HOME ＞ 特殊相対性理論 ＞ ガリレイの相対性原理 ＞マクスウェル方程式の低速極限
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本ページでは…
本ページでは、ガリレイ変換に加えて次のような電場·磁場の変換
\begin{align}E_x &= E’_x\\E_y &= E’_y + v\mu_0 H’_z \\
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
を行うことで、慣性系同士の相対速度\(v\)が光速度\(c\)と比べて十分小さい低速極限
\begin{align*}\frac{v}{c}\simeq0\end{align*}
で、マクスウェル方程式の形が不変となることを見る。
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前ページまで⋯
前ページでは、マイケルソン-モーリーの実験原理、理論的予想、そして物理学の歴史を変えた実験結果について解説した。19世紀末、多くの物理学者は、光が伝わるためには媒質「エーテル」が必要であると考えており、もし地球がそのエーテル中を運動しているなら、地球上ではエーテル風が観測されるはずであった。マイケルソン-モーリーは、この効果を干渉計によって直接検出しようとした。
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内容
マイケルソン・モーリーの実験の結果
前ページで述べたように、実際にマイケルソン-モーリーの実験を行なった結果、干渉縞のずれは全く観測されなかった。この結果は、光の速度が方向によらず一定であることを示唆し、電磁波の媒質であるエーテルの存在に強い疑問を投げかけるものであった。
電磁波が媒質に対して伝播すると考えるヘルツ方程式(以前のページを参照)では、観測者の運動状態によって光の速度が変化することが予測される。そのため、このマイケルソン–モーリーの実験の結果から、ヘルツ方程式の妥当性には疑問が生じた。
一方で、電磁波の媒質を考えないマクスウェル方程式の方が正しそうにも思えるが、ガリレイ変換の下ではその形が保たれないという問題がマクスウェル方程式にもあった(以前のページを参照)。このため、マクスウェル方程式の形を保ちつつ実験結果を説明するためには、従来のガリレイ変換そのものを見直す必要があると考えられるようになった。
マクスウェル方程式の低速極限
以後、マクスウェル方程式を成分ごとに見ていく必要があるため、4つの式からなるマクスウェル方程式
\begin{align*}\boldsymbol \nabla \cdot\boldsymbol H(\boldsymbol x,t)&=0\tag{1}\\\boldsymbol\nabla×\boldsymbol E(\boldsymbol x,t)&=-\mu_0\frac{\partial }{\partial t}\boldsymbol H(\boldsymbol x,t)\tag{2}\\\boldsymbol\nabla\cdot\boldsymbol E(\boldsymbol x,t)&=0\tag{3}\\\boldsymbol\nabla×\boldsymbol H(\boldsymbol x,t)&=\epsilon_0\frac{\partial }{\partial t}\boldsymbol E(\boldsymbol x,t)\tag{4}\end{align*}
を成分ごと
\begin{align}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\
\frac{\partial E_y}{\partial x}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}
\end{align}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\
\frac{\partial E_y}{\partial x}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}
\end{align}
に表す。
以前のページで見たように、マクスウェル方程式の形はガリレイ変換(電場·磁場は変わらないとする)
\begin{align}\frac{\partial}{\partial x}&=\frac{\partial}{\partial x'{}’}\\\frac{\partial}{\partial y}&=\frac{\partial}{\partial y'{}’}\\\frac{\partial}{\partial z}&=\frac{\partial}{\partial z'{}’}\\\frac{\partial}{\partial t}&=\frac{\partial}{\partial t'{}’}-v\frac{\partial}{\partial x'{}’}\\\boldsymbol E&=\boldsymbol E'{}’\\\boldsymbol H&=\boldsymbol H'{}’
\end{align}
\end{align}
の下で次の形
\begin{align}
\frac{\partial H'{}'{}_x}{\partial x'{}’}
+\frac{\partial H'{}'{}_y}{\partial y'{}’}
+\frac{\partial H'{}'{}_z}{\partial z'{}’}
&=0\tag{A$'{}’$}\\
\frac{\partial E'{}'{}_x}{\partial x'{}’}
+\frac{\partial E'{}'{}_y}{\partial y'{}’}
+\frac{\partial E'{}'{}_z}{\partial z'{}’}
&=0\tag{B$'{}’$}\\
\frac{\partial E'{}'{}_z}{\partial y'{}’}
-\frac{\partial E'{}'{}_y}{\partial z'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_x}{\partial t'{}’}-v\frac{\partial H'{}'{}_x}{\partial x'{}’}
\right)\tag{C$'{}’$}\\
\frac{\partial E'{}'{}_x}{\partial z'{}’}
-\frac{\partial E'{}'{}_z}{\partial x'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_y}{\partial t'{}’}-v\frac{\partial H'{}'{}_y}{\partial x'{}’}
\right)\tag{D$'{}’$}\\
\frac{\partial E'{}'{}_y}{\partial x'{}’}
-\frac{\partial E'{}'{}_x}{\partial y'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_z}{\partial t'{}’}-v\frac{\partial H'{}'{}_z}{\partial x'{}’}
\right)\tag{E$'{}’$}\\
\frac{\partial H'{}'{}_z}{\partial y'{}’}
-\frac{\partial H'{}'{}_y}{\partial z'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_x}{\partial t'{}’}-v\frac{\partial E'{}'{}_x}{\partial x'{}’}
\right)\tag{F$'{}’$}\\
\frac{\partial H'{}'{}_x}{\partial z'{}’}
-\frac{\partial H'{}'{}_z}{\partial x'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_y}{\partial t'{}’}-v\frac{\partial E'{}'{}_y}{\partial x'{}’}
\right)\tag{G$'{}’$}\\
\frac{\partial H'{}'{}_y}{\partial x'{}’}
-\frac{\partial H'{}'{}_x}{\partial y'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_z}{\partial t'{}’}-v\frac{\partial E'{}'{}_z}{\partial x'{}’}
\right)\tag{H$'{}’$}
\end{align}
\frac{\partial H'{}'{}_x}{\partial x'{}’}
+\frac{\partial H'{}'{}_y}{\partial y'{}’}
+\frac{\partial H'{}'{}_z}{\partial z'{}’}
&=0\tag{A$'{}’$}\\
\frac{\partial E'{}'{}_x}{\partial x'{}’}
+\frac{\partial E'{}'{}_y}{\partial y'{}’}
+\frac{\partial E'{}'{}_z}{\partial z'{}’}
&=0\tag{B$'{}’$}\\
\frac{\partial E'{}'{}_z}{\partial y'{}’}
-\frac{\partial E'{}'{}_y}{\partial z'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_x}{\partial t'{}’}-v\frac{\partial H'{}'{}_x}{\partial x'{}’}
\right)\tag{C$'{}’$}\\
\frac{\partial E'{}'{}_x}{\partial z'{}’}
-\frac{\partial E'{}'{}_z}{\partial x'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_y}{\partial t'{}’}-v\frac{\partial H'{}'{}_y}{\partial x'{}’}
\right)\tag{D$'{}’$}\\
\frac{\partial E'{}'{}_y}{\partial x'{}’}
-\frac{\partial E'{}'{}_x}{\partial y'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_z}{\partial t'{}’}-v\frac{\partial H'{}'{}_z}{\partial x'{}’}
\right)\tag{E$'{}’$}\\
\frac{\partial H'{}'{}_z}{\partial y'{}’}
-\frac{\partial H'{}'{}_y}{\partial z'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_x}{\partial t'{}’}-v\frac{\partial E'{}'{}_x}{\partial x'{}’}
\right)\tag{F$'{}’$}\\
\frac{\partial H'{}'{}_x}{\partial z'{}’}
-\frac{\partial H'{}'{}_z}{\partial x'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_y}{\partial t'{}’}-v\frac{\partial E'{}'{}_y}{\partial x'{}’}
\right)\tag{G$'{}’$}\\
\frac{\partial H'{}'{}_y}{\partial x'{}’}
-\frac{\partial H'{}'{}_x}{\partial y'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_z}{\partial t'{}’}-v\frac{\partial E'{}'{}_z}{\partial x'{}’}
\right)\tag{H$'{}’$}
\end{align}
となり不変ではなかった。しかし、変換後の式は元の式とかけ離れた全く別の形になっているわけではなく、ガリレイ変換
\begin{align}\frac{\partial}{\partial x}&=\frac{\partial}{\partial x’}\\\frac{\partial}{\partial y}&=\frac{\partial}{\partial y’}\\\frac{\partial}{\partial z}&=\frac{\partial}{\partial z’}\\\frac{\partial}{\partial t}&=\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}
\end{align}
\end{align}
に加えて電場·磁場にも次の変換
\begin{align}E_x &= E’_x\\E_y &= E’_y + v\mu_0 H’_z \\
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
を施すことで、慣性系同士の相対速度\(v\)が光速度\(c\)と比べて十分小さい低速極限
\begin{align*}\frac{v}{c}\simeq0\end{align*}
の下ではマクスウェル方程式の形を
\begin{align}
\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}
&=0\tag{A$’$}\\
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}
&=0\tag{B$’$}\\
\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{C$’$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}
&=-\mu_0\frac{\partial H’_y}{\partial t’}\tag{D$’$}\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}
&=-\mu_0\frac{\partial H’_z}{\partial t’}\tag{E$’$}\\
\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{F$’$}\\
\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}
&=\varepsilon_0\frac{\partial E’_y}{\partial t’}\tag{G$’$}\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}
&=\varepsilon_0\frac{\partial E’_z}{\partial t’}\tag{H$’$}
\end{align}
\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}
&=0\tag{A$’$}\\
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}
&=0\tag{B$’$}\\
\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{C$’$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}
&=-\mu_0\frac{\partial H’_y}{\partial t’}\tag{D$’$}\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}
&=-\mu_0\frac{\partial H’_z}{\partial t’}\tag{E$’$}\\
\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{F$’$}\\
\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}
&=\varepsilon_0\frac{\partial E’_y}{\partial t’}\tag{G$’$}\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}
&=\varepsilon_0\frac{\partial E’_z}{\partial t’}\tag{H$’$}
\end{align}
のように不変に保つことができる。まずは、このことを確かめて、次ページからこの変換則をより正確なものへと発展させていく。
はじめに、ファラデー-マクスウェルの式とアンペール-マクスウェルの式の\(x\)成分の形
\begin{align*}\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0
\frac{\partial H_x}{\partial t}\tag{C}\\\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0
\frac{\partial E_x}{\partial t}\tag{F}\end{align*}
-\frac{\partial E_y}{\partial z}
&=-\mu_0
\frac{\partial H_x}{\partial t}\tag{C}\\\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0
\frac{\partial E_x}{\partial t}\tag{F}\end{align*}
が変換によって不変
\begin{align*}\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0
\frac{\partial H’_x}{\partial t’}\tag{C$’$}\\\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0
\frac{\partial E_x’}{\partial t’}\tag{F$’$}\end{align*}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0
\frac{\partial H’_x}{\partial t’}\tag{C$’$}\\\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0
\frac{\partial E_x’}{\partial t’}\tag{F$’$}\end{align*}
となるとき、ガリレイ変換に加えて電場·磁場をどのように変換すればよいかを調べてみる。そのような変換を行なったとき、式(\(\text C’\)),(\(\text F’\))の形は電場·磁場を変換せずにガリレイ変換のみ行なった式(\(\text C'{}’\)),(\(\text F'{}’\))の形
\begin{align*}\frac{\partial E'{}'{}_z}{\partial y'{}’}
-\frac{\partial E'{}'{}_y}{\partial z'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_x}{\partial t'{}’}-v\frac{\partial H'{}'{}_x}{\partial x'{}’}
\right)\tag{C$'{}’$}\\\frac{\partial H'{}'{}_z}{\partial y'{}’}
-\frac{\partial H'{}'{}_y}{\partial z'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_x}{\partial t'{}’}-v\frac{\partial E'{}'{}_x}{\partial x'{}’}
\right)\tag{F$'{}’$}\end{align*}
-\frac{\partial E'{}'{}_y}{\partial z'{}’}
&=-\mu_0\left(
\frac{\partial H'{}'{}_x}{\partial t'{}’}-v\frac{\partial H'{}'{}_x}{\partial x'{}’}
\right)\tag{C$'{}’$}\\\frac{\partial H'{}'{}_z}{\partial y'{}’}
-\frac{\partial H'{}'{}_y}{\partial z'{}’}
&=\varepsilon_0\left(
\frac{\partial E'{}'{}_x}{\partial t'{}’}-v\frac{\partial E'{}'{}_x}{\partial x'{}’}
\right)\tag{F$'{}’$}\end{align*}
に近くなるはずである。そのため、式(\(\text C'{}’\)),(\(\text F'{}’\))に現れた余分な項\(v\mu_0\frac{\partial H'{}'{}_x}{\partial x'{}’}\),\(-v\varepsilon_0\frac{\partial E'{}'{}_x}{\partial x'{}’}\)と同じ形の\(v\mu_0\frac{\partial H’_x}{\partial x’}\),\(-v\varepsilon_0\frac{\partial E’_x}{\partial x’}\)を式(\(\text C’\)),(\(\text F’\))の両辺に足すと
\begin{align*}\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}+v\mu_0\frac{\partial H’_x}{\partial x’}
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}\right)\\\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}-v\varepsilon_0\frac{\partial E’_x}{\partial x’}
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\end{align*}
-\frac{\partial E’_y}{\partial z’}+v\mu_0\frac{\partial H’_x}{\partial x’}
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}\right)\\\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}-v\varepsilon_0\frac{\partial E’_x}{\partial x’}
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\end{align*}
となり、磁束密度の式とマクスウェル-ガウスの式
\begin{align*}\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}
&=0\tag{A$’$}\\\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}
&=0\tag{B$’$}\end{align*}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}
&=0\tag{A$’$}\\\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}
&=0\tag{B$’$}\end{align*}
を用いると
\begin{align*}\frac{\partial }{\partial y’}(E’_z- v\mu_0H’_y)
-\frac{\partial }{\partial z’}(E’_y+v\mu_0H’_z )
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}
\right)\tag{C$'{}'{}’$}\\\frac{\partial }{\partial y’}(H’_z+v\varepsilon_0E’_y)
-\frac{\partial }{\partial z’}(H’_y-v\varepsilon_0E’_z)
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\tag{F$'{}'{}’$}\end{align*}
-\frac{\partial }{\partial z’}(E’_y+v\mu_0H’_z )
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}
\right)\tag{C$'{}'{}’$}\\\frac{\partial }{\partial y’}(H’_z+v\varepsilon_0E’_y)
-\frac{\partial }{\partial z’}(H’_y-v\varepsilon_0E’_z)
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\tag{F$'{}'{}’$}\end{align*}
\begin{align*}\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}+v\mu_0\frac{\partial H’_x}{\partial x’}
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}\right)\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}- v\mu_0\frac{\partial H’_y}{\partial y’}- v\mu_0 \frac{\partial H’_z}{\partial z’}
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}
\right)\\\rightarrow\frac{\partial }{\partial y’}(E’_z- v\mu_0H’_y)
-\frac{\partial }{\partial z’}(E’_y+v\mu_0H’_z )
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}
\right)\end{align*}
-\frac{\partial E’_y}{\partial z’}+v\mu_0\frac{\partial H’_x}{\partial x’}
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}\right)\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}- v\mu_0\frac{\partial H’_y}{\partial y’}- v\mu_0 \frac{\partial H’_z}{\partial z’}
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}
\right)\\\rightarrow\frac{\partial }{\partial y’}(E’_z- v\mu_0H’_y)
-\frac{\partial }{\partial z’}(E’_y+v\mu_0H’_z )
&=-\mu_0\left(
\frac{\partial H’_x}{\partial t’}-v\frac{\partial H’_x}{\partial x’}
\right)\end{align*}
\begin{align*}\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}-v\varepsilon_0\frac{\partial E’_x}{\partial x’}
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}+v\varepsilon_0\frac{\partial E’_y}{\partial y’}+v\varepsilon_0\frac{\partial E’_z}{\partial z’}
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\\\rightarrow\frac{\partial }{\partial y’}(H’_z+v\varepsilon_0E’_y)
-\frac{\partial }{\partial z’}(H’_y-v\varepsilon_0E’_z)
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\end{align*}
-\frac{\partial H’_y}{\partial z’}-v\varepsilon_0\frac{\partial E’_x}{\partial x’}
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}+v\varepsilon_0\frac{\partial E’_y}{\partial y’}+v\varepsilon_0\frac{\partial E’_z}{\partial z’}
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\\\rightarrow\frac{\partial }{\partial y’}(H’_z+v\varepsilon_0E’_y)
-\frac{\partial }{\partial z’}(H’_y-v\varepsilon_0E’_z)
&=\varepsilon_0\left(
\frac{\partial E_x’}{\partial t’}-v\frac{\partial E_x}{\partial x’}
\right)\end{align*}
となる。これらの式と元の式(C),(F)
\begin{align*}\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0
\frac{\partial H_x}{\partial t}\tag{C}\\\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0
\frac{\partial E_x}{\partial t}\tag{F}\end{align*}
-\frac{\partial E_y}{\partial z}
&=-\mu_0
\frac{\partial H_x}{\partial t}\tag{C}\\\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0
\frac{\partial E_x}{\partial t}\tag{F}\end{align*}
を見比べると、ガリレイ変換
\begin{align}\frac{\partial}{\partial x}&=\frac{\partial}{\partial x’}\\\frac{\partial}{\partial y}&=\frac{\partial}{\partial y’}\\\frac{\partial}{\partial z}&=\frac{\partial}{\partial z’}\\\frac{\partial}{\partial t}&=\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}
\end{align}
\end{align}
に加えて次のような電場·磁場の変換
\begin{align}E_x &= E’_x\\E_y &= E’_y + v\mu_0 H’_z \\
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
を行なえば式(\(\text C\)),(\(\text F\))から式(\(\text C'{}'{}’\)),(\(\text F'{}'{}’\))に変換することができる。そして、式(\(\text C'{}'{}’\)),(\(\text F'{}'{}’\))は式(\(\text C’\)) ,(\(\text F’\))に等しいことから、これらの変換の下でファラデー-マクスウェルの式とアンペール-マクスウェルの式の\(x\)成分は不変となることがわかる。
ここで、全ての式にガリレイ変換と上記の電場·磁場の変換を行なうと
\begin{align}
\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=v \varepsilon_0\frac{\partial E’_z}{\partial y’}
-v \varepsilon_0\frac{\partial E’_y}{\partial z’} \tag{$\bar {\text A}$}\\
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\frac{\partial H’_z}{\partial y’}
+v\mu_0 \frac{\partial H’_y}{\partial z’}\tag{$\bar {\text B}$}\\
\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v\mu_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right)\tag{$\bar {\text C}$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial E’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial E’_z}{\partial x’}
\tag{$\bar {\text D}$}
\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial E’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial E’_y}{\partial x’}
\tag{$\bar {\text E}$}\\
\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}-v\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\tag{$\bar {\text F}$}\\\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial H’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial H’_z}{\partial x’}
\tag{$\bar {\text G}$}
\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial H’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial H’_y}{\partial x’}\tag{$\bar {\text H}$}
\end{align}
\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=v \varepsilon_0\frac{\partial E’_z}{\partial y’}
-v \varepsilon_0\frac{\partial E’_y}{\partial z’} \tag{$\bar {\text A}$}\\
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\frac{\partial H’_z}{\partial y’}
+v\mu_0 \frac{\partial H’_y}{\partial z’}\tag{$\bar {\text B}$}\\
\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v\mu_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right)\tag{$\bar {\text C}$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial E’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial E’_z}{\partial x’}
\tag{$\bar {\text D}$}
\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial E’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial E’_y}{\partial x’}
\tag{$\bar {\text E}$}\\
\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}-v\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\tag{$\bar {\text F}$}\\\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial H’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial H’_z}{\partial x’}
\tag{$\bar {\text G}$}
\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial H’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial H’_y}{\partial x’}\tag{$\bar {\text H}$}
\end{align}
\begin{align*}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial }{\partial y’}(H’_y -v \varepsilon_0 E’_z)
+\frac{\partial }{\partial z’}(H’_z + v\varepsilon_0 E’_y)
&=0\\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=v \varepsilon_0\frac{\partial E’_z}{\partial y’}
-v \varepsilon_0\frac{\partial E’_y}{\partial z’} \tag{$\bar {\text A}$}\end{align*}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial }{\partial y’}(H’_y -v \varepsilon_0 E’_z)
+\frac{\partial }{\partial z’}(H’_z + v\varepsilon_0 E’_y)
&=0\\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=v \varepsilon_0\frac{\partial E’_z}{\partial y’}
-v \varepsilon_0\frac{\partial E’_y}{\partial z’} \tag{$\bar {\text A}$}\end{align*}
\begin{align*}
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial }{\partial y’}( E’_y + v\mu_0 H’_z )
+\frac{\partial }{\partial z’}( E’_z {}- v\mu_0 H’_y)
&=0\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\frac{\partial H’_z}{\partial y’}
+v\mu_0 \frac{\partial H’_y}{\partial z’}\tag{$\bar {\text B}$}\end{align*}
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial }{\partial y’}( E’_y + v\mu_0 H’_z )
+\frac{\partial }{\partial z’}( E’_z {}- v\mu_0 H’_y)
&=0\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\frac{\partial H’_z}{\partial y’}
+v\mu_0 \frac{\partial H’_y}{\partial z’}\tag{$\bar {\text B}$}\end{align*}
\begin{align*}
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\\rightarrow\frac{\partial }{\partial y’}(E’_z {}- v\mu_0 H’_y)
-\frac{\partial }{\partial z’}(E’_y + v\mu_0 H’_z )
&=-\mu_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H’_x\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v\mu_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right)\tag{$\bar {\text C}$}\end{align*}
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\\rightarrow\frac{\partial }{\partial y’}(E’_z {}- v\mu_0 H’_y)
-\frac{\partial }{\partial z’}(E’_y + v\mu_0 H’_z )
&=-\mu_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H’_x\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v\mu_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right)\tag{$\bar {\text C}$}\end{align*}
\begin{align*}
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\\rightarrow\frac{\partial E’_x}{\partial z’}
-\frac{\partial}{\partial x’}( E’_z-v\mu_0 H’_y)
&=-\mu_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)(H’_y -v \varepsilon_0 E’_z)\\\rightarrow\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}+v\mu_0\frac{\partial H’_y}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+v\mu_0\frac{\partial H’_y}{\partial x’}
+v\mu_0\varepsilon_0\frac{\partial E’_z}{\partial t’}
-v^2\mu_0\varepsilon_0\frac{\partial E’_z}{\partial x’}\\\rightarrow\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial E’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial E’_z}{\partial x’}
\tag{$\bar {\text D}$}\end{align*}
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\\rightarrow\frac{\partial E’_x}{\partial z’}
-\frac{\partial}{\partial x’}( E’_z-v\mu_0 H’_y)
&=-\mu_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)(H’_y -v \varepsilon_0 E’_z)\\\rightarrow\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}+v\mu_0\frac{\partial H’_y}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+v\mu_0\frac{\partial H’_y}{\partial x’}
+v\mu_0\varepsilon_0\frac{\partial E’_z}{\partial t’}
-v^2\mu_0\varepsilon_0\frac{\partial E’_z}{\partial x’}\\\rightarrow\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial E’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial E’_z}{\partial x’}
\tag{$\bar {\text D}$}\end{align*}
\begin{align*}
\frac{\partial E_y}{\partial x}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\\rightarrow\frac{\partial}{\partial x’}(E’_y + v\mu_0 H’_z )
-\frac{\partial E’_x}{\partial y’}
&=-\mu_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)(H’_z + v\varepsilon_0 E’_y)\\\rightarrow
\frac{\partial E’_y}{\partial x’}
+v\mu_0\frac{\partial H’_z}{\partial x’}-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
+v\mu_0\frac{\partial H’_z}{\partial x’}
-v\mu_0\varepsilon_0\frac{\partial E’_y}{\partial t’}
+v^2\mu_0\varepsilon_0\frac{\partial E’_y}{\partial x’}\\\rightarrow\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial E’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial E’_y}{\partial x’}
\tag{$\bar {\text E}$}\end{align*}
\frac{\partial E_y}{\partial x}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\\rightarrow\frac{\partial}{\partial x’}(E’_y + v\mu_0 H’_z )
-\frac{\partial E’_x}{\partial y’}
&=-\mu_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)(H’_z + v\varepsilon_0 E’_y)\\\rightarrow
\frac{\partial E’_y}{\partial x’}
+v\mu_0\frac{\partial H’_z}{\partial x’}-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
+v\mu_0\frac{\partial H’_z}{\partial x’}
-v\mu_0\varepsilon_0\frac{\partial E’_y}{\partial t’}
+v^2\mu_0\varepsilon_0\frac{\partial E’_y}{\partial x’}\\\rightarrow\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial E’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial E’_y}{\partial x’}
\tag{$\bar {\text E}$}\end{align*}
\begin{align*}
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\\rightarrow\frac{\partial }{\partial y’}( H’_z + v\varepsilon_0 E’_y)
-\frac{\partial }{\partial z’}(H’_y -v \varepsilon_0 E’_z)
&=\varepsilon_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E’_x\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}-v\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\tag{$\bar {\text F}$}\end{align*}
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\\rightarrow\frac{\partial }{\partial y’}( H’_z + v\varepsilon_0 E’_y)
-\frac{\partial }{\partial z’}(H’_y -v \varepsilon_0 E’_z)
&=\varepsilon_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E’_x\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}-v\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\tag{$\bar {\text F}$}\end{align*}
\begin{align*}
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\\rightarrow\frac{\partial H’_x}{\partial z’}
-\frac{\partial}{\partial x’}(H’_z + v\varepsilon_0 E’_y)
&=\varepsilon_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)( E’_y + v\mu_0 H’_z )\\\rightarrow
\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}-v\varepsilon_0\frac{\partial E’_y}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
-v\varepsilon_0\frac{\partial E’_y}{\partial x’}+v\mu_0\varepsilon_0\frac{\partial H’_z}{\partial t’}
-v^2\mu_0\varepsilon_0\frac{\partial H’_z}{\partial x’}\\\rightarrow\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial H’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial H’_z}{\partial x’}
\tag{$\bar {\text G}$}\end{align*}
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\\rightarrow\frac{\partial H’_x}{\partial z’}
-\frac{\partial}{\partial x’}(H’_z + v\varepsilon_0 E’_y)
&=\varepsilon_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)( E’_y + v\mu_0 H’_z )\\\rightarrow
\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}-v\varepsilon_0\frac{\partial E’_y}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
-v\varepsilon_0\frac{\partial E’_y}{\partial x’}+v\mu_0\varepsilon_0\frac{\partial H’_z}{\partial t’}
-v^2\mu_0\varepsilon_0\frac{\partial H’_z}{\partial x’}\\\rightarrow\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial H’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial H’_z}{\partial x’}
\tag{$\bar {\text G}$}\end{align*}
\begin{align*}
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}\\\rightarrow\frac{\partial}{\partial x’}(H’_y -v \varepsilon_0 E’_z)
-\frac{\partial H’_x}{\partial y’}
&=\varepsilon_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)( E’_z{}- v\mu_0 H’_y)\\\rightarrow
\frac{\partial H’_y}{\partial x’}
-v\varepsilon_0\frac{\partial E’_z}{\partial x’}-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-v\varepsilon_0\frac{\partial E’_z}{\partial x’}-v\mu_0\varepsilon_0\frac{\partial H’_y}{\partial t’}
+v^2\mu_0\varepsilon_0\frac{\partial H’_y}{\partial x’}\\\rightarrow\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial H’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial H’_y}{\partial x’}\tag{$\bar {\text H}$}
\end{align*}
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}\\\rightarrow\frac{\partial}{\partial x’}(H’_y -v \varepsilon_0 E’_z)
-\frac{\partial H’_x}{\partial y’}
&=\varepsilon_0\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)( E’_z{}- v\mu_0 H’_y)\\\rightarrow
\frac{\partial H’_y}{\partial x’}
-v\varepsilon_0\frac{\partial E’_z}{\partial x’}-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-v\varepsilon_0\frac{\partial E’_z}{\partial x’}-v\mu_0\varepsilon_0\frac{\partial H’_y}{\partial t’}
+v^2\mu_0\varepsilon_0\frac{\partial H’_y}{\partial x’}\\\rightarrow\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial H’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial H’_y}{\partial x’}\tag{$\bar {\text H}$}
\end{align*}
ここで、次の関係を用いている。
\begin{align*}\mu_0\varepsilon_0=\frac{1}{c^2}\end{align*}
となり、式(\(\bar {\text A}\)),(\(\bar {\text B}\))に式(\(\bar {\text C}\)),(\(\bar {\text F}\))を代入し、式(\(\bar {\text C}\)),(\(\bar {\text F}\))に式(\(\bar {\text A}\)),(\(\bar {\text B}\))を代入すると
\begin{align}
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text A}’$}\\
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text B}’$}\\
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text C}’$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial E’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial E’_z}{\partial x’}
\tag{$\bar {\text D}$}
\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial E’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial E’_y}{\partial x’}
\tag{$\bar {\text E}$}\\
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text F}’$}\\\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial H’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial H’_z}{\partial x’}
\tag{$\bar {\text G}$}
\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial H’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial H’_y}{\partial x’}\tag{$\bar {\text H}$}
\end{align}
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text A}’$}\\
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text B}’$}\\
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text C}’$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}&=-\mu_0\frac{\partial H’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial E’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial E’_z}{\partial x’}
\tag{$\bar {\text D}$}
\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}&=-\mu_0\frac{\partial H’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial E’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial E’_y}{\partial x’}
\tag{$\bar {\text E}$}\\
\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text F}’$}\\\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}&=\varepsilon_0\frac{\partial E’_y}{\partial t’}
+\frac{v}{c^2}\frac{\partial H’_z}{\partial t’}
-\frac{v^2}{c^2}\frac{\partial H’_z}{\partial x’}
\tag{$\bar {\text G}$}
\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}&=\varepsilon_0\frac{\partial E’_z}{\partial t’}
-\frac{v}{c^2}\frac{\partial H’_y}{\partial t’}
+\frac{v^2}{c^2}\frac{\partial H’_y}{\partial x’}\tag{$\bar {\text H}$}
\end{align}
\begin{align*}\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=v \varepsilon_0\frac{\partial E’_z}{\partial y’}
-v \varepsilon_0\frac{\partial E’_y}{\partial z’} \tag{$\bar {\text A}$}\\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=-v\mu_0 \varepsilon_0\frac{\partial H’_x}{\partial t’}+v^2\mu_0\varepsilon_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right) \\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=-\frac{v}{c^2}\frac{\partial H’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right) \\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text A}’$}\end{align*}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=v \varepsilon_0\frac{\partial E’_z}{\partial y’}
-v \varepsilon_0\frac{\partial E’_y}{\partial z’} \tag{$\bar {\text A}$}\\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=-v\mu_0 \varepsilon_0\frac{\partial H’_x}{\partial t’}+v^2\mu_0\varepsilon_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right) \\\rightarrow\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_y}{\partial z’}
&=-\frac{v}{c^2}\frac{\partial H’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right) \\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text A}’$}\end{align*}
\begin{align*}
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\frac{\partial H’_z}{\partial y’}
+v\mu_0 \frac{\partial H’_y}{\partial z’}\tag{$\bar {\text B}$}\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\varepsilon_0\frac{\partial E’_x}{\partial t’}+v^2\mu_0\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-\frac{v}{c^2}\frac{\partial E’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text B}’$}\end{align*}
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\frac{\partial H’_z}{\partial y’}
+v\mu_0 \frac{\partial H’_y}{\partial z’}\tag{$\bar {\text B}$}\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-v\mu_0\varepsilon_0\frac{\partial E’_x}{\partial t’}+v^2\mu_0\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\\\rightarrow\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z }{\partial z’}
&=-\frac{v}{c^2}\frac{\partial E’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}\right)
&=-\frac{v}{c^2}\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text B}’$}\end{align*}
\begin{align*}\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v\mu_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right)\tag{$\bar {\text C}$}\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v^2\mu_0\epsilon_0\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)\\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text C}’$}\end{align*}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v\mu_0\left(\frac{\partial H’_x}{\partial x’}+\frac{\partial H’_y}{\partial y’}+\frac{\partial H’_z}{\partial z’}\right)\tag{$\bar {\text C}$}\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+v^2\mu_0\epsilon_0\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)\\\rightarrow\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)\\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}\right)
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{$\bar {\text C}’$}\end{align*}
\begin{align*}\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}-v\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\tag{$\bar {\text F}$}\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}+v^2\mu_0\varepsilon_0\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)\\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text F}’$}\end{align*}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}-v\varepsilon_0\left(\frac{\partial E’_x}{\partial x’}+\frac{\partial E’_y}{\partial y’}+\frac{\partial E’_z}{\partial z’}\right)\tag{$\bar {\text F}$}\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}+v^2\mu_0\varepsilon_0\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)\\\rightarrow\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}+\frac{v^2}{c^2}\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)\\\rightarrow\left(1-\frac{v^2}{c^2}\right)\left(\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}\right)
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{$\bar {\text F}’$}\end{align*}
となる。よって、ガリレイ変換に加えて電場·磁場の変換が
\begin{align}E_x &= E’_x\\E_y &= E’_y + v\mu_0 H’_z \\
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
であれば、\(\frac{v}{c}\)が十分小さい低速極限
\begin{align*}\frac{v}{c}\simeq0\end{align*}
のときに限って
\begin{align*}\frac{v^2}{c^2}\simeq0\end{align*}
も成り立ち、近似的にマクスウェル方程式の形は不変となることがわかる。
ここで、「低速極限では、なぜ \(\frac{v}{c^2}\) ではなく \(\frac{v}{c}\) を小さい量として考えるのか」と疑問に思ったかもしれない。しかし、\(\frac{v}{c^2}\) は
\begin{align*}\left[\frac{v}{c^2}\right]=\frac{\mathrm{m/s}}{\mathrm{m^2/s^2}}=\mathrm{s/m}\end{align*}
という次元を持つため、単独では「大きい」「小さい」を比較することができない。実際、次ページでわかるが、\(\frac{v}{c^2}\) は空間座標 \(x\) と組み合わさって
\begin{align*}\frac{vx}{c^2}\end{align*}
という時間補正として現れる。たとえば、\(v=30\,\mathrm{m/s}\) のとき
\begin{align*}\frac{v}{c^2}\approx3.3\times10^{-16}\,\mathrm{s/m}\end{align*}
であるが、\(x=1\,\mathrm{m}\) なら
\begin{align*}\frac{vx}{c^2}\approx3.3\times10^{-16}\,\mathrm{s}\end{align*}
と非常に小さい一方、\(x=10^{15}\,\mathrm{m}\) なら
\begin{align*}\frac{vx}{c^2}\approx0.33\,\mathrm{s}\end{align*}
となり、無視できない大きさになる。つまり、\(\frac{v}{c^2}\) 単独では大小を議論できないのである。
一方、\(\frac{v}{c}\) は
\begin{align*}\left[\frac{v}{c}\right]=1\end{align*}
であり無次元量であるため、純粋に「どれくらい小さいか」を比較できる。そのため、低速極限では \(\frac{v}{c}\) を小さい量として考えるのである。
次ページでは、さらに精度を上げて\(\frac{v^2}{c^2}\)が十分小さいときに近似的にマクスウェル方程式の形が保たれるように変換則を改良していくと、時間そのものを補正する「局所時間」の考えが現れることを見る。
スポンサーリンク
次ページから…
次ページでは、空間座標におけるガリレイ変換
\begin{align*}x’&=x-vt\\y’&=y\\z’&=z\end{align*}
に加えて局所時間
\begin{align*}t’&=t-\frac{v}{c^2}x\end{align*}
を導入すると、\(\frac{v^2}{c^2}\)が十分小さいときに近似的にマクスウェル方程式の形が保たれることをみる。これは、慣性系同士の相対速度\(v\)が光速度\(c\)と比べて十分小さい低速極限よりもさらに精度の高い変換である。
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