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Below I will use some simple formulas, that's why I must make a distinction between longitudinal and transverse relativistic mass. The transverse relativistic mass of an object has very much to do with the energy of the object: We just multiply the energy by a constant to get the transverse relativistic mass. The energy of an object has very much to do ...

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Is this concept of relativistic mass increase, related to the concept of Doppler effect of matter waves? No. Doppler's effect also happens for classical waves, including "classical matter wave", by which I meant Schroedinger's wave function. The effect is in fact trivial. When you change the reference frame, the momentum of the particle changes. By de ...

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No. In the wall frame, your sound is shifted down in frequency. It reflects at the frequency that strikes the wall. As you are receding from the wall, it is shifted down again. Two doppler shifts in total.

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Let's first take a look at what $x-ct$ means. Up to a phase, a light plane-wave can be written as $$\sim A^\mu \exp[ik_\nu x^\nu] \equiv A^\mu \exp[ik(x-ct)] \equiv A^\mu \exp[i\omega(x/c-t)]$$ I.e., along $x-ct=\rm const.$ the phase of the wave is constant. Hence we can define $x-ct \equiv \Delta \kappa(x,t)$ as a parametrization of "phase-time" along a ...

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