# Mass Variation By Special Relativity

According to Special Relativity, Mass of an object increases with increase in its velocity. I got a question in my test asking, will mass of a cannon ball fired from a cannon remain constant or increase? I answered increase which was rejected on the basis that these equations are only valid for velocities approaching to that of light. Theoretically mass should increase and even if the increment is of millionth factor, an increase is an increase. Granted that increment will be immeasurable and completely negligible but still it will increase; won't it?

• If one accepts that mass increases with speed, then the question should be "does the mass of the cannonball increase and, if so, in which direction?" – Alfred Centauri Dec 22 '16 at 22:49
• Consider an object with speed $v$ in the $x$ direction; according to SR, it requires more force to produce a given acceleration in the $x$ direction than it does in the $y$ and $z$ directions. If one interprets this as due to the mass changing with speed, it must be that the mass is different in the $x$ direction than in the $y$ and $z$ directions. – Alfred Centauri Dec 22 '16 at 23:42
• The search terms for the situation that @AlfredCentauri is describing are "longitudinal mass" and "transverse mass". The formula often given for the "relativistic mass" corresponds to the transverse mass. Nor does this issue go away in the modern idiom until you use four-force and four-acceleration, because the basic problem is trying to give interpretation to quantities with ill defined tensor nature. – dmckee --- ex-moderator kitten Dec 23 '16 at 1:50
• Thank you both of you guys for introducing this concept of directional mass. I studied the wikipedia article and got to know much – Danial Saleem Dec 23 '16 at 7:54
• See also physics.stackexchange.com/q/133376/50583 for a discussion of the now largely considered obsolete concept of increasing mass. – ACuriousMind Dec 23 '16 at 13:29

NO. The mass never increases or decreases with velocity. Mass is a Lorentz scalar, i.e., it remains invariant when you go from one frame to another in which the velocity of the particle can be generically different from the velocity in the original frame. The misleading and confusing concept of relativistic mass was introduced in early days of Special Relativity which tried to define the relativistic mass $M = \dfrac{m}{\sqrt{1-\dfrac{v^2}{c^2}}}$. Here, $m$ is the mass (or the so-called rest mass). But this concept is completely overthrown because of several reasons including the inconsistencies and unnecessary conceptual difficulties it gave rise to. Have a look at this fantastic article by Lev Okun for a detailed discussion.
• As long as you are aware that "relativistic mass" is synonymous with "energy" according to Einstein's formula $E=m_{rel}c^2$ and that it is not the same as "rest mass" $m_0$, there should be no problem in using this term, which you can (still) find in many basic textbooks. The concept of relativistic mass is not always disadvantageous, in certain contexts its use also has definite advantages. See e.g., T. R. Sandin,"In defense of relativistic mass", American Journal of Physics 59, 1032 (1991). – freecharly Dec 22 '16 at 19:48
• @DanialSaleem What it means is that there is no "structural" change in the object itself when its energy is higher because of a constant velocity as compared to an object which is at rest but would have the same energy if it were moving with a constant velocity equal to that of the actual object. This can be easily understood because one can imagine that even if the object is moving with a certain velocity wrt $O$, there exists an equivalent observer $O'$ wrt whom the object is at rest. Now, whatever observer I choose, it certainly doesn't make structural changes in the object itself..... – Dvij D.C. Dec 24 '16 at 12:55