Conservation of four momentum question

[vin300]

The question goes like this: Prove that conservation of four momentum forbids a reaction in which an electron and positron annihilate and produce a single photon(gamma ray). Prove that the production of two photons is not forbidden.
The solution is to work in the centre of momentum frame. I understand that, the electron and positron will travel in opposite directions in this frame since they both have the same mass, and the annihilation will conserve momentum by sending out even numbered photons, each pair in opposite directions of the same frequency.
My question is, why can't we not work in the centre of momentum frame. In the frame of the positron, the total momentum before collision is not zero, but after collision, if only one photon is produced with the same momentum as the sum of the four momenta of the electron and positron, things will still look good.
If two photons of the same frequency are emitted as seen in the C.M. frame, for an observer traveling in the MCRF of the positron traveling from right to left, the left going photon will be redshifted and the right going photon will be blueshifted. The resultant of this momentum has to be equivalent to the resultant momentum before collision.

 

[Ich]

if only one photon is produced with the same momentum as the sum of the four momenta of the electron and positron, things will still look good.

Why don't you try an example calculation? Remember, the norm of the photon four momentum ist zero.

 

[vin300]

Conservation of the zeroth component means conservation of energy. Does this mean all collisions in SR are elastic collisions?

 

[Matterwave]

Conservation of the zeroth component means conservation of energy. Does this mean all collisions in SR are elastic collisions?

Elastic vs inelastic collisions apply for macroscopic objects. If you are dealing with particles, all collisions are elastic because you keep track all the energy.

A macroscopic collision might be inelastic because some energy goes into deforming the material, or sound, or light, but if you are dealing with individual particles, the energy has nowhere else to go.