{"id":383,"date":"2022-12-30T14:31:54","date_gmt":"2022-12-30T05:31:54","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=383"},"modified":"2025-08-18T21:01:22","modified_gmt":"2025-08-18T12:01:22","slug":"post-383","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=383","title":{"rendered":"\u53cd\u5909\u30d9\u30af\u30c8\u30eb\u3068\u5171\u5909\u30d9\u30af\u30c8\u30eb"},"content":{"rendered":"\n

\u4e8c\u3064\u306e\u57fa\u5e95$e^{\\mu},e_{\\mu}$\u306b\u3064\u3044\u3066$\\eta_{\\mu\\nu}e^{\\mu}e_{\\nu}=\\delta^{\\mu}_{\\;\\nu}$\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u3001$e^{\\mu}$\u3092$e_{\\mu}$\u306e\u53cc\u5bfe\u57fa\u5e95<\/strong>(dual basis)\u3068\u8a00\u3044\u307e\u3059\u3002
\u6b63\u898f\u76f4\u4ea4\u57fa\u5e95\u306e\u5834\u5408\u306e\u53cc\u5bfe\u57fa\u5e95\u306f\u7c21\u5358\u306b\u6c42\u3081\u3089\u308c\u3066\u3001$(e^0,e^1,e^2,e^3)=(-e_0,e_1,e_2,e_3)$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

$$\n\\begin{gather*}\n \\eta_{\\mu\\nu}=\n \\begin{pmatrix}\n -1 & & & \\\\\n & 1 & & \\\\\n & & 1 & \\\\\n & & & 1\n \\end{pmatrix} \\\\\n e_0=\n \\begin{pmatrix}\n 1 \\\\\n 0 \\\\\n 0 \\\\\n 0\n \\end{pmatrix} \\\\\n e_1=\n \\begin{pmatrix}\n 0 \\\\\n 1 \\\\\n 0 \\\\\n 0\n \\end{pmatrix} \\\\\n e_2=\n \\begin{pmatrix}\n 0 \\\\\n 0 \\\\\n 1 \\\\\n 0\n \\end{pmatrix} \\\\\n e_3=\n \\begin{pmatrix}\n 0 \\\\\n 0 \\\\\n 0 \\\\\n 1\n \\end{pmatrix}\n\\end{gather*}\n$$<\/p>\n\n\n\n

\u306b\u5bfe\u3057\u3066$\\eta_{\\mu\\nu}e^{\\mu}e_{\\nu}=\\delta^{\\mu}_{\\;\\nu}$\u3068\u306a\u308b$e^{\\mu}$\u3092\u6c42\u3081\u308b\u3060\u3051\u3067\u3059\u3002<\/p>\n\n\n\n

\u5ea7\u6a19\u5909\u63db$x’^{\\mu}=x’^{\\mu}(x^{\\nu})$\u306b\u5bfe\u3057\u3066\u3001\u5ea7\u6a19\u57fa\u5e95\u30d9\u30af\u30c8\u30eb$\\frac{\\partial}{\\partial x^{\\mu}}$\u3068\u305d\u308c\u306b\u5bfe\u3057\u3066$dx^{\\mu}\\cdot\\frac{\\partial}{\\partial x^{\\nu}}=\\frac{\\partial x^{\\mu}}{\\partial x^{\\nu}}=\\delta^{\\mu}_{\\;\\nu}$\u3092\u6e80\u305f\u3059\u53cc\u5bfe\u57fa\u5e95$dx^{\\mu}$\u306f\u3001<\/p>\n\n\n\n

$$ \\left\\{ \\begin{aligned} \\frac{\\partial}{\\partial x^{\\mu}}&=\\frac{\\partial x’^{\\nu}}{\\partial x^{\\mu}}\\frac{\\partial}{\\partial x’^{\\nu}} \\\\ dx^{\\mu}&=\\frac{\\partial x^{\\mu}}{\\partial x’^{\\nu}}dx’^{\\nu} \\end{aligned} \\right. $$<\/p>\n\n\n\n

\u3068\u8868\u305b\u307e\u3059\u3002<\/p>\n\n\n\n

\u5b9f\u969b\u306b\u3001<\/p>\n\n\n\n

$$\n\\begin{align*}\n dx^{\\mu}\\cdot \\frac{\\partial}{\\partial x^{\\nu}}&=\\frac{\\partial x^{\\mu}}{\\partial x’^{\\nu}}dx’^{\\nu}\\cdot \\frac{\\partial x’^{\\rho}}{\\partial x^{\\nu}}\\frac{\\partial}{\\partial x’^{\\rho}} \\\\\n &=\\frac{\\partial x^{\\mu}}{\\partial x^{\\nu}}\\cdot\\frac{dx’^{\\nu}}{\\partial x’^{\\nu}} \\\\\n &=\\frac{\\partial x^{\\mu}}{\\partial x^{\\nu}} \\\\\n &=\\delta^{\\mu}_{\\;\\nu}\n\\end{align*}\n$$<\/p>\n\n\n\n

\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3042\u308b\u30d9\u30af\u30c8\u30eb$V$\u3092\u3053\u308c\u3089\u306e\u57fa\u5e95\u3092\u7528\u3044\u3066\u8868\u3059\u3068\u3001<\/p>\n\n\n\n

$$\nV=V^{\\mu}(x)\\frac{\\partial}{\\partial x^{\\mu}}=V_{\\mu}(x)dx^{\\mu}\n$$<\/p>\n\n\n\n

\u3088\u308a\u3001<\/p>\n\n\n\n

$$\nV=V^{\\mu}(x)\\frac{\\partial x’^{\\nu}}{\\partial x^{\\mu}}\\frac{\\partial}{\\partial x’^{\\nu}}=V_{\\mu}(x)\\frac{\\partial x^{\\mu}}{\\partial x’^{\\nu}}dx’^{\\nu}\n$$<\/p>\n\n\n\n

\u4e00\u65b9\u3001\u540c\u3058\u3082\u306e\u3092\u65b0\u5ea7\u6a19$x’^{\\mu}$\u306e\u57fa\u5e95\u3067\u8868\u3059\u3068\u3001<\/p>\n\n\n\n

$$\nV=V’^{\\mu}(x’)\\frac{\\partial}{\\partial x’^{\\mu}}=V’_{\\mu}(x’)dx’^{\\mu}\n$$<\/p>\n\n\n\n

\u3053\u308c\u3089\u3092\u6bd4\u8f03\u3059\u308b\u3053\u3068\u3067\u3001\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206$V^{\\mu},V_{\\mu}$\u306e\u5909\u63db\u5247\u306f\u3001<\/p>\n\n\n\n

$$ \\left\\{ \\begin{aligned} V’^{\\mu}(x’)&=\\frac{\\partial x’^{\\mu}}{\\partial x^{\\nu}}V^{\\nu}(x) \\\\ V’_{\\mu}(x’)&=\\frac{\\partial x^{\\nu}}{\\partial x’^{\\mu}}V_{\\nu}(x) \\end{aligned} \\right. $$<\/p>\n\n\n\n

\u3053\u306e\u3088\u3046\u306b\u5909\u63db\u3059\u308b$V^{\\mu}$\u3092\u53cd\u5909\u30d9\u30af\u30c8\u30eb<\/strong>(contravariant vector)\u3001$V_{\\mu}$\u3092\u5171\u5909\u30d9\u30af\u30c8\u30eb<\/strong>(covariant vector)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4e8c\u3064\u306e\u57fa\u5e95$e^{\\mu},e_{\\mu}$\u306b\u3064\u3044\u3066$\\eta_{\\mu\\nu}e^{\\mu}e_{\\nu}=\\delta^{\\mu}_{\\;\\nu}$\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u3001$e^{\\mu}$\u3092$e_{\\mu}$\u306e\u53cc\u5bfe\u57fa\u5e95(d […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,10],"tags":[],"class_list":["post-383","post","type-post","status-publish","format-standard","hentry","category-9","category-10"],"_links":{"self":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/383","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=383"}],"version-history":[{"count":11,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/383\/revisions"}],"predecessor-version":[{"id":1602,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/383\/revisions\/1602"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=383"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=383"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=383"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}