Page:BatemanElectrodynamical.djvu/35

The formulae of transformation of the type used by Einstein and Laub in the case of Lorentz's transformation are obtained by putting θ = φ = 1. If we use a spherical wave transformation with positive Jacobian, we can deduce the relation

${\begin{array}{l}E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-B_{x}dx\ dt-B_{y}dy\ dt-B_{z}dz\ dt\\\qquad =E'_{x}dy'dz'+E'_{y}dz'dx'+E'_{z}dx'dy'-B'_{x}dx'dt'-B'_{y}dy'dt'-B'_{z}dz'dt',\end{array}}$

(VII)

and this, combined with (V), gives

${\begin{array}{l}P_{x}dy\ dz+P_{y}dz\ dx+P_{z}dx\ dy-Q_{x}dx\ dt-Q_{y}dy\ dt-Q_{z}dz\ dt\\\qquad =P'_{x}dy'dz'+P'_{y}dz'dx'+P'_{z}dx'dy'-Q'_{x}dx'dt'-Q'_{y}dy'dt'-Q'_{z}dz'dt',\end{array}}$

(VIII)

which enables us to obtain the formulae of transformation of the polarisation vectors.

The constitutive relations are obtained by Einstein and Laub by assuming that all the bodies in the dashed system of coordinates are at rest, and that in this system

$D'=\epsilon G',\ B'=\mu H',\ s'=\sigma E'.$

This gives a set of constitutive relations for the case in which a system of bodies are moving with constant velocity w.

The constitutive relations that are obtained in this way may be written in the form^[1]

$\left.{\begin{array}{rl}D+[wH]=&\epsilon \{E+[wB]\}\\\\B-[wE]=&\mu \{H-[wD]\}\\\\{\frac {s_{w}-\left|w\right|\rho }{\sqrt {1-w^{2}}}}=&\sigma \{E+[wB]\}_{w}\\\\s_{\overline {w}}=&{\frac {\sigma \{E+[wB]\}_{\overline {w}}}{\sqrt {1-w^{2}}}}\end{array}}\right\},$

(IX)

where the suffix w denotes that the component in the direction of w, ${\overline {w}}$, a component in a direction perpendicular to w is to be taken.

The first two equations are seen to be invariant for the group of spherical wave transformations when we obtain them in the following way. The expression

${\frac {\lambda \left[dt-w_{x}dx+w_{y}dy-w_{z}dz\right]}{\sqrt {1-w^{2}}}}$

(X)

1. Minkowski, Göttinger Nachrichten (1908); Einstein and Laub, Ann. d. Phys. (1908).