Center of momentum frame with photons?

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Is it possible to have a center of momentum frame in a system with a photon and a massive particle?

If you were to do so in a system with two massive particles, this would involve redefining the velocity of the particles in the CM frame. E.g. if A moves towards B with speed 2v, in the CM frame, A and B would move towards each other at v.

Since the photon will have speed c in all frames, is it possible to have a unique frame in which the massive particle has momentum -E/c (where E is the energy of the photon)?
 
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Yes, it is possible. Remember that as you change reference frames the photon will red-shift or blue-shift. The Doppler shift will change the energy and therefore the momentum of the photon. There will be a unique frame where it is opposite that of the massive particle.
 
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