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Although I'm loathe to get into one of those infernal discussions about the definition of mass I will say this. One has to be careful about when the relationship P = mU where m = proper mass, P = 4-momentum and U = 4-velocity. Suppose you have an ion in an electric field. This relationship doesn't hold because it doesn't take into account the inertia of the stress imposed in the ion due to the electric field.
Ahem, think it's your turn to present arguments Simplified. Just use your logic to show me how you think it is done.And try to be as clear as you can so we get how you think there.