Page:BatemanElectrodynamical.djvu/40

With these constitutive relations the equations of electrodynamics are invariant for a very large class of space time transformations. The relations simplify considerably when the quadratic form is of a simple type.

The vanishing of the quadratic form may be supposed to represent the condition that two neighbouring particles are in a position to act on one another.

We shall now verify that in a particular case the constitutive relations obtained in this way agree with the ones obtained by Minkowski, Einstein, and Laub.

Starting with a system at rest, we assume that the invariant quadratic form is

$dx^{2}+dy^{2}+dz^{2}-\epsilon \mu dt^{2}.$

This gives

$\Delta =-\epsilon \mu ,$

and if we take

$\kappa ={\sqrt {-{\frac {\epsilon }{\mu }}}},$

the relations (2) take the form

$B=\mu H,\ E=\epsilon ^{-1}D.$

Now make the Lorentzian transformation

$x={\frac {x'-ut'}{\sqrt {1-u^{2}}}},\ y=y',\ z=z',\ t={\frac {t'-ux'}{\sqrt {1-u^{2}}}};$

the quadratic form then becomes

$dx'^{2}\left[{\frac {1-\epsilon \mu \cdot u^{2}}{1-u^{2}}}\right]+dy'^{2}+dz'^{2}+dt'^{2}{\frac {u^{2}-\epsilon u}{1-u^{2}}}-2u{\frac {1-\epsilon \mu }{1-u^{2}}}dx'dt'$

We again have $\Delta =-\epsilon \mu$, and the relations (2) take the form

${\begin{array}{l}B_{x}=\mu H_{x},\\\\B_{y}={\frac {\epsilon \mu -u^{2}}{\epsilon \left(1-u^{2}\right)}}H_{y}+u{\frac {\epsilon \mu -1}{\epsilon \left(1-u^{2}\right)}}D_{z},\\\\B_{z}={\frac {\epsilon \mu -u^{2}}{\epsilon \left(1-u^{2}\right)}}H_{z}-u{\frac {\epsilon \mu -1}{\epsilon \left(1-u^{2}\right)}}D_{y},\\\\E_{x}={\frac {1}{\epsilon }}D_{x},\\\\E_{y}={\frac {1-u^{2}\epsilon \mu }{\epsilon \left(1-u^{2}\right)}}D_{y}+u{\frac {\epsilon \mu -1}{\epsilon \left(1-u^{2}\right)}}H_{z},\\\\E_{z}={\frac {1-u^{2}\epsilon \mu }{\epsilon \left(1-u^{2}\right)}}D_{z}-u{\frac {\epsilon \mu -1}{\epsilon \left(1-u^{2}\right)}}H_{y}.\end{array}}$