Page:TolmanFundamental.djvu/2

electromagnetic field

$\mathrm {curl} \ {\mathsf {H}}=4\pi \rho {\mathsf {u}}+{\frac {1}{c}}{\frac {\partial {\mathsf {E}}}{\partial t}}$

(5)

$\mathrm {curl} \ {\mathsf {E}}=-{\frac {1}{c}}{\frac {\partial {\mathsf {H}}}{\partial t}}$

(6)

$\mathrm {div} .\ {\mathsf {E}}=4\pi \rho$

(7)

$\mathrm {div} .\ {\mathsf {H}}=0.$

(8)

If these equations are true descriptions of electromagnetic phenomena, it is evident then by the first postulate of relativity that similar equations

$\mathrm {curl} \ {\mathsf {H}}'=4\pi \rho '{\mathsf {u}}'+{\frac {1}{c}}{\frac {\partial {\mathsf {E}}'}{\partial t'}}$

(9)

$\mathrm {curl} \ {\mathsf {E}}'=-{\frac {1}{c}}{\frac {\partial {\mathsf {H'}}}{\partial t'}}$

(10)

$\mathrm {div} .\ {\mathsf {E}}'=4\pi \rho '$

(11)

$\mathrm {div} .\ {\mathsf {H}}'=0.$

(12)

must hold when the phenomena are described by an observer moving with the system S'.

It has been shown by Einstein that the following equations, together with the kinematic relations (1-4) already given are necessary and sufficient for transforming equations (5, 6, 7 and 8) into (9, 10, 11 and 12)^[1].

${\mathsf {E}}_{x}'={\mathsf {E}}_{x}$

(13)

${\mathsf {E}}_{y}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {E}}_{y}-{\frac {v}{c}}{\mathsf {H}}_{z}\right)$

(14)

${\mathsf {E}}_{z}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {E}}_{z}+{\frac {v}{c}}{\mathsf {H}}_{y}\right)$

(15)

${\mathsf {H}}_{x}'={\mathsf {H}}_{x}$

(16)

1. For the purposes of this transformation, it is necessary to use not only the simple kinematic relations (1-4), but also the following relations which can be directly derived from them:-

   $u_{x'}={\frac {u_{x}-v}{1-{\frac {u_{x}v}{c^{2}}}}},\ u_{y'}={\frac {u_{y}{\sqrt {1-\beta ^{2}}}}{1-{\frac {u_{x}v}{c^{2}}}}},\ u_{z'}={\frac {u_{z}{\sqrt {1-\beta ^{2}}}}{1-{\frac {u_{x}v}{c^{2}}}}},$ $\rho '={\frac {\partial {\mathsf {E}}_{x}'}{\partial x'}}+{\frac {\partial {\mathsf {E}}_{y}'}{\partial y'}}+{\frac {\partial {\mathsf {E}}_{z}'}{\partial z'}}={\frac {\left(1-{\frac {u_{x}v}{c^{2}}}\right)}{\left(1-\beta ^{2}\right)}}\rho .$