{"id":1412,"date":"2022-01-24T11:48:31","date_gmt":"2022-01-24T02:48:31","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1412"},"modified":"2024-07-10T17:51:48","modified_gmt":"2024-07-10T08:51:48","slug":"%e8%86%a8%e5%bc%b5%e5%ae%87%e5%ae%99%e3%81%ae%e8%a8%88%e9%87%8f%e3%81%ae%e5%b0%8e%e5%87%ba%e3%81%a8%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%ae%87%e5%ae%99%e8%ab%96\/%e8%86%a8%e5%bc%b5%e5%ae%87%e5%ae%99%e3%81%ae%e8%a8%88%e9%87%8f%e3%81%ae%e5%b0%8e%e5%87%ba%e3%81%a8%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f\/","title":{"rendered":"\u81a8\u5f35\u5b87\u5b99\u306e\u8a08\u91cf\u306e\u5c0e\u51fa\u3068\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f"},"content":{"rendered":"<p>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u3046\u305a\u306a\u3057\u306e\u30c0\u30b9\u30c8\u6d41\u4f53\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u89e3\u304d\uff0c\u81a8\u5f35\u5b87\u5b99\u306e\u89e3\u3067\u3042\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc\u8a08\u91cf<\/strong><\/span>\uff08\u4ee5\u4e0b\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>FLRW <\/strong><\/span>\u8a08\u91cf\uff09\u3092\u5c0e\u51fa\u3059\u308b\u3002\uff08\u4ee5\u4e0b\uff0c\\(c = 1\\) \u3068\u3059\u308b\u3002\uff09<\/p>\n<p>\u306f\u306a\u304b\u3089\u4e00\u69d8\u7b49\u65b9\u6027\u3092\u6c7a\u3081\u6253\u3061\u3067\u4eee\u5b9a\u3059\u308b\u306e\u3082\u82b8\u304c\u306a\u3044\u306e\u3067\uff0c\u5c11\u3057\u3060\u3051\u4e00\u822c\u7684\u72b6\u6cc1\u8a2d\u5b9a\u304b\u3089\u306f\u3058\u3081\u3066\uff0c\u6700\u7d42\u7684\u306b\u4e00\u69d8\u7b49\u65b9\u306a3\u6b21\u5143\u7a7a\u9593\u3067\u3042\u308b\u3053\u3068\u304c\u5c0e\u304b\u308c\u308b\u3088\u3046\u306a\u66f8\u304d\u632f\u308a\u306b\u3057\u3066\u307f\u308b\u3002<!--more--><\/p>\n<ul>\n<li><a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%83%95%E3%83%AA%E3%83%BC%E3%83%89%E3%83%9E%E3%83%B3%E3%83%BB%E3%83%AB%E3%83%A1%E3%83%BC%E3%83%88%E3%83%AB%E3%83%BB%E3%83%AD%E3%83%90%E3%83%BC%E3%83%88%E3%82%BD%E3%83%B3%E3%83%BB%E3%82%A6%E3%82%A9%E3%83%BC%E3%82%AB%E3%83%BC%E8%A8%88%E9%87%8F\">\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc\u8a08\u91cf &#8211; Wikipedia<\/a><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3>\u3046\u305a\u306a\u3057\u306e\u30c0\u30b9\u30c8\u6d41\u4f53<\/h3>\n<p>\u7269\u8cea\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u3068\u3057\u3066\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u3046\u305a\u306a\u3057<\/strong><\/span>\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30c0\u30b9\u30c8\u6d41\u4f53<\/strong><\/span>\u3092\u4eee\u5b9a\u3057\uff0c<br \/>\n$$T^{\\mu\\nu} = \\rho u^{\\mu} u^{\\nu}$$\u3068\u3059\u308b\u3002<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30c0\u30b9\u30c8\u6d41\u4f53<\/strong><\/span>\u3068\u306f\uff0c\u5727\u529b \\(P\\) \u304c\u30bc\u30ed\uff08\u7121\u8996\u3067\u304d\u308b\u307b\u3069\u5c0f\u3055\u3044\uff09\u3067\u3042\u308b\u3088\u3046\u306a<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5b8c\u5168\u6d41\u4f53<\/strong><\/span>\u306e\u3053\u3068\u3067\uff0c\\(\\rho\\) \u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6<\/strong><\/span>\uff08\\(c=1\\) \u3067\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8cea\u91cf\u5bc6\u5ea6<\/strong><\/span>\u3068\u8a00\u3044\u63db\u3048\u3066\u3082\u53ef\uff09\uff0c\\(u^{\\mu}\\) \u306f\u6d41\u4f53\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u901f\u5ea6<\/strong><\/span>\u3002<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u304c\u6e80\u305f\u3059\u3079\u304d\u5f0f$$T^{\\mu\\nu}_{\\ \\ \\\u00a0 \\ ;\\nu} = 0$$<br \/>\n\u306f \\(u^{\\mu}\\) \u306b\u5e73\u884c\u306a\u6210\u5206\u3068\u76f4\u4ea4\u3059\u308b\u6210\u5206\u306b\u5206\u89e3\u3067\u304d\u3066\uff0c\u5e73\u884c\u306a\u6210\u5206\u306f<br \/>\n$$\\rho_{;\\nu} u^{\\nu} + \\rho u^{\\nu}_{\\ \\ ;\\nu} = 0$$<br \/>\n\u3068\u3044\u3046<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u8cea\u91cf\uff09\u4fdd\u5b58\u5247<\/strong><\/span>\u3092\u4e0e\u3048\uff0c\u76f4\u4ea4\u3059\u308b\u6210\u5206\u306f<br \/>\n$$u^{\\mu}_{\\ \\ ; \\nu} u^{\\nu} = 0, \\quad\\mbox{i.e.,} \\quad \\frac{du^{\\mu}}{d\\tau} + \\varGamma^{\\mu}_{\\ \\ \\ \\nu\\lambda} u^{\\nu} u^{\\lambda} = 0$$\u3068\u3044\u3046<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306b\u306a\u308b\u3002<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u3046\u305a\u306a\u3057<\/strong><\/span>\uff0c\u3068\u306f<br \/>\n$$u_{\\mu, \\nu}\u00a0 -u_{\\nu, \\mu} = 0$$<br \/>\n\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308a\uff0c\u3053\u308c\u306b\u3088\u308a \\(u^{\\mu}\\) \u30923\u6b21\u5143\u8d85\u66f2\u9762\u306e\u6cd5\u7dda\u30d9\u30af\u30c8\u30eb\u306b\u3067\u304d\u308b\u3002\u3053\u306e\u8d85\u66f2\u9762\u3092 \\(x^0 = t = \\mbox{const.}\\) \u8d85\u66f2\u9762\u3068\u3059\u308c\u3070\uff0c\u4ee5\u4e0b\u306b\u8ff0\u3079\u308b\u3088\u3046\u306b\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u540c\u671f\u5316\u3055\u308c\u305f\u5171\u52d5\u5ea7\u6a19\u7cfb<\/strong><\/span>\u3092\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n<h3>\u540c\u671f\u5316\u3055\u308c\u305f\u5171\u52d5\u5ea7\u6a19\u7cfb<\/h3>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u540c\u671f\u5316\u3055\u308c\u305f\u5ea7\u6a19\u7cfb<\/strong><\/span>\u3068\u306f\uff0c\\(g_{00} = -1, \\ g_{0i} = 0\\) \u3067\u3042\u308b\u5ea7\u6a19\u7cfb\u306e\u3053\u3068\u3067\u7dda\u7d20\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<br \/>\n$$ds^2 = -dt^2 + g_{ij} dx^i dx^j$$<\/p>\n<p>\u307e\u305f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5171\u52d5\u5ea7\u6a19\u7cfb<\/strong><\/span>\u3068\u306f4\u5143\u901f\u5ea6\u306e\u7a7a\u9593\u6210\u5206\u304c\u30bc\u30ed\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u4e0a\u8a18\u306e\u540c\u671f\u5316\u3055\u308c\u305f\u5ea7\u6a19\u7cfb\u3067\u306f\uff08\\(g_{\\mu\\nu} u^{\\mu} u^{\\nu} = g_{00} \\left(u^0\\right)^2 = -1\\) \u3067\u3042\u308b\u304b\u3089\uff09<br \/>\n$$u^{\\mu} = (u^0, 0, 0, 0) = (1, 0, 0, 0)$$<\/p>\n<p>\u3061\u306a\u307f\u306b\uff0c\\(u^{\\mu}= (1, 0, 0, 0)\\) \u304c\u78ba\u304b\u306b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u3053\u3068\u306f\u5bb9\u6613\u3067\u3042\u308b\u3002\u540c\u671f\u5316\u3055\u308c\u305f\u5ea7\u6a19\u7cfb\u3067\u306f \\(\\varGamma^{\\mu}_{\\ \\ \\ 00} =0 \\) \u3067\u3042\u308b\u3053\u3068\uff0c\u304a\u3088\u3073 \\(u^{\\mu}\\) \u306e\u5168\u3066\u306e\u6210\u5206\u304c\u5b9a\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089<br \/>\n$$\\frac{du^{\\mu}}{d\\tau} + \\varGamma^{\\mu}_{\\ \\ \\ \\nu\\lambda} u^{\\nu} u^{\\lambda} = \\varGamma^{\\mu}_{\\ \\ \\ 00} =0$$<br \/>\n\u3068\u306a\u308a\uff0c\u78ba\u304b\u306b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3088\u306d\u3047\u3002<\/p>\n<h3><span id=\"i-2\">\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/span><\/h3>\n<p>\\(t = \\mbox{const.}\\) \u3067\u3042\u308b3\u6b21\u5143\u8d85\u66f2\u9762\u4e0a\u306e\u8fd1\u63a5\u3057\u305f2\u70b9\u9593\u306e\u8ddd\u96e2 \\(\\ell\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<br \/>\n$$\\ell \\equiv \\sqrt{g_{ij} dx^i dx^j}$$<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u30922\u70b9\u9593\u306e\u8ddd\u96e2\u306e\u6642\u9593\u5909\u5316\u304c\u8ddd\u96e2\u306b\u6bd4\u4f8b\u3059\u308b<br \/>\n$$\\frac{\\partial \\ell}{\\partial t} = \\dot{\\ell}\u00a0 \\propto \\ell$$<br \/>\n\u3068\u3044\u3046\u5f62\u306b\u66f8\u304f\u3068\uff0c\u3053\u306e\u95a2\u4fc2\u304c3\u6b21\u5143\u8d85\u66f2\u9762\u4e0a\u306e\u3069\u306e\u5730\u70b9\u306b\u304a\u3044\u3066\u3082\u6210\u308a\u7acb\u3064\u305f\u3081\u306b\u306f\uff0c\uff08\u5171\u52d5\u5ea7\u6a19\u5909\u4f4d \\(dx^i\\) \u306f\u6642\u9593\u7684\u306b\u5909\u5316\u3057\u306a\u3044\u304b\u3089\uff09<br \/>\n$$\\dot{g}_{ij} \\propto g_{ij} $$<br \/>\n\u3067\u3042\u308b\u3053\u3068\u304c\u5fc5\u8981\u3067\u3042\u308a\uff0c\u5f93\u3063\u3066<br \/>\n$$g_{ij} = a^2(t, \\vec{x}) \\gamma_{ij}(\\vec{x})$$<br \/>\n\u3068\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3053\u3053\u3067 \\(\\vec{x} = (x^1, x^2, x^3)\\) \u306f\u7a7a\u9593\u5ea7\u6a19\u306e\u307f\u3092\u8868\u3057\uff0c\u3057\u305f\u304c\u3063\u3066<br \/>\n$$\\dot{\\gamma}_{ij} = 0$$<\/p>\n<h3>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e \\(3 + 1\\) \u5206\u89e3<\/h3>\n<p>\\(g_{ij}\\) \u304a\u3088\u3073\u305d\u306e\u9006\u884c\u5217\u3067\u3042\u308b \\(g^{ij}\\) \u304b\u3089\u4ee5\u4e0b\u306e\u91cf\u3092\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n<p>$$K_{ij} \\equiv\u00a0 \\frac{1}{2} \\dot{g}_{ij},<br \/>\n\\quad K^i_{\\ \\ j} = g^{ik} K_{kj}$$<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u306e\u7d50\u679c\u3092\u4f7f\u3046\u3068<br \/>\n$$K^i_{\\ \\ j} = \\frac{\\dot{a}(t,\\vec{x})}{a(t,\\vec{x})} \\delta^i_{\\ \\ j} \\equiv H(t,\\vec{x}) \\delta^i_{\\ \\ j} $$<\/p>\n<p>\u307e\u305f\uff0c\\(g_{ij}\\) \u304b\u3089\u8a08\u7b97\u3055\u308c\u308b3\u6b21\u5143\u7a7a\u9593\u306e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092 \\({}^{(3)}\\! \\varGamma^{i}_{\\ \\\u00a0 jk}\\) \u3068\u66f8\u304f\u3068\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e \\(0i\\) \u6210\u5206 \\(G_{0i} = 8\\pi G T_{0i}\\) \u306f<br \/>\n$$K^{j}_{\\ \\ i, j} + {}^{(3)}\\! \\varGamma^{j}_{\\ \\\u00a0 ki} K^k_{\\ \\ j} -{}^{(3)}\\! \\varGamma^{j}_{\\ \\\u00a0 kj} K^k_{\\ \\ i} -K^{j}_{\\ \\ j,i} = 0$$<\/p>\n<p>\u3068\u66f8\u3051\u308b\u3002<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u306e\u7d50\u679c \\(K^i_{\\ \\ j} = H \\delta^i_{\\ \\ j}\\) \u3092\u4f7f\u3046\u3068<br \/>\n$$H_{, i} = 0 \\quad\\Rightarrow\\quad H = H(t), \\ \\ \\therefore\\ \\ a= a(t)$$<\/p>\n<p>\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u540c\u671f\u5316\u3055\u308c\u305f\u8a08\u91cf\u306f\uff0c\u6642\u9593 \\(t\\) \u306e\u307f\u306b\u4f9d\u5b58\u3059\u308b<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50<\/strong><\/span> \\(a(t)\\) \u3068\u7a7a\u9593\u5ea7\u6a19\u306e\u307f\u306b\u4f9d\u5b58\u3059\u308b \\(\\gamma_{ij}\\) \u3092\u4f7f\u3063\u3066<br \/>\n$$ds^2 = -dt^2 + a^2(t) \\,\\gamma_{ij} dx^i dx^j$$<br \/>\n\u3068\u66f8\u304b\u308c\u308b\u3068\u3053\u308d\u307e\u3067\u304d\u305f\u3002<\/p>\n<p>\\(\\gamma_{ij}\\)\u30923\u6b21\u5143\u7a7a\u9593\u306e\u8a08\u91cf\u3068\u3057\u3066\u8a08\u7b97\u3055\u308c\u308b3\u6b21\u5143\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u3092 \\({}^{(3)}\\! R^{i}_{\\ \\\u00a0 j}\\) \u3068\u66f8\u304f\u3068\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e \\(00\\) \u6210\u5206 \\(G_{00} = 8\\pi G T_{00}\\) \u306f<\/p>\n<p>$$\\frac{1}{2} \\left\\{\\left(K^i_{\\ \\ i}\\right)^2 -K^i_{\\ \\ j}K^j_{\\ \\ i} + \\frac{1}{a^2} {}^{(3)}\\! R^{i}_{\\ \\\u00a0 i}\\right\\} = 8\\pi G \\rho$$<br \/>\n<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u304b\u3089\u306e\u5e30\u7d50\u3067\u3042\u308b \\(\\displaystyle K^i_{\\ \\ j} = \\frac{\\dot{a}}{a} \\delta^i_{\\ \\ j}\\) \u3092\u4f7f\u3046\u3068<br \/>\n$$\\therefore\\ \\ \\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{1}{6a^2} {}^{(3)}\\! R^{i}_{\\ \\\u00a0 i} = \\frac{8\\pi G}{3} \\rho \\tag{1}$$<\/p>\n<p>\u307e\u305f trace-reversed \u306a\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e \\(ij\\) \u6210\u5206 \\(\\displaystyle R^i_{\\ \\ j} = 8\\pi G\\left(T^i_{\\ \\ j} -\\frac{1}{2} \\delta^i_{\\ \\ j} T^{\\mu}_{\\ \\ \\mu}\\right)\\) \u306f<br \/>\n$$\\dot{K}^i_{\\ \\ j} + K^k_{\\ \\ k} K^i_{\\ \\ j} +\\frac{1}{a^2}\u00a0 {}^{(3)}\\! R^{i}_{\\ \\\u00a0 j} = 4\\pi G \\rho \\delta^i_{\\ \\ j}$$<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u304b\u3089\u306e\u5e30\u7d50\u3067\u3042\u308b \\(\\displaystyle K^i_{\\ \\ j} = \\frac{\\dot{a}}{a} \\delta^i_{\\ \\ j}\\) \u3092\u4f7f\u3046\u3068<\/p>\n<p>$$\\frac{\\ddot{a}}{a}\\delta^i_{\\\u00a0 j} + 2 \\left(\\frac{\\dot{a}}{a}\\right)^2\\delta^i_{\\\u00a0 j} +\\frac{1}{a^2}\u00a0 {}^{(3)}\\! R^{i}_{\\\u00a0\u00a0 j} = 4\\pi G \\rho \\delta^i_{\\\u00a0 j} \\tag{2}$$<\/p>\n<p>\u3042\u3068\u5c11\u3057\u3002<\/p>\n<p>\\((1)\\) \u5f0f\u3068 \\((2)\\) \u5f0f\u306e\u30c8\u30ec\u30fc\u30b9\u3092\u4f7f\u3063\u3066 \\({}^{(3)}\\! R^{i}_{\\ \\\u00a0 i}\\)\u00a0 \u3092\u6d88\u53bb\u3059\u308b\u3068\uff0c\u7269\u8cea\u5bc6\u5ea6 \\(\\rho\\) \u306f\u6642\u9593\u3060\u3051\u306e\u95a2\u6570 \\(a(t)\\) \u3060\u3051\u3067\u66f8\u304b\u308c\u308b\u304b\u3089\uff0c \\(\\rho\\) \u3082\u307e\u305f\u6642\u9593\u3060\u3051\u306e\u95a2\u6570\u3067\u3042\u308b\u304c\u308f\u304b\u308b\u3002<br \/>\n$$\\rho = \\rho(t)$$<\/p>\n<p>\u305d\u3046\u3059\u308b\u3068\u00a0 \\((2)\\) \u5f0f\u304b\u3089\uff0c\u7a7a\u9593\u5ea7\u6a19\u3060\u3051\u306e\u95a2\u6570\u3067\u3042\u308b \\({}^{(3)}\\! R^{i}_{\\ \\\u00a0 j}\\) \u304c\u305d\u308c\u4ee5\u5916\u306f\u5168\u3066\u6642\u9593\u3060\u3051\u306e\u95a2\u6570\u3068\u91e3\u308a\u5408\u3046\u3053\u3068\u306b\u306a\u308b\u304b\u3089\uff0c\u8003\u3048\u3089\u308c\u308b\u53ef\u80fd\u6027\u306f\uff0c \\({}^{(3)}\\! R^{i}_{\\ \\\u00a0 j}\\) \u304c\u5b9a\u6570\u306e\u307f\u3092\u4f7f\u3063\u3066\u66f8\u3051\u308b\u3060\u308d\u3046\u3068\u3044\u3046\u3053\u3068\u3002\u307e\u305f\uff0c\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u306e\u30c7\u30eb\u30bf \\(\\delta^i_{\\ j}\\) \u306b\u3082\u6bd4\u4f8b\u3057\u3066\u3044\u308b\u306f\u305a\u3067\uff0c\u305d\u3046\u3067\u306a\u3044\u3068\u4ed6\u306e\u9805\u3068\u3064\u308a\u3042\u308f\u306a\u3044\u3002\u305d\u3053\u3067\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7a7a\u9593\u66f2\u7387\u5b9a\u6570<\/strong><\/span> \\(k\\) \u3092\u4f7f\u3063\u3066<br \/>\n$${}^{(3)}\\! R^{i}_{\\ \\\u00a0 j} = 2 k \\delta^i_{\\ j} \\tag{3}$$<br \/>\n\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u306f\u305a\u3060\uff0c\u3068\u3044\u3046\u3068\u3053\u308d\u307e\u3067\u308f\u304b\u3063\u305f\u3002<\/p>\n<p>\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u304c\u3053\u306e\u3088\u3046\u306b\u66f8\u3051\u308b3\u6b21\u5143\u7a7a\u9593\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5b9a\u66f2\u7387\u7a7a\u9593<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u306e\u3060\u304c\uff0c\u305d\u308c\u306f\u3068\u3082\u304b\u304f\uff0c\u3053\u306e \\((3)\\) \u5f0f\u3092\u3042\u3089\u305f\u3081\u3066 \\((1)\\) \u5f0f\u3068 \\((2)\\) \u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n<p>$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho \\tag{\\(1&#8217;\\)}$$<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{\\ddot{a}}{a} + 2 \\left(\\frac{\\dot{a}}{a}\\right)^2 +2 \\frac{k}{a^2} &amp;=&amp; 4\\pi G \\rho \\\\<br \/>\n\\frac{\\ddot{a}}{a} + 2 \\left(\\frac{8\\pi G}{3} \\rho \\right) &amp;=&amp; 4\\pi G \\rho \\\\<br \/>\n\\therefore \\ \\ \\frac{\\ddot{a}}{a} &amp;=&amp; -\\frac{4\\pi G}{3} \\rho \\tag{\\(2&#8217;\\)}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u307e\u305f\uff0c\\(\\rho_{;\\nu} u^{\\nu} + \\rho u^{\\nu}_{\\ ;\\nu} = 0 \\) \u306f<br \/>\n$$\\dot{\\rho} + 3\\frac{\\dot{a}}{a} \\rho = 0 \\tag{4}$$<br \/>\n\u3068\u306a\u308b\u3002<\/p>\n<h3>\u3053\u3053\u307e\u3067\u306e\u307e\u3068\u3081<\/h3>\n<p>\\((4)\\) \u5f0f\uff0c\\((2^{\\prime})\\) \u5f0f\uff0c\\((1^{\\prime})\\) \u5f0f\u3092\u3042\u3089\u305f\u3081\u3066\u307e\u3068\u3081\u3066\u66f8\u304f\u3068<\/p>\n<p>$$\\dot{\\rho} + 3\\frac{\\dot{a}}{a} \\rho = 0 \\tag{A}$$<br \/>\n$$\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} \\rho \\tag{B}$$<br \/>\n$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho \\tag{C}$$<br \/>\n\u6700\u5f8c\u306e\u5f0f\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n<p><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%83%8b%e3%83%a5%e3%83%bc%e3%83%88%e3%83%b3%e5%ae%87%e5%ae%99%e8%ab%96\/#i-5\">\u30cb\u30e5\u30fc\u30c8\u30f3\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u5f97\u3089\u308c\u305f\u5f0f\u3068\u5168\u304f\u540c\u3058<\/a>\u3067\u3042\u308b\u3053\u3068\u306b\u522e\u76ee\u305b\u3088\u3002<\/p>\n<h3>\u5727\u529b\u304c\u3042\u308b\u5b8c\u5168\u6d41\u4f53\u306e\u5834\u5408<\/h3>\n<p>\u4e0a\u8a18\u3067\u4eee\u5b9a\u3057\u305f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30c0\u30b9\u30c8\u6d41\u4f53<\/strong><\/span>\u306f\u5727\u529b \\(P\\) \u304c\u30bc\u30ed\u306e\u5b8c\u5168\u6d41\u4f53\u3067\u3042\u3063\u305f\u3002\u5727\u529b\u304c\u3042\u308b\u5b8c\u5168\u6d41\u4f53\u306e\u5834\u5408\u306f\uff0c\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb \\(T^{\\mu\\nu}\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n<p>$$T^{\\mu\\nu} = \\rho\u00a0 u^{\\mu} u^{\\nu} {\\color{red}{+ P( g^{\\mu\\nu} + u^{\\mu} u^{\\nu})}}$$<\/p>\n<p>\\((A)\\) \u5f0f\uff0c\\((B)\\) \u5f0f\uff0c\\((C)\\) \u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n<p>$$\\dot{\\rho} + 3\\frac{\\dot{a}}{a} (\\rho {\\color{red}{ + P}})= 0 \\tag{A&#8217;}$$<br \/>\n$$\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} (\\rho {\\color{red}{ + 3P}}) \\tag{B&#8217;}$$<br \/>\n$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho \\tag{C}$$<\/p>\n<h3>\u5b87\u5b99\u5b9a\u6570\u304c\u3042\u308b\u5834\u5408<\/h3>\n<p>\u30c0\u30b9\u30c8\u6d41\u4f53\u30d7\u30e9\u30b9\u5b87\u5b99\u5b9a\u6570 \\(\\Lambda\\) \u304c\u3042\u308b\u5834\u5408\u306e\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306f<br \/>\n$$G_{\\mu\\nu} {\\color{red}{+ \\Lambda g_{\\mu\\nu}}} = 8\\pi G T_{\\mu\\nu}$$<\/p>\n<p>$$\\therefore\\ \\\u00a0 G_{\\mu\\nu} = 8\\pi G \\left(T_{\\mu\\nu} {\\color{red}{-\\frac{\\Lambda}{8\\pi G} g_{\\mu\\nu} }}\\right)$$<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c\u5b87\u5b99\u5b9a\u6570\u306e\u5bc4\u4e0e\u3092\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6 \\(\\rho_{\\Lambda}\\)\uff0c\u5727\u529b \\(P_{\\Lambda}\\) \u306e\u5b8c\u5168\u6d41\u4f53\u3068\u307f\u306a\u3059\u3068<br \/>\n$$(\\rho_{\\Lambda} + P_{\\Lambda}) u_{\\mu} u_{\\nu} + P_{\\Lambda} g_{\\mu\\nu} = -\\frac{\\Lambda}{8\\pi G} g_{\\mu\\nu}$$\u3088\u308a<br \/>\n$$P_{\\Lambda} = -\\frac{\\Lambda}{8\\pi G}, \\ \\ \\rho_{\\Lambda}\u00a0 = \\frac{\\Lambda}{8\\pi G}$$<\/p>\n<p>\u3053\u306e\u5bc4\u4e0e\u3092\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e\u53f3\u8fba\u306b\u52a0\u3048\u308c\u3070\u3088\u3044\u3002\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n<p>$$\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} \\rho {\\color{red}{+ \\frac{\\Lambda}{3}}} \\tag{B&#8221;}$$<br \/>\n$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho {\\color{red}{+ \\frac{\\Lambda}{3}}} \\tag{C&#8217;}$$<\/p>\n<h3>\u53c2\u8003\uff1a\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u4ee3\u6570\u30b7\u30b9\u30c6\u30e0\u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b<\/h3>\n<p>&#8230; \u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e \\(3+1\\) \u5206\u89e3\u3092\u6d3b\u7528\u3057\u3066\u4eba\u529b\u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u4f8b\u3092\u6bb5\u968e\u3092\u8ffd\u3063\u3066\u8aac\u660e\u3057\u3066\u304d\u305f\u308f\u3051\u3060\u304c\uff0c\u53c2\u8003\u307e\u3067\u306b\uff0c\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u4ee3\u6570\u30b7\u30b9\u30c6\u30e0\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3059\u308b\u4f8b\u3092\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u306b\u307e\u3068\u3081\u3066\u304a\u3044\u305f\u3002<\/p>\n<ul>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e6%99%82%e7%a9%ba%e3%81%ae%e8%a1%a8%e3%81%97%e6%96%b9\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e4%bb%a3%e6%95%b0%e3%82%b7%e3%82%b9%e3%83%86%e3%83%a0%e3%81%a7%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9\/maxima-%e3%81%ae-ctensor-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/\">Maxima \u306e ctensor \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b<\/a><\/li>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e6%99%82%e7%a9%ba%e3%81%ae%e8%a1%a8%e3%81%97%e6%96%b9\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e4%bb%a3%e6%95%b0%e3%82%b7%e3%82%b9%e3%83%86%e3%83%a0%e3%81%a7%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/\">EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b<\/a><\/li>\n<\/ul>\n<h3>\u307e\u3068\u3081\uff1aFLRW \u306e\u5b9a\u7fa9<\/h3>\n<p>FLRW \u306e\u5b9a\u7fa9\u3068\u306f\u4f55\u304b\uff0c\u3069\u3093\u306a\u6761\u4ef6\u3092\u8ab2\u3057\u305f\u3089 FLRW \u306b\u306a\u308b\u306e\u304b\uff0c\u3068\u3044\u3046\u554f\u984c\u306b\u3064\u3044\u3066\uff0c\u304f\u3069\u304f\u3069\u3068\u8aac\u660e\u3057\u3066\u304d\u305f\u304c\uff0c\u7d50\u5c40\uff0c\u4e00\u8a00\u3067\u307e\u3068\u3081\u308b\u3068<\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u540c\u671f\u5316\u3055\u308c\u305f\u5171\u52d5\u5ea7\u6a19\u7cfb\u3067<\/strong><\/span><br \/>\n<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>$$\\dot{g}_{ij} \\propto g_{ij}$$\u3067\u3042\u308b\u3088\u3046\u306a\u6642\u7a7a\u306f FLRW \u6642\u7a7a\u3067\u3042\u308b<\/strong><\/span><\/p>\n<p>\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002\u7d50\u5c40\uff0cFLRW \u8a08\u91cf\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u3002<\/p>\n<p>$$ds^2 = -dt^2 + a^2(t) \\,\\gamma_{ij} dx^i dx^j$$<\/p>\n<p>\u3053\u3053\u3067\uff0c$a(t)$ \u306f\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\uff0c$\\gamma_{ij}$ \u306f3 \u6b21\u5143\u5b9a\u66f2\u7387\u7a7a\u9593\u306e\u8a08\u91cf\u3002\u5b9a\u66f2\u7387\u7a7a\u9593\u306e\u8a08\u91cf\u306e\u5177\u4f53\u7684\u306a\uff0c\u3044\u304f\u3064\u304b\u306e\u8868\u793a\u4f8b\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u3092\u53c2\u7167\uff1a<\/p>\n<ul>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%ae%87%e5%ae%99%e8%ab%96\/%e5%ae%9a%e6%9b%b2%e7%8e%87%e7%a9%ba%e9%96%93%e3%81%ae%e8%a8%88%e9%87%8f\/#i-2\" target=\"_blank\" rel=\"noopener\">\u5b9a\u66f2\u7387\u7a7a\u9593\u8a08\u91cf\u306e\u3044\u304f\u3064\u304b\u306e\u8868\u793a\u4f8b<\/a><\/li>\n<\/ul>\n<hr \/>\n<p>\u3061\u306a\u307f\u306b\uff0c\u3053\u306e\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\uff08\u79c1\u304c\u601d\u3046\u306b\uff0c\u3082\u3063\u3068\u3082\u4e00\u822c\u7684\u306a\u8a00\u3044\u56de\u3057\u3067\u3042\u308b\u3068\u601d\u308f\u308c\u308b\uff09\u4ee5\u4e0b\u306e\u5b9a\u7fa9\u3092\u79c1\u306a\u308a\u306b\u5c11\u3057\u5358\u7d14\u5316\u3057\uff0c\u5177\u4f53\u7684\u306a\u72b6\u6cc1\u3092\u8a2d\u5b9a\u3057\u3066\u8aac\u660e\u3057\u305f\u3082\u306e\u306b\u306a\u3063\u3066\u3044\u308b\u3002\uff08\u304b\u3048\u3063\u3066\u308f\u304b\u308a\u306b\u304f\u304f\u306a\u3063\u3066\u305f\u3089\uff0c\u3054\u3081\u3093\u306a\u3055\u3044\u3002\uff09<\/p>\n<p style=\"padding-left: 40px;\"><span style=\"color: #ff0000;\"><strong>The following set of properties is a necessary and sufficient condition for a space-time to be FLRW: <\/strong><\/span><\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li><span style=\"color: #ff0000;\"><strong>The metric obeys the Einstein equations with a perfect fluid source. <\/strong><\/span><\/li>\n<li><span style=\"color: #ff0000;\"><strong>The velocity field of the perfect fluid source has zero rotation, shear and acceleration.<\/strong><\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p style=\"padding-left: 40px;\">(&#8220;<a href=\"https:\/\/www.cambridge.org\/core\/books\/inhomogeneous-cosmological-models\/5262510EFFBD4CD0E23879ED87DE9765\">Inhomogeneous Cosmological Models<\/a>&#8221; by A. Krasinski \u3088\u308a\u5f15\u7528)<\/p>\n<hr \/>\n<p>\u4ee5\u4e0b\u306f\u6982\u8981\u8aac\u660e\u3002<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>FLRW<\/strong><\/span> \u3068\u306f\uff0c\u5b87\u5b99\u7a7a\u9593\u3092\u6e80\u305f\u3059\u5b8c\u5168\u6d41\u4f53\u306e4\u5143\u901f\u5ea6 \\(u^{\\mu}\\) \u306b\u3064\u3044\u3066<\/p>\n<p>\\begin{eqnarray}<br \/>\nu_{\\mu ;\\nu} &amp;\\equiv&amp; a_{\\mu} u_{\\nu} + \u00a0 u_{(\\mu ;\\nu)} + u_{[\\mu ;\\nu]} \\\\<br \/>\n&amp;=&amp; a_{\\mu} u_{\\nu} + \\frac{\\theta}{3}\u00a0 (g_{\\mu\\nu} + u_{\\mu} u_{\\nu}) + \\sigma_{\\mu\\nu} + \\omega_{\\mu\\nu}<br \/>\n\\end{eqnarray}<br \/>\n\u3068\u3057\u305f\u3068\u304d\u306b\uff0c<\/p>\n<ul>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>acceleration<\/strong><\/span> \\(a_{\\mu} = 0\\)<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>shear<\/strong><\/span> \\(\\sigma_{\\mu\\nu} = 0, \\ \\ (\\sigma_{\\mu\\nu} = \\sigma_{\\nu\\mu}, \\ \\ \\sigma^{\\mu}_{\\ \\ \\mu} = 0)\\)<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>vorticity<\/strong><\/span> \\(\\omega_{\\mu\\nu} = 0, \\ \\ (\\omega_{\\mu\\nu} = -\\omega_{\\nu\\mu})\\)<\/li>\n<\/ul>\n<p>\u3067\u3042\u308a\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>expansion<\/strong><\/span> \\(\\theta = u^{\\mu}_{\\ \\\u00a0 ;\\mu}\\) \u306e\u307f\u304c\u5b58\u5728\u3059\u308b\u6642\u7a7a\u306e\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u306e\u5b9a\u7fa9\u304b\u3089\u4f55\u304c\u8a00\u3048\u308b\u304b\u3068\u3044\u3046\u3068\uff0c<\/p>\n<ol>\n<li>\u307e\u305a acceleration \u304c\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u3067 \\(u^{\\mu}\\) \u306f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\u3002\n<ul>\n<li>\u79c1\u306e\u4e0a\u8a18\u306e\u8a18\u8ff0\u3067\u306f\u7c21\u5358\u306e\u305f\u3081\u30c0\u30b9\u30c8\u6d41\u4f53\u306b\u3057\u305f\u306e\u3067 \\(u^{\\mu}\\) \u306f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306b\u5f93\u3063\u305f\u3002<\/li>\n<li>\u8f3b\u5c04\u6d41\u4f53\u306e\u3088\u3046\u306b\u5727\u529b\u304c\u3042\u308b\u5834\u5408\u306f\uff0c\u6e2c\u5730\u7dda\u306b\u306a\u308b\u305f\u3081\u306b\u306f\u5727\u529b\u52fe\u914d\u304c\u30bc\u30ed\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308a\uff0c\u3053\u3053\u304b\u3089\u5727\u529b\u306e\u4e00\u69d8\u6027\u304c\u5c0e\u304b\u308c\uff0c\u72b6\u614b\u65b9\u7a0b\u5f0f\u304b\u3089\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u306e\u4e00\u69d8\u6027\u3082\u5c0e\u304b\u308c\u308b\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\u6b21\u306b vorticity \u304c\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u3067 \\(u^{\\mu}\\) \u306f3\u6b21\u5143\u8d85\u66f2\u9762\u306e\u6cd5\u7dda\u30d9\u30af\u30c8\u30eb\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/li>\n<li>\u4e0a\u8a182\u3064\u304b\u3089\uff0c\u540c\u671f\u5316\u3055\u308c\u305f\u5171\u52d5\u5ea7\u6a19\u7cfb\u3092\u3068\u308c\u308b\u3002<\/li>\n<li>expansion \u306e\u307f\u304c\u5b58\u5728\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u304b\u3089 extrinsic curvature \\(K^i_{\\ \\ j}\\) \u30823\u6b21\u5143\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u3082\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u306e\u30c7\u30eb\u30bf\u306b\u6bd4\u4f8b\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/li>\n<li>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u304b\u3089 \\(g_{ij} = a^2(t) \\gamma_{ij}\\) \u3068\u66f8\u3051\u3066<br \/>\n$$K^i_{\\ \\ j} = \\frac{\\dot{a}}{a} \\delta^i_{\\ \\ j}$$\u3068\u306a\u308b\u3002<\/li>\n<li>\u6700\u7d42\u7684\u306b \\(\\gamma_{ij}\\) \u3067\u8a18\u8ff0\u3055\u308c\u308b3\u6b21\u5143\u7a7a\u9593\u304c\u5b9a\u66f2\u7387\u7a7a\u9593\u306b\u306a\u308a\uff0c\u66f2\u7387\u5b9a\u6570 \\(k\\) \u3092\u4f7f\u3063\u3066<br \/>\n$${}^{(3)}\\! R^{i}_{\\ \\\u00a0 j} = 2 k \\delta^i_{\\ j} $$\u3068\u66f8\u3051\u308b\u3002<br \/>\n\u3053\u306e\u3088\u3046\u306a3\u6b21\u5143\u5b9a\u66f2\u7387\u7a7a\u9593\u306f\uff0c\u4e00\u69d8\u7b49\u65b9\u7a7a\u9593\u3068\u8a00\u3063\u305f\u308a\u307e\u305f maximally symmetric space \u306a\u3069\u3068\u8a00\u3063\u305f\u308a\u3059\u308b\u3002<\/li>\n<\/ol>\n<p>\u307e\u3041\uff0c\u3060\u3044\u305f\u3044\u3053\u306e\u3088\u3046\u306a\u9806\u756a\u304b\u3068\u3002<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u3046\u305a\u306a\u3057\u306e\u30c0\u30b9\u30c8\u6d41\u4f53\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u89e3\u304d\uff0c\u81a8\u5f35\u5b87\u5b99\u306e\u89e3\u3067\u3042\u308b\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc\u8a08\u91cf\uff08\u4ee5\u4e0b\uff0cFLRW \u8a08\u91cf\uff09\u3092\u5c0e\u51fa\u3059\u308b\u3002\uff08\u4ee5\u4e0b\uff0c\\(c = 1\\) \u3068\u3059\u308b\u3002\uff09<\/p>\n<p>\u306f\u306a\u304b\u3089\u4e00\u69d8\u7b49\u65b9\u6027\u3092\u6c7a\u3081\u6253\u3061\u3067\u4eee\u5b9a\u3059\u308b\u306e\u3082\u82b8\u304c\u306a\u3044\u306e\u3067\uff0c\u5c11\u3057\u3060\u3051\u4e00\u822c\u7684\u72b6\u6cc1\u8a2d\u5b9a\u304b\u3089\u306f\u3058\u3081\u3066\uff0c\u6700\u7d42\u7684\u306b\u4e00\u69d8\u7b49\u65b9\u306a3\u6b21\u5143\u7a7a\u9593\u3067\u3042\u308b\u3053\u3068\u304c\u5c0e\u304b\u308c\u308b\u3088\u3046\u306a\u66f8\u304d\u632f\u308a\u306b\u3057\u3066\u307f\u308b\u3002<\/p><p><a class=\"more-link btn\" href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%ae%87%e5%ae%99%e8%ab%96\/%e8%86%a8%e5%bc%b5%e5%ae%87%e5%ae%99%e3%81%ae%e8%a8%88%e9%87%8f%e3%81%ae%e5%b0%8e%e5%87%ba%e3%81%a8%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":1430,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1412","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1412","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=1412"}],"version-history":[{"count":48,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1412\/revisions"}],"predecessor-version":[{"id":1517,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1412\/revisions\/1517"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1430"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=1412"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}