XV

General Results of the Theory

It is clear from our previous considerations that the (special) theory of relativity has grown out of electrodynamics

and optics. In these fields it has not

appreciably altered the predictions of theory, but it has considerably simplified the theoretical structure, i.e. the derivation of laws, and–-what is incomparably

more important–-it has considerably reduced the number of independent hypotheses forming the basis of

theory. The special theory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latter

would have been generally accepted by physicists even if experiment had decided less unequivocally in its favour.

Classical mechanics required to be modified before it

could come into line with the demands of the special theory of relativity. For the main part, however, this modification affects only the laws for rapid motions, in which the velocities of matter $v$ are not very small as compared with the velocity of light. We have experience of such rapid motions only in the case of electrons

and ions; for other motions the variations from the laws

of classical mechanics are too small to make themselves evident in practice. We shall not consider the motion

of stars until we come to speak of the general theory of relativity. In accordance with the theory of relativity

the kinetic energy of a material point of mass $m$ is no

longer given by the well-known expression $m\frac{v^2}{2},$ but by the expression $\frac{m{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}.$ This expression approaches infinity as the velocity $v$ approaches the velocity of light $c$. The velocity must therefore always remain less than $c$, however great may be the energies used to produce the acceleration. If we develop the expression for the kinetic energy in the form of a series, we obtain $m{c}^2+m\frac{v^2}{2}+\frac{3}{8}m\frac{v^4}{c^2}+\dots .$

When ${\displaystyle \frac{v^2}{c^2}}$ is small compared with unity, the third of these terms is always small in comparison with the second, which last is alone considered in classical mechanics. The first term $m{c}^2$ does not contain the velocity, and requires no consideration if we are only dealing with the question as to how the energy of a point-mass depends on the velocity. We shall speak

of its essential significance later.