{"id":4439,"date":"2022-12-19T10:20:25","date_gmt":"2022-12-19T01:20:25","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4439"},"modified":"2023-04-18T16:17:21","modified_gmt":"2023-04-18T07:17:21","slug":"flrw-%e5%ae%87%e5%ae%99%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%85%89%e5%ad%a6%e3%82%b9%e3%82%ab%e3%83%a9%e3%83%bc%e3%81%ae%e8%a8%88%e7%ae%97%e4%be%8b","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4439\/","title":{"rendered":"FLRW \u5b87\u5b99\u306b\u304a\u3051\u308b\u5149\u5b66\u30b9\u30ab\u30e9\u30fc\u306e\u8a08\u7b97\u4f8b"},"content":{"rendered":"

FLRW \u5b87\u5b99<\/p>\n

$$ds^2 = a^2(\\eta) \\left\\{ -d\\eta^2 + d\\chi^2 + \\sigma^2(\\chi)\\left( d\\vartheta^2 + \\sin^2\\vartheta d\\phi^2\\right)\\right\\}$$<\/p>\n

\u306b\u304a\u3051\u308b\u52d5\u5f84\u65b9\u5411\u306b\u4f1d\u64ad\u3059\u308b\u5149\u7dda<\/p>\n

$$ k^{\\mu} = \\left( k^0, k^1, 0, 0\\right)$$<\/p>\n

\u306e\u5149\u5b66\u30b9\u30ab\u30e9\u30fc $\\theta, \\ \\sigma$ \u306e\u8a08\u7b97\u4f8b\u3002\u6f14\u7fd2\u554f\u984c\u3068\u3057\u3066\u3002<\/p>\n

\u30cc\u30eb\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068<\/a> $k_0 = a^2 k^0 = \\mbox{const.}, \\\u00a0 k_1 = a^2 k^1 = \\mbox{const.}$\u00a0 \u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\u3053\u306e\u7d50\u679c\u3092\u6d3b\u7528\u3057\u3088\u3046\u3002$k_{\\mu} = (\\mbox{const.}, \\mbox{const.}, 0, 0)$ \u3067\u3042\u308b\u304b\u3089<\/p>\n

$$k_{\\alpha, \\beta} = 0, \\quad\\therefore\\ \\ k_{\\alpha; \\beta} = – \\varGamma^{\\lambda}_{\\ \\ \\ \\alpha\\beta} k_{\\lambda}= – \\frac{1}{2} k^{\\lambda} \\left( g_{\\lambda\\alpha, \\beta} + g_{\\lambda\\beta, \\alpha} – g_{\\alpha\\beta, \\lambda}\\right)$$ \u3068\u306a\u308b\u3002<\/p>\n

\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 $u^{\\mu}$ \u3092<\/p>\n

$$u^{\\mu} = \\left( u^0, 0, 0, 0\\right)$$<\/p>\n

\u3068\u3059\u308b\u3068\uff0c$u^{\\mu}$\u306b\u76f4\u4ea4\u3059\u308b\uff0c\u5149\u306e\u4f1d\u64ad\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb $\\gamma^{\\mu}$ \u306f<\/p>\n

$$\\gamma^{\\mu} = \\left(0, \\gamma^1, 0, 0\\right)$$<\/p>\n

\u5c04\u5f71\u6f14\u7b97\u5b50 ${}^{(2)}\\!P^{\\mu}_{\\ \\ \\nu} \\equiv \\delta^{\\mu}_{\\ \\ \\nu} + u^{\\mu} u_{\\nu} -\\gamma^{\\mu}\\gamma_{\\nu}$ \u306f<\/p>\n

$${}^{(2)}\\!P^{\\mu}_{\\ \\ 0} = 0, \\quad {}^{(2)}\\!P^{\\mu}_{\\ \\ 1} = 0,
\n\\quad {}^{(2)}\\!P^{\\mu}_{\\ \\ 2} = \\delta^{\\mu}_{\\ \\ 2}, \\quad {}^{(2)}\\!P^{\\mu}_{\\ \\ 3} = \\delta^{\\mu}_{\\ \\ 3}$$<\/p>\n

\u3067\u3042\u308b\u3053\u3068\u3092\u7c21\u5358\u306b\u78ba\u8a8d\u3067\u304d\u308b\u304b\u3089\uff0c\u5149\u5b66\u30b9\u30ab\u30e9\u30fc<\/a><\/strong><\/span>\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a<\/p>\n

\\begin{eqnarray}
\n{}^{(2)}\\!k_{\\mu ; \\nu} &\\equiv& k_{\\alpha ; \\beta} {}^{(2)}\\!P^{\\alpha}_{\\ \\ \\mu}{}^{(2)}\\!P^{\\beta}_{\\ \\ \\nu} \\\\
\nk_{\\alpha ; \\beta} &=& k_{\\alpha , \\beta} – \\varGamma^{\\lambda}_{\\ \\ \\alpha\\beta} k_{\\lambda} \\\\
\n&=& – \\frac{1}{2} k^{\\lambda} \\left( g_{\\lambda\\alpha, \\beta} + g_{\\lambda\\beta, \\alpha} – g_{\\alpha\\beta, \\lambda}\\right)
\n\\end{eqnarray}<\/p>\n

\u306e\u3046\u3061\uff0c<\/p>\n

$${}^{(2)}\\!k_{0 ; \\nu} = {}^{(2)}\\!k_{\\nu ; 0} = 0, \\quad {}^{(2)}\\!k_{1 ; \\nu} = {}^{(2)}\\!k_{\\nu ; 1} = 0$$<\/p>\n

\u3068\u306a\u308b\u304b\u3089\uff0c\u5fc5\u8981\u306a\u306e\u306f\u4ee5\u4e0b\u306e3\u3064\u3060\u3051<\/strong><\/span>\u3002\u8a08\u7b97\u6642\u9593\u306e\u7bc0\u7d04\uff01<\/p>\n

\\begin{eqnarray}
\n{}^{(2)}\\!k_{2;2}
\n&=& -\\frac{1}{2} k^{\\mu} \\left( g_{\\mu 2, 2} + g_{\\mu 2, 2} – g_{22, \\mu} \\right) \\\\
\n&=& \\frac{1}{2} \\left( k^0 g_{22, 0} + k^{1} g_{22, 1} \\right)\\\\
\n&=& \\frac{1}{2} \\frac{d}{dv} \\left(a(\\eta) \\sigma(\\chi)\\right)^2 \\\\
\n{}^{(2)}\\!k_{2;3} &=& \u00a0{}^{(2)}\\!k_{3;2} = -\\frac{1}{2} k^{\\mu} \\left( g_{\\mu 2, 3} + g_{\\mu 3, 2} – g_{23, \\mu} \\right) = 0 \\\\
\n{}^{(2)}\\!k_{3;3} &=& -\\frac{1}{2} k^{\\mu} \\left( g_{\\mu 3, 3} + g_{\\mu 3, 3} – g_{33, \\mu} \\right) \\\\
\n&=& \\frac{1}{2} \\left( k^0 g_{33, 0} + k^1 g_{33, 1}\\right) \\\\
\n&=& \\frac{1}{2} \\frac{d}{dv} \\left(a(\\eta) \\sigma(\\chi)\\right)^2 \\cdot \\sin^2\\vartheta
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066<\/p>\n

\\begin{eqnarray}
\n{}^{(2)}\\!k^2_{\\ \\ ;2}
\n&=& \\frac{1}{\\left(a(\\eta) \\sigma(\\chi)\\right)^2} \\frac{1}{2} \\frac{d}{dv} \\left(a(\\eta) \\sigma(\\chi)\\right)^2 \\\\
\n&=& \\frac{1}{a \\sigma}\u00a0 \\frac{d}{dv} \\left(a \\sigma\\right)\\\\
\n{}^{(2)}\\!k^3_{\\ \\ ;3} &=& \\frac{1}{a \\sigma}\u00a0 \\frac{d}{dv} \\left(a \\sigma\\right)
\n\\end{eqnarray}<\/p>\n

\u5149\u5b66\u30b9\u30ab\u30e9\u30fc\u306f<\/p>\n

\\begin{eqnarray}
\n\\theta &=& \\frac{1}{2} \u00a0{}^{(2)}\\!k^{\\mu}_{\\ \\ ;\\mu} = \\frac{1}{2}\\left(\u00a0{}^{(2)}\\!k^2_{\\ \\ ;2} + \u00a0{}^{(2)}\\!k^3_{\\ \\ ;3}\\right) \\\\
\n&=& \\frac{1}{a \\sigma} \\frac{d}{dv} \\left(a \\sigma\\right) \\\\
\n\\sigma^2 &=& \\frac{1}{2} \u00a0{}^{(2)}\\!k^{\\mu}_{\\ \\ ;\\nu}\u00a0{}^{(2)}\\!k^{\\nu}_{\\ \\ ;\\mu} – \\theta^2 \\\\
\n&=& \\frac{1}{2} \\left(\u00a0{}^{(2)}\\!k^2_{\\ \\ ;2} \u00a0{}^{(2)}\\!k^2_{\\ \\ ;2} +\u00a0{}^{(2)}\\!k^3_{\\ \\ ;3}\u00a0{}^{(2)}\\!k^3_{\\ \\ ;3} \\right)
\n– \\left\\{ \\frac{1}{a \\sigma} \\frac{d}{dv} \\left(a \\sigma\\right)\\right\\}^2\\\\
\n&=& \\left\\{ \\frac{1}{a \\sigma} \\frac{d}{dv} \\left(a \\sigma\\right)\\right\\}^2
\n-\\left\\{ \\frac{1}{a \\sigma} \\frac{d}{dv} \\left(a \\sigma\\right)\\right\\}^2\\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\u3061\u306a\u307f\u306b\uff0c\\(\\theta\\) \u3060\u3051\u3092\u6c42\u3081\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u884c\u5217\u5f0f<\/p>\n

\\begin{eqnarray}
\ng &\\equiv& \\det(g_{\\mu\\nu})\\\\
\n&=& g_{00}\\cdot g_{11}\\cdot g_{22}\\cdot g_{33} \\\\
\n&=& – a^8 \\sigma^4 \\sin^2\\vartheta
\n\\end{eqnarray}<\/p>\n

\u3092\u4f7f\u3063\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\nk^{\\mu}_{\\ \\ ;\\mu} &=& \\frac{1}{\\sqrt{-g}} \\left( \\sqrt{-g} k^{\\mu}\\right)_{, \\mu} \\\\
\n&=& \\frac{1}{a^4\\sigma^2 \\sin\\vartheta}
\n\\left\\{ \\left( a^4\\sigma^2 \\sin\\vartheta \\,k^0\\right)_{,0} +
\n\\left( a^4\\sigma^2 \\sin\\vartheta \\,k^1\\right)_{,1}\\right\\} \\\\
\n&=& \\frac{1}{a^4\\sigma^2} \\left\\{ a^2 k^0 \\left( a^2\\sigma^2\u00a0 \\right)_{,0} +
\na^2 k^1\\left( a^2\\sigma^2\\right)_{,1}\\right\\} \\\\
\n&=& \\frac{1}{a^2 \\sigma^2} \\frac{d}{dv} \\left(a \\sigma \\right)^2\\\\
\n&=& 2 \\frac{1}{a \\sigma} \\frac{d}{dv} \\left(a \\sigma \\right) \\\\
\n\\ \\\\
\n\\therefore\\ \\ \\theta &=& \\frac{1}{2}k^{\\mu}_{\\ \\ ;\\mu} \\\\
\n&=& \\frac{1}{a \\sigma} \\frac{d}{dv} \\left(a \\sigma \\right)
\n\\end{eqnarray}<\/p>\n

\u3068\u8a08\u7b97\u3067\u304d\u308b\u3002\u4ee5\u4e0b\u306e\u95a2\u4fc2\u3092\u4f7f\u3063\u3066\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u5fae\u5206\u306b\u3059\u308b\u306e\u304c\u5409\u3002\uff08\u3060\u304c\uff0c$\\sigma$ \u3082\u8a08\u7b97\u3059\u308b\u3068\u306a\u308b\u3068\uff0c\u3053\u308c\u3060\u3051\u3067\u306f\u3060\u3081\u3002\uff09<\/p>\n

$$k^0 \\frac{\\partial}{\\partial x^0} + k^1 \\frac{\\partial}{\\partial x^1} = \\frac{dx^{\\mu}}{dv} \\frac{\\partial}{\\partial x^{\\mu}} = \\frac{d}{dv}$$<\/p>\n

\n

\u8ffd\u8a18\uff1a<\/p>\n

$2 + 1+1$ \u5206\u89e3<\/a>\u306e\u3053\u3068\u3092\u77e5\u3089\u306a\u3051\u308c\u3070\uff0c<\/p>\n

\\begin{eqnarray}
\n\\theta &\\equiv& \\frac{1}{2} k^{\\mu}_{\\ \\ ;\\mu} \\\\
\n\\sigma^2 &\\equiv& \\frac{1}{2} k^{\\mu}_{\\ \\ ;\\nu}k^{\\nu}_{\\ \\ ;\\mu} – \\theta^2
\n\\end{eqnarray}<\/p>\n

\u3067\u3042\u308b\u304b\u3089\uff0c$k_{\\mu; \\nu}$ \u306e\u5168\u6210\u5206\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u306a\u308d\u3046\u3002\u5b9f\u76f4\u306b\u3084\u308b\u3068\u8a08\u7b97\u3059\u308b\u91cf\u3082\u591a\u304f\u306a\u308a\uff0c\u8a08\u7b97\u6642\u9593\u3082\u304b\u304b\u308b\u3002<\/p>\n

\\begin{eqnarray}
\nk_{\\alpha ; \\beta} &=& k_{\\alpha , \\beta} \u2013 \\varGamma^{\\lambda}_{\\ \\ \\alpha\\beta} k_{\\lambda} \\\\
\n&=& \u2013 \\frac{1}{2} k^{\\lambda} \\left( g_{\\lambda\\alpha, \\beta} + g_{\\lambda\\beta, \\alpha} \u2013 g_{\\alpha\\beta, \\lambda}\\right)
\n\\end{eqnarray}<\/p>\n

\u306e\u3046\u3061\uff0c\u4e0a\u3067\u8a08\u7b97\u3057\u3066\u3044\u306a\u3044\u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
\nk_{0;0} &=& -\\frac{1}{2} k^{\\mu} \\left(g_{\\mu 0, 0} +\u00a0 g_{\\mu 0, 0} – g_{0 0, \\mu}\\right) \\\\
\n&=& -\\frac{1}{2} k^0 g_{00, 0} \\\\
\n&=& k^0 a a’ \\\\
\nk_{0;1} &=& -\\frac{1}{2} k^{\\mu} \\left(g_{\\mu 0, 1} +\u00a0 g_{\\mu 1, 0} – g_{0 1, \\mu}\\right)\\\\
\n&=& -\\frac{1}{2} k^1 g_{11, 0} \\\\
\n&=& – k^1 a a’ \\\\
\nk_{0;2} &=&0 \\\\
\nk_{0;3} &=&0 \\\\
\nk_{1;1} &=& -\\frac{1}{2} k^{\\mu} \\left(g_{\\mu 1, 1} +\u00a0 g_{\\mu 1, 1} – g_{1 1, \\mu}\\right)\\\\
\n&=& \\frac{1}{2} k^0 g_{11,0} \\\\
\n&=& k^0 a a’ \\\\
\nk_{1;2} &=& 0\\\\
\nk_{1;3} &=& 0
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u3089\u306e\u6210\u5206\u3092\u4f7f\u3063\u3066 $\\mu, \\nu = 0, \\dots 3$ \u3067\u5149\u5b66\u30b9\u30ab\u30e9\u30fc\u3092\u8a08\u7b97\u3057\u3066\u3084\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n2 \\theta &=& k^{\\mu}_{\\ \\ ;\\mu} \\\\
\n&=& k^{0}_{\\ \\ ;0} +k^{1}_{\\ \\ ;1} + k^{2}_{\\ \\ ;2} + k^{3}_{\\ \\ ;3}\u00a0 \\\\
\n&=& – \\frac{1}{a^2} k^0 a a’ + \\frac{1}{a^2} k^0 a a’+ k^{2}_{\\ \\ ;2} + k^{3}_{\\ \\ ;3}\u00a0 \\\\
\n&=&\u00a0 k^{2}_{\\ \\ ;2} + k^{3}_{\\ \\ ;3}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n2\\sigma^2 &=& k^{\\mu}_{\\ \\ ;\\nu}k^{\\nu}_{\\ \\ ;\\mu} – 2 \\theta^2\\\\
\n&=& \\left(k^0_{\\ \\ ;0}\\right)^2 + 2 k^0_{\\ \\ ;1}k^1_{\\ \\ ;0} + \\left(k^1_{\\ \\ ;1}\\right)^2 + \\left(k^2_{\\ \\ ;2}\\right)^2 + \\left(k^3_{\\ \\ ;3}\\right)^2 – 2 \\theta^2 \\\\
\n&=& 2 \\left( \\frac{a’}{a}\\right)^2\\left\\{ (k^0)^2 – (k^1)^2\\right\\} + \\left(k^2_{\\ \\ ;2}\\right)^2 + \\left(k^3_{\\ \\ ;3}\\right)^2 – 2 \\theta^2 \\\\
\n&=&\u00a0 \\left(k^2_{\\ \\ ;2}\\right)^2 + \\left(k^3_{\\ \\ ;3}\\right)^2 – 2 \\theta^2 \\\\
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c\u5b9f\u8cea\u7684\u306b $k_{2;2}, k_{2;3}, k_{3;3}$ \u3060\u3051\u306e\u8a08\u7b97\u3067\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

FLRW \u5b87\u5b99<\/p>\n

$$ds^2 = a^2(\\eta) \\left\\{ -d\\eta^2 + d\\chi^2 + \\sigma^2(\\chi)\\left( d\\vartheta^2 + \\sin^2\\vartheta d\\phi^2\\right)\\right\\}$$<\/p>\n

\u306b\u304a\u3051\u308b\u52d5\u5f84\u65b9\u5411\u306b\u4f1d\u64ad\u3059\u308b\u5149\u7dda<\/p>\n

$$ k^{\\mu} = \\left( k^0, k^1, 0, 0\\right)$$<\/p>\n

\u306e\u5149\u5b66\u30b9\u30ab\u30e9\u30fc $\\theta, \\ \\sigma$ \u306e\u8a08\u7b97\u4f8b\u3002\u6f14\u7fd2\u554f\u984c\u3068\u3057\u3066\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[20],"tags":[],"class_list":["post-4439","post","type-post","status-publish","format-standard","hentry","category-rel-cosmo","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4439","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"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=4439"}],"version-history":[{"count":32,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4439\/revisions"}],"predecessor-version":[{"id":4759,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4439\/revisions\/4759"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=4439"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=4439"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=4439"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}