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My one sentence version of the difference is as follows:
Do you prefer to write your relativistically correct momentum formula
as:
a) p = gamma*m*v
or
b) p = m*v
If you like (b) you are preferring a relativistic mass approach.
If you like (a) you are preferring an invariant mass approach.
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OTOH, there probably is some reason why most current GR and SR (general
relativity and special relativity) practitioners follow approach (a),
and its not simply a matter of faddism.