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I've been working through R.E. Turner's "Relativity Physics," and have a question about the example he gives of Compton scattering, in which the quantity of interest is the post-collision gamma particle. All the examples I've seen elsewhere assume that the electron is at rest before the collision. In Turner's example, the electron is in motion before the collision. When Turner expands the right-hand side of the 4-vector conservation equation (equation 5.29 if you happen to have a copy), he gets the following terms:

--the dot-product of the pre-collision electron with itself,

--the energy/time-like term from the dot-product of the electron with the pre-collision gamma particle

--the energy/time-like term from the dot-product of the electron with the post-collision gamma particle

--the energy/time-like term from the dot-product of pre- and post-collision gamma particles

--the dot-product of the momenta of pre-collision electron and the post-collision gamma ray

But since both the pre-collision electron and gamma ray have momentum, shouldn't there also be a term for the dot-product of the momenta of these two particles?

Similarly, shouldn't there be a term for the dot-product of the pre-collision electron and the post-collision gamma ray?