{"id":3570,"date":"2022-08-25T14:37:31","date_gmt":"2022-08-25T05:37:31","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=3570"},"modified":"2023-03-14T16:52:04","modified_gmt":"2023-03-14T07:52:04","slug":"%e5%ae%87%e5%ae%99%e8%ab%96%e7%9a%84%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%82%92%e5%85%b1%e5%bd%a2%e6%99%82%e9%96%93%e3%82%92%e4%bd%bf%e3%82%8f%e3%81%9a%e3%81%ab%e5%b0%8e%e3%81%8f","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/3570\/","title":{"rendered":"\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u3092\u5171\u5f62\u6642\u9593\u3092\u4f7f\u308f\u305a\u306b\u5c0e\u304f"},"content":{"rendered":"

\u3053\u306e\u30da\u30fc\u30b8<\/strong><\/span><\/a>\u3068\u304b\uff0c\u3053\u306e\u30da\u30fc\u30b8<\/strong><\/span><\/a>\u3068\u304b\u3067\u306f\u5171\u5f62\u6642\u9593<\/strong><\/span> (conformal time) \\(\\eta\\) \u3067\u8868\u3057\u305f FLRW \u8a08\u91cf<\/p>\n

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

\u3092\u4f7f\u3063\u305f\u304c\uff0c\u901a\u5e38\u306e\u6642\u9593\u5ea7\u6a19 (cosmic time) \\(t\\) \u3067\u8868\u3057\u305f FLRW \u8a08\u91cf<\/p>\n

$$ ds^2 =\u00a0\u00a0 -dt^2 + g_{ij} \\,dx^i dx^j = -dt^2 + a^2(t) \\left\\{\\frac{dr^2}{1- k r^2} + r^2 (d\\theta^2+\\sin^2\\theta d\\phi^2)\\right\\}$$<\/p>\n

\u3092\u4f7f\u3063\u3066\u3082\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3051\u308b\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u304c<\/p>\n

\\begin{eqnarray}
\n\\omega &=& – k_{\\mu} u^{\\mu} \\\\
\n&=& – k_0\\, u^0 = \\frac{\\omega_c}{a} \\propto \\frac{1}{a(t)}
\n\\end{eqnarray}<\/p>\n

\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002\uff08\u3053\u308c\u306f\u6388\u696d\u306e\u7df4\u7fd2\u554f\u984c\u3002\uff09<\/p>\n

\n

\u307e\u305a\uff0c\u30cc\u30eb\u6761\u4ef6\uff1a<\/p>\n

$$g_{\\alpha\\beta} \\,k^{\\alpha} k^{\\beta} = – \\left(k^0\\right)^2 + g_{ij}\\, k^i k^j = 0$$<\/p>\n

\u307e\u305f\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{dk_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu}\\, k^{\\alpha} k^{\\beta}$$<\/p>\n

\u306e \\(\\mu = 0\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{dk_{0}}{dv} &=& \\frac{1}{2} g_{ij, 0}\\, k^i k^j \\\\
\n&=& \\frac{\\dot{a}}{a} g_{ij} \\,k^i k^j \\\\
\n&=& \\frac{\\dot{a}}{a} \\left(k^0\\right)^2\\\\
\n&=& \\frac{1}{a} \\frac{da}{dt} \\frac{dt}{dv} k^{{\\color{red}{0}}} \\\\
\n&=& – \\frac{1}{a} \\frac{da}{dv} k_{{\\color{blue}{0}}}
\n\\quad \\because\u00a0 k_{{\\color{blue}{0}}} = g_{00} k^{{\\color{red}{0}}} = -k^{{\\color{red}{0}}}\\\\
\n\\therefore\\ \\ \\frac{d}{dv} \\left( k_0\\, a \\right) &=& 0\\\\
\n\\therefore\\ \\ k_0\\, a &=& \\mbox{const.} \\equiv – \\omega_c \\\\
\n\\therefore\\ \\ k_0 &=& – \\frac{\\omega_c}{a}
\n\\end{eqnarray}<\/p>\n

\uff08\u5b66\u751f\u306e\u56de\u7b54\u3092\u307f\u308b\u3068\uff0c\u53f3\u8fba\u306e $\\frac{d}{dt}$ \u3092\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u5fae\u5206 $\\frac{d}{dv}$ \u306b\u306a\u304a\u3059\u3068\u3053\u308d\u306b\u6c17\u3065\u304b\u306a\u3044\u5b50\u304c\u591a\u3044\u3088\u3046\u3060\u3002\uff09<\/p>\n

\u5171\u52d5\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 \\(u^{\\mu}\\) \u306f<\/p>\n

$$u^{\\mu} = (1, 0,\u00a0 0, 0)$$\u3067\u3042\u308b\u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\omega &=& – k_{\\mu} u^{\\mu} \\\\
\n&=& – k_0\\, u^0 = \\frac{\\omega_c}{a} \\propto \\frac{1}{a(t)}
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u6642\u523b $t$ \u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u3092\u6642\u523b $t_0$ \u306b\u89b3\u6e2c\u3057\u305f\u6642\u306e\u8d64\u65b9\u504f\u79fb $z$ \u306f\u4ee5\u4e0b\u306e\u5f0f\uff1a<\/p>\n

$$1 + z = \\frac{\\omega(t)}{\\omega(t_0)} = \\frac{a(t_0)}{a(t)}= \\frac{a_0}{a(t)}$$<\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

\u3053\u306e\u30da\u30fc\u30b8\u3068\u304b\uff0c\u3053\u306e\u30da\u30fc\u30b8\u3068\u304b\u3067\u306f\u5171\u5f62\u6642\u9593 (conformal time) \\(\\eta\\) \u3067\u8868\u3057\u305f FLRW \u8a08\u91cf<\/p>\n

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

\u3092\u4f7f\u3063\u305f\u304c\uff0c\u901a\u5e38\u306e\u6642\u9593\u5ea7\u6a19 (cosmic time) \\(t\\) \u3067\u8868\u3057\u305f FLRW \u8a08\u91cf<\/p>\n

$$ ds^2 =\u00a0\u00a0 -dt^2 + g_{ij} \\,dx^i dx^j = -dt^2 + a^2(t) \\left\\{\\frac{dr^2}{1- k r^2} + r^2 (d\\theta^2+\\sin^2\\theta d\\phi^2)\\right\\}$$<\/p>\n

\u3092\u4f7f\u3063\u3066\u3082\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3051\u308b\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u304c<\/p>\n

\\begin{eqnarray} \\omega &=& – k_{\\mu} u^{\\mu} \\\\ &=& – k_0\\, u^0 = \\frac{\\omega_c}{a} \\propto \\frac{1}{a(t)} \\end{eqnarray}<\/p>\n

\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002\uff08\u3053\u308c\u306f\u6388\u696d\u306e\u7df4\u7fd2\u554f\u984c\u3002\uff09<\/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-3570","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\/3570","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=3570"}],"version-history":[{"count":13,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3570\/revisions"}],"predecessor-version":[{"id":3594,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3570\/revisions\/3594"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=3570"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=3570"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=3570"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}