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This is the Twin frog paradox. I have two questions.

First, what does it mean by there is no rigid body in special relativity? What I have known is that the length can be changed from one frame to another so there is no rigid body. However, to solve this paradox, I have to assume that there is no rigid body within one frame too, that is given a frame, any object can be observed in the frame to be elastic. What does this exactly mean?

Second the solution I found states that the center of the cylinder(which is at rest in the inertial frame of the cylinder) does not jerk back and forth in any other inertial frame. However, in the second picture below, the two ends of the cylinder pulsate at different phases so the center of the cylinder must jerk back and forth... Could anyone help me with this discrepancy?

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What he means by “there is no rigid body in special relativity” is that srt forbids the existence of any object that is not elastic to some degree. Of course, no such thing has ever been found or theorized. When you push on an object, the effect will normally travel at the speed of sound for that material, so there will be no effect, and no motion of that part of the object, until enough time has passed for the pulse to travel at the speed of sound to that location in the object. We don’t notice this in normal life because such speeds are very high, for example the speed of sound in steel is 5,000 m/s.

In the figures, the jagged lines are the effects from the frog collisions traveling down the length of the can. See in the top figure how they start when the frog hits that end and then through time go to the other end of the can, presumably at the speed of sound for the can’s material, which is very slow compared to the speed of light, and also probably slow compared to the relative speeds between the frame of the can and the other observer (I say this because the figures show significant relativistic effects and hence seem like the relative speed must be a meaningful fraction of the speed of light, which the speed of sound is not). I’ll explain the motion of the ends of the can in a comment to keep this shorter.

In the lower figure, the frogs travel faster toward the bottom of the can, but the tension wave from hitting the bottom then travels slower back to the top. No effect is seen on the center of the can until a tension wave gets there. The net forces on the center of the can never go above zero even in the moving frame. The tension waves cross there, canceling each other out (see figure)

In all frames, the effects from the two frogs reach the center of mass at the same time even though one frog hits first.

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  • $\begingroup$ The reason the top of the can moves to the right from point 2 to 4 is because it received an impulse impact velocity from frog A hitting it and the effect from frog B has not yet reached the top so it moves to the right until that frog-B tension wave arrives and delivers both the frog-B impulse and the can’s spring tension which combine to stop the top and accelerate it in the other direction until frog B hits the top. We can see similar deformation in the other figure; the ends only change speed from a) impacts or b) tension waves arriving. $\endgroup$
    – Al Brown
    Jul 27 at 7:20
  • $\begingroup$ Incidentally, if the can is metal, and the time axis is to scale, the subsonic frogs seem to be traveling somewhere 2,500 m/s inside the can, based on the speed of sound, and of tension waves, in metal. And the second figure really should have the can as a whole going much faster (slanted more horizontally) because the magnitude of the relativistic effects mean that observer is going idk 5- 30% of the speed of light relative to the can, not five or ten thousand m/s as it appears. $\endgroup$
    – Al Brown
    Jul 27 at 7:31
  • $\begingroup$ Im NOT absolutely 100% certain now the impact tension waves reach the center at the same time in the moving frame. For one thing, they are of different magnitude - the frog hitting the bottom hits with a higher speed and hence greater impulse. So they wouldn’t cancel each other out by arriving at the same time. We do know that the center of mass does not change speed (because no external force acts on the frogs/can system), so no net force ever affects the center. The complex dynamics of impulse tension waves and spring tension fluctuate in such a way as to make that true in all frames $\endgroup$
    – Al Brown
    Jul 27 at 8:31
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any object can be observed in the frame to be elastic. What does this exactly mean?

It means forces at one end of a body are transmitted to the other side usually via EM interactions (the electrons bonding the atoms together). When you push on one side, it reacts and starts moving before the force is transmitted to the other side. For the body to be truly inelastic, both sides would have to move at the same time. This doesn't happen.

In the second frame, the pulses don't move through the medium at the same relative speed. So even though they don't begin simultaneously, they reach the center of the cylinder simultaneously. Neither pulse reaches the center before the other.

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  • $\begingroup$ Im not so sure the impact tension waves reach the center at the same time in the moving frame. For one thing, they are of different magnitude - the frog hitting the bottom hits with a higher speed and hence greater impulse. So they wouldn’t cancel each other out by arriving at the same time. We do know that the center of mass does not change speed (because no external force acts on the frogs/can system), so no net force ever affects the center. The complex dynamics of impulse tension waves and spring tension fluctuate in such a way as to make that true in all frames. What do you think? $\endgroup$
    – Al Brown
    Jul 27 at 7:47
  • $\begingroup$ If the waves did not reach the center at the same time, then during the interval between the two, there would be a non-zero net force on the center of the cylinder, resulting in an acceleration. Wave speed is due to material it travels through, not impact speed/impulse of initiation. $\endgroup$
    – BowlOfRed
    Jul 27 at 7:59
  • $\begingroup$ Wave speed is not due to impact, but wave magnitude is. A frog hits the top at a high speed transferring a lot of acceleration and speed to the can top. That tension waves reaches the center putting a lot of force on the center. The other frog hits the can bottom slower and that wave reaches the center, putting a little bit of force on the center. They dont cancel out. Different magnitudes because different collision speeds. $\endgroup$
    – Al Brown
    Jul 27 at 8:14

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