Page:The Meaning of Relativity - Albert Einstein (1922).djvu/58

46 {{MathForm2|(34)| $$ \left. \begin{matrix} \mathbf{e}_x' = \mathbf{e}_x & \mathbf{h}_x' = \mathbf{h}_x \\ \mathbf{e}_y' = \frac{\mathbf{e}_y - v\mathbf{h}_z}{\sqrt{1 - v^2}} & \mathbf{h}_y' = \frac{\mathbf{h}_y + v\mathbf{e}_z}{\sqrt{1 - v^2}} \\ \mathbf{e}_z' = \frac{\mathbf{e}_z - v\mathbf{h}_y}{\sqrt{1 - v^2}} & \mathbf{h}_z' = \frac{\mathbf{h}_z + v\mathbf{e}_y}{\sqrt{1 - v^2}} \\ \end{matrix} \right\} $$}}

If there exists with respect to $$K$$ only a magnetic field, $$\mathbf{h}$$, but no electric field, $$\mathbf{e}$$, then with respect to $$K'$$ there exists an electric field $$\mathbf{e'}$$ as well, which would act upon an electric particle at rest relatively to $$K'$$. An observer at rest relatively to $$K$$ would designate this force as the Biot-Savart force, or the Lorentz electromotive force. It therefore appears as if this electromotive force had become fused with the electric field intensity into a single entity.

In order to view this relation formally, let us consider the expression for the force acting upon unit volume of electricity,

in which $$\mathbf{i}$$ is the vector velocity of electricity, with the velocity of light as the unit. If we introduce $$J_\mu$$ and $$\phi_\mu$$ according to (30a) and (31), we obtain for the first component the expression

Observing that $$\phi_{11}$$ vanishes on account of the skew-symmetry of the tensor ($$\phi$$), the components of $$k$$ are given by the first three components of the four-dimensional vector

and the fourth component is given by