\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306b\u51fa\u3066\u304f\u308b\u4fc2\u6570$\\kappa$\u3092\u5177\u4f53\u7684\u306b\u6c42\u3081\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002
\u3053\u306e\u3068\u304d\u3001\u4fc2\u6570$\\kappa$\u306b\u3088\u3063\u3066\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb$\\phi$\u304c\u6b63\u3057\u304f\u8a18\u8ff0\u3055\u308c\u308b\u3088\u3046\u306b\u3057\u305f\u3044\u3067\u3059\u3002<\/p>\n\n\n\n
\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a\u6761\u4ef6\u4e0b\u3067\u306f\u3001\u5fae\u5c0f\u91cf$h_{\\mu\\nu}$\u3092\u7528\u3044\u3066\u3001\u8a08\u91cf$g_{\\mu\\nu}$\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002
\u5e73\u5766\u6642\u7a7a$\\eta_{\\mu\\nu}$\u304b\u3089\u306e\u305a\u308c\u304c\u5c0f\u3055\u304f\u3001\u5fae\u5c0f\u91cf$h_{\\mu\\nu}$\u3092\u7528\u3044\u3066\u8a08\u91cf$g_{\\mu\\nu}$\u3092\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3001\u3068\u3044\u3046\u8003\u3048\u3067\u3059\u3002<\/p>\n\n\n\n
$$\ng_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}\n$$<\/p>\n\n\n\n
\u5929\u4e0b\u308a\u7684\u3067\u3059\u304c\u3001\u4fc2\u6570$\\kappa$\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f$h_{00}$\u306e\u5024\u304c\u5fc5\u8981\u3067\u3059\u3002
\u3053\u3053\u3067\u306f\u3001$h_{00}$\u3092\u6c42\u3081\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n
\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n\n\n\n
$$\n\\frac{d^2x^{\\sigma}}{d\\tau^2}+\\Gamma^{\\sigma}_{\\rho\\nu}\\frac{dx^{\\rho}}{d\\tau}\\frac{dx^{\\nu}}{d\\tau}=0\n$$<\/p>\n\n\n\n
\u30924\u5143\u901f\u5ea6\u3092\u7528\u3044\u3066\u8868\u3059\u3068\u3001<\/p>\n\n\n\n
$$\n\\frac{du^{\\sigma}}{d\\tau}+\\Gamma^{\\sigma}_{\\rho\\nu}u^{\\rho}u^{\\nu}=0\n$$<\/p>\n\n\n\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
4\u5143\u901f\u5ea6\u306b$mc$\u3092\u304b\u3051\u308b\u30684\u5143\u904b\u52d5\u91cf\u306b\u306a\u308b\u306e\u3067\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n &(mc)^2\\frac{du^{\\sigma}}{d\\tau}+(mc)^2\\Gamma^{\\sigma}_{\\rho\\nu}u^{\\rho}u^{\\nu}=0 \\\\\n &\\Leftrightarrow \\frac{dp^{\\sigma}}{d\\tau}\\cdot mc+\\Gamma^{\\sigma}_{\\rho\\nu}p^{\\rho}p^{\\nu}=0 \\\\\n &\\Leftrightarrow \\frac{dp^{\\sigma}}{d\\tau}=-\\frac{1}{mc}\\Gamma^{\\sigma}_{\\rho\\nu}p^{\\rho}p^{\\nu}\n\\end{align*}\n$$<\/p>\n\n\n\n
\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u7a7a\u9593\u3067\u306f\u3001<\/p>\n\n\n\n
$$\nds^2=\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}=-(cdt)^2+dx^2+dy^2+dz^2\n$$<\/p>\n\n\n\n
\u3068\u8868\u3055\u308c\u307e\u3059\u3002
\u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6\u3067\u8003\u3048\u308b\u6b6a\u3093\u3060\u6642\u7a7a\u3067\u306f\u8a08\u91cf$g_{\\mu\\nu}$\u304c\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u8a08\u91cf$\\eta_{\\mu\\nu}$\u304b\u3089\u305a\u308c\u307e\u3059\u304c\u3001\u8cea\u70b9\u306e\u901f\u5ea6\u304c\u5341\u5206\u306b\u9045\u304f\u3001\u6642\u9593$cdt$\u306e\u9593\u306e\u79fb\u52d5\u8ddd\u96e2\u304c\u7121\u8996\u3067\u304d\u308b\u5834\u5408\u306b\u306f\u3001<\/p>\n\n\n\n
$$\nds^2\\approx g_{00}(cdt)^2\n$$<\/p>\n\n\n\n
\u3068\u8fd1\u4f3c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n
$ds$\u3068\u56fa\u6709\u6642\u9593$d\\tau$\u306e\u9593\u306b\u306f$d\\tau^2=-ds^2$\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308b\u306e\u3067\u3001$d\\tau^2 \\approx -g_{00}(cdt)^2$\u3088\u308a\u3001$d\\tau=\\sqrt{-g_{00}}cdt$<\/p>\n\n\n\n
\u3053\u308c\u3092\u7528\u3044\u308b\u3068\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\u3001<\/p>\n\n\n\n
$$\n\\frac{dp^{\\sigma}}{dt}=-\\frac{\\sqrt{-g_{00}}}{m}\\Gamma^{\\sigma}_{\\rho\\nu}p^{\\rho}p^{\\nu}\n$$<\/p>\n\n\n\n
\u3068\u8868\u305b\u307e\u3059\u3002
\u901f\u5ea6\u304c\u5341\u5206\u9045\u3044\u5834\u5408\u306b\u306f\u3001\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306b\u4f3c\u305f\u5f62\u306b\u306a\u308a\u3001\u53f3\u8fba\u304c\u529b\u3092\u8868\u3057\u3066\u3044\u308b\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n
\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u3092\u884c\u3046\u305f\u3081\u306b\u3055\u3089\u306b\u6761\u4ef6\u3092\u8ffd\u52a0\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u6761\u4ef61\u306f$g_{\\mu\\nu}$\u3068$\\eta_{\\mu\\nu}$\u306e\u5dee\u304c\u308f\u305a\u304b\u3067\u3001\u5fae\u5c0f\u91cf$h_{\\mu\\nu}$\u3092\u7528\u3044\u3066$g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$\u3068\u8868\u305b\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u6761\u4ef62\u306f$g_{\\mu\\nu}$\u306e\u6642\u9593\u5fae\u5206\u304c0\u3001\u3059\u306a\u308f\u3061$\\frac{\\partial g_{\\mu\\nu}}{\\partial x^0}=0$\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u6761\u4ef63\u3088\u308a\u30014\u5143\u901f\u5ea6\u306f$u^{\\mu}=(\\gamma,\\gamma\\frac{v^i}{c})=(1,0)$\u30014\u5143\u904b\u52d5\u91cf$p^{\\mu}=mcu^{\\mu}=(mc,0)$\u3068\u8868\u305b\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u308c\u3089\u306e\u6761\u4ef6\u3092\u7528\u3044\u308b\u3068\u3001\u5148\u7a0b\u306e\u5f0f\u306e\u53f3\u8fba\u306f\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n F^i&=-\\frac{\\sqrt{-g_{00}}}{m}\\Gamma^{\\sigma}_{\\rho\\nu}p^{\\rho}p^{\\nu} \\\\\n &=-\\frac{\\sqrt{-g_{00}}}{m}\\Gamma^{\\sigma}_{00}p^0p^0 \\\\\n &=-\\sqrt{-g_{00}}mc^2\\Gamma^{\\sigma}_{00} \\\\\n &=-\\sqrt{-g_{00}}mc^2\\cdot\\frac{1}{2}g^{\\sigma\\alpha}(\\frac{\\partial g_{0\\alpha}}{\\partial x^0}+\\frac{\\partial g_{0\\alpha}}{\\partial x^0}-\\frac{\\partial g_{00}}{\\partial x^{\\alpha}}) \\\\\n &=-\\sqrt{-g_{00}}mc^2\\cdot\\frac{1}{2}g^{\\sigma\\alpha}(-\\frac{\\partial g_{00}}{\\partial x^{\\alpha}}) \\\\\n &=\\sqrt{-g_{00}}mc^2\\cdot \\frac{1}{2}(\\eta^{\\sigma\\alpha}+h^{\\sigma\\alpha})\\frac{\\partial(\\eta_{00}+h_{00})}{\\partial x^{\\alpha}} \\\\\n &=\\sqrt{-g_{00}}mc^2\\cdot \\frac{1}{2}(\\eta^{\\sigma\\alpha}+h^{\\sigma\\alpha})\\frac{\\partial h_{00}}{\\partial x^{\\alpha}} \\\\\n &=\\sqrt{-g_{00}}mc^2\\cdot\\frac{1}{2}(\\frac{\\partial h_{00}}{\\partial x^{\\sigma}}+h^{\\sigma\\alpha}\\frac{\\partial h_{00}}{\\partial x^{\\alpha}})\n\\end{align*}\n$$<\/p>\n\n\n\n
$h^{\\sigma\\alpha}\\frac{\\partial h_{00}}{\\partial x^{\\alpha}}$\u306f2\u6b21\u306e\u5fae\u5c0f\u91cf\u306b\u306a\u308b\u306e\u3067\u6368\u3066\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n\n\n\n
$$\n\\begin{align*}\n F^i&=\\frac{1}{2}\\sqrt{-g_{00}}mc^2\\frac{\\partial h_{00}}{\\partial x^{\\sigma}} \\\\\n &=\\frac{1}{2}\\sqrt{-\\eta_{00}-h_{00}}mc^2\\frac{\\partial h_{00}}{\\partial x^{\\sigma}} \\\\\n &=\\frac{1}{2}\\sqrt{1-h_{00}}mc^2\\frac{\\partial h_{00}}{\\partial x^{\\sigma}}\n\\end{align*}\n$$<\/p>\n\n\n\n
$\\sqrt{1-h_{00}}$\u3092$h_{00}$\u306e1\u6b21\u306e\u9805\u307e\u3067\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$$\nF^i=\\frac{1}{2}(1-\\frac{1}{2}h_{00})mc^2\\frac{\\partial h_{00}}{\\partial x^{\\sigma}}\n$$<\/p>\n\n\n\n
2\u6b21\u306e\u5fae\u5c0f\u91cf\u306f\u6368\u3066\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n\n\n\n
$$\nF^i=\\frac{1}{2}mc^2\\frac{\\partial h_{00}}{\\partial x^{\\sigma}}\n$$<\/p>\n\n\n\n
\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u3067\u306f\u3001\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092$\\phi$\u3067\u8868\u3057\u305f\u3068\u304d\u306b\u3001<\/p>\n\n\n\n
$$\nF^i=-m\\frac{\\partial \\phi}{\\partial x^i}\n$$<\/p>\n\n\n\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u308c\u3092\u5148\u7a0b\u306e\u5f0f\u3068\u6bd4\u8f03\u3059\u308b\u3068\u3001<\/p>\n\n\n\n
$$\n\\frac{1}{2}mc^2\\frac{\\partial h_{00}}{\\partial x^i}=-m\\frac{\\partial \\phi}{\\partial x^i}\n$$<\/p>\n\n\n\n
\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n
$$\nh_{00}=-\\frac{2}{c^2}\\phi\n$$<\/p>\n\n\n\n
\u3068\u3059\u308b\u3053\u3068\u3067\u3001\u4e8c\u3064\u306e\u5f0f\u304c\u4e00\u81f4\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306b\u51fa\u3066\u304f\u308b\u4fc2\u6570$\\kappa$\u3092\u5177\u4f53\u7684\u306b\u6c42\u3081\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u4fc2\u6570$\\kappa$\u306b\u3088\u3063\u3066\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb$\\phi$\u304c\u6b63\u3057\u304f\u8a18\u8ff0\u3055\u308c\u308b\u3088\u3046\u306b\u3057\u305f\u3044\u3067\u3059\u3002 \u30cb\u30e5\u30fc\u30c8\u30f3 […]<\/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-498","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\/498","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=498"}],"version-history":[{"count":5,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/498\/revisions"}],"predecessor-version":[{"id":796,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/498\/revisions\/796"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=498"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=498"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}