XV

The most important result of a general character to

which the special theory of relativity has led is concerned with the conception of mass. Before the advent of

relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the conservation of energy and the law of the conservation of mass; these two fundamental laws appeared to be quite

independent of each other. By means of the theory of relativity they have been united into one law. We shall now briefly consider how this unification came about, and what meaning is to be attached to it.

The principle of relativity requires that the law of the conservation of energy should hold not only with reference to a co-ordinate system $K$, but also with respect to every co-ordinate system $K^{\prime }$ which is in a state of uniform motion of translation relative to $K$, or, briefly, relative to every "Galileian" system of co-ordinates.

In contrast to classical mechanics, the Lorentz transformation is the deciding factor in the transition from one such system to another.

By means of comparatively simple considerations we are led to draw the following conclusion from these premises, in conjunction with the fundamental equations of the electrodynamics of Maxwell: A body

moving with the velocity $v$, which absorbs$E_0$ is the energy taken up, as judged from a co-ordinate system moving with the body. an amount of energy $E_0$ in the form of radiation without suffering

an alteration in velocity in the process, has, as a consequence, its energy increased by an amount $\frac{E_0}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}.$

In consideration of the expression given above for the kinetic energy of the body, the required energy of the body comes out to be $\frac{(m+{\displaystyle \frac{E_0}{c^2}})c^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}.$

Thus the body has the same energy as a body of mass $(m+{\displaystyle \frac{E_0}{c^2}})$ moving with the velocity $v$. Hence we can say: If a body takes up an amount of energy $E_0$, then its inertial mass increases by an amount ${\displaystyle \frac{E_0}{c^2}}$; the

inertial mass of a body is not a constant, but varies according to the change in the energy of the body. The inertial mass of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the mass of a system becomes identical with the law of the conservation of energy, and is only

valid provided that the system neither takes up nor sends out energy. Writing the expression for the energy in the form $\frac{m{c}^2+E_0}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}},$ we see that the term $m{c}^2$, which has hitherto attracted our attention, is nothing else than the energy possessed by the bodyAs judged from a co-ordinate system moving with the body. before it absorbed the energy $E_0$.