Page:The Meaning of Relativity - Albert Einstein (1922).djvu/63

$\left.{\begin{aligned}I_{x}=&{\frac {m\mathbf {q} _{x}}{\sqrt {1-q^{2}}}}\\&\vdots \\E=&{\frac {m}{\sqrt {1-q^{2}}}}\end{aligned}}\right\}$

(43)

We recognize, in fact, that these components of momentum agree with those of classical mechanics for velocities which are small compared to that of light. For large velocities the momentum increases more rapidly than linearly with the velocity, so as to become infinite on approaching the velocity of light.

If we apply the last of equations (43) to a material particle at rest ($q=0$), we see that the energy, $E_{0}$, of a body at rest is equal to its mass. Had we chosen the second as our unit of time, we would have obtained

$E_{0}=mc^{2}$

(44)

Mass and energy are therefore essentially alike; they are only different expressions for the same thing. The mass of a body is not a constant; it varies with changes in its energy.^[1] We see from the last of equations (43) that $E$ becomes infinite when $q$ approaches 1, the velocity of light. If we develop $E$ in powers of $q^{2}$, we obtain,

$E=m+{\frac {m}{2}}q^{2}+{\frac {3}{8}}mq^{4}+\cdots$

(45)

1. The emission of energy in radioactive processes is evidently connected with the fact that the atomic weights are not integers. Attempts have been made to draw conclusions from this concerning the structure and stability of the atomic nuclei.