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- Summary
- relativistic spinning cylinder resembles a wave function

Hello, I am curious if I have this correct and if it has a name.

A thin walled cylinder is spinning on its axis along its length in a closed system. It begins to draw itself in converting its invariant mass to kinetic energy. In polar coordinates ##E=\gamma_\theta m c^2, L=\gamma_\theta m v_\theta r, \omega r = v_\theta##. Assuming* ##v_\theta## can approach ##c## , all of the mass is converted and ##E \to L \omega##.

It is very long. An observer is at rest with its axis along ##z## in reference frame ##S##. It translates at ##v_z## with respect ##S'##. To the observer in ##S'## the cylinder is twisted. A line down the surface that is straight in ##S## is a helix in ##S'##. The magnitude of the helix twist is given by ## \frac{d\theta}{dz} = \gamma_z \omega v_z/c^2##. Sorry I don't have a proof for this but references state that ##\omega## is for the non translating frame. So the helix twists ##2\pi## radians in ##\frac{2\pi L c^2}{\gamma_z E v_z}## units of length. Note ##E## is the constant energy in ##S## and all of it contributes to the momentum in ##S'##. So expressed in ##p_z## the "wavelength" is ##\lambda=\frac{2\pi L}{p_z}##.

One problem that arises is the translating observer in ##S'## sees a slower angular velocity ##\omega'=\omega/\gamma_z## and that is not proportional to the total energy for that observer. But, the helix propagates wave crests at a velocity of ##\frac{\lambda \omega'}{2\pi}=\frac{c^2}{v_z}-v_z##. These are not physically moving objects but coordinates moving along a moving object. So the straight addition to the velocity of the cylinder gives ##c^2/v_z##, resembling the phase velocity of a wavefunction. The helix line can then look like a wave that is not translating, but just spinning at ##\gamma_z \omega## which is proportional to the energy in ##S'##, by ##L##.

Lastly both observers see the surface of the cylinder moving at ##c## and the observed energy is proportional to momentum for that observer. In ##S'## it is moving in a spiral. In ##S'## if you break this momentum into components of ##p_\theta## and ##p_z## their Pythagorean sum is the relativistic energy-momentum equation (taking advantage of ##v_\theta' \gamma_z = c##).

*clearly all the energy cannot be converted due to atomic structure, but is there a problem before then? Weak energy condition perhaps?

Thanks for reading. Any comments or help pointing out mistakes are welcome.

A thin walled cylinder is spinning on its axis along its length in a closed system. It begins to draw itself in converting its invariant mass to kinetic energy. In polar coordinates ##E=\gamma_\theta m c^2, L=\gamma_\theta m v_\theta r, \omega r = v_\theta##. Assuming* ##v_\theta## can approach ##c## , all of the mass is converted and ##E \to L \omega##.

It is very long. An observer is at rest with its axis along ##z## in reference frame ##S##. It translates at ##v_z## with respect ##S'##. To the observer in ##S'## the cylinder is twisted. A line down the surface that is straight in ##S## is a helix in ##S'##. The magnitude of the helix twist is given by ## \frac{d\theta}{dz} = \gamma_z \omega v_z/c^2##. Sorry I don't have a proof for this but references state that ##\omega## is for the non translating frame. So the helix twists ##2\pi## radians in ##\frac{2\pi L c^2}{\gamma_z E v_z}## units of length. Note ##E## is the constant energy in ##S## and all of it contributes to the momentum in ##S'##. So expressed in ##p_z## the "wavelength" is ##\lambda=\frac{2\pi L}{p_z}##.

One problem that arises is the translating observer in ##S'## sees a slower angular velocity ##\omega'=\omega/\gamma_z## and that is not proportional to the total energy for that observer. But, the helix propagates wave crests at a velocity of ##\frac{\lambda \omega'}{2\pi}=\frac{c^2}{v_z}-v_z##. These are not physically moving objects but coordinates moving along a moving object. So the straight addition to the velocity of the cylinder gives ##c^2/v_z##, resembling the phase velocity of a wavefunction. The helix line can then look like a wave that is not translating, but just spinning at ##\gamma_z \omega## which is proportional to the energy in ##S'##, by ##L##.

Lastly both observers see the surface of the cylinder moving at ##c## and the observed energy is proportional to momentum for that observer. In ##S'## it is moving in a spiral. In ##S'## if you break this momentum into components of ##p_\theta## and ##p_z## their Pythagorean sum is the relativistic energy-momentum equation (taking advantage of ##v_\theta' \gamma_z = c##).

*clearly all the energy cannot be converted due to atomic structure, but is there a problem before then? Weak energy condition perhaps?

Thanks for reading. Any comments or help pointing out mistakes are welcome.

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