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My question is basically about four different versions of mass from the early years of special relativity when the concept of relativistic mass was acceptable. I'd appreciate it if you try to keep your answer as simple as possible.

Rest mass: $m_{0}$

Relativistic mass: $m_{rel}=\frac{m_{0}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}$

Longitudinal mass: $m_{L}=\frac{m_{0}}{\left( \root{2}\of{1-\frac{v^{2}}{c^{2}}}\right) ^{3}}$

Transverse mass: $m_{T}=\frac{m_{0}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}$

I was under the impression that the concepts of longitudinal mass and transverse mass are no longer valid but it seems like I was wrong. The text indirectly points out that the concepts are okay but there might be some other reasons for not using them.

Sometimes people like to use the terminology ‘longitudinal mass’ $\gamma ^{3}m_{0}$ and ‘transverse mass’ $\gamma m_{0}$. This can be useful, but we will not adopt it. The main point is that there is a greater inertial resistance to velocity changes (whether an increase or a decrease) along the direction of motion, compared to the inertial resistance to picking up a velocity component transverse to the current motion (and both exceed the inertia of the rest mass).

Source: Relativity Made Relatively Easy By Andrew M. Steane · 2012

Thus, at the turn of the century, there arose, through, as we now understand, the incorrect use of nonrelativistic equations to describe relativistic objects, a family of “masses” that increase with the energy of the body:

“relativistic mass” $m = \frac{E}{c^{2}}$,

“transverse mass” $m_{t}=m\gamma $,

“longitudinal mass” $m_{l}=m\gamma ^{3}$.

Note that for m ≠ 0 the relativistic mass is equal to the transverse mass, but, in contrast to the latter, it also exists for massless bodies, for which m = 0. We here use the letter m in the usual sense, since we used it in the first part of this paper. But all physicists during the first five years of this century, i.e., before the creation of the theory of relativity, and many after the creation of that theory called the relativistic mass the mass and denoted it by the letter m, as did Poincaré in his 1900 paper. And then there must necessarily arise, and did arise, a further, fourth term: the “rest mass,“ which was denoted by $m_{0}$. The term “rest mass“ came to be used for the ordinary mass, which, in a consistent exposition of the theory of relativity, is denoted m.

Source: Energy And Mass In Relativity Theory 2009

The pages #305 and #306 from "Classical Dynamics (1977)" by Donald T. Greenwood are also relevant to this discussion.

Einstein also used longitudinal mass and transverse mass in his 1905 special relativity paper. Longitudinal mass is greater than the transverse mass, and transverse mass is equal to relativistic mass. When Force, F, and velocity, v, are in the same direction, longitudinal mass would be used and acceleration would be in the direction of v. When F and v are perpendicular to each other, transverse mass would come into play, and v changes in direction but not in magnitude.

Now coming to the question.

I had though that relativistic mass is a combination of some sort of transverse mass and longitudinal mass but I think I was wrong. Assuming that the relativistic mass and transverse mass are the same, I'd say that longitudinal mass should have been used more frequently because force and velocity are mostly in the same direction. Phrasing it differently, if longitudinal mass is (or, was) real (please check edit #1 below) then it should have been used more than the transverse or relativistic mass but in actuality relativistic/transverse mass was the one which was mostly used in calculations. Personally, I have never seen longitudinal mass being used. Could you please help me with it.

Edit #1: I'd say that using the word 'real' was a poor of choice of words. I was trying to say that if it is true that more force is required to accelerate a mass in the direction of velocity, then using longitudinal mass would have been a better choice for doing calculations. I think that if you are trying to accelerate an electron, which is already moving toward positive x-axis at constant velocity, then one would rather choose its longitudinal mass to calculate acceleration for a given force.

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    $\begingroup$ Transverse and longitudinal masses predate Einstein, they were introduced by Lorentz. Adler's paper discusses their use and reinterpretation by Einstein. Transverse and relativistic masses, as physically characterized, are not the same, although the expressions are identical, Lewis and Tolman argued for replacing transverse and longitudinal masses with a single relativistic mass from 1909, which is what happened. $\endgroup$ – Conifold Sep 11 at 20:53

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