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

$\left.{\begin{matrix}\mathbf {e} _{x}'=\mathbf {e} _{x}&\mathbf {h} _{x}'=\mathbf {h} _{x}\\\mathbf {e} _{y}'={\frac {\mathbf {e} _{y}-v\mathbf {h} _{z}}{\sqrt {1-v^{2}}}}&\mathbf {h} _{y}'={\frac {\mathbf {h} _{y}+v\mathbf {e} _{z}}{\sqrt {1-v^{2}}}}\\\mathbf {e} _{z}'={\frac {\mathbf {e} _{z}-v\mathbf {h} _{y}}{\sqrt {1-v^{2}}}}&\mathbf {h} _{z}'={\frac {\mathbf {h} _{z}+v\mathbf {e} _{y}}{\sqrt {1-v^{2}}}}\\\end{matrix}}\right\}$

(34)

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,

$\mathbf {k} =\rho \mathbf {e} +[\mathbf {i} ,\mathbf {h} ]$

(35)

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

$\phi _{12}J_{2}+\phi _{13}J_{3}+\phi _{14}J_{4}.$

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

$K_{\mu }=\phi _{\mu \nu }J_{\nu }$

(36)

and the fourth component is given by

$K_{4}=\phi _{41}J_{1}+\phi _{42}J_{2}+\phi _{43}J_{3}=\mathbf {i} (\mathbf {e_{x}i_{x}} +\mathbf {e} _{y}\mathbf {i} _{y}+\mathbf {e} _{z}\mathbf {i} _{z})=\mathbf {i} \lambda .$

(37)