Page:LewisTolmanMechanics.djvu/12

From equations I and II we may derive one of the most interesting consequences of the principle of relativity. If E is the total energy (including internal energy) of a body in motion, and E₀ is its energy at rest, the kinetic energy E' is equal to E—E₀, and equation II may be written,

$E-E_{0}=m_{0}c^{2}\left({\frac {1}{\sqrt {1-\beta ^{2}}}}-1\right)$.

III

Moreover, we may write equation I in the form,

$m-m_{0}=m_{0}\left({\frac {1}{\sqrt {1-\beta ^{2}}}}-1\right)$,

IV

and dividing III by IV

${\frac {E-E_{0}}{m-m_{0}}}=c^{2}$.

V

In other words, when a body is in motion its energy and mass are both increased, and the increase in energy is equal to the increase in mass multiplied by the square of the velocity of light. From the fundamental conservation laws we know that when a body is set in motion and thus gains mass and energy, these must come from the environment. So also when a moving body is brought to rest, it must give up mass as well as energy to the environment. The mass thus acquired by the environment is independent of the particular form which the energy may assume, and we are thus forced to the important conclusion that when a system acquires energy in any form it acquires mass in proportion, the ratio of the energy to the mass being equal to the square of the velocity of light. We might go further and assume that if a system should lose all its energy it would lose all its mass. If we admit this plausible although unproved assumption, then we may regard the mass of every body as a measure of its total energy according to the equation,

$m={\frac {E}{c^{2}}}$.

VI

For a body at rest,

$m_{0}={\frac {E_{0}}{c^{2}}}$.