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OK, if anyone is so inclined, let's continue the discussion here, since I think we are distracting from the main issues in https://www.physicsforums.com/showthread.php?p=3652601#post3652601
Anyway, may I suggest that the real thing we want to get rid of is not relativistic mass. Rather it is the concept of force and point particle. If we take Maxwell's equations as the reason for special relativity, then we can't really make the Lorentz force law and point particles work, can we? Without those, we don't need relativistic mass. OTOH, since force and point particles are useful in some regime, we keep relativistic mass around as a moral link between the new theories which deal primarily in fields, and the old theories in which point particles and forces were ok. This link is more moral than quantitative, since it has to be generalized to a form applicable to fields, but it is the historical route that indicated what sits on the right-hand side of the Einstein field equation.
Anyway, may I suggest that the real thing we want to get rid of is not relativistic mass. Rather it is the concept of force and point particle. If we take Maxwell's equations as the reason for special relativity, then we can't really make the Lorentz force law and point particles work, can we? Without those, we don't need relativistic mass. OTOH, since force and point particles are useful in some regime, we keep relativistic mass around as a moral link between the new theories which deal primarily in fields, and the old theories in which point particles and forces were ok. This link is more moral than quantitative, since it has to be generalized to a form applicable to fields, but it is the historical route that indicated what sits on the right-hand side of the Einstein field equation.