<\/p>\n
\\(S\\) \u7cfb\u306b\u5bfe\u3057\u3066\u901f\u5ea6 \\(\\boldsymbol{V}\\) \u3067\u904b\u52d5\u3059\u308b \\(S’\\) \u7cfb\u3067\u307f\u308b\u3068\uff0c\u96fb\u78c1\u5834\u306e \\(\\boldsymbol{V}\\) \u306b\u5e73\u884c\u306a\u6210\u5206\uff08\\({\\ }_{\/\\!\/}\\) \u3092\u3064\u3051\u3066\u8868\u3059\uff09\uff0c\u304a\u3088\u3073\u5782\u76f4\u306a\u6210\u5206\uff08\\({\\ }_{\\perp}\\) \u3092\u3064\u3051\u3066\u8868\u3059\uff09\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u63db\u3055\u308c\u308b\u3002
\n$$\\boldsymbol{E}’_{\/\\!\/} = \\boldsymbol{E}_{\/\\!\/} , \\quad
\n\\boldsymbol{E}’_{\\perp} =\u00a0 \\frac{\\boldsymbol{E}_{\\perp} + (\\boldsymbol{V}\\times\\boldsymbol{B})_{\\perp}}{\\sqrt{1-V^2}}$$
\n$$\\boldsymbol{B}’_{\/\\!\/} = \\boldsymbol{B}_{\/\\!\/}, \\quad
\n\\boldsymbol{B}’_{\\perp} = \\frac{\\boldsymbol{B}_{\\perp} – (\\boldsymbol{V}\\times\\boldsymbol{E})_{\\perp} }{\\sqrt{1-V^2}}$$
\n\u4ee5\u4e0a\u306e\u7d50\u679c\u306f\uff0c\u901a\u5e38\u306f\u96fb\u78c1\u5834\u30c6\u30f3\u30bd\u30eb\u306e\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u304b\u3089\u5c0e\u304f\u306e\u3067\u3042\u308b\u304c\uff0c\u3053\u308c\u3092\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u308f\u305a\u306b\u5c0e\u3053\u3046\u3068\u3044\u3046\u306e\u304c\u3053\u3053\u306e\u672c\u984c\u3067\u3042\u308b\u3002<\/p>\n
\u96fb\u78c1\u5834\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\uff08\u307e\u305f\u306f\u96fb\u78c1\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\uff09\\(F_{\\mu\\nu}\\) \u306f\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb<\/strong><\/span> \\(A_{\\mu}\\) \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u53cd\u5bfe\u79f0\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u3067\u3042\u308b\u3002<\/p>\n $$F_{\\mu\\nu} \\equiv \\partial_{\\mu} A_{\\nu} – \\partial_{\\nu} A_{\\mu}$$<\/p>\n 4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u6210\u5206\u304c \\(u^{\\mu} = (1, 0, 0, 0)\\) \u3068\u306a\u308b\u9759\u6b62\u7cfb\u3067\u306f\uff0c\u96fb\u78c1\u30c6\u30f3\u30bd\u30eb \\(F_{\\mu\\nu}\\) \u306e\u6210\u5206\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b3\u6b21\u5143\u96fb\u5834\u30d9\u30af\u30c8\u30eb \\(\\vec{E} = (E_x, E_y, E_z)\\) \u3084\u78c1\u5834\u30d9\u30af\u30c8\u30eb\uff08\u78c1\u675f\u5bc6\u5ea6\u30d9\u30af\u30c8\u30eb\uff09\\(\\vec{B} = (B_x, B_y, B_z)\\) \u306e\u6210\u5206\u3067\u3042\u308f\u3089\u3055\u308c\u308b\u3002<\/p>\n \\begin{eqnarray} \u3053\u3053\u3067\uff0c4\u5143\u901f\u5ea6 \\( u^{\\mu} \\) \u306e\u89b3\u6e2c\u8005 \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u96fb\u5834\uff0c\u78c1\u5834\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002 \u4e0a\u8a18\u306e\u5b9a\u7fa9\u304b\u3089\uff0c\\(E^{\\mu}, \\ B_{\\nu} \\) \u304c4\u5143\u901f\u5ea6 \\( u^{\\mu} \\) \u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u300c\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb\u300d\u3067\u3042\u308b\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308b\u3002 \u89b3\u6e2c\u8005 \\(A\\) \u306e<\/strong>4\u5143\u901f\u5ea6<\/strong>: \\(\\boldsymbol{u}\\)<\/p>\n \\(\\boldsymbol{u}\\) \u306b\u76f4\u4ea4\u3059\u308b<\/strong>\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/strong>: \\(\\boldsymbol{e}\\) \uff08\u904b\u52d5\u3059\u308b\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb\uff09 \\(\\boldsymbol{u}\\) \u306b\u3082\\(\\boldsymbol{e}\\)\u306b\u3082\u76f4\u4ea4\u3059\u308b<\/strong>\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/strong>: \\(\\boldsymbol{n}\\) \uff08\u904b\u52d5\u306b\u76f4\u4ea4\u3059\u308b\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb\uff09<\/p>\n \\begin{eqnarray} \u89b3\u6e2c\u8005 \\(A\\) \u306b\u5bfe\u3057\u3066\uff0c\\(\\boldsymbol{e}\\) \u65b9\u5411\u306b\u901f\u3055 \\(V\\) \u3067\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6<\/strong>: \\(\\bar{\\boldsymbol{u}}\\)<\/p>\n \\(\\bar{\\boldsymbol{u}}\\) \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/strong>: \\(\\bar{\\boldsymbol{e}}\\) \uff08\u904b\u52d5\u3059\u308b\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb\uff09<\/p>\n \\begin{eqnarray} \\(\\bar{\\boldsymbol{u}}\\) \u306b\u3082\\(\\bar{\\boldsymbol{e}}\\)\u306b\u3082\u76f4\u4ea4\u3059\u308b<\/strong>\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/strong>: \\(\\bar{\\boldsymbol{n}}\\) \uff08\u904b\u52d5\u306b\u76f4\u4ea4\u3059\u308b\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb\uff09<\/p>\n \\begin{eqnarray} \\begin{eqnarray} \u89b3\u6e2c\u8005 \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u96fb\u5834\u30d9\u30af\u30c8\u30eb \\(E_{\\mu} = F_{\\mu\\nu} u^{\\nu}\\) \u306e\uff0c\u9032\u884c\u65b9\u5411\u3092\u3042\u3089\u308f\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(e^{\\mu}\\) \u306b\u5e73\u884c\u306a\u6210\u5206 \\(E_{\/\\!\/} \\) \u306f \u89b3\u6e2c\u8005 \\(B\\) \u306b\u3068\u3063\u3066\u306f\uff0c\u81ea\u8eab\u306e4\u5143\u901f\u5ea6 \\(\\bar{u}^{\\mu}\\) \u3068\u9032\u884c\u65b9\u5411\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\bar{e}^{\\mu}\\) \u3092\u4f7f\u3063\u3066\uff0c \\(\\bar{E}_{\/\\!\/} = \\bar{e}^{\\mu} F_{\\mu\\nu} \\bar{u}^{\\nu}\\)\u00a0 \u3067\u3042\u308a\uff0c<\/p>\n \\begin{eqnarray} <\/p>\n \u89b3\u6e2c\u8005 \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u96fb\u5834\u30d9\u30af\u30c8\u30eb \\(E_{\\mu} = F_{\\mu\\nu} u^{\\nu}\\) \u306e\uff0c\u9032\u884c\u65b9\u5411\u306b\u5782\u76f4\u306a\u6210\u5206\u306f $$E_{\\perp} = n^{\\mu}\u00a0 E_{\\mu} = n^{\\mu} F_{\\mu\\nu} u^{\\nu}$$\u3067\u3042\u308b\u3002<\/p>\n \u89b3\u6e2c\u8005 \\(B\\) \u306b\u3068\u3063\u3066\u306f\uff0c\u81ea\u8eab\u306e4\u5143\u901f\u5ea6 \\(\\bar{u}^{\\mu}\\) \u3092\u4f7f\u3063\u3066\uff0c \\(\\bar{E}_{\\perp} = \\bar{n}^{\\mu} F_{\\mu\\nu} \\bar{u}^{\\nu} \\)\u00a0 \u3067\u3042\u308a\uff0c<\/p>\n \\begin{eqnarray} \u5ff5\u306e\u305f\u3081\u306b\u8a73\u7d30\u3092\u66f8\u304f\u3068\uff0c\u89b3\u6e2c\u8005 \\(A\\) \u306e\u9759\u6b62\u7cfb\u3067\u306f \u89b3\u6e2c\u8005 \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u96fb\u5834\u30d9\u30af\u30c8\u30eb \\(B_{\\mu} = {}^{\\ast}\\!F_{\\nu\\mu} u^{\\nu}\\) \u306e\uff0c\u9032\u884c\u65b9\u5411\u3092\u3042\u3089\u308f\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(e^{\\mu}\\) \u306b\u5e73\u884c\u306a\u6210\u5206 \\(B_{\/\\!\/} \\) \u306f \u89b3\u6e2c\u8005 \\(B\\) \u306b\u3068\u3063\u3066\u306f\uff0c\u81ea\u8eab\u306e4\u5143\u901f\u5ea6 \\(\\bar{u}^{\\mu}\\) \u3068\u9032\u884c\u65b9\u5411\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\bar{e}^{\\mu}\\) \u3092\u4f7f\u3063\u3066\uff0c \\(\\bar{B}_{\/\\!\/} = \\bar{e}^{\\mu}\\, {}^{\\ast}\\!F_{\\nu\\mu} \\bar{u}^{\\nu}\\)\u00a0 \u3067\u3042\u308a\uff0c<\/p>\n \\begin{eqnarray} <\/p>\n \u89b3\u6e2c\u8005 \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u78c1\u5834\u30d9\u30af\u30c8\u30eb \\(B_{\\mu} = {}^{\\ast}\\!F_{\\nu\\mu} u^{\\nu}\\) \u306e\uff0c\u9032\u884c\u65b9\u5411\u306b\u5782\u76f4\u306a\u6210\u5206\u306f $$B_{\\perp} = n^{\\mu} B_{\\mu} = n^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} u^{\\nu}$$\u3067\u3042\u308b\u3002<\/p>\n \u89b3\u6e2c\u8005 \\(B\\) \u306b\u3068\u3063\u3066\u306f\uff0c\u81ea\u8eab\u306e4\u5143\u901f\u5ea6 \\(\\bar{u}^{\\mu}\\) \u3092\u4f7f\u3063\u3066\uff0c \\(\\bar{B}_{\\perp} = \\bar{n}^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} \\bar{u}^{\\nu}\\)\u00a0 \u3067\u3042\u308a\uff0c<\/p>\n \\begin{eqnarray} \u5ff5\u306e\u305f\u3081\u306b\u8a73\u7d30\u3092\u66f8\u304f\u3068\uff0c\u89b3\u6e2c\u8005 \\(A\\) \u306e\u9759\u6b62\u7cfb\u3067\u306f \u89b3\u6e2c\u8005 \\(A\\) \u306b\u5bfe\u3057\u3066\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e\u904b\u52d5\u65b9\u5411\u306b\u5e73\u884c\u306a\u6210\u5206\u6210\u5206\uff08\\({\\ }_{\/\\!\/}\\) \u3092\u3064\u3051\u3066\u8868\u3059\uff09\uff0c\u304a\u3088\u3073\u5782\u76f4\u306a\u6210\u5206\uff08\\({\\ }_{\\perp}\\) \u3092\u3064\u3051\u3066\u8868\u3059\uff09\u306b\u3064\u3044\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5909\u63db\u5247\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u308f\u305a\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u305f\u3002<\/p>\n \\begin{eqnarray} \u89b3\u6e2c\u8005 \\(A\\)\u306b\u3068\u3063\u3066\u306e\u96fb\u78c1\u5834\u306f \u89b3\u6e2c\u8005 \\(B\\)\u306b\u3068\u3063\u3066\u306e\u96fb\u78c1\u5834\u306f
\nF_{\\mu\\nu} &=&
\n\\left(
\n\\begin{array}{cccc}
\nF_{00}& F_{01} & F_{02}& F_{03}\\\\
\nF_{10}& F_{11} & F_{12}& F_{13}\\\\
\nF_{20}& F_{21} & F_{22}& F_{23}\\\\
\nF_{30}& F_{31} & F_{32}& F_{33}
\n\\end{array} \\right) \\\\
\n&=&
\n\\left(
\n\\begin{array}{cccc}
\n0& -E_x & -E_y& -E_z\\\\
\nE_x&0 & B_z& -B_y\\\\
\nE_y& -B_z & 0& B_x\\\\
\nE_z& B_y & -B_x& 0
\n\\end{array} \\right)\\
\n\\end{eqnarray}<\/p>\n
\n\\begin{eqnarray}
\nE_{\\mu} &\\equiv& F_{\\mu\\nu}\\, u^{\\nu} \\\\
\nB_{\\nu} &\\equiv& {}^{\\ast}\\!F_{\\mu\\nu}\\, u^{\\mu}
\n\\end{eqnarray} \u3053\u3053\u3067\uff0c\\( {}^{\\ast}\\!F_{\\mu\\nu} \\) \u306f\u5b8c\u5168\u53cd\u5bfe\u79f0\u306a Levi-Civita \u30c6\u30f3\u30bd\u30eb<\/strong><\/span> \\(\\varepsilon_{\\mu \\nu \\alpha \\beta} \\) \u3092\u7528\u3044\u3066
\n$$ {}^{\\ast}\\!F_{\\mu\\nu} \\equiv \\frac{1}{2} \\varepsilon_{\\mu \\nu \\alpha \\beta}\\,F^{\\alpha \\beta} $$ \u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b2\u968e\u53cd\u5bfe\u79f0\u30c6\u30f3\u30bd\u30eb\u3067\uff0c\\( F^{\\mu \\nu}\\) \u306b\u53cc\u5bfe (dual)<\/strong> <\/span>\u306a\u30c6\u30f3\u30bd\u30eb\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n
\n$$ E_{\\mu} u^{\\mu} = 0, \\quad B_{\\nu} u^{\\nu} = 0$$\u00a0 \u7279\u306b\u89b3\u6e2c\u8005 \\(A\\) \u304c \\(S\\) \u7cfb\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u3068\u3059\u308b\u3068
\n$$ u^{\\mu} = (1, 0, 0, 0)$$ \u3068\u306a\u308b\u3002\u3053\u308c\u3092\u4f7f\u3063\u3066\u5404\u6210\u5206\u3092\u5177\u4f53\u7684\u306b\u8868\u3059\u3068
\n\\begin{eqnarray}
\nE_0 &=& F_{0 0} u^{0} = 0\\\\
\nE_1 &=& F_{1 0} u^{0} =\u00a0 E_x \\\\
\nE_2 &=&\u00a0 F_{2 0} u^{0} = E_y \\\\
\nE_3 &=& F_{3 0} u^{0} = E_z \\\\
\nB_0 &=& \\frac{1}{2} u^0\u00a0 \\varepsilon_{0 0 \\alpha \\beta} F^{\\alpha \\beta} = 0\\\\
\nB_1 &=& \\frac{1}{2} u^0\u00a0 \\varepsilon_{0 1 \\alpha \\beta} F^{\\alpha \\beta} = \\frac{1}{2} \\left(\\varepsilon_{0 1 23} F^{23}+\\varepsilon_{0 1 32} F^{32} \\right) = F^{23} =F_{23} = B_x\\\\
\nB_2 &=& \\frac{1}{2} u^0\u00a0 \\varepsilon_{0 2 \\alpha \\beta} F^{\\alpha \\beta} = \\frac{1}{2} \\left(\\varepsilon_{0 2 31} F^{31}+\\varepsilon_{0 2 13} F^{13} \\right) = F^{31} =F_{31} = B_y\\\\
\nB_3 &=& \\frac{1}{2} u^0\u00a0 \\varepsilon_{0 3 \\alpha \\beta} F^{\\alpha \\beta} = \\frac{1}{2} \\left(\\varepsilon_{0 3 12} F^{12}+\\varepsilon_{0 3 21} F^{21} \\right) = F^{12} =F_{12} = B_z
\n\\end{eqnarray} \u3068\u306a\u308a\uff0c4\u5143\u30d9\u30af\u30c8\u30eb \\( E^{\\mu}, B_{\\nu} \\) \u306e\u7a7a\u9593\u6210\u5206\u304c\u78ba\u304b\u306b3\u6b21\u5143\u306e\u96fb\u5834\u30d9\u30af\u30c8\u30eb \\(\\vec{E}\\) \u304a\u3088\u3073\u78c1\u5834\uff08\u78c1\u675f\u5bc6\u5ea6\uff09\u30d9\u30af\u30c8\u30eb \\(\\vec{B} \\) \u306e\u6210\u5206\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h3>\n
\u89b3\u6e2c\u8005 \\(A\\)<\/h4>\n
\n\\begin{eqnarray}
\n{\\boldsymbol{u}}\\cdot{\\boldsymbol{e}} &=&\u00a0 \\eta_{\\mu\\nu} {u}^{\\mu}{e}^{\\nu} = 0\\\\
\n{\\boldsymbol{e}}\\cdot{\\boldsymbol{e}} &=&\\eta_{\\mu\\nu} {e}^{\\mu}{e}^{\\nu} = 1
\n\\end{eqnarray}<\/p>\n
\n{\\boldsymbol{u}}\\cdot{\\boldsymbol{n}} &=&\u00a0 \\eta_{\\mu\\nu} {u}^{\\mu}{n}^{\\nu} = 0\\\\
\n{\\boldsymbol{e}}\\cdot{\\boldsymbol{n}} &=&\\eta_{\\mu\\nu} {e}^{\\mu}{n}^{\\nu} = 0\\\\
\n{\\boldsymbol{n}}\\cdot{\\boldsymbol{n}} &=&\\eta_{\\mu\\nu} {n}^{\\mu}{n}^{\\nu} = 1
\n\\end{eqnarray}<\/p>\n\u89b3\u6e2c\u8005 \\(B\\)<\/h4>\n
\n\\bar{\\boldsymbol{u}}\\cdot\\bar{\\boldsymbol{e}} &=&\u00a0 \\eta_{\\mu\\nu} \\bar{u}^{\\mu}\\bar{e}^{\\nu} = 0\\\\
\n\\bar{\\boldsymbol{e}}\\cdot\\bar{\\boldsymbol{e}} &=&\\eta_{\\mu\\nu} \\bar{e}^{\\mu}\\bar{e}^{\\nu} = 1
\n\\end{eqnarray}<\/p>\n
\n{\\bar{\\boldsymbol{u}}}\\cdot{\\bar{\\boldsymbol{n}} }&=&\u00a0 \\eta_{\\mu\\nu} {\\bar{u}}^{\\mu}{\\bar{n}}^{\\nu} = 0\\\\
\n{\\bar{\\boldsymbol{e}}}\\cdot{\\bar{\\boldsymbol{n}}} &=&\\eta_{\\mu\\nu} {\\bar{e}}^{\\mu}{\\bar{n}}^{\\nu} = 0\\\\
\n{\\bar{\\boldsymbol{n}}}\\cdot{\\bar{\\boldsymbol{n}}} &=&\\eta_{\\mu\\nu} {\\bar{n}}^{\\mu}{\\bar{n}}^{\\nu} = 1
\n\\end{eqnarray}<\/p>\n4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u3068\u305d\u306e\u95a2\u9023<\/h4>\n
\n\\bar{\\boldsymbol{u}} &=& \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{u} + \\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{e}\\\\
\n\\bar{\\boldsymbol{e}} &=& \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{e} + \\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{u}\\\\
\n\\bar{\\boldsymbol{n}} &=& \\boldsymbol{n}
\n\\end{eqnarray}<\/p>\n\u89b3\u6e2c\u8005 \\(A\\) \u304a\u3088\u3073 \\(B\\) \u306e\u89b3\u6e2c\u3059\u308b\u96fb\u78c1\u5834\u306e\u6210\u5206<\/h3>\n
\u96fb\u5834<\/h4>\n
\n$$E_{\/\\!\/} = e^{\\mu}\u00a0 E_{\\mu} = e^{\\mu} F_{\\mu\\nu} u^{\\nu}$$\u3067\u3042\u308b\u3002<\/p>\n
\n\\bar{{E}}_{\/\\!\/} &\\equiv& \\bar{e}^{\\mu} F_{\\mu\\nu} \\bar{u}^{\\nu}\\\\
\n&=& \\frac{e^{\\mu} + V u^{\\mu}}{\\sqrt{1-V^2}} F_{\\mu\\nu} \\frac{u^{\\nu} + V e^{\\nu}}{\\sqrt{1-V^2}}\\\\
\n&=& \\frac{e^{\\mu} F_{\\mu\\nu}u^{\\nu} + V^2 u^{\\mu}F_{\\mu\\nu}e^{\\nu}}{1-V^2}\\\\
\n&=& \\frac{e^{\\mu} F_{\\mu\\nu}u^{\\nu} – V^2 e^{\\nu}F_{\\nu\\mu}u^{\\mu}}{1-V^2}\\\\
\n&=& e^{\\mu} F_{\\mu\\nu}u^{\\nu} \\\\ &=& E_{\/\\!\/}
\n\\end{eqnarray}<\/p>\n
\n\\bar{E}_{\\perp} &\\equiv& \\bar{n}^{\\mu} F_{\\mu\\nu} \\bar{u}^{\\nu} = {n}^{\\mu} F_{\\mu\\nu} \\bar{u}^{\\nu}\\\\
\n&=& {n}^{\\mu} F_{\\mu\\nu} \\frac{u^{\\nu} + V e^{\\nu}}{\\sqrt{1 – V^2}}\\\\
\n&=& \\frac{{n}^{\\mu} F_{\\mu\\nu} u^{\\nu} + {n}^{\\mu} F_{\\mu\\nu} V e^{\\nu}}{\\sqrt{1 – V^2}}\\\\
\n&=& \\frac{E_{\\perp}+ (\\boldsymbol{V}\\times\\boldsymbol{B})_{\\perp}}{\\sqrt{1 – V^2}}
\n\\end{eqnarray}<\/p>\n
\n$$u^{\\mu} = (1, 0, 0, 0), \\quad e^{\\mu} = (0, 1, 0, 0), \\quad n^{\\mu} = (0, 0, 1, 0)$$\u3068\u3057\u3066\u3088\u3044\u306e\u3067\uff0c
\n\\begin{eqnarray}
\n{n}^{\\mu} F_{\\mu\\nu} V e^{\\nu}
\n&=& n^2 F_{21} V e_1 \\\\
\n&=& – B_z V = (\\boldsymbol{V}\\times\\boldsymbol{B})_y\\\\
\n&=& (\\boldsymbol{V}\\times\\boldsymbol{B})_{\\perp}
\n\\end{eqnarray}\u4e00\u65e6\u7b54\u3048\u304c\u3067\u305f\u3089\uff0c\u3053\u306e\u5f0f\u306f\u5de6\u8fba\u304c4\u5143\u30b9\u30ab\u30e9\u30fc\u3067\u3042\u308b\u304b\u3089\uff0c\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u305a\u306b\u6210\u308a\u7acb\u3064\u3002<\/p>\n\u78c1\u5834<\/h4>\n
\n$$B_{\/\\!\/} = e^{\\mu}\u00a0 B_{\\mu} = e^{\\mu}\\, {}^{\\ast}\\!F_{\\nu\\mu} u^{\\nu}$$\u3067\u3042\u308b\u3002<\/p>\n
\n\\bar{B}_{\/\\!\/}&\\equiv& \\bar{e}^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} \\bar{u}^{\\nu}\\\\
\n&=& \\frac{e^{\\mu} + V u^{\\mu}}{\\sqrt{1-V^2}}\\, {}^{\\ast}\\!F_{\\nu\\mu} \\frac{u^{\\nu} + V e^{\\nu}}{\\sqrt{1-V^2}}\\\\
\n&=& \\frac{e^{\\mu}\\, {}^{\\ast}\\!F_{\\nu\\mu} u^{\\nu} + V^2 u^{\\mu}\\, {}^{\\ast}\\!F_{\\nu\\mu}e^{\\nu}}{1-V^2}\\\\
\n&=& \\frac{e^{\\mu}\\, {}^{\\ast}\\!F_{\\nu\\mu}u^{\\nu} – V^2 e^{\\nu} {}^{\\ast}\\!F_{\\mu\\nu}u^{\\mu}}{1-V^2}\\\\
\n&=& e^{\\mu}\\, {}^{\\ast}\\!F_{\\nu\\mu}u^{\\nu} \\\\ &=& B_{\/\\!\/}
\n\\end{eqnarray}<\/p>\n
\n\\bar{B}_{\\perp} &\\equiv& \\bar{n}^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} \\bar{u}^{\\nu} =n^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} \\bar{u}^{\\nu}\\\\
\n&=& {n}^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} \\frac{u^{\\nu} + V e^{\\nu}}{\\sqrt{1 – V^2}}\\\\
\n&=& \\frac{{n}^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} u^{\\nu} + {n}^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} V e^{\\nu}}{\\sqrt{1 – V^2}}\\\\
\n&=& \\frac{B_{\\perp}- (\\boldsymbol{V}\\times\\boldsymbol{E})_{\\perp}}{\\sqrt{1 – V^2}}
\n\\end{eqnarray}<\/p>\n
\n$$u^{\\mu} = (1, 0, 0, 0), \\quad e^{\\mu} = (0, 1, 0, 0), \\quad n^{\\mu} = (0, 0, 1, 0)$$\u3068\u3057\u3066\u3088\u3044\u306e\u3067\uff0c
\n\\begin{eqnarray}
\n{n}^{\\mu} \\,{}^{\\ast}\\!F_{\\nu\\mu} V e^{\\nu}
\n&=& n^2 \\,{}^{\\ast}\\!F_{21} V e_1 \\\\
\n&=&\u00a0 F_{30} V \\\\
\n&=& E_z V = -(\\boldsymbol{V}\\times\\boldsymbol{E})_y\\\\
\n&=& -(\\boldsymbol{V}\\times\\boldsymbol{E})_{\\perp}
\n\\end{eqnarray}\u4e00\u65e6\u7b54\u3048\u304c\u3067\u305f\u3089\uff0c\u3053\u306e\u5f0f\u306f\u5de6\u8fba\u304c4\u5143\u30b9\u30ab\u30e9\u30fc\u3067\u3042\u308b\u304b\u3089\uff0c\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u305a\u306b\u6210\u308a\u7acb\u3064\u3002<\/p>\n\u307e\u3068\u3081<\/h3>\n
\n\\bar{{E}}_{\/\\!\/}\u00a0 &=& E_{\/\\!\/} \\\\
\n\\bar{E}_{\\perp} &=& \\frac{E_{\\perp}+ (\\boldsymbol{V}\\times\\boldsymbol{B})_{\\perp}}{\\sqrt{1 – V^2}}\\\\
\n\\bar{{B}}_{\/\\!\/} &=& B_{\/\\!\/}\\\\
\n\\bar{B}_{\\perp} &=& \\frac{B_{\\perp}- (\\boldsymbol{V}\\times\\boldsymbol{E})_{\\perp}}{\\sqrt{1 – V^2}}
\n\\end{eqnarray}<\/p>\n
\n$$E_{\\mu} = F_{\\mu\\nu} u^{\\nu}, \\quad E_{\/\\!\/} = e^{\\mu}\u00a0 E_{\\mu}, \\quad E_{\\perp} = n^{\\mu}\u00a0 E_{\\mu}$$
\n$$B_{\\mu} = {}^{\\ast}\\!F_{\\nu\\mu} u^{\\nu}, \\quad B_{\/\\!\/} = e^{\\mu}\u00a0 B_{\\mu}, \\quad B_{\\perp} = n^{\\mu}\u00a0 B_{\\mu}$$<\/p>\n
\n$$\\bar{E}_{\\mu} = F_{\\mu\\nu} \\bar{u}^{\\nu}, \\quad \\bar{E}_{\/\\!\/} = \\bar{e}^{\\mu}\u00a0 \\bar{E}_{\\mu}, \\quad \\bar{E}_{\\perp} = \\bar{n}^{\\mu}\u00a0 \\bar{E}_{\\mu}= n^{\\mu}\u00a0 \\bar{E}_{\\mu}$$
\n$$\\bar{B}_{\\mu} = {}^{\\ast}\\!F_{\\nu\\mu} \\bar{u}^{\\nu}, \\quad \\bar{B}_{\/\\!\/} = \\bar{e}^{\\mu}\u00a0 \\bar{B}_{\\mu}, \\quad \\bar{B}_{\\perp} = \\bar{n}^{\\mu}\u00a0 \\bar{B}_{\\mu}= n^{\\mu}\u00a0 \\bar{B}_{\\mu}$$<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":71,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1077","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1077","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=1077"}],"version-history":[{"count":23,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1077\/revisions"}],"predecessor-version":[{"id":1243,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1077\/revisions\/1243"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/71"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=1077"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}