Variable-mass system events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass "m" moves at a velocity v at an initial time "t". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time "t" + d"t", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d"m"dv is the product of two small values, it can be ignored, meaning during d"t"
Mass in special relativity derived form formula_44 is not valid because formula_45 in formula_46 is generally not a constant (see the section above on transverse and longitudinal mass). Even though Einstein initially used the expressions "longitudinal" and "transverse" mass in two papers (see previous section), in his first paper on formula_47 (1905) he treated "m" as what would now be called the "rest mass". Einstein never derived an equation for "relativistic mass", and in later years he expressed his dislike of the idea: Okun and followers reject the concept of relativistic mass. Also Arnold B. Arons has argued against teaching the concept of relativistic