Page:The Meaning of Relativity - Albert Einstein (1922).djvu/60

Mass and Energy. An important conclusion can be drawn from the existence and significance of the 4-vector $K_{\mu }$. Let us imagine a body upon which the electromagnetic field acts for a time. In the symbolic figure (Fig. 2) $Ox_{1}$ designates the $x_{1}$-axis, and is at the same time a substitute for the three space axes $Ox_{1},Ox_{2},Ox_{3}$; $Ol$ designates the real time axis. In this diagram a body of finite extent is represented, at a definite time $l$, by the interval AB; the whole space-time existence of the body is represented by a strip whose boundary is everywhere inclined less than 45° to the $l$-axis. Between the time sections, $l=l_{1}$ and $l=l_{2}$, but not extending to them, a portion of the strip is shaded. This represents the portion of the space-time manifold in which the electromagnetic field acts upon the body, or upon the electric charges contained in it, the action upon them being transmitted to the body. We shall now consider the changes which take place in the momentum and energy of the body as a result of this action.

We shall assume that the principles of momentum and energy are valid for the body. The change in momentum, $\Delta I_{x},\Delta I_{y},\Delta I_{z}$, and the change in energy, $\Delta E$, are then given by the expressions

${\begin{aligned}\Delta I_{x}=\int _{l_{0}}^{l_{1}}dl\int \mathbf {k} _{x}dxdydz&={\frac {1}{\mathbf {i} }}\int K_{1}dx_{1}dx_{2}dx_{3}dx_{4}\\&\vdots \\\Delta E=\int _{l_{0}}^{l_{1}}dl\int \lambda dxdydz&={\frac {1}{\mathbf {i} }}\int {\frac {1}{\mathbf {i} }}K_{4}dx_{1}dx_{2}dx_{3}dx_{4}\end{aligned}}$