One of the issues surround the concept of mass and the Einstein equation is the idea on what "mass" really means in relativity, and the validity of the concept of "relativistic mass". There have been many articles written to address this issue, but it is obvious that, even today, many media, including textbooks and popular writings, continue to use the term "relativistic mass" to mean an increase in the measured mass when an entity is moving at relativistic speeds. Whether the faulty understanding of such a concept can create a stumbling block in understanding relativity or not is an entirely different issue. But can there be a simpler approach to such a concept without invoking the name "relativistic mass"?
Lev Okun seems to think so. In a highly compact 2-page paper in the Am. Journal of Physics[1], he wrote a very concise explanation of what "mass" is, and why there is really only ONE concept of mass as defined in terms of momentum and energy by what he called the most fundamental equation of relativity theory:
m^2 = (E/c^2)^2 - (p/c)^2,
where E is the energy, p is the momentum. There is nothing new here that someone who has gone through an intro class in relativity/Modern Physics would not have seen. But it is put in such a compact and clear form that it summarizes Special Relativity in almost 1 1/2 pages!
What is as interesting is his commentary on how this issue has been treated in the media and in textbooks.
Unfortunately, sometimes and especially in his popular writings Einstein was careless about the subscript 0 and spoke about the equivalence of mass and energy and omitted the attribute “rest” for the energy. As a result Einstein's equation E0=mc^2 became known in its famous but misleading form E=mc^2. One of the most unfortunate consequences is the concept that the mass of a relativistic body increases with its velocity. This velocity dependent mass is known as “relativistic mass.” Another consequence is the term “rest mass” and the corresponding symbol m0. These confusing concepts and notations prevail in such classic texts as the ones by Born and Feynman. Moreover, in these texts the dependence of mass on velocity is presented as an experimental fact predicted by relativity theory and proving its correctness.
To substantiate the formula m=E/c^2 some authors use the connection between momentum and velocity in Newtonian mechanics, p=mv, forgetting that this relation is valid only when v (is significantly less than) c and that it contradicts the basic equation m^2=(E/c^2)^2−(p/c)^2. Einstein's tolerance of E=mc^2 is related to the fact that he never used in his writings the basic equation of relativity theory. However, in 1948 he forcefully warned against the concept of mass increasing with velocity. Unfortunately this warning was ignored. The formula E=mc^2, the concept relativistic mass, and the term rest mass are widely used even in the recent popular science literature, and thus create serious stumbling blocks for beginners in relativity.
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[1] L.B. Okun Am. J. Phys. v.77, p.430 (2009).
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