電磁場のローレンツ変換

真空中のマクスウェル方程式は
$$\begin{array}{rcl}\mathrm{div}\mathit{D}& =& 0\\ \mathrm{div}\mathit{B}& =& 0\\ \mathrm{rot}\mathit{H}& =& \frac{\partial \mathit{D}}{\partial t}\\ \mathrm{rot}\mathit{E}& =& -\frac{\partial \mathit{B}}{\partial t}\end{array}$$
であるが、
$$\begin{array}{rcl}\mathit{D}& =& {\epsilon }_0\mathit{E}\\ \mathit{B}& =& {\mu }_0\mathit{H}\\ c^2& =& \frac{1}{{\epsilon }_0{\mu }_0}\end{array}$$
より、$\mathit{E}$と$\mathit{B}$だけを使って成分で書くと
$$\begin{array}{rcl}\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z}& =& 0\\ \frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}& =& 0\\ \frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z}& =& \frac{1}{c^2}\frac{\partial E_x}{\partial t}\\ \frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}& =& \frac{1}{c^2}\frac{\partial E_y}{\partial t}\\ \frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y}& =& \frac{1}{c^2}\frac{\partial E_z}{\partial t}\\ \frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}& =& -\frac{\partial B_x}{\partial t}\\ \frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}& =& -\frac{\partial B_y}{\partial t}\\ \frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}& =& -\frac{\partial B_z}{\partial t}\end{array}$$
となる。これにローレンツ変換
$$\begin{array}{rcl}{t}^{\prime }& =& \frac{t-vx/c^2}{\sqrt{1-(v/c{)}^2}}\\ x^{\prime }& =& \frac{x-vt}{\sqrt{1-(v/c{)}^2}}\\ y^{\prime }& =& y\\ z^{\prime }& =& z\end{array}$$
を適用することを考える。まず、$K$系の電磁場を$K^{\prime }$系の座標で表す。例えば、
$$f(t,x,y,z)\to f(t^{\prime },x^{\prime },y^{\prime },z^{\prime })$$
の時、全微分を使って
$$\begin{array}{rcl}\frac{\partial f}{\partial t}& =& \frac{\partial f}{\partial t^{\prime }}\frac{\partial t^{\prime }}{\partial t}+\frac{\partial f}{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial t}+\frac{\partial f}{\partial y^{\prime }}\frac{\partial y^{\prime }}{\partial t}+\frac{\partial f}{\partial z^{\prime }}\frac{\partial z^{\prime }}{\partial t}=\frac{\partial f}{\partial t^{\prime }}\frac{1}{\sqrt{1-(v/c{)}^2}}+\frac{\partial f}{\partial x^{\prime }}\frac{-v}{\sqrt{1-(v/c{)}^2}}+0+0=\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})f\\ \frac{\partial f}{\partial x}& =& \frac{\partial f}{\partial t^{\prime }}\frac{\partial t^{\prime }}{\partial x}+\frac{\partial f}{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial x}+\frac{\partial f}{\partial y^{\prime }}\frac{\partial y^{\prime }}{\partial x}+\frac{\partial f}{\partial z^{\prime }}\frac{\partial z^{\prime }}{\partial x}=\frac{\partial f}{\partial t^{\prime }}\frac{-v/c^2}{\sqrt{1-(v/c{)}^2}}+\frac{\partial f}{\partial x^{\prime }}\frac{1}{\sqrt{1-(v/c{)}^2}}+0+0=\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial x^{\prime }}-\frac{v}{c^2}\frac{\partial }{\partial t^{\prime }})f\\ \frac{\partial f}{\partial y}& =& \frac{\partial f}{\partial t^{\prime }}\frac{\partial t^{\prime }}{\partial y}+\frac{\partial f}{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial y}+\frac{\partial f}{\partial y^{\prime }}\frac{\partial y^{\prime }}{\partial y}+\frac{\partial f}{\partial z^{\prime }}\frac{\partial z^{\prime }}{\partial y}=0+0+\frac{\partial f}{\partial y^{\prime }}\frac{\partial y}{\partial y}+0=\frac{\partial f}{\partial y^{\prime }}\end{array}$$
となるから、マクスウェル方程式は
$$\begin{array}{rcl}\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial x^{\prime }}-\frac{v}{c^2}\frac{\partial }{\partial t^{\prime }})E_x+\frac{\partial E_y}{\partial y^{\prime }}+\frac{\partial E_z}{\partial z^{\prime }}& =& 0\\ \frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial x^{\prime }}-\frac{v}{c^2}\frac{\partial }{\partial t^{\prime }})B_x+\frac{\partial B_y}{\partial y^{\prime }}+\frac{\partial B_z}{\partial z^{\prime }}& =& 0\\ \frac{\partial B_z}{\partial y^{\prime }}-\frac{\partial B_y}{\partial z^{\prime }}& =& \frac{1}{c^2}\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})E_x\\ \frac{\partial B_x}{\partial z^{\prime }}-\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial x^{\prime }}-\frac{v}{c^2}\frac{\partial }{\partial t^{\prime }})B_z& =& \frac{1}{c^2}\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})E_y\\ \frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial x^{\prime }}-\frac{v}{c^2}\frac{\partial }{\partial t^{\prime }})B_y-\frac{\partial B_x}{\partial y^{\prime }}& =& \frac{1}{c^2}\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})E_z\\ \frac{\partial E_z}{\partial y^{\prime }}-\frac{\partial E_y}{\partial z^{\prime }}& =& -\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})B_x\\ \frac{\partial E_x}{\partial z^{\prime }}-\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial x^{\prime }}-\frac{v}{c^2}\frac{\partial }{\partial t^{\prime }})E_z& =& -\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})B_y\\ \frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial x^{\prime }}-\frac{v}{c^2}\frac{\partial }{\partial t^{\prime }})E_y-\frac{\partial E_x}{\partial y^{\prime }}& =& -\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})B_z\end{array}$$
となる。最初の2式は
$$\begin{array}{rcl}v\frac{\partial E_y}{\partial y^{\prime }}+v\frac{\partial E_z}{\partial z^{\prime }}& =& -\frac{1}{\sqrt{1-(v/c{)}^2}}(1-\frac{v^2}{c^2})\frac{\partial E_x}{\partial t^{\prime }}+\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})E_x=-\sqrt{1-(v/c{)}^2}\frac{\partial E_x}{\partial t^{\prime }}+\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})E_x\\ v\frac{\partial B_y}{\partial y^{\prime }}+v\frac{\partial B_z}{\partial z^{\prime }}& =& -\frac{1}{\sqrt{1-(v/c{)}^2}}(1-\frac{v^2}{c^2})\frac{\partial B_x}{\partial t^{\prime }}+\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})B_x=-\sqrt{1-(v/c{)}^2}\frac{\partial B_x}{\partial t^{\prime }}+\frac{1}{\sqrt{1-(v/c{)}^2}}(\frac{\partial }{\partial t^{\prime }}-v\frac{\partial }{\partial x^{\prime }})B_x\end{array}$$
となるので、これを3番目の式と6番目の式の右辺に代入して、残り6式をまとめると
$$\begin{array}{rcl}\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial y^{\prime }}(B_z-\frac{v}{c^2}{E}_y)-\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial z^{\prime }}(B_y+\frac{v}{c^2}{E}_z)& =& \frac{1}{c^2}\frac{\partial E_x}{\partial t^{\prime }}\\ \frac{\partial B_x}{\partial z^{\prime }}-\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial x^{\prime }}(B_z-\frac{v}{c^2}{E}_y)& =& \frac{1}{c^2}\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial t^{\prime }}(E_y-v{B}_z)\\ \frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial x^{\prime }}(B_y+\frac{v}{c^2}{E}_z)-\frac{\partial H_x}{\partial y^{\prime }}& =& \frac{1}{c^2}\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial t^{\prime }}(E_z+v{B}_y)\\ \frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial y^{\prime }}(E_z+v{B}_y)-\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial z^{\prime }}(E_y-v{B}_z)& =& -\frac{\partial B_x}{\partial t^{\prime }}\\ \frac{\partial E_x}{\partial z^{\prime }}-\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial x^{\prime }}(E_z+v{B}_y)& =& -\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial t^{\prime }}(B_y+\frac{v}{c^2}{E}_z)\\ \frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial x^{\prime }}(E_y-v{B}_z)-\frac{\partial E_x}{\partial y^{\prime }}& =& -\frac{1}{\sqrt{1-(v/c{)}^2}}\frac{\partial }{\partial t^{\prime }}(B_z-\frac{v}{c^2}{E}_y)\end{array}$$
となる。さて、相対性理論では、マクスウェル方程式はどの系でも同様に成り立つ筈であるから、$K^{\prime }$系の電磁場も
$$\begin{array}{rcl}\mathrm{div}{\mathit{D}}^{\prime }& =& 0\\ \mathrm{div}{\mathit{B}}^{\prime }& =& 0\\ \mathrm{rot}{\mathit{H}}^{\prime }& =& \frac{\partial {\mathit{D}}^{\prime }}{\partial t}\\ \mathrm{rot}{\mathit{E}}^{\prime }& =& -\frac{\partial {\mathit{B}}^{\prime }}{\partial t}\end{array}$$
を満たす筈である。したがって、先程求めた6式と
$$\begin{array}{rcl}\frac{\partial B_z^{\prime }}{\partial y}-\frac{\partial B_y^{\prime }}{\partial z}& =& \frac{1}{c^2}\frac{\partial E_x^{\prime }}{\partial t}\\ \frac{\partial B_x^{\prime }}{\partial z}-\frac{\partial B_z^{\prime }}{\partial x}& =& \frac{1}{c^2}\frac{\partial E_y^{\prime }}{\partial t}\\ \frac{\partial B_y^{\prime }}{\partial x}-\frac{\partial B_x^{\prime }}{\partial y}& =& \frac{1}{c^2}\frac{\partial E_z^{\prime }}{\partial t}\\ \frac{\partial E_z^{\prime }}{\partial y}-\frac{\partial E_y^{\prime }}{\partial z}& =& -\frac{\partial B_x^{\prime }}{\partial t}\\ \frac{\partial E_x^{\prime }}{\partial z}-\frac{\partial E_z^{\prime }}{\partial x}& =& -\frac{\partial B_y^{\prime }}{\partial t}\\ \frac{\partial E_y^{\prime }}{\partial x}-\frac{\partial E_x^{\prime }}{\partial y}& =& -\frac{\partial B_z^{\prime }}{\partial t}\end{array}$$
を比較すれば、電磁場のローレンツ変換は、
$$\boxed{\begin{array}{rcl}{E}_x^{\prime }& =& E_x\\ E_y^{\prime }& =& \frac{1}{\sqrt{1-(v/c{)}^2}}(E_y-v{B}_z)\\ E_z^{\prime }& =& \frac{1}{\sqrt{1-(v/c{)}^2}}(E_z+v{B}_y)\\ B_x^{\prime }& =& B_x\\ B_y^{\prime }& =& \frac{1}{\sqrt{1-(v/c{)}^2}}(B_y+\frac{v}{c^2}{E}_z)\\ B_z^{\prime }& =& \frac{1}{\sqrt{1-(v/c{)}^2}}(B_z-\frac{v}{c^2}{E}_y)\end{array}}$$
となる。