\u5e73\u5766\u306a\u6642\u7a7a($ds^2=\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}$)\u3067\u5916\u529b\u306e\u304b\u304b\u3063\u3066\u3044\u306a\u3044\u7c92\u5b50\u306e\u4f4d\u7f6e\u5ea7\u6a19$x^{\\mu}(\\tau)$\u306f\u3001<\/p>\n\n\n\n
$$\n\\frac{d^2x^{\\mu}(\\tau)}{d\\tau^2}=0\n$$<\/p>\n\n\n\n
\u306b\u5f93\u3063\u3066\u6642\u9593\u767a\u5c55\u3057\u307e\u3059\u3002
($\\tau$\u306f\u7c92\u5b50\u306e\u56fa\u6709\u6642\u9593\u3067\u3059)<\/p>\n\n\n\n
\u3053\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u5225\u306e\u5ea7\u6a19$x^{\\mu}(\\tau)=x^{\\mu}(x’^{\\nu}(\\tau))$\u3067\u8868\u3059\u3068\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n 0&=\\frac{d^2x^{\\mu}(\\tau)}{d\\tau^2} \\\\\n &=\\frac{d}{d\\tau}(\\frac{dx’^{\\nu}(\\tau)}{d\\tau}\\frac{\\partial x^{\\mu}}{\\partial x’^{\\nu}}) \\\\\n &=\\frac{d^2x’^{\\nu}(\\tau)}{d\\tau^2}\\frac{\\partial x^{\\mu}}{\\partial x’^{\\nu}}+\\frac{dx’^{\\rho}(\\tau)}{d\\tau}\\frac{dx’^{\\nu}(\\tau)}{d\\tau}\\frac{\\partial^2x^{\\mu}}{\\partial x’^{\\rho}\\partial x’^{\\nu}}\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3088\u308a\u3001<\/p>\n\n\n\n
$$\n\\frac{d^2x’^{\\sigma}(\\tau)}{d\\tau^2}+\\frac{\\partial x’^{\\sigma}}{\\partial x^{\\mu}}\\frac{\\partial^2x^{\\mu}}{\\partial x’^{\\rho}\\partial x’^{\\nu}}\\frac{dx’^{\\rho}(\\tau)}{d\\tau}\\frac{dx’^{\\nu}(\\tau)}{d\\tau}=0\n$$<\/p>\n\n\n\n
\u65b0\u5ea7\u6a19$x’^{\\mu}$\u306b\u304a\u3051\u308b\u8a08\u91cf$g’_{\\mu\\nu}$\u306f\u5143\u306e\u5ea7\u6a19$x^{\\mu}$\u306b\u304a\u3051\u308b\u8a08\u91cf$\\eta_{\\mu\\nu}$\u3092<\/p>\n\n\n\n
$$\ng’_{\\mu\\nu}=\\frac{\\partial x^{\\alpha}}{\\partial x’^{\\mu}}\\frac{\\partial x^{\\beta}}{\\partial x’^{\\nu}}\\eta_{\\alpha\\beta}\n$$<\/p>\n\n\n\n
\u3068\u5909\u63db\u3057\u305f\u3082\u306e\u3067\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u308c\u3092\u504f\u5fae\u5206\u3057\u305f\u5f0f<\/p>\n\n\n\n
$$\n\\partial_{\\rho}g’_{\\mu\\nu}=\\eta_{\\alpha\\beta}(\\frac{\\partial^2 x^{\\alpha}}{\\partial x’^{\\rho}\\partial x’^{\\mu}}\\frac{\\partial x^{\\beta}}{\\partial x’^{\\nu}}+\\frac{\\partial x^{\\alpha}}{\\partial x’^{\\mu}}\\frac{\\partial^2x^{\\beta}}{\\partial x’^{\\rho}\\partial x’^{\\nu}})\n$$<\/p>\n\n\n\n
\u3092\u6dfb\u5b57\u3092\u7f6e\u63db\u3057\u3064\u3064\u8db3\u3057\u5408\u308f\u305b\u308b\u3053\u3068\u3067\u3001<\/p>\n\n\n\n
$$\n\\frac{\\partial x’^{\\sigma}}{\\partial x^{\\mu}}\\frac{\\partial^2 x^{\\mu}}{\\partial x’^{\\rho}\\partial x’^{\\nu}}=\\frac{1}{2}g’^{\\sigma\\lambda}(\\partial_{\\rho}g’_{\\nu\\lambda}+\\partial_{\\nu}g’_{\\rho\\lambda}-\\partial_{\\lambda}g’_{\\rho\\nu})=\\Gamma’^{\\sigma}_{\\rho\\nu}\n$$<\/p>\n\n\n\n
\u3068\u3044\u3046\u3088\u3046\u306b\u3001\u8a08\u91cf$g’_{\\mu\\nu}$\u306b\u3064\u3044\u3066\u306e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u308c\u3092\u7528\u3044\u308b\u3068\u3001\u5148\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u3001<\/p>\n\n\n\n
$$\n\\frac{d^2x’^{\\sigma}(\\tau)}{d\\tau^2}+\\Gamma’^{\\sigma}_{\\rho\\nu}\\frac{dx’^{\\rho}(\\tau)}{d\\tau}\\frac{dx’^{\\nu}(\\tau)}{d\\tau}=0\n$$<\/p>\n\n\n\n
\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\u3001\u5916\u529b\u306e\u304b\u304b\u3063\u3066\u3044\u306a\u3044\u7c92\u5b50\u304c\u8a08\u91cf$g’_{\\mu\\nu}$\u306e\u6642\u7a7a\u4e0a\u3067\u904b\u52d5\u3059\u308b\u3068\u304d\u306e\u7d4c\u8def\u3092\u4e0e\u3048\u308b\u5f0f\u3067\u3059\u3002 \u5e73\u5766\u306a\u6642\u7a7a($ds^2=\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}$)\u3067\u5916\u529b\u306e\u304b\u304b\u3063\u3066\u3044\u306a\u3044\u7c92\u5b50\u306e\u4f4d\u7f6e\u5ea7\u6a19$x^{\\mu}(\\tau)$\u306f\u3001 $$ \\frac{d^2x^{\\mu}(\\tau)}{d\\ta […]<\/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-439","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\/439","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=439"}],"version-history":[{"count":8,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/439\/revisions"}],"predecessor-version":[{"id":1169,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/439\/revisions\/1169"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=439"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=439"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=439"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u3053\u306e\u5f0f\u3092\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong>(geodesic equation)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u3053\u3067\u306f\u5e73\u5766\u306a\u6642\u7a7a$\\eta_{\\mu\\nu}$\u3092\u60f3\u5b9a\u3057\u3066\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u5c0e\u51fa\u3057\u307e\u3057\u305f\u304c\u3001\u3088\u308a\u4e00\u822c\u306e\u66f2\u304c\u3063\u305f\u6642\u7a7a$g_{\\mu\\nu}$\u306b\u3064\u3044\u3066\u3082\u3053\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"