{"id":3080,"date":"2022-06-23T14:52:17","date_gmt":"2022-06-23T05:52:17","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3080"},"modified":"2024-08-05T11:32:31","modified_gmt":"2024-08-05T02:32:31","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%81%ae%e3%81%8a%e3%81%95%e3%82%89%e3%81%84","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%81%ae%e7%b5%b1%e4%b8%80%e7%9a%84%e7%90%86%e8%a7%a3\/%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%81%ae%e3%81%8a%e3%81%95%e3%82%89%e3%81%84\/","title":{"rendered":"\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u306e\u304a\u3055\u3089\u3044"},"content":{"rendered":"

\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u306f\uff0c\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u306f\u3069\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u304b\u3002\u4f8b\u3048\u3070\uff0c\u300c\u30ef\u30a4\u30f3\u30d0\u30fc\u30b0\u306e\u5b87\u5b99\u8ad6\uff08\u4e0a\uff09\u300d\uff08\u30ef\u30a4\u30f3\u30d0\u30fc\u30af\u8457\uff0c\u65e5\u672c\u8a55\u8ad6\u793e\uff09<\/a>\u306e 1.2 \u7bc0\u3092\u53c2\u8003\u306b\uff0cnotation \u3092\u82e5\u5e72\u5909\u66f4\u3057\uff0c\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u306e\u8aac\u660e\u3068\u30d1\u30e9\u30ec\u30eb\u306b\u306a\u308b\u3088\u3046\u306b\u8ad6\u8abf\u3092\u3042\u308f\u305b\u3066\u304a\u3055\u3089\u3044\u3059\u308b\u3002<\/p>\n

FLRW \uff08\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\uff09\u8a08\u91cf\u306f<\/p>\n

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

\u5149\u306e\u9032\u8def\u306f\u30cc\u30eb\u3067\u3042\u308b\u304b\u3089\uff0cFLRW \u6642\u7a7a\u306b\u304a\u3044\u3066\u52d5\u5f84\u65b9\u5411\u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u306b\u3064\u3044\u3066<\/p>\n

$$ds^2 = -dt^2 + a^2(t) \\frac{dr^2}{1 -k r^2} = 0$$
\n$$\\therefore\\ \\ dt = \\pm a(t) \\frac{dr}{\\sqrt{1 -k r^2}}$$<\/p>\n

\\(dt > 0\\) \u3067 \\(dr > 0\\) \u3068\u3057\uff0c\u300c\u5909\u6570\u5206\u96e2\u300d\u3057\u3066\u304a\u304f\u3068<\/p>\n

$$\\frac{dt}{a(t)} = \\frac{dr}{\\sqrt{1 -k r^2}}$$<\/p>\n

\\(r = r_1\\) \u304b\u3089 \\(t = t_1\\) \u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u304c\uff0c\\(r= r_2 (> r_1) \\) \u3078 \\(t=t_2 (> t_1)\\) \u306b\u5230\u7740\u3059\u308b\uff0c\u3068\u3059\u308b\u3068\uff0c<\/p>\n

$$\\int_{t_1}^{t_2} \\frac{dt}{a(t)} = \\int_{r_1}^{r_2} \\frac{dr}{\\sqrt{1 -k r^2}}$$<\/p>\n

\\(r = r_1\\) \u3067 \\(n\\) \u500b\u306e\u6ce2\u3092\u653e\u51fa\u3059\u308b\u3002\u6700\u5f8c\u306e\u6ce2\u304c\u653e\u51fa\u3055\u308c\u305f\u6642\u523b\u304c \\(t = t_1 + \\Delta t_1\\)\u3002<\/p>\n

\\(r = r_2\\) \u3078\u6700\u5f8c\u306e\u6ce2\u304c\u5230\u7740\u3059\u308b\u6642\u523b\u3092 \\(t = t_2 + \\Delta t_2\\) \u3068\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\int_{t_1+ \\Delta t_1}^{t_2+ \\Delta t_2} \\frac{dt}{a(t)} &=& \\int_{r_1}^{r_2} \\frac{dr}{\\sqrt{1 -k r^2}}\\\\
\n\\therefore\\ \\ \\int_{t_1}^{t_2} \\frac{dt}{a(t)} &=& \\int_{t_1+ \\Delta t_1}^{t_2+ \\Delta t_2} \\frac{dt}{a(t)} \\\\
\n&=& \\int_{t_1}^{t_2} \\frac{dt}{a(t)} + \\int_{t_2}^{t_2+ \\Delta t_2} \\frac{dt}{a(t)}\u00a0 -\\int_{t_1}^{t_1+ \\Delta t_1} \\frac{dt}{a(t)}\\\\
\n\\therefore\\ \\ \\int_{t_1}^{t_1+ \\Delta t_1} \\frac{dt}{a(t)} &=& \\int_{t_2}^{t_2+ \\Delta t_2} \\frac{dt}{a(t)}
\n\\end{eqnarray}<\/p>\n

\u6700\u5f8c\u306e\u5f0f\u306f \\(\\Delta t_1, \\Delta t_2\\) \u304c\u5341\u5206\u5c0f\u3055\u3044\u3068\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002\uff08\u5b9a\u7a4d\u5206\u304c\u9762\u7a4d\u306b\u306a\u308b\u3053\u3068\uff0c\u7a4d\u5206\u7bc4\u56f2\u304c\u5341\u5206\u72ed\u3051\u308c\u3070\u307b\u307c\u9577\u65b9\u5f62\u306e\u77ed\u518a\u306e\u9762\u7a4d\u3067\u3088\u3044\u3053\u3068\u3092\u8868\u3057\u3066\u3044\u308b\u3002\uff09<\/p>\n

$$\\frac{\\Delta t_1}{a(t_1)} = \\frac{\\Delta t_2}{a(t_2)} \\equiv \\Delta T$$<\/p>\n

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

$$\\Delta t_1 = a(t_1) \\Delta T, \\quad \\Delta t_2 = a(t_2) \\Delta T$$<\/p>\n

FLRW \u8a08\u91cf\u3067\u306f \\(g_{00} = -1\\) \u3067\u3042\u308b\u305f\u3081\uff0c\u7a7a\u9593\u5ea7\u6a19\u304c\u4e00\u5b9a\u306e\u5171\u52d5\u89b3\u6e2c\u8005\u306e\u56fa\u6709\u6642\u9593\u306f<\/p>\n

$$ds^2 = -d\\tau^2 = -dt^2$$<\/p>\n

\u3088\u308a \\(d\\tau = dt\\) \u3067\u3042\u308b\u3002<\/p>\n

\uff08\u91cd\u529b\u8d64\u65b9\u504f\u79fb<\/strong><\/span><\/a>\u306e\u5834\u5408\u3068\u540c\u69d8\u306b\uff09\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u306f\uff0c\u6ce2\u306e\u500b\u6570\u3092\u56fa\u6709\u6642\u9593 \\(d\\tau\\) \u3067\u5272\u3063\u305f\u3082\u306e\u3067\u3042\u308b<\/strong><\/span>\uff0c\u3068\u3059\u308b\u3002\uff08\u305f\u3060\u3057\uff0c\u3053\u3053\u3067\u306f \\(g_{00} = -1\\) \u3067\u3042\u308b\u305f\u3081 \\({\\color{red}{d\\tau}} = \\color{blue}{dt}\\) \u306a\u3093\u3067\u3059\u3051\u3069\u306d\u3002\u307e\u305f\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306b\u304a\u3051\u308b\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u306e\u3068\u304d\u306f \\(\\Delta t_1 = \\Delta t_2\\) \u3067\u3042\u3063\u305f\u304c\uff0c\u81a8\u5f35\u5b87\u5b99\u3067\u306f\u4e0a\u8a18\u306e\u3088\u3046\u306b\u4e39\u5ff5\u306b\u5c0e\u304f\u3068 \\(\\Delta t_1 \\neq \\Delta t_2\\) \u3067\u3042\u308b\u3053\u3068\u306b\u7559\u610f\u3059\u308b\u3002\uff09<\/p>\n

\\(r = r_1\\) \u306e\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570 \\(\\nu_1\\) \u306f<\/p>\n

$$\\nu_1 = \\frac{n}{\\Delta \\tau_1} = \\frac{n}{\\Delta t_1} = \\frac{n}{a(t_1) \\Delta T}$$<\/p>\n

\u3053\u306e\u540c\u3058\u5149\u3092 \\(r = r_2\\) \u306e\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u3068\uff0c\u305d\u306e\u632f\u52d5\u6570 \\(\\nu_2\\) \u306f<\/p>\n

$$\\nu_2 = \\frac{n}{\\Delta \\tau_2} = \\frac{n}{\\Delta t_2} = \\frac{n}{a(t_2) \\Delta T}$$<\/p>\n

$$\\therefore\\ \\ \\frac{\\nu_2}{\\nu_1} = \\frac{a(t_1)}{a(t_2)}$$<\/p>\n

\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3044\u3066\u306f \\(\\dot{a} > 0\\) \u3067\u3042\u308b\u304b\u3089\uff0c\\(a(t_1) < a(t_2)\\)\u3002\u5f93\u3063\u3066\uff0c<\/p>\n

$$\\frac{\\nu_2}{\\nu_1} = \\frac{a(t_1)}{a(t_2)} < 1$$<\/p>\n

\u3068\u306a\u308b\u3002\u4e00\u822c\u306b\u6642\u523b \\(t\\) \u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u3092\u73fe\u5728\u6642\u523b \\(t_0\\) \u306b\u89b3\u6e2c\u3059\u308b\u3068\uff0c\\(t_1 \\rightarrow t, \\ t_2 \\rightarrow t_0\\) \u3068\u7f6e\u304d\u63db\u3048\u3066<\/p>\n

$$\\frac{\\nu_0}{\\nu} = \\frac{a(t)}{a(t_0)} < 1$$<\/p>\n

\u3053\u306e\u5f0f\u3092\u6ce2\u9577\u3092\u7528\u3044\u3066\u66f8\u304d\u63db\u3048\u308b\u3002\u305d\u306e\u969b\uff0c\u52d5\u5f84\u65b9\u5411\u306e\u5ea7\u6a19\u5909\u4f4d \\(dr\\) \u306b\u5bfe\u5fdc\u3059\u308b\u7a7a\u9593\u7684\u9577\u3055 \\(d\\ell\\) \u306f<\/p>\n

$$d\\ell = \\sqrt{g_{11}} dr$$<\/p>\n

\u307e\u305f\u30cc\u30eb\u3067\u3042\u308b\u304b\u3089<\/p>\n

$$d\\ell = \\sqrt{g_{11}} dr = \\sqrt{-g_{00}} dt = dt$$<\/p>\n

\u3092\u7528\u3044\u308b\u3068\uff0c\u52d5\u5f84\u65b9\u5411\u306b\u9032\u3080\u5149\u306e\u901f\u3055 \\(v\\) \u306f<\/p>\n

$$v = \\frac{d\\ell}{d\\tau} = \\frac{dt}{d\\tau} = \\frac{dt}{dt} = 1 = c$$
\n\uff08\u3053\u3053\u3067\u306f \\(c = 1\\) \u3068\u3057\u3066\u3044\u305f\u3053\u3068\u306b\u6ce8\u610f\u3002\uff09<\/p>\n

\u3064\u307e\u308a\uff0c\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3044\u3066\u3082\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u901f\u3055\u306f \\(v = c\\) \u3067\u4e00\u5b9a\u3068\u306a\u308b\u3002<\/p>\n

\u3053\u308c\u3092\u4f7f\u3046\u3068\uff0c\u6ce2\u9577 \\(\\lambda\\) \u3068\u632f\u52d5\u6570 \\(\\nu\\) \u306e\u95a2\u4fc2\u306f<\/p>\n

$$\\lambda \\nu = c, \\quad \\therefore\\ \\ \\lambda \\propto \\frac{1}{\\nu}$$<\/p>\n

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

$$\\frac{\\lambda_0}{\\lambda} = \\frac{\\nu}{\\nu_0} = \\frac{a(t_0)}{a(t)}$$<\/p>\n

\u8d64\u65b9\u504f\u79fb \\(z\\) \u3092<\/p>\n

$$z \\equiv \\frac{\\lambda_0 -\\lambda}{\\lambda} = \\frac{\\lambda_0}{\\lambda} -1$$<\/p>\n

\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3068\uff0c\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb \\(z\\) \u306f<\/p>\n

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

\u3068\u306a\u308b\u3002\u3061\u306a\u307f\u306b\uff0cconformal time \u5171\u5f62\u6642\u9593 \\(\\eta\\) \u3092\u4f7f\u3063\u305f\u8a08\u91cf<\/p>\n

$$ds^2 = a^2(\\eta) \\left\\{-d\\eta^2 +\u00a0 \\frac{dr^2}{1 -k r^2} + r^2 \\left(d\\theta^2 + \\sin^2\\theta \\,d\\phi^2\\right)\\right\\}$$<\/p>\n

\u304b\u3089\u59cb\u3081\u3066\u3082\uff0c\u30cc\u30eb\u6761\u4ef6\u304b\u3089<\/p>\n

$$d\\eta = \\frac{dr}{1-kr^2} \\quad\\Rightarrow\\ \\ \\ \\Delta \\eta_1 = \\Delta \\eta_2$$\u304a\u3088\u3073\uff0c\u56fa\u6709\u6642\u9593\u304c \\(d\\tau = a(\\eta) d\\eta\\) \u3068\u306a\u308b\u3053\u3068\u304b\u3089\uff0c\u540c\u69d8\u306b\u3057\u3066\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u306e\u5f0f<\/p>\n

$$1 + z = \\frac{a(\\eta_0)}{a(\\eta)}$$<\/p>\n

\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u306e\u8aac\u660e\u306b\u304a\u3044\u3066\u3082\uff0c\u5149\u306e\u4f1d\u64ad\u306f\u30cc\u30eb\u6e2c\u5730\u7dda<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u306f\u5fc5\u9808\u304b\u3068\u601d\u3046\u304c\uff0c\u4e0a\u8a18\u306e\u8aac\u660e\u3067\u306f\u300c\u30cc\u30eb\u6761\u4ef6<\/strong><\/span>\u300d\u306e\u307f\u3092\u4f7f\u7528\u3057\uff0c\u300c\u6e2c\u5730\u7dda<\/strong><\/span>\u300d\u3067\u3042\u308b\u3053\u3068\u306f\u3042\u304b\u3089\u3055\u307e\u306b\u306f\u4f7f\u3063\u3066\u3044\u306a\u3044\u3088\u3046\u306b\u601d\u308f\u308c\u308b\u3002\uff08\u6e2c\u5730\u7dda\u3067\u3042\u308b\u3053\u3068\u306f\u5149\u306e\u4f1d\u64ad\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u3067\u3042\u308b\u3068\u601d\u3046\u306e\u3060\u304c\uff0c\u3069\u3053\u3067\u6e2c\u5730\u7dda\u6761\u4ef6\u3092\u4f7f\u3063\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u308b\u306e\u304b\uff0c\u308f\u304b\u308a\u307e\u3059\u304b\uff1f\uff09<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u306f\uff0c\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb\u306f\u3069\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u304b\u3002\u4f8b\u3048\u3070\uff0c\u300c\u30ef\u30a4\u30f3\u30d0\u30fc\u30b0\u306e\u5b87\u5b99\u8ad6\uff08\u4e0a\uff09\u300d\uff08\u30ef\u30a4\u30f3\u30d0\u30fc\u30af\u8457\uff0c\u65e5\u672c\u8a55\u8ad6\u793e\uff09\u306e 1.2 \u7bc0\u3092\u53c2\u8003\u306b\uff0cnotation \u3092\u82e5\u5e72\u5909\u66f4\u3057\uff0c\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u306e\u8aac\u660e\u3068\u30d1\u30e9\u30ec\u30eb\u306b\u306a\u308b\u3088\u3046\u306b\u8ad6\u8abf\u3092\u3042\u308f\u305b\u3066\u304a\u3055\u3089\u3044\u3059\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":3059,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-3080","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\/3080","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=3080"}],"version-history":[{"count":17,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3080\/revisions"}],"predecessor-version":[{"id":9330,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3080\/revisions\/9330"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3059"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=3080"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}