{"id":446,"date":"2023-01-01T13:05:37","date_gmt":"2023-01-01T04:05:37","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=446"},"modified":"2024-04-27T08:52:38","modified_gmt":"2024-04-26T23:52:38","slug":"post-446","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=446","title":{"rendered":"\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb"},"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\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/daba-no-heya.com\/?p=446\/#%E7%9B%AE%E7%9A%84\" >\u76ee\u7684<\/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=446\/#%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E3%82%92%E5%B9%B3%E8%A1%8C%E7%A7%BB%E5%8B%95%E3%81%95%E3%81%9B%E3%81%9F%E5%A0%B4%E5%90%88%E3%81%AE%E5%B7%AE%E5%88%86%E3%82%92%E8%A8%88%E7%AE%97%E3%81%99%E3%82%8B\" >\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3055\u305b\u305f\u5834\u5408\u306e\u5dee\u5206\u3092\u8a08\u7b97\u3059\u308b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/daba-no-heya.com\/?p=446\/#nabla_rhonabla_mu-nabla_munabla_rhoV_nu%E3%82%92%E8%A8%88%E7%AE%97%E3%81%97%E3%81%A6%E3%81%BF%E3%82%8B\" >$(\\nabla_{\\rho}\\nabla_{\\mu}-\\nabla_{\\mu}\\nabla_{\\rho})V_{\\nu}$\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/daba-no-heya.com\/?p=446\/#%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%BE%A9\" >\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5b9a\u7fa9<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%9B%AE%E7%9A%84\"><\/span>\u76ee\u7684<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6\u3067\u306f\u6b6a\u3093\u3060\u6642\u7a7a\u3092\u8003\u3048\u308b\u305f\u3081\u3001\u6642\u7a7a\u304c\u3069\u308c\u304f\u3089\u3044\u6b6a\u3093\u3067\u3044\u308b\u306e\u304b\u8868\u3059\u91cf\u3092\u69cb\u6210\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<br>\u3053\u306e\u305f\u3081\u306b\u306f\u3001\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3055\u305b\u305f\u3068\u304d\u306b\u751f\u3058\u308b\u5411\u304d\u306e\u305a\u308c\u3092\u8003\u3048\u308c\u3070\u3088\u3055\u305d\u3046\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u305f\u3068\u3048\u3070\u3001\u5730\u7403\u306e\u8d64\u9053\u4e0a\u306e\u3042\u308b\u4e00\u70b9\u3067\u5730\u9762\u306b\u5317\u5411\u304d\u306e\u30d9\u30af\u30c8\u30eb\u304c\u3042\u308b\u3068\u3057\u307e\u3059\u3002<br>\u3053\u306e\u30d9\u30af\u30c8\u30eb\u3092\u8d64\u9053\u306b\u6cbf\u3063\u3066\u5e73\u884c\u79fb\u52d5\u3057\u3066\u5730\u7403\u306e\u88cf\u5074\u307e\u3067\u3082\u3063\u3066\u3044\u3063\u3066\u3082\u5317\u5411\u304d\u306e\u307e\u307e\u3067\u3059\u3002<br>\u3057\u304b\u3057\u3001\u305d\u3053\u304b\u3089\u5317\u6975\u306b\u5411\u304b\u3063\u3066\u5e73\u884c\u79fb\u52d5\u3057\u3066\u305d\u306e\u307e\u307e\u5143\u306e\u5834\u6240\u306b\u623b\u3063\u3066\u304f\u308b\u3068\u3001\u5357\u5411\u304d\u306e\u30d9\u30af\u30c8\u30eb\u306b\u5909\u308f\u3063\u3066\u3044\u307e\u3059\u3002<br>\u3082\u3057\u3082\u5730\u7403\u304cFlat Earther\u306e\u4fe1\u3058\u308b\u3088\u3046\u306a\u6b6a\u307f\u306e\u306a\u3044\u5e73\u9762\u306a\u3089\u3053\u306e\u3088\u3046\u306a\u5909\u5316\u306f\u8d77\u3053\u3089\u306a\u3044\u306e\u3067\u3059\u304c\u3001\u5730\u9762\u304c\u66f2\u304c\u3063\u3066\u3044\u308b\u5834\u5408\u306b\u306f\u5e73\u884c\u79fb\u52d5\u306b\u3088\u3063\u3066\u30d9\u30af\u30c8\u30eb\u306e\u5411\u304d\u306b\u5909\u5316\u304c\u751f\u3058\u307e\u3059\u3002<br>\u3053\u306e\u5909\u5316\u3092\u6570\u5f0f\u3067\u8868\u3059\u3053\u3068\u3092\u8003\u3048\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E3%82%92%E5%B9%B3%E8%A1%8C%E7%A7%BB%E5%8B%95%E3%81%95%E3%81%9B%E3%81%9F%E5%A0%B4%E5%90%88%E3%81%AE%E5%B7%AE%E5%88%86%E3%82%92%E8%A8%88%E7%AE%97%E3%81%99%E3%82%8B\"><\/span>\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3055\u305b\u305f\u5834\u5408\u306e\u5dee\u5206\u3092\u8a08\u7b97\u3059\u308b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3042\u308b\u30d9\u30af\u30c8\u30eb\u3092\u7570\u306a\u308b\u7d4c\u8def\u3067\u540c\u3058\u5834\u6240\u307e\u3067\u5e73\u884c\u79fb\u52d5\u3055\u305b\u3001\u7d4c\u8def\u306e\u9055\u3044\u306b\u3088\u3063\u3066\u751f\u3058\u308b\u30d9\u30af\u30c8\u30eb\u306e\u5dee\u5206\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u307e\u305a\u3001$x_0$\u306b\u3042\u308b\u30d9\u30af\u30c8\u30eb\u3092$x_1$\u307e\u3067\u5e73\u884c\u79fb\u52d5\u3057\u307e\u3059\u3002<br>\u30d9\u30af\u30c8\u30eb\u306e\u5e73\u884c\u79fb\u52d5\u306b\u3064\u3044\u3066\u306f\u3001\u300c\u5171\u5909\u5fae\u5206\u300d\u306e\u3068\u3053\u308d\u3067\u7d39\u4ecb\u3057\u305f\u3082\u306e\u3068\u540c\u3058\u64cd\u4f5c\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\nV_{\/\/\\nu}(x_1)=V_{\\nu}(x_0)+\\Gamma^{\\rho}_{\\nu\\sigma}(x_0)V_{\\rho}(x_0)\\Delta x^{\\sigma}\n$$<\/p>\n\n\n\n<p>\u3053\u306e\u30d9\u30af\u30c8\u30eb\u3092\u3055\u3089\u306b$x_2$\u307e\u3067\u5e73\u884c\u79fb\u52d5\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\nV_{\/\/\\nu}(x_2)=V_{\/\/\\nu}(x_1)+\\Gamma^{\\beta}_{\\nu\\alpha}(x_1)V_{\/\/\\beta}(x_1)\\Delta x^{\\alpha}\n$$<\/p>\n\n\n\n<p>\u6700\u521d\u306e\u5f0f\u3092\u4e8c\u3064\u76ee\u306e\u5f0f\u306b\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    V_{\/\/\\nu}(x_2)&amp;=(V_{\\nu}(x_0)+\\Gamma^{\\rho}_{\\nu\\sigma}(x_0)V_{\\rho}(x_0)\\Delta x^{\\sigma}) \\\\\n    &amp;\\qquad +\\Gamma^{\\beta}_{\\nu\\alpha}(x_1)(V_{\\beta}(x_0)+\\Gamma^{\\rho}_{\\beta\\sigma}(x_0)V_{\\rho}(x_0)\\Delta x^{\\sigma})\\Delta x^{\\alpha}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>$\\Gamma^{\\beta}_{\\nu\\alpha}(x_1)$\u3060\u3051\u304c$x_1$\u3067\u306e\u5024\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3001$x_0$\u3067\u306e\u5024\u3092\u7528\u3044\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\Gamma^{\\beta}_{\\nu\\alpha}(x_1)=\\Gamma^{\\beta}_{\\nu\\alpha}(x_0)+\\frac{\\partial \\Gamma^{\\beta}_{\\nu\\alpha}(x_0)}{\\partial x^{\\gamma}}\\Delta x^{\\gamma}\n$$<\/p>\n\n\n\n<p>\u3053\u308c\u3092\u7528\u3044\u308b\u3068\u3001<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    V_{\/\/\\nu}(x_2)&amp;=(V_{\\nu}(x_0)+\\Gamma^{\\rho}_{\\nu\\sigma}(x_0)V_{\\rho}(x_0)\\Delta x^{\\sigma}) \\\\\n    &amp;\\qquad +(\\Gamma^{\\beta}_{\\nu\\alpha}(x_0)+\\frac{\\partial \\Gamma^{\\beta}_{\\nu\\alpha}(x_0)}{\\partial x^{\\gamma}}\\Delta x^{\\gamma}) \\\\\n    &amp;\\qquad (V_{\\beta}(x_0)+\\Gamma^{\\rho}_{\\beta\\sigma}(x_0)V_{\\rho}(x_0)\\Delta x^{\\sigma})\\Delta x^{\\alpha}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u3059\u3079\u3066$x_0$\u3067\u306e\u5024\u3092\u7528\u3044\u3066\u8868\u73fe\u3067\u304d\u305f\u306e\u3067\u3001\u4ee5\u964d\u306f$(x_0)$\u3068\u3044\u3046\u8868\u8a18\u306f\u7701\u7565\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    V_{\/\/\\nu}(x_2)&amp;=(V_{\\nu}+\\Gamma^{\\rho}_{\\nu\\sigma}V_{\\rho}\\Delta x^{\\sigma}) \\\\\n    &amp;\\qquad +(\\Gamma^{\\beta}_{\\nu\\alpha}+\\frac{\\partial \\Gamma^{\\beta}_{\\nu\\alpha}}{\\partial x^{\\gamma}}\\Delta x^{\\gamma})(V_{\\beta}+\\Gamma^{\\rho}_{\\beta\\sigma}V_{\\rho}\\Delta x^{\\sigma})\\Delta x^{\\alpha} \\\\\n    &amp;=V_{\\nu}+\\Gamma^{\\rho}_{\\nu\\sigma}V_{\\rho}\\Delta x^{\\sigma}+\\Gamma^{\\beta}_{\\nu\\alpha}V_{\\beta}\\Delta x^{\\alpha}+\\Gamma^{\\beta}_{\\nu\\alpha}\\Gamma^{\\rho}_{\\beta\\sigma}V_{\\rho}\\Delta x^{\\sigma}\\Delta x^{\\alpha} \\\\\n    &amp;\\qquad +\\frac{\\partial \\Gamma^{\\beta}_{\\nu\\alpha}}{\\partial x^{\\gamma}}\\Delta x^{\\gamma}V_{\\beta}\\Delta x^{\\alpha}+\\frac{\\partial \\Gamma^{\\beta}_{\\nu\\alpha}}{\\partial x^{\\gamma}}\\Delta x^{\\gamma}\\Gamma^{\\rho}_{\\beta\\sigma}V_{\\rho}\\Delta x^{\\sigma}\\Delta x^{\\alpha}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u6700\u5f8c\u306e\u9805\u306f\u5fae\u5c0f\u91cf\u304c3\u6b21\u306b\u306a\u3063\u3066\u3044\u3066\u5c0f\u3055\u3059\u304e\u308b\u306e\u3067\u6368\u3066\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<br>2\u6b21\u306e\u9805\u3092\u6d88\u3057\u3066\u3057\u307e\u3046\u3068\u5f8c\u306e\u8a08\u7b97\u3067\u4f55\u3082\u6b8b\u3089\u306a\u304f\u306a\u308b\u306e\u3067\u30012\u6b21\u306e\u9805\u306f\u6b8b\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    V_{\/\/\\nu}(x_2)&amp;=V_{\\nu}+\\Gamma^{\\rho}_{\\nu\\sigma}V_{\\rho}\\Delta x^{\\sigma}+\\Gamma^{\\beta}_{\\nu\\alpha}V_{\\beta}\\Delta x^{\\alpha} \\\\\n    &amp;\\qquad +\\Gamma^{\\beta}_{\\nu\\alpha}\\Gamma^{\\rho}_{\\beta\\sigma}V_{\\rho}\\Delta x^{\\sigma}\\Delta x^{\\alpha}+\\frac{\\partial \\Gamma^{\\beta}_{\\nu\\alpha}}{\\partial x^{\\gamma}}\\Delta x^{\\gamma}V_{\\beta}\\Delta x^{\\alpha} \\\\\n    &amp;=V_{\\nu}+\\Gamma^{\\rho}_{\\nu\\sigma}V_{\\rho}\\Delta x^{\\sigma}+\\Gamma^{\\beta}_{\\nu\\alpha}V_{\\beta}\\Delta x^{\\alpha} \\\\\n    &amp;\\qquad +(\\Gamma^{\\beta}_{\\nu\\alpha}\\Gamma^{\\rho}_{\\beta\\sigma}V_{\\rho}+\\frac{\\partial \\Gamma^{\\beta}_{\\nu\\alpha}}{\\partial x^{\\sigma}}V_{\\beta})\\Delta x^{\\sigma}\\Delta x^{\\alpha} \\\\\n    &amp;=V_{\\nu}+\\Gamma^{\\rho}_{\\nu\\sigma}V_{\\rho}\\Delta x^{\\sigma}+\\Gamma^{\\beta}_{\\nu\\alpha}V_{\\beta}\\Delta x^{\\alpha} \\\\\n    &amp;\\qquad +(\\Gamma^{\\beta}_{\\nu\\alpha}\\Gamma^{\\rho}_{\\beta\\sigma}+\\frac{\\partial \\Gamma^{\\rho}_{\\nu\\alpha}}{\\partial x^{\\sigma}})V_{\\rho}\\Delta x^{\\sigma}\\Delta x^{\\alpha}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u3053\u3053\u3067$\\sigma$\u3068$\\alpha$\u3092\u5165\u308c\u66ff\u3048\u308b\u3068\u3001\u5225\u306e\u7d4c\u8def\u3092\u901a\u3063\u3066$x_2$\u307e\u3067\u5e73\u884c\u79fb\u52d5\u3057\u305f\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    V&#8217;_{\/\/\\nu}(x_2)&amp;=V_{\\nu}+\\Gamma^{\\rho}_{\\nu\\alpha}V_{\\rho}\\Delta x^{\\alpha}+\\Gamma^{\\beta}_{\\nu\\sigma}V_{\\beta}\\Delta x^{\\sigma} \\\\\n    &amp;\\qquad +(\\Gamma^{\\beta}_{\\nu\\sigma}\\Gamma^{\\rho}_{\\beta\\alpha}+\\frac{\\partial \\Gamma^{\\rho}_{\\nu\\sigma}}{\\partial x^{\\alpha}})V_{\\rho}\\Delta x^{\\alpha}\\Delta x^{\\sigma}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>$V_{\/\/\\nu}(x_2)$\u3068$V&#8217;_{\/\/\\nu}(x_2)$\u306e\u5dee\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    &amp;V_{\/\/\\nu}(x_2)-V&#8217;_{\/\/\\nu}(x_2) \\\\\n    &amp;=(\\Gamma^{\\beta}_{\\nu\\alpha}\\Gamma^{\\rho}_{\\beta\\sigma}+\\frac{\\partial \\Gamma^{\\rho}_{\\nu\\alpha}}{\\partial x^{\\sigma}})V_{\\rho}\\Delta x^{\\sigma}\\Delta x^{\\alpha}-(\\Gamma^{\\beta}_{\\nu\\sigma}\\Gamma^{\\rho}_{\\beta\\alpha}+\\frac{\\partial \\Gamma^{\\rho}_{\\nu\\sigma}}{\\partial x^{\\alpha}})V_{\\rho}\\Delta x^{\\alpha}\\Delta x^{\\sigma} \\\\\n    &amp;=(\\Gamma^{\\beta}_{\\nu\\alpha}\\Gamma^{\\rho}_{\\beta\\sigma}+\\frac{\\partial \\Gamma^{\\rho}_{\\nu\\alpha}}{\\partial x^{\\sigma}}-\\Gamma^{\\beta}_{\\nu\\sigma}\\Gamma^{\\rho}_{\\beta\\alpha}-\\frac{\\partial \\Gamma^{\\rho}_{\\nu\\sigma}}{\\partial x^{\\alpha}})V_{\\rho}\\Delta x^{\\alpha}\\Delta x^{\\sigma}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u62ec\u5f27\u306e\u4e2d\u306e\u5024\u306f\u5e73\u884c\u79fb\u52d5\u3059\u308b\u7d4c\u8def\u3084\u30d9\u30af\u30c8\u30eb\u306e\u53d6\u308a\u65b9\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3001\u6642\u7a7a\u306e\u6b6a\u307f\u3092\u8868\u3059\u91cf\u3060\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"nabla_rhonabla_mu-nabla_munabla_rhoV_nu%E3%82%92%E8%A8%88%E7%AE%97%E3%81%97%E3%81%A6%E3%81%BF%E3%82%8B\"><\/span>$(\\nabla_{\\rho}\\nabla_{\\mu}-\\nabla_{\\mu}\\nabla_{\\rho})V_{\\nu}$\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u7a81\u7136\u3067\u3059\u304c\u3053\u3053\u3067$(\\nabla_{\\rho}\\nabla_{\\mu}-\\nabla_{\\mu}\\nabla_{\\rho})V_{\\nu}$\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<br>$(\\nabla_{\\rho}\\nabla_{\\mu}-\\nabla_{\\mu}\\nabla_{\\rho})V_{\\nu}=\\nabla_{\\rho}(\\nabla_{\\mu}V_{\\nu})-\\nabla_{\\mu}(\\nabla_{\\rho}V_{\\nu})$\u306a\u306e\u3067\u3001$\\nabla_{\\rho}(\\nabla_{\\mu}V_{\\nu})$\u3068$\\nabla_{\\mu}(\\nabla_{\\rho}V_{\\nu})$\u3092\u305d\u308c\u305e\u308c\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    \\nabla_{\\rho}(\\nabla_{\\mu}V_{\\nu})&amp;=\\partial_{\\rho}(\\nabla_{\\mu}V_{\\nu})-\\Gamma^{\\sigma}_{\\nu\\rho}(\\nabla_{\\mu}V_{\\sigma})-\\Gamma^{\\sigma}_{\\mu\\rho}(\\nabla_{\\sigma}V_{\\nu}) \\\\\n    &amp;=\\partial_{\\rho}(\\partial_{\\mu}V_{\\nu}-\\Gamma^{\\alpha}_{\\nu\\mu}V_{\\alpha}) \\\\\n    &amp;\\qquad -\\Gamma^{\\sigma}_{\\nu\\rho}(\\partial_{\\mu}V_{\\sigma}-\\Gamma^{\\beta}_{\\sigma\\mu}V_{\\beta})-\\Gamma^{\\sigma}_{\\mu\\rho}(\\partial_{\\sigma}V_{\\nu}-\\Gamma^{\\gamma}_{\\nu\\sigma}V_{\\gamma}) \\\\\n    &amp;=\\partial_{\\rho}\\partial_{\\mu}V_{\\nu}-(\\partial_{\\rho}\\Gamma^{\\alpha}_{\\nu\\mu})V_{\\alpha}-\\Gamma^{\\alpha}_{\\nu\\mu}(\\partial_{\\rho}V_{\\alpha}) \\\\\n    &amp;\\qquad -\\Gamma^{\\sigma}_{\\nu\\rho}(\\partial_{\\mu}V_{\\sigma}-\\Gamma^{\\beta}_{\\sigma\\mu}V_{\\beta})-\\Gamma^{\\sigma}_{\\mu\\rho}(\\partial_{\\sigma}V_{\\nu}-\\Gamma^{\\gamma}_{\\nu\\sigma}V_{\\gamma}) \\\\\n    &amp;=\\partial_{\\rho}\\partial_{\\mu}V_{\\nu}-(\\partial_{\\rho}\\Gamma^{\\alpha}_{\\nu\\mu})V_{\\alpha}-\\Gamma^{\\alpha}_{\\nu\\mu}(\\partial_{\\rho}V_{\\alpha})-\\Gamma^{\\sigma}_{\\nu\\rho}(\\partial_{\\mu}V_{\\sigma}) \\\\\n    &amp;\\qquad +\\Gamma^{\\sigma}_{\\nu\\rho}\\Gamma^{\\beta}_{\\sigma\\mu}V_{\\beta}-\\Gamma^{\\sigma}_{\\mu\\rho}(\\partial_{\\sigma}V_{\\nu})+\\Gamma^{\\sigma}_{\\mu\\rho}\\Gamma^{\\gamma}_{\\nu\\sigma}V_{\\gamma}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>$\\nabla_{\\mu}(\\nabla_{\\rho}V_{\\nu})$\u306f$\\mu$\u3068$\\rho$\u3092\u5165\u308c\u66ff\u3048\u308b\u3060\u3051\u3067\u3059\u3002<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    \\nabla_{\\mu}(\\nabla_{\\rho}V_{\\nu})&amp;=\\partial_{\\mu}\\partial_{\\rho}V_{\\nu}-(\\partial_{\\mu}\\Gamma^{\\alpha}_{\\nu\\rho})V_{\\alpha}-\\Gamma^{\\alpha}_{\\nu\\rho}(\\partial_{\\mu}V_{\\alpha})-\\Gamma^{\\sigma}_{\\nu\\mu}(\\partial_{\\rho}V_{\\sigma}) \\\\\n    &amp;\\qquad +\\Gamma^{\\sigma}_{\\nu\\mu}\\Gamma^{\\beta}_{\\sigma\\rho}V_{\\beta}-\\Gamma^{\\sigma}_{\\rho\\mu}(\\partial_{\\sigma}V_{\\nu})+\\Gamma^{\\sigma}_{\\rho\\mu}\\Gamma^{\\gamma}_{\\nu\\sigma}V_{\\gamma}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n    (\\nabla_{\\rho}\\nabla_{\\mu}-\\nabla_{\\mu}\\nabla_{\\rho})V_{\\nu}&amp;=\\nabla_{\\rho}(\\nabla_{\\mu}V_{\\nu})-\\nabla_{\\mu}(\\nabla_{\\rho}V_{\\nu}) \\\\\n    &amp;=\\Gamma^{\\sigma}_{\\nu\\rho}\\Gamma^{\\beta}_{\\sigma\\mu}V_{\\beta}+(\\partial_{\\mu}\\Gamma^{\\alpha}_{\\nu\\rho})V_{\\alpha} \\\\\n    &amp;\\qquad -\\Gamma^{\\sigma}_{\\nu\\mu}\\Gamma^{\\beta}_{\\sigma\\rho}V_{\\beta}-(\\partial_{\\rho}\\Gamma^{\\alpha}_{\\nu\\mu})V_{\\alpha}\n\\end{align*}\n$$<\/p>\n\n\n\n<p>$\\alpha$\u3068$\\beta$\u306f\u540c\u3058\u9805\u306b\u540c\u6642\u306b\u73fe\u308c\u306a\u3044\u72ec\u7acb\u3057\u305f\u5909\u6570\u306a\u306e\u3067\u3001$\\gamma=\\alpha=\\beta$\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\n(\\nabla_{\\rho}\\nabla_{\\mu}-\\nabla_{\\mu}\\nabla_{\\rho})V_{\\nu}=(\\Gamma^{\\sigma}_{\\nu\\rho}\\Gamma^{\\gamma}_{\\sigma\\mu}+\\partial_{\\mu}\\Gamma^{\\gamma}_{\\nu\\rho}-\\Gamma^{\\sigma}_{\\nu\\mu}\\Gamma^{\\gamma}_{\\sigma\\rho}-\\partial_{\\rho}\\Gamma^{\\gamma}_{\\nu\\mu})V_{\\gamma}\n$$<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u6dfb\u5b57\u3092$\\gamma\\rightarrow\\rho,\\mu\\rightarrow\\sigma,\\rho\\rightarrow\\alpha,\\sigma\\rightarrow\\beta$\u3068\u7f6e\u63db\u3059\u308b\u3068\u3001\u62ec\u5f27\u306e\u4e2d\u304c\u5148\u7a0b\u8a08\u7b97\u3057\u305f\u3001\u5e73\u884c\u79fb\u52d5\u5f8c\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5dee\u3092\u8868\u3059\u5f0f\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%BE%A9\"><\/span>\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5b9a\u7fa9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3053\u3053\u3067\u3001$(\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu})V^{\\rho}=R_{\\mu\\nu\\sigma}^{\\quad\\rho}V^{\\sigma}$\u3068\u66f8\u304d\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002<br>\u3053\u306e$R_{\\mu\\nu\\sigma}^{\\quad\\rho}$\u3092<strong>\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong>(Riemann tensor)\u3068\u547c\u3073\u3001\u3053\u308c\u306f\u6642\u7a7a\u306e\u6b6a\u307f\u3092\u8868\u3059\u91cf\u3067\u3059\u3002<br>\u5148\u306b\u5c0e\u51fa\u3057\u305f\u3068\u304a\u308a\u3001<\/p>\n\n\n\n<p>$$\nR_{\\mu\\nu\\rho}^{\\quad\\sigma}=\\frac{\\partial \\Gamma^{\\sigma}_{\\mu\\rho}}{\\partial x^{\\nu}}-\\frac{\\partial \\Gamma^{\\sigma}_{\\nu\\rho}}{\\partial x^{\\mu}}+\\Gamma^{\\alpha}_{\\mu\\rho}\\Gamma^{\\sigma}_{\\alpha\\nu}-\\Gamma^{\\alpha}_{\\nu\\rho}\\Gamma^{\\sigma}_{\\alpha\\mu}\n$$<\/p>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<br>(\u6dfb\u5b57\u306f<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>\u306b\u5408\u308f\u305b\u307e\u3057\u305f)<\/p>\n\n\n\n<p>\u81ea\u5206\u306f\u6700\u521d\u3001\u3044\u304d\u306a\u308a$\\nabla_{\\mu}\\nabla_{\\nu}-\\nabla_{\\nu}\\nabla_{\\mu}$\u3092\u6642\u7a7a\u306e\u6b6a\u307f\u3092\u8868\u3059\u91cf\u3068\u3057\u3066\u7528\u3044\u308b\u3001\u3068\u8a00\u308f\u308c\u3066\u3088\u304f\u308f\u304b\u3089\u306a\u304b\u3063\u305f\u306e\u3067\u3059\u304c\u3001\u5148\u306b\u7d39\u4ecb\u3057\u305f\u3068\u304a\u308a\u3001\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3057\u305f\u969b\u306e\u305a\u308c\u304c\u8003\u3048\u65b9\u306e\u6839\u5e95\u306b\u3042\u308b\u308f\u3051\u3067\u3059\u306d\u3002<\/p>\n\n\n\n<p>$R_{\\mu\\nu\\rho\\sigma} \\equiv R_{\\mu\\nu\\rho}^{\\quad\\sigma&#8217;}g_{\\sigma&#8217;\\sigma}$\u3068\u5b9a\u7fa9\u3059\u308b\u3068\u3001\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306f\u305d\u306e\u6dfb\u5b57\u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u5bfe\u79f0\u6027\u3092\u3082\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\nR_{\\mu\\nu\\rho\\sigma}=-R_{\\nu\\mu\\rho\\sigma}=-R_{\\mu\\nu\\sigma\\rho}=R_{\\sigma\\rho\\mu\\nu}\n$$<\/p>\n\n\n\n<p>\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u3092\u7e2e\u7d04\u3057\u3066\u5f97\u3089\u308c\u308b$R_{\\mu\\nu} \\equiv g^{\\rho\\sigma}R_{\\mu\\rho\\nu\\sigma}$\u3092<strong>\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb<\/strong>(Ricci tensor)\u3001$R \\equiv g^{\\mu\\nu}R_{\\mu\\nu}$\u3092<strong>\u30ea\u30c3\u30c1\u30b9\u30ab\u30e9\u30fc<\/strong>(Ricci scalar)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u76ee\u7684 \u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6\u3067\u306f\u6b6a\u3093\u3060\u6642\u7a7a\u3092\u8003\u3048\u308b\u305f\u3081\u3001\u6642\u7a7a\u304c\u3069\u308c\u304f\u3089\u3044\u6b6a\u3093\u3067\u3044\u308b\u306e\u304b\u8868\u3059\u91cf\u3092\u69cb\u6210\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u3053\u306e\u305f\u3081\u306b\u306f\u3001\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3055\u305b\u305f\u3068\u304d\u306b\u751f\u3058\u308b\u5411\u304d\u306e\u305a\u308c\u3092\u8003\u3048\u308c\u3070\u3088\u3055\u305d\u3046\u3067\u3059\u3002 \u305f\u3068\u3048\u3070\u3001\u5730\u7403\u306e\u8d64\u9053\u4e0a\u306e\u3042 [&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-446","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\/446","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=446"}],"version-history":[{"count":33,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/446\/revisions"}],"predecessor-version":[{"id":1170,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/446\/revisions\/1170"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=446"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=446"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}