4\u6b21\u5143\u6642\u7a7a\u3067\u306f\uff0c \\(x^{\\mu} = (x^0, x^1, x^2, x^3) \\) \u306e\u3088\u3046\u306b\uff0c\u30ae\u30ea\u30b7\u30e3\u6587\u5b57\u306e\u6dfb\u5b57\u30674\u3064\u306e\u5ea7\u6a19\u3092\u4e00\u6319\u306b\u8868\u3059\u3002<\/p>\n
\u307e\u305a\uff0c\u8fd1\u63a5\u3057\u305f2\u70b9 \\(P(x^{\\mu})\\) \u3068 \\(Q(x^{\\mu} +dx^{\\mu})\\) \u9593\u306e<\/p>\n
\\begin{equation} \u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{dx}\\) \u306e\u300c\u5927\u304d\u3055\u300d\u306e2\u4e57 \uff08\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\uff09<\/strong><\/span>\u3092\u7dda\u7d20<\/strong><\/span> \\(ds^2\\) \u3068\u547c\u3073\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002 \u3053\u3053\u3067\uff0c<\/p>\n \u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206$$g_{\\mu\\nu} \\equiv\\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu}$$ \u306f\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u3092\u8868\u3059<\/strong><\/span><\/p>\n \u3053\u3068\u3092\u3042\u3089\u305f\u3081\u3066\u8a00\u660e\u3057\u3066\u304a\u304f\u3002<\/p>\n \u7279\u6b8a\u76f8\u5bfe\u8ad6\u3067\u306f\uff0c\u3044\u308f\u3086\u308b\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u5ea7\u6a19\u7cfb<\/strong><\/span>\uff08\u7a7a\u9593\u5ea7\u6a19\u3068\u3057\u3066\u76f4\u4ea4\u76f4\u7dda\u5ea7\u6a19\u7cfb\u3067\u3042\u308b\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u3092\u3068\u308a\uff0c\u305d\u308c\u306b\u7b2c\u30bc\u30ed\u6210\u5206\u3068\u3057\u3066 \\(ct\\) \u3092\u52a0\u3048\u305f4\u6b21\u5143\u5ea7\u6a19\u7cfb\uff09 $$x^{\\mu} = (x^0, x^i) = (ct, x, y, z)$$ \u30674\u6b21\u5143\u6642\u7a7a\u3092\u5927\u57df\u7684\u306b\u5f35\u308b\u3053\u3068\u304c\u53ef\u80fd\u3067\u3042\u308a\uff0c\u305d\u306e\u5834\u5408\u306b\uff08\u305d\u306e\u5834\u5408\u306b\u306e\u307f\uff09<\/p>\n \\begin{eqnarray} \\begin{eqnarray} \u3053\u306e \\(\\eta_{\\mu\\nu}\\) \u3092\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u6642\u7a7a\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\uff0c\u307e\u305f\u306f\u77ed\u304f\u3057\u3066\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u8a08\u91cf<\/strong><\/span>\u306e\u6210\u5206\u3068\u547c\u3076\u3002<\/p>\n \u4e00\u822c\u306e\u66f2\u304c\u3063\u305f\u6642\u7a7a\u3067\u306f\uff0c\u5927\u57df\u7684\u306b\u76f4\u4ea4\u76f4\u7dda\u5ea7\u6a19\u7cfb\u3092\u5f35\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u305f\u3081\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306f\u5b9a\u6570\u306e\u307e\u307e\u3068\u306f\u3044\u304b\u305a\uff0c\u4e00\u822c\u306b\u5ea7\u6a19\u4f9d\u5b58\u6027\u3092\u3082\u3064\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n
\nd\\boldsymbol{x} = \\sum_{\\mu=0}^3 dx^{\\mu}\\,\\boldsymbol{e}_{\\mu}
\n\\equiv dx^{\\mu}\\,\\boldsymbol{e}_{\\mu}
\n\\end{equation}
\n\u3068\u66f8\u304f\u3002\u4e0a\u6dfb\u5b57\u3068\u4e0b\u6dfb\u5b57\u306b\u540c\u3058\u30ae\u30ea\u30b7\u30e3\u6587\u5b57\u304c\u3042\u308b\u5834\u5408\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u306e\u898f\u7d04<\/strong><\/span>\u3067\u306f\u305f\u3068\u3048 \\(\\displaystyle \\sum\\) \u304c\u66f8\u3044\u3066\u3044\u306a\u304f\u3066\u3082\u81ea\u52d5\u7684\u306b 0 \u304b\u3089 3 \u307e\u3067\u8db3\u3057\u3042\u3052\u308b\u3002<\/p>\n\u7dda\u7d20<\/h3>\n
\n\\begin{eqnarray}
\nds^2 &=& d\\boldsymbol{x}\\cdot d\\boldsymbol{x}=
\n\\left(dx^{\\mu}\\,\\boldsymbol{e}_{\\mu}\\right)\\cdot
\n\\left(dx^{\\nu}\\,\\boldsymbol{e}_{\\nu}\\right) \\nonumber\\\\
\n&=&\\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu}\\,dx^{\\mu}dx^{\\nu}
\n\\nonumber\\\\
\n&\\equiv& g_{\\mu\\nu}\\,dx^{\\mu}dx^{\\nu}.
\n\\end{eqnarray}<\/p>\n\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/h3>\n
\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u8a08\u91cf<\/h3>\n
\n\\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu} &=& \\eta_{\\mu\\nu} \\\\
\n&=& \\mbox{diag}(-1, 1, 1, 1)
\n\\end{eqnarray}
\n\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\\(\\mbox{diag}\\) \u3068\u306f\u5bfe\u89d2\u6210\u5206\u3092\u3042\u3089\u308f\u3059\u3002\u4e01\u5be7\u306b\u66f8\u304f\u3068<\/p>\n
\n\\eta_{\\mu\\nu} &\\equiv&
\n\\left(\u00a0 \\begin{array}{cccc}
\n\\eta_{00} & \\eta_{01} & \\eta_{02} & \\eta_{03} \\\\
\n\\eta_{10} & \\eta_{11} & \\eta_{12} & \\eta_{13} \\\\
\n\\eta_{20} & \\eta_{21} & \\eta_{22} & \\eta_{23} \\\\
\n\\eta_{30} & \\eta_{31} & \\eta_{32} & \\eta_{33}
\n\\end{array} \\right) \\\\ \\ \\\\
\n&=&
\n\\left(\\begin{array}{cccc}
\n-1 & 0 & 0 & 0 \\\\
\n0 & 1 & 0 & 0 \\\\
\n0 &0 & 1 & 0 \\\\
\n0 & 0 & 0 & 1
\n\\end{array}\\right)
\n\\end{eqnarray}<\/p>\n