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If a particle and antiparticle annihilate they produce photons of certain frequency. My question is about the frequency of the photons if the particles move at relativistic speeds with let say gamma=5. So how is the frequency dependent on relativistic masses of the particles if the intuition suggests that the frequency should emerge from amount of motion wich cannot excede the speed of light and nevertheless the gamma is 2 or 5 the speeds are almost equal close to c?

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In a particle-antiparticle annihilation, you can always find a “rest frame” where the total momentum is zero. In that rest frame, the photons created will carry away the total energy of the particles. In a threshold capture, like the annihilation of positronium, the energy will just be the masses of the particle and antiparticle. If the pair collide with significant kinetic energy in the rest frame, that extra energy is also carried away by the decay radiation.

If your particle-antiparticle pair is not in your rest frame, the photons in the lab frame will also carry away the center-of-mass kinetic energy. You can find the energy by analyzing the collision in the rest frame, then applying a Lorentz transformation.

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