In Newtonian physics, mass is an invariant, by which I mean that any observer would measure the same value for the mass of an object, irrespective of how fast the observer was moving. Also, the relationship between momentum (p) and mass (m) was given by momentum equals mass times velocity (v). i.e., p=mv.
When relativity came along, the relationship between momentum and mass became more complicated by the presence of an additional velocity dependent multiplicative factor denoted by the Greek letter gamma γ. Now the expression for the momentum was p=γmv.
But the old Newtonian form could be retrieved by introducing the concept of ‘relativistic mass’ in which the multiplicative factor could be tacked onto the old mass to give a new relativistic mass M=γm, thus resulting in p=Mv. This relativistic mass is now velocity dependent and this led to a spate of popular descriptions of how an object’s mass depended on its velocity.
The idea of relativistic mass created a mess because while it had a certain heuristic appeal, it coexisted along with those who deplored its use and retained the term mass to refer only to the old invariant mass m, which the advocates of relativistic mass now referred to as the ‘rest mass’. It was argued that which approach one adopted was largely a matter of taste.
The eminent Russian physicist Lev Okun argued strongly that the idea of relativistic mass should be abandoned because whatever heuristic value it had was strongly overcome by its lack of consistency and the misconceptions it created. (His 1989 article in Physics Today was one such effort.) He was so persuasive that I was convinced and never referred to it again. I also thought that the issue was now dead.
So I was surprised to be told about this 2005 paper by Gary Oas who did an exhaustive survey of physics textbooks and popular books and found that relativistic mass is alive and well and still being used by people who I feel really should know better. Like Okun, Oas argues that this is a mistake and that we would be far better off burying the concept of relativistic mass once and for all.
I totally agree.