{"id":415,"date":"2023-01-01T13:05:14","date_gmt":"2023-01-01T04:05:14","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=415"},"modified":"2025-08-18T21:04:51","modified_gmt":"2025-08-18T12:04:51","slug":"post-415","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=415","title":{"rendered":"\u5171\u5909\u5fae\u5206"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" 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href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AE%E5%AE%9A%E7%BE%A9\" >\u5171\u5909\u5fae\u5206\u306e\u5b9a\u7fa9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AB%E8%A6%81%E8%AB%8B%E3%81%95%E3%82%8C%E3%82%8B%E6%9D%A1%E4%BB%B6\" >\u5171\u5909\u5fae\u5206\u306b\u8981\u8acb\u3055\u308c\u308b\u6761\u4ef6<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E3%82%B9%E3%82%AB%E3%83%A9%E3%83%BCphi%E3%81%AE%E4%BA%8C%E9%9A%8E%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AF%E4%BA%A4%E6%8F%9B%E3%81%99%E3%82%8B\" >\u30b9\u30ab\u30e9\u30fc$\\phi$\u306e\u4e8c\u968e\u5171\u5909\u5fae\u5206\u306f\u4ea4\u63db\u3059\u308b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E3%82%B9%E3%82%AB%E3%83%A9%E3%83%BCphi%E3%81%AE%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AF%E5%81%8F%E5%BE%AE%E5%88%86%E3%81%A8%E4%B8%80%E8%87%B4%E3%81%99%E3%82%8B\" >\u30b9\u30ab\u30e9\u30fc$\\phi$\u306e\u5171\u5909\u5fae\u5206\u306f\u504f\u5fae\u5206\u3068\u4e00\u81f4\u3059\u308b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E5%B9%B3%E8%A1%8C%E7%A7%BB%E5%8B%95%E3%81%AB%E3%82%88%E3%81%A3%E3%81%A6%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E3%81%AE%E3%83%8E%E3%83%AB%E3%83%A0%E3%81%AF%E4%BF%9D%E5%AD%98%E3%81%99%E3%82%8B\" >\u5e73\u884c\u79fb\u52d5\u306b\u3088\u3063\u3066\u30d9\u30af\u30c8\u30eb\u306e\u30ce\u30eb\u30e0\u306f\u4fdd\u5b58\u3059\u308b<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AE%E6%80%A7%E8%B3%AA\" >\u5171\u5909\u5fae\u5206\u306e\u6027\u8cea<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E3%82%AF%E3%83%AA%E3%82%B9%E3%83%88%E3%83%83%E3%83%95%E3%82%A7%E3%83%AB%E8%A8%98%E5%8F%B7%E3%81%AE%E4%B8%8B%E4%BA%8C%E3%81%A4%E3%81%AE%E6%B7%BB%E5%AD%97%E3%81%AF%E5%AF%BE%E7%A7%B0%E3%81%A7%E3%81%82%E3%82%8B\" >\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u4e0b\u4e8c\u3064\u306e\u6dfb\u5b57\u306f\u5bfe\u79f0\u3067\u3042\u308b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#%E3%82%AF%E3%83%AA%E3%82%B9%E3%83%88%E3%83%83%E3%83%95%E3%82%A7%E3%83%AB%E8%A8%98%E5%8F%B7Gammamu_nurho%E3%82%92%E8%A8%88%E9%87%8Fg_munu%E3%81%A7%E8%A1%A8%E3%81%9B%E3%82%8B\" >\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7$\\Gamma^{\\mu}_{\\nu\\rho}$\u3092\u8a08\u91cf$g_{\\mu\\nu}$\u3067\u8868\u305b\u308b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/daba-no-heya.com\/?p=415\/#nabla_rhog_munu0\" >$\\nabla_{\\rho}g_{\\mu\\nu}=0$<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AE%E5%AE%9A%E7%BE%A9\"><\/span>\u5171\u5909\u5fae\u5206\u306e\u5b9a\u7fa9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u7269\u7406\u65b9\u7a0b\u5f0f\u3092\u66f8\u304d\u4e0b\u3059\u305f\u3081\u306b\u306f\u7269\u7406\u91cf\u306e\u5fae\u5206\u3092\u3068\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u7269\u7406\u91cf\u3092\u30c6\u30f3\u30bd\u30eb\u3068\u3057\u3066\u8868\u3059\u3068\u304d\u306b\u3001\u305d\u308c\u3092\u5fae\u5206\u3057\u305f\u91cf\u3082\u30c6\u30f3\u30bd\u30eb\u306b\u306a\u3063\u3066\u3044\u306a\u3044\u3068\u6271\u3044\u306b\u304f\u3044\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u307e\u305a\u306f\u3001\u30d9\u30af\u30c8\u30eb\u3092\u5358\u7d14\u306b\u504f\u5fae\u5206\u3059\u308b\u5834\u5408\u3092\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    \\frac{\\partial V_{\\nu}(x)}{\\partial x^{\\mu}}&amp;=\\frac{\\partial x&#8217;^{\\rho}}{\\partial x^{\\mu}}\\frac{\\partial}{\\partial x&#8217;^{\\rho}}(\\frac{\\partial x&#8217;^{\\sigma}}{\\partial x^{\\nu}}V&#8217;_{\\sigma}(x&#8217;)) \\\\\n    &amp;=\\frac{\\partial x&#8217;^{\\rho}}{\\partial x^{\\mu}}\\frac{\\partial x&#8217;^{\\sigma}}{\\partial x^{\\nu}}\\frac{\\partial V&#8217;_{\\sigma}(x&#8217;)}{\\partial x&#8217;^{\\rho}}+\\frac{\\partial x&#8217;^{\\rho}}{\\partial x^{\\mu}}V&#8217;_{\\sigma}(x&#8217;)\\frac{\\partial}{\\partial x&#8217;^{\\rho}}(\\frac{\\partial x&#8217;^{\\sigma}}{\\partial x^{\\nu}})\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u4f59\u8a08\u306a\u7b2c\u4e8c\u9805\u304c\u51fa\u3066\u304f\u308b\u306e\u3067\u3001\u30c6\u30f3\u30bd\u30eb\u306e\u5909\u63db\u5247\u3092\u6e80\u305f\u3055\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30d9\u30af\u30c8\u30eb\u306e\u504f\u5fae\u5206\u3092\u5358\u7d14\u306b\u3068\u308b\u3068\u3001\u7570\u306a\u308b\u5730\u70b9\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5024\u540c\u58eb\u306e\u5dee\u3092\u3068\u308b\u3053\u3068\u306b\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\frac{\\partial}{\\partial x^{\\mu}}V_{\\nu}(x)=\\lim_{\\Delta x^{\\mu} \\to 0}\\frac{V_{\\nu}(x+\\Delta x)-V_{\\nu}(x)}{\\Delta x^{\\mu}}\n$$<\/p>\n\n\n\n<p>\u7570\u306a\u308b\u5730\u70b9\u306e\u30d9\u30af\u30c8\u30eb\u91cf\u306f\u5ea7\u6a19\u5909\u63db\u306b\u5bfe\u3057\u3066\u7570\u306a\u3063\u305f\u5909\u63db\u3092\u3059\u308b\u306e\u3067\u3001\u305d\u308c\u3089\u306e\u5dee\u3067\u3042\u308b\u504f\u5fae\u5206\u5024\u3082\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u632f\u308b\u821e\u308f\u306a\u304f\u306a\u308a\u3001\u3053\u306e\u5206\u306e\u305a\u308c\u304c\u4f59\u5206\u306a\u9805\u306e\u539f\u56e0\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u306e\u305a\u308c\u3092\u53d6\u308a\u9664\u304f\u305f\u3081\u306b\u306f\u3001\u5730\u70b9$x^{\\mu}$\u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u3092\u5730\u70b9$x^{\\mu}+\\Delta x^{\\mu}$\u307e\u3067\u5e73\u884c\u79fb\u52d5\u3057\u3066\u304b\u3089\u8a08\u7b97\u3092\u884c\u3048\u3070\u3088\u3055\u305d\u3046\u3067\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\nabla_{\\mu}V_{\\nu}(x) \\equiv \\lim_{\\Delta x^{\\mu} \\to 0}\\frac{V_{\\nu}(x+\\Delta x)-V_{\/\/\\nu}(x+\\Delta x)}{\\Delta x^{\\mu}}\n$$<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001$V_{\/\/\\nu}(x+\\Delta x) \\equiv V_{\\nu}(x)+\\Gamma^{\\rho}_{\\nu\\sigma}(x)V_{\\rho}(x)\\Delta x^{\\sigma}$\u3068\u8868\u3055\u308c\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002<br>\u3053\u306e\u3068\u304d\u3001<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    \\nabla_{\\mu}V_{\\nu}(x)&amp;=\\lim_{\\Delta x^{\\mu} \\to 0}\\frac{V_{\\nu}(x+\\Delta x)-V_{\\nu}(x)-\\Gamma^{\\rho}_{\\nu\\mu}(x)V_{\\rho}(x)\\Delta x^{\\mu}}{\\Delta x^{\\mu}} \\\\\n    &amp;=\\lim_{\\Delta x^{\\mu} \\to 0}\\frac{V_{\\nu}(x+\\Delta x)-V_{\\nu}(x)}{\\Delta x^{\\mu}}-\\Gamma^{\\rho}_{\\nu\\mu}(x)V_{\\rho}(x) \\\\\n    &amp;=\\frac{\\partial V_{\\nu}(x)}{\\partial x^{\\mu}}-\\Gamma^{\\rho}_{\\nu\\mu}(x)V_{\\rho}(x)\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<br>\u5171\u901a\u3057\u3066\u73fe\u308c\u308b$(x)$\u306f\u5197\u9577\u306a\u306e\u3067\u7701\u7565\u3057\u3066\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\nabla_{\\mu}V_{\\nu}=\\frac{\\partial V_{\\nu}}{\\partial x^{\\mu}}-\\Gamma^{\\rho}_{\\nu\\sigma}V_{\\rho}\n$$<\/p>\n\n\n\n<p>\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b$\\nabla_{\\mu}$\u3092<strong>\u5171\u5909\u5fae\u5206<\/strong>(covariant derivative)\u3001$\\Gamma^{\\rho}_{\\nu\\sigma}$\u3092<strong>\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/strong>(Christoffel symbols)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AB%E8%A6%81%E8%AB%8B%E3%81%95%E3%82%8C%E3%82%8B%E6%9D%A1%E4%BB%B6\"><\/span>\u5171\u5909\u5fae\u5206\u306b\u8981\u8acb\u3055\u308c\u308b\u6761\u4ef6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u5171\u5909\u5fae\u5206$\\nabla_{\\mu}$\u3068\u5e73\u884c\u79fb\u52d5\u3057\u305f\u30d9\u30af\u30c8\u30eb$V^{\\mu}_{\/\/}$\u306b\u3064\u3044\u3066\u3001\u4ee5\u4e0b\u3092\u6e80\u305f\u3059\u3053\u3068\u304c\u8981\u8acb\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u30b9\u30ab\u30e9\u30fc$\\phi$\u306e\u4e8c\u968e\u5171\u5909\u5fae\u5206\u306f\u4ea4\u63db\u3059\u308b<\/li>\n\n\n\n<li>\u30b9\u30ab\u30e9\u30fc$\\phi$\u306e\u5171\u5909\u5fae\u5206\u306f\u504f\u5fae\u5206\u3068\u4e00\u81f4\u3059\u308b<\/li>\n\n\n\n<li>\u5e73\u884c\u79fb\u52d5\u306b\u3088\u3063\u3066\u30d9\u30af\u30c8\u30eb\u306e\u30ce\u30eb\u30e0\u306f\u4fdd\u5b58\u3059\u308b<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%82%B9%E3%82%AB%E3%83%A9%E3%83%BCphi%E3%81%AE%E4%BA%8C%E9%9A%8E%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AF%E4%BA%A4%E6%8F%9B%E3%81%99%E3%82%8B\"><\/span>\u30b9\u30ab\u30e9\u30fc$\\phi$\u306e\u4e8c\u968e\u5171\u5909\u5fae\u5206\u306f\u4ea4\u63db\u3059\u308b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u30b9\u30ab\u30e9\u30fc$\\phi$\u306b\u3064\u3044\u3066\u3001$(\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu})\\phi=0$\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u5f8c\u3067\u307e\u305f\u7d39\u4ecb\u3059\u308b\u3068\u601d\u3044\u307e\u3059\u304c\u3001$\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu}$\u306f\u6642\u7a7a\u306e\u6b6a\u307f\u3092\u8868\u3059\u91cf\u3067\u3059\u3002<br>\u6b6a\u3093\u3060\u6642\u7a7a\u3067\u30d9\u30af\u30c8\u30eb\u3092\u9069\u5f53\u306b\u5e73\u884c\u79fb\u52d5\u3057\u3066\u5143\u306e\u5834\u6240\u306b\u623b\u3059\u3068\u3001\u5143\u306e\u30d9\u30af\u30c8\u30eb\u3068\u306e\u305a\u308c\u304c\u751f\u3058\u307e\u3059\u3002<br>\u3053\u306e\u305a\u308c\u3092\u8868\u3059\u91cf\u304c$\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu}$\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u8907\u6570\u6210\u5206\u3092\u3082\u305f\u306a\u3044\u95a2\u6570$\\phi(x)$\u306f\u30d9\u30af\u30c8\u30eb\u3068\u7570\u306a\u308a\u3001\u6642\u7a7a\u304c\u6b6a\u3093\u3067\u3044\u3066\u3082\u3001\u5e73\u884c\u79fb\u52d5\u306b\u3088\u3063\u3066\u5143\u306e\u5024\u304b\u3089\u305a\u308c\u308b\u3053\u3068\u306f\u3042\u308a\u307e\u305b\u3093\u3002<br>\u3053\u306e\u3088\u3046\u306a\u5024\u3092<strong>\u30b9\u30ab\u30e9\u30fc<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002<br>\u30b9\u30ab\u30e9\u30fc\u3092\u5e73\u884c\u79fb\u52d5\u3057\u3066\u3082\u5024\u304c\u305a\u308c\u308b\u3053\u3068\u306f\u306a\u3044\u306e\u3067\u3001\u524d\u8ff0\u306e\u3068\u304a\u308a\u3001$(\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu})\\phi=0$\u3068\u306a\u308b\u3053\u3068\u304c\u8981\u8acb\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%82%B9%E3%82%AB%E3%83%A9%E3%83%BCphi%E3%81%AE%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AF%E5%81%8F%E5%BE%AE%E5%88%86%E3%81%A8%E4%B8%80%E8%87%B4%E3%81%99%E3%82%8B\"><\/span>\u30b9\u30ab\u30e9\u30fc$\\phi$\u306e\u5171\u5909\u5fae\u5206\u306f\u504f\u5fae\u5206\u3068\u4e00\u81f4\u3059\u308b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u30b9\u30ab\u30e9\u30fc$\\phi$\u306e\u5171\u5909\u5fae\u5206\u306f\u504f\u5fae\u5206\u3068\u4e00\u81f4\u3059\u308b\u3053\u3068\u304c\u8981\u8acb\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\nabla_{\\mu}\\phi=\\frac{\\partial \\phi}{\\partial x^{\\mu}}\n$$<\/p>\n\n\n\n<p>\u30b9\u30ab\u30e9\u30fc\u306f\u5e73\u884c\u79fb\u52d5\u306b\u3088\u3063\u3066\u5024\u304c\u305a\u308c\u308b\u3053\u3068\u304c\u306a\u3044\u306e\u3067\u3001\u5358\u7d14\u306b\u504f\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br>\u3057\u305f\u304c\u3063\u3066\u3001\u30b9\u30ab\u30e9\u30fc\u3092\u5171\u5909\u5fae\u5206\u3057\u305f\u3082\u306e\u304c\u30b9\u30ab\u30e9\u30fc\u3092\u5358\u7d14\u306b\u504f\u5fae\u5206\u3057\u305f\u3082\u306e\u306b\u4e00\u81f4\u3059\u308b\u3088\u3046\u306b\u3059\u308b\u306e\u304c\u81ea\u7136\u3067\u306f\u306a\u3044\u3067\u3057\u3087\u3046\u304b\u3002(\u8981\u691c\u8a3c)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%B9%B3%E8%A1%8C%E7%A7%BB%E5%8B%95%E3%81%AB%E3%82%88%E3%81%A3%E3%81%A6%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E3%81%AE%E3%83%8E%E3%83%AB%E3%83%A0%E3%81%AF%E4%BF%9D%E5%AD%98%E3%81%99%E3%82%8B\"><\/span>\u5e73\u884c\u79fb\u52d5\u306b\u3088\u3063\u3066\u30d9\u30af\u30c8\u30eb\u306e\u30ce\u30eb\u30e0\u306f\u4fdd\u5b58\u3059\u308b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3057\u3066\u3082\u30ce\u30eb\u30e0\u306f\u4fdd\u5b58\u3059\u308b\u3053\u3068\u304c\u8981\u8acb\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\ng_{\\mu\\nu}(x+\\Delta x)V^{\\mu}_{\/\/}(x+\\Delta x)V^{\\nu}_{\/\/}(x+\\Delta x)=g_{\\mu\\nu}(x)V^{\\mu}(x)V^{\\nu}(x)\n$$<\/p>\n\n\n\n<p>\u3053\u308c\u306b\u3064\u3044\u3066\u306f\u3001\u3053\u3046\u3044\u3046\u3075\u3046\u306b\u5b9a\u3081\u3066\u304a\u304f\u3068\u8272\u3005\u90fd\u5408\u304c\u3044\u3044\u304b\u3089\u3053\u3046\u3059\u308b\u3001\u3068\u3044\u3046\u611f\u3058\u306a\u6c17\u304c\u3057\u307e\u3059\u3002(\u8981\u691c\u8a3c)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%85%B1%E5%A4%89%E5%BE%AE%E5%88%86%E3%81%AE%E6%80%A7%E8%B3%AA\"><\/span>\u5171\u5909\u5fae\u5206\u306e\u6027\u8cea<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u5171\u5909\u5fae\u5206\u306b\u8981\u8acb\u3055\u308c\u308b\u6761\u4ef6\u3088\u308a\u3001\u4ee5\u4e0b\u306e\u6027\u8cea\u304c\u5c0e\u304b\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u4e0b\u4e8c\u3064\u306e\u6dfb\u5b57\u306f\u5bfe\u79f0\u3067\u3042\u308b<\/li>\n\n\n\n<li>\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7$\\Gamma^{\\mu}_{\\nu\\rho}$\u3092\u8a08\u91cf$g_{\\mu\\nu}$\u3067\u8868\u305b\u308b<\/li>\n\n\n\n<li>$\\nabla_{\\rho}g_{\\mu\\nu}=0$<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%82%AF%E3%83%AA%E3%82%B9%E3%83%88%E3%83%83%E3%83%95%E3%82%A7%E3%83%AB%E8%A8%98%E5%8F%B7%E3%81%AE%E4%B8%8B%E4%BA%8C%E3%81%A4%E3%81%AE%E6%B7%BB%E5%AD%97%E3%81%AF%E5%AF%BE%E7%A7%B0%E3%81%A7%E3%81%82%E3%82%8B\"><\/span>\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u4e0b\u4e8c\u3064\u306e\u6dfb\u5b57\u306f\u5bfe\u79f0\u3067\u3042\u308b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>$$\n\\begin{align*}\n    (\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu})\\phi&amp;=(\\partial_{\\mu}\\partial_{\\nu}-\\partial_{\\nu}\\partial_{\\mu})\\phi-\\Gamma^{\\rho}_{\\nu\\mu}\\nabla_{\\rho}\\phi+\\Gamma^{\\rho}_{\\mu\\nu}\\nabla_{\\rho}\\phi \\\\\n    &amp;=(-\\Gamma^{\\rho}_{\\nu\\mu}+\\Gamma^{\\rho}_{\\mu\\nu})\\nabla_{\\rho}\\phi\n\\end{align*}\n$$<\/p>\n\n\n\n<p>$(\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu})\\phi=0$\u306a\u306e\u3067\u3001$\\Gamma^{\\rho}_{\\mu\\nu}=\\Gamma^{\\rho}_{\\nu\\mu}$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%82%AF%E3%83%AA%E3%82%B9%E3%83%88%E3%83%83%E3%83%95%E3%82%A7%E3%83%AB%E8%A8%98%E5%8F%B7Gammamu_nurho%E3%82%92%E8%A8%88%E9%87%8Fg_munu%E3%81%A7%E8%A1%A8%E3%81%9B%E3%82%8B\"><\/span>\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7$\\Gamma^{\\mu}_{\\nu\\rho}$\u3092\u8a08\u91cf$g_{\\mu\\nu}$\u3067\u8868\u305b\u308b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>$$\n\\begin{align*}\n    &amp;g_{\\mu\\nu}(x+\\Delta x)V^{\\mu}_{\/\/}(x+\\Delta x)V^{\\nu}_{\/\/}(x+\\Delta x) \\\\\n    &amp;=g_{\\mu\\nu}(x+\\Delta x)(V^{\\mu}(x)+\\Gamma^{\\mu}_{\\alpha\\beta}(x)V^{\\alpha}(x)\\Delta x^{\\beta}) \\\\\n    &amp;\\qquad (V^{\\nu}(x)+\\Gamma^{\\nu}_{\\gamma\\delta}(x)V^{\\gamma}(x)\\Delta x^{\\delta}) \\\\\n    &amp;=(g_{\\mu\\nu}(x)+\\partial_{\\rho}g_{\\mu\\nu}(x)\\Delta x^{\\rho}) \\\\\n    &amp;\\qquad (V^{\\mu}(x)+\\Gamma^{\\mu}_{\\alpha\\beta}(x)V^{\\alpha}(x)\\Delta x^{\\beta})(V^{\\nu}(x)+\\Gamma^{\\nu}_{\\gamma\\delta}(x)V^{\\gamma}(x)\\Delta x^{\\delta}) \\\\\n    &amp;=(g_{\\mu\\nu}(x)+\\partial_{\\rho}g_{\\mu\\nu}(x)\\Delta x^{\\rho})V^{\\mu}(x)V^{\\nu}(x) \\\\\n    &amp;\\qquad +g_{\\mu\\nu}(x)V^{\\mu}(x)\\Gamma^{\\nu}_{\\gamma\\delta}V^{\\gamma}(x)\\Delta x^{\\delta} \\\\\n    &amp;\\qquad +g_{\\mu\\nu}(x)V^{\\nu}(x)\\Gamma^{\\mu}_{\\alpha\\beta}V^{\\alpha}(x)\\Delta x^{\\beta}+\\mathcal{O}(\\Delta x^2)\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u3053\u308c\u304c$g_{\\mu\\nu}(x)V^{\\mu}(x)V^{\\nu}(x)$\u306b\u7b49\u3057\u3044\u306e\u3067\u3001$\\Delta x^{\\mu}$\u306b\u3064\u3044\u3066\u4e00\u6b21\u306e\u90e8\u5206\u3092\u53d6\u308a\u51fa\u3059\u3068\u3001<\/p>\n\n\n\n<p>$$\n\\partial_{\\rho}g_{\\mu\\nu}-g_{\\mu\\sigma}\\Gamma^{\\sigma}_{\\nu\\rho}-g_{\\nu\\sigma}\\Gamma^{\\sigma}_{\\mu\\rho}=0\n$$<\/p>\n\n\n\n<p>\u3068\u306a\u3063\u3066\u3044\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001$\\Gamma_{\\rho,\\mu\\nu} \\equiv g_{\\rho\\sigma}\\Gamma^{\\sigma}_{\\mu\\nu}$\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002<br>\u5148\u306e\u5f0f\u306e\u6dfb\u5b57\u3092$\\mu,\\nu,\\rho$\u306b\u3064\u3044\u3066\u5de1\u56de\u7f6e\u63db\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u4e09\u3064\u306e\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$ \\left\\{ \\begin{aligned} \\partial_{\\rho}g_{\\mu\\nu}&amp;=\\Gamma_{\\mu,\\nu\\rho}+\\Gamma_{\\nu,\\mu\\rho} \\\\ \\partial_{\\mu}g_{\\nu\\rho}&amp;=\\Gamma_{\\nu,\\rho\\mu}+\\Gamma_{\\rho,\\nu\\mu} \\\\ \\partial_{\\nu}g_{\\rho\\mu}&amp;=\\Gamma_{\\rho,\\mu\\nu}+\\Gamma_{\\mu,\\rho\\nu} \\end{aligned} \\right. $$<\/p>\n\n\n\n<p>$\\partial_{\\rho}g_{\\mu\\nu}+\\partial_{\\mu}g_{\\nu\\rho}-\\partial_{\\nu}g_{\\rho\\mu}=2\\Gamma_{\\nu,\\mu\\rho}$\u3088\u308a\u3001<\/p>\n\n\n\n<p>$$\n\\Gamma_{\\nu,\\mu\\rho}=\\frac{1}{2}(\\partial_{\\mu}g_{\\nu\\rho}+\\partial_{\\rho}g_{\\mu\\nu}-\\partial_{\\nu}g_{\\mu\\rho})\n$$<\/p>\n\n\n\n<p>$\\Gamma_{\\nu,\\mu\\rho}=g_{\\nu\\sigma}\\Gamma^{\\sigma}_{\\mu\\rho}$\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$\ng_{\\nu\\sigma}\\Gamma^{\\sigma}_{\\mu\\rho}=\\frac{1}{2}(\\partial_{\\mu}g_{\\nu\\rho}+\\partial_{\\rho}g_{\\mu\\nu}-\\partial_{\\nu}g_{\\mu\\rho})\n$$<\/p>\n\n\n\n<p>$g_{\\nu\\sigma}$\u306e\u9006\u884c\u5217$g^{\\sigma\\nu}$\u3092\u5de6\u5074\u304b\u3089\u304b\u3051\u3066\u3001<\/p>\n\n\n\n<p>$$\n\\Gamma^{\\sigma}_{\\mu\\rho}=\\frac{1}{2}g^{\\sigma\\nu}(\\partial_{\\mu}g_{\\nu\\rho}+\\partial_{\\rho}g_{\\mu\\nu}-\\partial_{\\nu}g_{\\mu\\rho})\n$$<\/p>\n\n\n\n<p><a href=\"https:\/\/www2.yukawa.kyoto-u.ac.jp\/~norihiro.tanahashi\/pdf\/relativity\/note_relativity_2019.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">\u76f8\u5bfe\u6027\u7406\u8ad6\u6982\u8aac<\/a>\u306e\u8a18\u8f09\u306b\u5408\u308f\u305b\u3066\u6dfb\u5b57\u3092\u5909\u66f4\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\Gamma^{\\rho}_{\\mu\\nu}=\\frac{1}{2}g^{\\rho\\sigma}(\\partial_{\\mu}g_{\\nu\\sigma}+\\partial_{\\nu}g_{\\mu\\sigma}-\\partial_{\\sigma}g_{\\mu\\nu})\n$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"nabla_rhog_munu0\"><\/span>$\\nabla_{\\rho}g_{\\mu\\nu}=0$<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u307e\u305a\u3001\u5171\u5909\u5fae\u5206\u306e\u5b9a\u7fa9\u3092(0,2)-\u30c6\u30f3\u30bd\u30eb\u306b\u62e1\u5f35\u3057\u307e\u3059\u3002<br>\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb$V_{\\mu}$\u3068$W_{\\mu}$\u3092\u8003\u3048\u308b\u3068\u3001\u3053\u308c\u3089\u306e\u7a4d\u306f(0,2)-\u30c6\u30f3\u30bd\u30eb\u306b\u306a\u308b\u306e\u3067\u3001$\\nabla_{\\rho}V_{\\mu}W_{\\nu}$\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    \\nabla_{\\rho}V_{\\mu}W_{\\nu}&amp;=V_{\\mu}(\\nabla_{\\rho}W_{\\nu})+(\\nabla_{\\rho}V_{\\mu})W_{\\nu} \\\\\n    &amp;=V_{\\mu}(\\frac{\\partial W_{\\nu}}{\\partial x^{\\rho}}-\\Gamma^{\\sigma}_{\\nu\\rho}W_{\\sigma})+(\\frac{\\partial V_{\\mu}}{\\partial x^{\\rho}}-\\Gamma^{\\sigma}_{\\mu\\rho}V_{\\sigma})W_{\\nu} \\\\\n    &amp;=V_{\\mu}\\frac{\\partial W_{\\nu}}{\\partial x^{\\rho}}+\\frac{\\partial V_{\\mu}}{\\partial x^{\\rho}}W_{\\nu}-\\Gamma^{\\sigma}_{\\nu\\rho}V_{\\mu}W_{\\sigma}-\\Gamma^{\\sigma}_{\\nu\\rho}V_{\\sigma}W_{\\nu} \\\\\n    &amp;=\\frac{\\partial (V_{\\mu}W_{\\nu})}{\\partial x^{\\rho}}-\\Gamma^{\\sigma}_{\\nu\\rho}V_{\\mu}W_{\\sigma}-\\Gamma^{\\sigma}_{\\mu\\rho}V_{\\sigma}W_{\\nu}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u3088\u308a\u3001\u30c6\u30f3\u30bd\u30eb$T_{\\mu\\nu}$\u306e\u5171\u5909\u5fae\u5206\u306f\u3001<\/p>\n\n\n\n<p>$$\n\\nabla_{\\rho}T_{\\mu\\nu}=\\frac{\\partial T_{\\mu\\nu}}{\\partial x^{\\rho}}-\\Gamma^{\\sigma}_{\\nu\\rho}T_{\\mu\\sigma}-\\Gamma^{\\sigma}_{\\mu\\rho}T_{\\sigma\\nu}\n$$<\/p>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u8a08\u91cf$g_{\\mu\\nu}$\u306e\u5171\u5909\u5fae\u5206\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\nabla_{\\rho}g_{\\mu\\nu}=\\partial_{\\rho}g_{\\mu\\nu}-\\Gamma^{\\sigma}_{\\nu\\rho}g_{\\mu\\sigma}-\\Gamma^{\\sigma}_{\\mu\\rho}g_{\\sigma\\nu}\n$$<\/p>\n\n\n\n<p>\u3053\u306e\u5f0f\u306b<\/p>\n\n\n\n<p>$$ \\left\\{ \\begin{aligned} \\Gamma^{\\sigma}_{\\nu\\rho}&amp;=\\frac{1}{2}g^{\\sigma\\alpha}(\\partial_{\\nu}g_{\\rho\\alpha}+\\partial_{\\rho}g_{\\nu\\alpha}-\\partial_{\\alpha}g_{\\nu\\rho}) \\\\ \\Gamma^{\\sigma}_{\\mu\\rho}&amp;=\\frac{1}{2}g^{\\sigma\\beta}(\\partial_{\\mu}g_{\\rho\\beta}+\\partial_{\\rho}g_{\\mu\\beta}-\\partial_{\\beta}g_{\\mu\\rho}) \\\\ \\end{aligned} \\right. $$<\/p>\n\n\n\n<p>\u3092\u4ee3\u5165\u3057\u3066\u3001<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    \\nabla_{\\rho}g_{\\mu\\nu}&amp;=\\partial_{\\rho}g_{\\mu\\nu}-\\frac{1}{2}g^{\\sigma\\alpha}(\\partial_{\\nu}g_{\\rho\\alpha}+\\partial_{\\rho}g_{\\nu\\alpha}-\\partial_{\\alpha}g_{\\nu\\rho})g_{\\mu\\sigma} \\\\\n    &amp;\\qquad -\\frac{1}{2}g^{\\sigma\\beta}(\\partial_{\\mu}g_{\\rho\\beta}+\\partial_{\\rho}g_{\\mu\\beta}-\\partial_{\\beta}g_{\\mu\\rho})g_{\\sigma\\nu} \\\\\n    &amp;=\\partial_{\\rho}g_{\\mu\\nu}-\\frac{1}{2}\\delta^{\\alpha}_{\\;\\mu}(\\partial_{\\nu}g_{\\rho\\alpha}+\\partial_{\\rho}g_{\\nu\\alpha}-\\partial_{\\alpha}g_{\\nu\\rho}) \\\\\n    &amp;\\qquad -\\frac{1}{2}\\delta^{\\beta}_{\\;\\nu}(\\partial_{\\mu}g_{\\rho\\beta}+\\partial_{\\rho}g_{\\mu\\beta}-\\partial_{\\beta}g_{\\mu\\rho}) \\\\\n    &amp;=\\partial_{\\rho}g_{\\mu\\nu}-\\frac{1}{2}(\\partial_{\\nu}g_{\\rho\\mu}+\\partial_{\\rho}g_{\\nu\\mu}-\\partial_{\\mu}g_{\\nu\\rho}) \\\\\n    &amp;\\qquad -\\frac{1}{2}(\\partial_{\\mu}g_{\\rho\\nu}+\\partial_{\\rho}g_{\\mu\\nu}-\\partial_{\\nu}g_{\\mu\\rho}) \\\\\n    &amp;=0\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u3057\u305f\u304c\u3063\u3066\u3001$\\nabla_{\\rho}g_{\\mu\\nu}=0$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5171\u5909\u5fae\u5206\u306e\u5b9a\u7fa9 \u7269\u7406\u65b9\u7a0b\u5f0f\u3092\u66f8\u304d\u4e0b\u3059\u305f\u3081\u306b\u306f\u7269\u7406\u91cf\u306e\u5fae\u5206\u3092\u3068\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u7269\u7406\u91cf\u3092\u30c6\u30f3\u30bd\u30eb\u3068\u3057\u3066\u8868\u3059\u3068\u304d\u306b\u3001\u305d\u308c\u3092\u5fae\u5206\u3057\u305f\u91cf\u3082\u30c6\u30f3\u30bd\u30eb\u306b\u306a\u3063\u3066\u3044\u306a\u3044\u3068\u6271\u3044\u306b\u304f\u3044\u3067\u3059\u3002 \u307e\u305a\u306f\u3001\u30d9\u30af\u30c8\u30eb\u3092\u5358\u7d14\u306b\u504f\u5fae\u5206\u3059\u308b\u5834\u5408\u3092\u8003\u3048\u3066 [&hellip;]<\/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-415","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\/415","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=415"}],"version-history":[{"count":26,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/415\/revisions"}],"predecessor-version":[{"id":1606,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/415\/revisions\/1606"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}