r.t lab observer won't arrive at speed of light, air itself won't let him do that).

On the other hand, from lab observer's perspective who is at the rest w.r.t air, when propeller tries to rotates faster, say $d\omega=1\,\,rad/s$ from pilot's point of view, it will just become $d\omega'=10^{-8}\,\,rad/s$ faster in lab observer perspective due to the time dilation which is not that much, and that's why aircraft itself won't arrive at speed of light, because $d\omega'$ will become zero in physical sense, when aircraft speed is very near to speed of light.

Surely as propeller rotates faster, aircraft will move faster from lab observer point of view. This is correct, and it has nothing to do with time dilation. Time dilation just doesn't let propeller rotation goes above certain speed. $\omega$ will increase in lab observer's perspective as propeller rotation become faster in pilot's point of view, time dilation just changes "how much propeller becomes faster" in lab's observer frame. At high speed, propeller will rotate a little faster in lab's point of view as pilot tries to make it a lot faster.

Paradoxy
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  • Mumbo jumbos! ;) Air density is infinite at exactly the speed of light, whereas I am speaking about speeds less than light speed. Moreover, if the angular speed is $\omega=10^{-8} \space rad/s$, the airplane must move very slow, and not at a velocity close to that of light from the standpoint of the lab observer. – Mohammad Javanshiry Nov 14 '19 at 10:53
  • @MohammadJavanshiry I guess you didn't get my point. You see, as airplane moves faster, the magnitude of angular velocity won't become less than before due to the higher time dilation. Rather, propeller rotation will become faster as you said your self "the faster a propeller rotates, the faster the attached vehicle moves". But how much will it become faster from lab observer's point of view? That's where time dilation plays a big role. – Paradoxy Nov 14 '19 at 11:14
  • As I said, the propeller would nearly stop spinning when the aircraft approaches light speed. Stop spinning means no angular velocity, how can this be reconciled to your sentence: You see, as airplane moves faster, the magnitude of angular velocity won't become less than before due to the higher time dilation.? – Mohammad Javanshiry Nov 14 '19 at 11:21
  • @MohammadJavanshiry Resistance of air is proportional to $\text{velocity}^2$. However this formula doesn't work when you go beyond a certain speed. Propeller is used to speed up an airplane. Or it will just keep airplane at constant speed, due to the resistance of air. At a certain speed which is very close to speed of light, you are right angular velocity will indeed drop to zero but note that mass of airplane will become infinitely large and that means (in my opinion) resistance of air will become zero for that airplane. – Paradoxy Nov 14 '19 at 11:58
  • Now in this case if propeller were to rotate in lab observer's point of view, he would expect that airplane goes beyond speed of light, which is clearly wrong. The problem is, we don't know mechanism of a certain medium such as air at very high velocities. And "as airplane moves faster, the magnitude of angular velocity won't become less than before due to the higher time dilation" is right as long as you don't go beyond a certain speed. This certain speed however, should be calculated. – Paradoxy Nov 14 '19 at 11:59
  • The problem is, we don't know the mechanism of a certain medium such as air at very high velocities. Yes, this is possible. – Mohammad Javanshiry Nov 14 '19 at 15:26
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There was a similar problem involving a relativistic traincar with toothed wheels moving on toothed rail.

Due to time dilation, in the reference frame of the rail rotation of the wheel must slow down as speed of the train approaches that of light, that seems as a paradox. It seems that the faster the train moves, the slower its wheels must rotate.

If the train moves at velocity close to $c$, rotation of the wheels must slow down until almost complete stop. This way the train appears to be “slipping” on the rail in the reference frame of the rail, though it is impossible, since “smooth rotation” is an absolute effect and it cannot depend on chosen reference frame.

Resolution of that „paradox“ was in relativistic kinematics. The rim of the wheel Lorentz – contracts as velocity of the train increases.

The rest length of the rim of the wheel must remain constant. This means that the rim Lorentz contracts, and that the radial extension of the wheel contracts accordingly. The result is that the wheel becomes infinitely small in the limit that the train moves with the velocity of light.

If $v$ is velocity on the rim in the rest frame $K$ of the wheel, we have $\Omega=v/R$, where $R=R_0/\gamma$ is the contracted radius of the rotating wheel, and $R_0$ is their radius when they are at rest. The angular velocity of the rotating wheel is then

$$\Omega = \gamma v /R_0$$

Hence, in this case the angular velocity $\Omega$ must approach an infinitely great value in $K$ when the speed of the train approaches that of light.

Through a gear wheel rotation can be transmitted from the wheel to the "propeller"; apparently in the reference system of the traincar angular velocity of this "propeller" must also approach infinitely large value as speed of the train comes close to that of light.