\u91cd\u529b\u5834\u304c\u5f31\u3044\u5834\u5408\u306b\u306f\u8a08\u91cf$g_{\\mu\\nu}$\u3092\u5e73\u5766\u8a08\u91cf$\\eta_{\\mu\\nu}$\u3067\u8fd1\u4f3c\u3067\u304d\u308b\u306e\u3067\u3001$g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$\u3068\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092$h_{\\mu\\nu}$\u306e1\u6b21\u306e\u9805\u307e\u3067\u5c55\u958b\u3057\u307e\u3059\u3002
\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e\u4fc2\u6570\u3092\u6c42\u3081\u305f\u3068\u304d\u306b\u540c\u3058\u8a08\u7b97\u3092\u3057\u305f\u306e\u3067\u3001\u8a73\u7d30\u306f\u7701\u7565\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$$\n\\begin{align*}\n \\Gamma^{\\rho}_{\\mu\\nu}&=\\frac{1}{2}g^{\\rho\\sigma}(\\partial_{\\mu}g_{\\nu\\sigma}+\\partial_{\\nu}g_{\\mu\\sigma}-\\partial_{\\sigma}g_{\\mu\\nu}) \\\\\n &\\approx\\frac{1}{2}\\eta^{\\rho\\sigma}(\\partial_{\\mu}h_{\\nu\\sigma}+\\partial_{\\nu}h_{\\mu\\sigma}-\\partial_{\\sigma}h_{\\mu\\nu})\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3053\u308c\u3092\u3082\u3068\u306b\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb$R_{\\mu\\nu}$\u3092\u5c55\u958b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$$\nR_{\\mu\\nu}=g^{\\rho\\sigma}R_{\\mu\\rho\\nu\\sigma}=g^{\\sigma\\rho}R_{\\mu\\rho\\nu}^{\\quad\\sigma’}g_{\\sigma’\\sigma}=R_{\\mu\\rho\\nu}^{\\quad\\rho}\n$$<\/p>\n\n\n\n
\u306a\u306e\u3067\u3001<\/p>\n\n\n\n
\n$$\n\\begin{align*} R_{\\mu\\nu}&=R_{\\mu\\rho\\nu}^{\\quad\\rho} \\\\ \n&\\approx \\partial_{\\rho}\\Gamma^{\\rho}_{\\mu\\nu}-\\partial_{\\mu}\\Gamma^{\\rho}_{\\rho\\nu} \\\\\n&=\\partial_{\\rho}\\left(\\frac{1}{2}\\eta^{\\rho\\sigma}(\\partial_{\\mu}h_{\\nu\\sigma}+\\partial_{\\nu}h_{\\mu\\sigma}-\\partial_{\\sigma}h_{\\mu\\nu})\\right) \\\\ \n&\\qquad -\\partial_{\\mu}\\left(\\frac{1}{2}\\eta^{\\rho\\sigma}(\\partial_{\\rho}h_{\\nu\\sigma}+\\partial_{\\nu}h_{\\rho\\sigma}-\\partial_{\\sigma}h_{\\rho\\nu})\\right) \\\\ &=\\frac{1}{2}\\eta^{\\rho\\sigma}(\\partial_{\\rho}\\partial_{\\mu}h_{\\nu\\sigma}+\\partial_{\\rho}\\partial_{\\nu}h_{\\mu\\sigma}-\\partial_{\\rho}\\partial_{\\sigma}h_{\\mu\\nu} \\\\ \n&\\qquad -\\partial_{\\mu}\\partial_{\\rho}h_{\\nu\\sigma}-\\partial_{\\mu}\\partial_{\\nu}h_{\\rho\\sigma}+\\partial_{\\mu}\\partial_{\\sigma}h_{\\rho\\nu}) \\\\\n&=\\frac{1}{2}\\eta^{\\rho\\sigma}(\\partial_{\\rho}\\partial_{\\nu}h_{\\mu\\sigma}-\\partial_{\\rho}\\partial_{\\sigma}h_{\\mu\\nu}-\\partial_{\\mu}\\partial_{\\nu}h_{\\rho\\sigma}+\\partial_{\\mu}\\partial_{\\sigma}h_{\\rho\\nu}) \\\\ \n&=\\frac{1}{2}(\\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}-\\Box h_{\\mu\\nu}-\\partial_{\\mu}\\partial_{\\nu}h+\\partial^{\\mu}\\partial^{\\rho}h_{\\rho\\nu}) \n\\end{align*}\n$$\n<\/p>\n\n\n\n
\u3053\u3053\u3067\u767b\u5834\u3057\u305f$\\partial^{\\mu}\\partial^{\\nu}h_{\\alpha\\beta}=\\eta^{\\mu\\nu}\\partial_{\\mu}\\partial_{\\nu}h_{\\alpha\\beta}$\u3001$h=\\eta^{\\mu\\nu}h_{\\mu\\nu}$\u3067\u3059\u3002
\u307e\u305f\u3001$\\Box \\equiv \\eta^{\\mu\\nu}\\partial_{\\mu}\\partial_{\\nu}=-\\partial_{t}^2+\\partial_i\\partial^i$\u3092\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u6f14\u7b97\u5b50\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n
\u30ea\u30c3\u30c1\u30b9\u30ab\u30e9\u30fc$R$\u306f\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n R&=\\eta^{\\mu\\nu}R_{\\mu\\nu} \\\\\n &=\\frac{1}{2}\\eta^{\\mu\\nu}(\\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}-\\Box h_{\\mu\\nu}-\\partial_{\\mu}\\partial_{\\nu}h+\\partial^{\\mu}\\partial^{\\rho}h_{\\rho\\nu})\n\\end{align*}\n$$<\/p>\n\n\n\n
$\\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}=\\eta^{\\rho\\nu}\\partial_{\\rho}\\partial_{\\nu}h_{\\mu\\rho}$\u306a\u306e\u3067\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n \\eta^{\\mu\\nu}\\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}&=\\eta^{\\mu\\nu}\\eta^{\\rho\\nu}\\partial_{\\rho}\\partial_{\\nu}h_{\\mu\\rho} \\\\\n &=\\delta^{\\mu\\rho}\\partial_{\\rho}\\partial_{\\mu}h_{\\mu\\rho} \\\\\n &=\\partial^{\\mu}\\partial_{\\rho}h_{\\mu\\rho}\n\\end{align*}\n$$<\/p>\n\n\n\n
\u540c\u69d8\u306b\u3001$\\eta^{\\mu\\nu}\\partial^{\\mu}\\partial^{\\rho}h_{\\rho\\nu}=\\partial^{\\nu}\\partial_{\\rho}h_{\\nu\\rho}$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\nR&=\\frac{1}{2}\\eta^{\\mu\\nu}(\\partial^{\\rho}\\partial_{\\nu}h_{\\mu\\rho}-\\Box h_{\\mu\\nu}-\\partial_{\\mu}\\partial_{\\nu}h+\\partial^{\\mu}\\partial_{\\rho}h_{\\rho\\nu}) \\\\\n&=\\frac{1}{2}(\\partial^{\\mu}\\partial_{\\rho}h_{\\mu\\rho}-\\Box h-\\Box h+\\partial^{\\nu}\\partial_{\\rho}h_{\\nu\\rho}) \\\\\n&=\\frac{1}{2}(2\\partial^{\\mu}\\partial_{\\rho}h_{\\mu\\rho}-2\\Box h) \\\\\n&=\\partial^{\\mu}\\partial_{\\rho}h_{\\mu\\rho}-\\Box h\n\\end{align*}\n$$<\/p>\n\n\n\n
\u5f8c\u306e\u8a08\u7b97\u306e\u305f\u3081\u306b\u6dfb\u5b57\u3092$\\alpha,\\beta$\u306b\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n
$$\nR=\\partial^{\\alpha}\\partial_{\\beta}h_{\\alpha\\beta}-\\Box h\n$$<\/p>\n\n\n\n
\u3053\u308c\u3088\u308a\u3001\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb$G_{\\mu\\nu}$\u306f\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n G_{\\mu\\nu}&=R_{\\mu\\nu}-\\frac{1}{2}Rg_{\\mu\\nu} \\\\\n &\\approx \\frac{1}{2}(\\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}-\\Box h_{\\mu\\nu}-\\partial_{\\mu}\\partial_{\\nu}h+\\partial^{\\mu}\\partial^{\\rho}h_{\\rho\\nu}) \\\\\n &\\qquad -\\frac{1}{2}(\\partial^{\\alpha}\\partial_{\\beta}h_{\\alpha\\beta}-\\Box h)\\eta_{\\mu\\nu} \\\\\n &=\\frac{1}{2}(\\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}-\\Box h_{\\mu\\nu}-\\partial_{\\mu}\\partial_{\\nu}h \\\\\n &\\qquad +\\partial^{\\mu}\\partial^{\\rho}h_{\\rho\\nu}-(\\partial^{\\alpha}\\partial_{\\beta}h_{\\alpha\\beta})\\eta_{\\mu\\nu}+(\\Box h)\\eta_{\\mu\\nu})\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001$\\psi_{\\mu\\nu}=h_{\\mu\\nu}-\\frac{1}{2}h\\eta_{\\mu\\nu}$\u3068\u304a\u304f\u3068\u3001$h_{\\mu\\nu}=\\psi_{\\mu\\nu}-\\frac{1}{2}\\psi\\eta_{\\mu\\nu}$\u3068\u306a\u308a\u307e\u3059\u3002
\u7ba1\u7406\u4eba\u306f$h_{\\mu\\nu}=\\psi_{\\mu\\nu}-\\frac{1}{2}\\psi\\eta_{\\mu\\nu}$\u3068\u306a\u308b\u7406\u7531\u304c\u308f\u304b\u3089\u306a\u304f\u3066\u3057\u3070\u3089\u304f\u60a9\u3093\u3067\u3044\u307e\u3057\u305f\u304c\u3001\u3053\u308c\u306f\u8a66\u3057\u306b$\\psi$\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
$$\n\\begin{align*}\n \\psi&=\\eta^{\\mu\\nu}\\psi_{\\mu\\nu} \\\\\n &=\\eta^{\\mu\\nu}(h_{\\mu\\nu}-\\frac{1}{2}h\\eta_{\\mu\\nu}) \\\\\n &=h-\\frac{1}{2}h\\cdot 4 \\\\\n &=-h\n\\end{align*}\n$$<\/p>\n\n\n\n
\u306a\u306e\u3067\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n h_{\\mu\\nu}&=\\psi_{\\mu\\nu}+\\frac{1}{2}h\\eta_{\\mu\\nu} \\\\\n &=\\psi_{\\mu\\nu}-\\frac{1}{2}\\psi\\eta_{\\mu\\nu}\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u308c\u3092\u7528\u3044\u3066\u3001\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u305d\u308c\u305e\u308c\u306e\u9805\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n
$\\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}$\u306f\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n \\partial^{\\rho}\\partial^{\\nu}h_{\\mu\\rho}&=\\partial^{\\rho}\\partial^{\\nu}(\\psi_{\\mu\\rho}-\\frac{1}{2}\\psi\\eta_{\\mu\\rho}) \\\\\n &=\\partial^{\\rho}\\partial^{\\nu}\\psi_{\\mu\\rho}-\\frac{1}{2}\\eta_{\\mu\\rho}\\partial^{\\rho}\\partial^{\\nu}\\psi \\\\\n &=\\partial^{\\rho}\\partial^{\\nu}\\psi_{\\mu\\rho}-\\frac{1}{2}\\partial^{\\mu}\\partial_{\\mu}\\psi\n\\end{align*}\n$$<\/p>\n\n\n\n
\u540c\u69d8\u306e\u8a08\u7b97\u3067\u3001$\\partial^{\\mu}\\partial^{\\rho}h_{\\rho\\nu}=\\partial^{\\rho}\\partial^{\\nu}\\psi_{\\mu\\rho}-\\frac{1}{2}\\partial^{\\rho}\\partial_{\\rho}\\psi$<\/p>\n\n\n\n
$\\Box h_{\\mu\\nu}$\u306f\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n \\Box h_{\\mu\\nu}&=\\Box(\\psi_{\\mu\\nu}-\\frac{1}{2}\\psi\\eta_{\\mu\\nu}) \\\\\n &=\\Box \\psi_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\mu\\nu}\\Box\\psi \\\\\n &=\\Box\\psi_{\\mu\\nu}-\\frac{1}{2}\\partial^{\\mu}\\partial_{\\mu}\\psi\n\\end{align*}\n$$<\/p>\n\n\n\n
$(\\partial^{\\alpha}\\partial_{\\beta}h_{\\alpha\\beta})\\eta_{\\mu\\nu}$\u306f\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n (\\partial^{\\alpha}\\partial_{\\beta}h_{\\alpha\\beta})\\eta_{\\mu\\nu}&=\\partial^{\\alpha}\\partial_{\\beta}(\\psi_{\\alpha\\beta}-\\frac{1}{2}\\psi\\eta_{\\alpha\\beta})\\eta_{\\mu\\nu} \\\\\n &=\\partial^{\\alpha}\\partial_{\\beta}\\psi_{\\alpha\\beta}\\eta_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\alpha\\beta}\\eta_{\\mu\\nu}\\partial^{\\alpha}\\partial_{\\beta}\\psi \\\\\n &=\\partial^{\\mu}\\partial^{\\nu}\\partial_{\\mu\\nu}-\\frac{1}{2}\\partial^{\\mu}\\partial_{\\mu}\\psi\n\\end{align*}\n$$<\/p>\n\n\n\n
$(\\Box h)\\eta_{\\mu\\nu}$\u306f\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n (\\Box h)\\eta_{\\mu\\nu}&=(\\eta^{\\mu\\nu}\\partial_{\\mu}\\partial_{\\nu}h)\\eta_{\\mu\\nu} \\\\\n &=\\partial_{\\mu}\\partial_{\\nu}h\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n G_{\\mu\\nu}&=\\frac{1}{2}((\\partial^{\\rho}\\partial^{\\nu}\\psi_{\\mu\\rho}-\\frac{1}{2}\\partial^{\\mu}\\partial_{\\mu}\\psi)-(\\Box\\psi_{\\mu\\nu}-\\frac{1}{2}\\partial^{\\mu}\\partial_{\\mu}\\psi) \\\\\n &\\qquad -\\partial_{\\mu}\\partial_{\\nu}h+(\\partial^{\\mu}\\partial^{\\rho}\\psi_{\\rho\\nu}-\\frac{1}{2}\\partial^{\\rho}\\partial_{\\rho}\\psi) \\\\\n &\\qquad -(\\partial^{\\mu}\\partial^{\\nu}\\psi_{\\mu\\nu}-\\frac{1}{2}\\partial^{\\mu}\\partial_{\\mu}\\psi)+\\partial_{\\mu}\\partial_{\\nu}h) \\\\\n &=\\frac{1}{2}(\\partial^{\\rho}\\partial^{\\nu}\\psi_{\\mu\\rho}-\\Box\\psi_{\\mu\\nu}+\\partial^{\\mu}\\partial^{\\rho}\\psi_{\\rho\\nu}-\\partial^{\\mu}\\partial^{\\nu}\\psi_{\\mu\\nu})\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3068\u306a\u308a\u3001\u5f0f\u304b\u3089$h$\u3092\u6d88\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001$\\partial^{\\mu}\\psi_{\\mu\\nu}=0$\u3068\u3044\u3046\u5236\u7d04\u3092\u304b\u3051\u308b\u3068\u3001$G_{\\mu\\nu}=-\\frac{1}{2}\\Box\\psi_{\\mu\\nu}$\u3088\u308a\u3001\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306f\u3001<\/p>\n\n\n\n
$$\n-\\frac{1}{2}\\Box\\psi_{\\mu\\nu}=\\frac{8\\pi G}{c^4}T_{\\mu\\nu}\n$$<\/p>\n\n\n\n
\u3068\u306a\u308a\u307e\u3059\u3002
\u3053\u308c\u304c$\\psi_{\\mu\\nu}$\u306b\u3064\u3044\u3066\u7dda\u5f62\u5316\u3055\u308c\u305f\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u91cd\u529b\u5834\u304c\u5f31\u3044\u5834\u5408\u306b\u306f\u8a08\u91cf$g_{\\mu\\nu}$\u3092\u5e73\u5766\u8a08\u91cf$\\eta_{\\mu\\nu}$\u3067\u8fd1\u4f3c\u3067\u304d\u308b\u306e\u3067\u3001$g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$\u3068\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002 \u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18 […]<\/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-518","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\/518","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=518"}],"version-history":[{"count":6,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/518\/revisions"}],"predecessor-version":[{"id":1619,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/518\/revisions\/1619"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=518"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=518"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}