同語反復

まるで意味のない計算をしてみた。

 

任意の4次元量 j^{\mu}に対して

\begin{equation} \begin{aligned} \bf{j} = (j^{1}, j^{2}, j^{3}) \end{aligned} \end{equation}

\begin{equation} \begin{aligned} \bf{E} = \partial^{0} \bf{j} - \nabla j^{0} \end{aligned} \end{equation}

\begin{equation} \begin{aligned} \bf{B} = \nabla \times \bf{j} \end{aligned} \end{equation}

\begin{equation} \begin{aligned} \sigma = \partial_{\mu} j^{\mu} \end{aligned} \end{equation} 

とする。

ただし計量の符号は(-,+,+,+)である。

 

このとき、どんな関係式がなりたつだろうか。

まず \bf{B} の定義から

\begin{equation} \begin{aligned}  \nabla \cdot \bf{B} =0 \end{aligned} \end{equation}

が成り立つ。

また

{ \begin{aligned}   \nabla \times \bf{E} + \partial_{0} \bf{B} \end{aligned} }

{ \begin{aligned} = \nabla \times (\partial^{0} \bf{j} - \nabla j^{0} ) + \partial_{0} \nabla \times \bf{j} \end{aligned} }

{ \begin{aligned} = \partial^{0} \nabla \times \bf{j} + \partial_{0} \nabla \times \bf{j} \end{aligned} }

{ \begin{aligned} =  - \partial_{0} \nabla \times \bf{j} + \partial_{0} \nabla \times \bf{j} \end{aligned} }

{ \begin{aligned} = 0 \end{aligned} }

 

{ \begin{aligned}   \nabla \cdot \bf{E} \end{aligned} }はどうだろうか。

{ \begin{aligned}   \nabla \cdot \bf{E} \end{aligned} }

{ \begin{aligned}   = \nabla \cdot (\partial^{0} \bf{j} - \nabla  j^{0}) \end{aligned} }

{ \begin{aligned}   = \partial^{0} \nabla \cdot \bf{j} - \nabla^{2}  j^{0} \end{aligned} }

{ \begin{aligned}   = \partial^{0} ( \partial_{0} j^{0} + \nabla \cdot \bf{j} ) - \partial_{0}  j^{0} - \nabla^{2}  j^{0} \end{aligned} }

{ \begin{aligned}   = \partial^{0} ( \partial_{\mu} j^{\mu} ) - \partial_{0} \partial^{0}  j^{0} - \nabla^{2}  j^{0} \end{aligned} }

{ \begin{aligned}   = \partial^{0} \sigma - \partial_{\mu} \partial^{\mu} j^{0}  \end{aligned} }

最後は

{ \begin{aligned}   \nabla \times \bf{B} - \partial_{0} \bf{E} \end{aligned} }

{ \begin{aligned} = \nabla \times (\nabla \times \bf{j}) - \partial_{0} (\partial^{0} \bf{j} - \nabla j^{0} ) \end{aligned} }

{ \begin{aligned} = \nabla ( \nabla \cdot \bf{j}) - \nabla^{2} \bf{j} - \partial_{0} \partial^{0} \bf{j}  + \nabla \partial_{0}  j^{0} \end{aligned} }

{ \begin{aligned} =  \nabla ( \partial_{0} j^{0} + \nabla \cdot \bf{j}  )  - \partial_{\mu} \partial^{\mu} \bf{j} \end{aligned} }

{ \begin{aligned} =  \nabla ( \partial_{\mu} j^{\mu})  - \partial_{\mu} \partial^{\mu} \bf{j} \end{aligned} }

{ \begin{aligned} =  \nabla \sigma - \partial_{\mu} \partial^{\mu} \bf{j} \end{aligned} }

 

ここまで得られた式を並べてみると

{ \begin{aligned}   \nabla \times \bf{E} + \partial_{0} \bf{B} = 0 \end{aligned} }

{ \begin{aligned}  \nabla \cdot \bf{B} =0 \end{aligned} }

{ \begin{aligned}   \nabla \times \bf{B} - \partial_{0} \bf{E} =  \nabla \sigma - \partial_{\mu} \partial^{\mu} \bf{j} \end{aligned} }

{ \begin{aligned}   \nabla \cdot \bf{E} = \partial^{0} \sigma - \partial_{\mu} \partial^{\mu} j^{0}  \end{aligned} }

 

Maxwell方程式っぽく見えて一瞬だけほっこりしてしまうのだが

冒頭のような量を定義すれば必ず成り立つ関係式であって

意味はないのだよな。

 

しかし表記ゆれがひどいね。