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Please give a qualitative explanation, as opposed to $E = mc^{2}$ or the Lorentz equation of mass. Does the matter of body, i.e. the number of atoms, increase?

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    $\begingroup$ The mass of the object does not increase: it is and always stays $m_0$. A corrective factor $\gamma$ for the velocity is present though in front of the equations of motion, but the mass remains the same. $\endgroup$
    – gented
    Aug 30 '16 at 7:44
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What changes isn't its actual mass - that is defined by its rest mass. What increases is something called its relativistic mass. As the velocity of a body asymptotically approaches $c$, it may be helpful to think of an "acceleration" preventing the velocity from every reaching $c$. Naturally, this "acceleration" may be expressed as a "force" divided by a "mass". This "mass" is the relativistic mass.

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Here, number of atoms doesn't change, but the scale itself changes. When you measure something, you compare it with a predefined scale. e.g. You put a ruler next to a rod to measure its length. If you change the markings on this ruler, you are going to get a different result. It doesn't mean that dimension of the rod was changed from underlying matter perspective.

What happens in relativistic physics is: Due to changes devised in Space and Time itself, our measurement standards are affected.

Focus not on increased mass, but measurement done by different observers. Speed is observer dependent quantity. So, a body in motion can have different speeds depending on observers. Each observer would measure a different mass of the same body (observer seeing higher speed would measure higher mass than the other observer's measured mass). I hope, now you get it as you can't apply increase in atoms logic in this situation.

Whenever you check mass, length, time, or any speed-dependent physical quantity in Relativistic realm, always see everything from at least two observers perspective because the original concepts always involve two observers. And, people get confusion by seeing from one observer perspective.

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