{"id":3072,"date":"2022-06-23T14:39:36","date_gmt":"2022-06-23T05:39:36","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3072"},"modified":"2022-08-09T15:51:35","modified_gmt":"2022-08-09T06:51:35","slug":"%e9%87%8d%e5%8a%9b%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\/%e9%87%8d%e5%8a%9b%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":"\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u306e\u304a\u3055\u3089\u3044"},"content":{"rendered":"<p>\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u306f\uff0c\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u306f\u3069\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u304b\u3002\u4f8b\u3048\u3070<a id=\"yui_3_17_2_1_1655961827632_1394\" href=\"https:\/\/www.shokabo.co.jp\/mybooks\/ISBN978-4-7853-2315-8.htm\" target=\"_blank\" rel=\"noopener\">\u300c\u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6\u300d\uff08\u5185\u5c71\u9f8d\u96c4\u8457\uff0c\u88f3\u83ef\u623f\uff09<\/a>\u306e\u00a737. \u3092\u53c2\u8003\u306b\uff0cnotation \u3092\u82e5\u5e72\u5909\u66f4\u3057\u3066\u304a\u3055\u3089\u3044\u3059\u308b\u3002\u307e\u305f\uff0c\u3053\u306e\u30c6\u30ad\u30b9\u30c8\u4e2d\u306e\u9593\u9055\u3044\u306b\u3064\u3044\u3066\u3082\u6307\u6458\u3057\u3066\u304a\u304f\u3002<!--more--><\/p>\n<p dir=\"ltr\">\u3042\u3089\u304b\u3058\u3081\u5ff5\u306e\u305f\u3081\u306b\u8a00\u3063\u3066\u304a\u304f\u304c\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4ee5\u4e0b\u306e\u6a19\u6e96\u7684\u306a\u8aac\u660e\u306f<span style=\"color: #ff0000;\">\u91cd\u529b\u5834\u4e2d\u306b\u304a\u3044\u3066\u5149\u6e90\u3082\u89b3\u6e2c\u8005\u3082\u9759\u6b62\u3057\u3066\u3044\u308b\u5834\u5408\u306b\u306e\u307f\u6709\u52b9\u3067\u3042\u308b<\/span>\u3002\u5149\u6e90\u3042\u308b\u3044\u306f\u89b3\u6e2c\u8005\u304c\u91cd\u529b\u5834\u4e2d\u3092\u904b\u52d5\u3059\u308b\u5834\u5408\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8aac\u660e\u306e\u307e\u307e\u3067\u306f\u6709\u52b9\u6027\u3092\u793a\u3059\u3053\u3068\u304c\u56f0\u96e3\u3067\u3042\u308a\uff0c\u5225\u9014\u7406\u5c48\u304c\u5fc5\u8981\u306b\u306a\u308b\u3002<\/strong><\/span><\/p>\n<p dir=\"ltr\">\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306f<\/p>\n<p id=\"yui_3_17_2_1_1655961827632_1420\" dir=\"ltr\">\\begin{eqnarray}<br \/>\nds^2 &amp;=&amp; g_{00} dt^2 + g_{11} dr^2 + r^2 \\left(d\\theta^2 + \\sin^2 \\theta \\,d\\phi^2 \\right)\\\\<br \/>\n&amp;=&amp; &#8211; \\left(1 &#8211; \\frac{r_g}{r}\\right)dt^2 + \\frac{1}{1 &#8211; \\frac{r_g}{r}}dr^2 + r^2 \\left(d\\theta^2 + \\sin^2 \\theta \\,d\\phi^2 \\right)<br \/>\n\\end{eqnarray}<\/p>\n<p dir=\"ltr\">\u5185\u5c71\u672c\u3067\u306f\uff0c\u592a\u967d\u8868\u9762 \\(r = r_{\\odot}\\) \u304b\u3089\u306e\u5149\u3092\u5730\u7403\u4f4d\u7f6e \\(r = r_E\\) \u3067\u89b3\u6e2c\u3059\u308b\uff0c\u3068\u6c7a\u3081\u6253\u3061\u3057\u3066\u3044\u308b\u304c\uff0c\u3053\u3053\u3067\u306f\u4e00\u822c\u306b\u91cd\u529b\u6e90\u8fd1\u508d \\(r = r_1\\) \u304b\u3089\u306e\u5149\u3092\u96e2\u308c\u305f\u5834\u6240 \\(r = r_2\\,\u00a0 (&gt; r_1)\\) \u3067\u89b3\u6e2c\u3059\u308b\uff0c\u3068\u82e5\u5e72 notation \u3092\u5909\u66f4\u3057\u3066\u8aac\u660e\u3092\u7d9a\u3051\u308b\u3002<\/p>\n<p dir=\"ltr\">\u307e\u305f\uff0c\u5185\u5c71\u306f\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5149\u306e\u9032\u8def\u306f\u30cc\u30eb\u6e2c\u5730\u7dda<\/strong><\/span>\u300d\u3068\u660e\u8a00\u3057\u3066\u3044\u308b\u304c\uff0c\u5b9f\u969b\u306b\u306f\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30cc\u30eb\u6761\u4ef6<\/strong><\/span>\u300d\u306e\u307f\u3092\u4f7f\u7528\u3057\uff0c\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\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\u3061\u3087\u3063\u3068\u898b\u629c\u3051\u305a\u306b\u3044\u307e\u3059\u3002\uff09<\/p>\n<p dir=\"ltr\">\u30cc\u30eb\u3067\u3042\u308b\u304b\u3089\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306b\u304a\u3044\u3066\u52d5\u5f84\u65b9\u5411\u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u306b\u3064\u3044\u3066<\/p>\n<p dir=\"ltr\">$$ds^2 = &#8211; \\left(1 &#8211; \\frac{r_g}{r}\\right) dt^2 + \\frac{1}{1 &#8211; \\frac{r_g}{r}} dr^2 = 0$$<br \/>\n$$ \\therefore\\ \\ dt = \\pm \\frac{1}{1 &#8211; \\frac{r_g}{r}} dr$$<\/p>\n<p dir=\"ltr\">\u5149\u306e\u9032\u8def\u304c\u5916\u5411\u304d\u3067\u3042\u308b\u3068\u3059\u308b\u3068\uff0c\\(dt &gt; 0\\)\uff08\u6642\u9593\u304c\u7d4c\u904e\u3059\u308b\uff09\u306e\u3068\u304d \\(dr &gt; 0\\) \uff08\\(r\\) \u306e\u5024\u304c\u5927\u304d\u304f\u306a\u308b\uff09\u306e\u3067\u5fa9\u53f7\u306e \\(+\\) \u5074\u306b\u306a\u308a\uff0c<\/p>\n<p dir=\"ltr\">$$dt = \\frac{1}{1 &#8211; \\frac{r_g}{r}} dr$$<\/p>\n<p dir=\"ltr\">\\(r = r_1\\) \u304b\u3089 \\(t = t_1\\) \u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u304c\uff0c\\(r = r_2\\) \u306b \\(t = t_2\\) \u306b\u5230\u7740\u3059\u308b\uff0c\u3068\u3059\u308b\u3068\uff0c<\/p>\n<p dir=\"ltr\">$$\\int_{t_1}^{t_2} dt = \\int_{r_1}^{r_2} \\frac{1}{1 &#8211; \\frac{r_g}{r}} dr$$<\/p>\n<p dir=\"ltr\">\\(r = r_1\\) \u3067 \\(n\\) \u500b\uff08\\(n\\) \u56de\u632f\u52d5\u5206\uff0c\u3042\u308b\u3044\u306f \\(n\\) \u6ce2\u9577\u5206\uff09\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<p dir=\"ltr\">\\(r = r_2\\) \u306b\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<p dir=\"ltr\"><span id=\"selectionBoundary_1654738621317_6068952409311168\"><\/span>\\begin{eqnarray}<br \/>\n\\int_{t_1+ \\Delta t_1}^{t_2+ \\Delta t_2} dt &amp;=&amp; \\int_{r_1}^{r_2} \\frac{1}{1 &#8211; \\frac{r_g}{r}} dr\\\\<br \/>\n\\therefore\\ \\ \\int_{t_1}^{t_2} dt &amp;=&amp; \\int_{t_1+ \\Delta t_1}^{t_2+ \\Delta t_2} dt \\\\<br \/>\nt_2 &#8211; t_1 &amp;=&amp; \\left(t_2+ \\Delta t_2 \\right) &#8211; \\left(t_1+ \\Delta t_1 \\right) \\\\<br \/>\n\\therefore\\ \\ \\Delta t_1 &amp;=&amp; \\Delta t_2 \\equiv \\Delta t<br \/>\n\\end{eqnarray}<\/p>\n<p dir=\"ltr\">\u5ea7\u6a19\u6642\u9593\u306e\u9593\u9694\u304c\u5909\u308f\u3089\u305a\u7b49\u3057\u304f\u306a\u308b\uff08\\( \\Delta t_1 = \\Delta t_2\\)\uff09\u306e\u306f\uff0c\u76f4\u63a5\u7684\u306b\u306f \\(g_{00}\\) \u304a\u3088\u3073 \\(g_{11}\\)\uff0c\u3072\u308b\u304c\u3048\u3063\u3066\u306f\u5168\u3066\u306e\u8a08\u91cf \\(g_{\\mu\\nu}\\) \u304c\u6642\u9593\u5ea7\u6a19 \\(t\\) \u306b\u3088\u3089\u306a\u3044\u3053\u3068\uff0c\u3064\u307e\u308a<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9759\u7684\u306a\u6642\u7a7a<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u306b\u3088\u308b\u3002\u307e\u305f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong><span style=\"color: #ff0000;\">\u5149\u6e90\u304a\u3088\u3073\u89b3\u6e2c\u8005\u304c\u9759\u6b62\u3057\u3066\u3044\u308b\uff08\u6642\u9593\u304c\u7d4c\u904e\u3057\u3066\u3082 \\(r = r_1\\) \u304a\u3088\u3073 \\(r = r_2\\) \u306b\u5c45\u7d9a\u3051\u308b\uff09\u3068\u3044\u3046\u4eee\u5b9a\u304c\u91cd\u8981\u3067\u3042\u308b<\/span>\u3002<\/strong><\/span><\/p>\n<p dir=\"ltr\">\u5149\u6e90\u307e\u305f\u306f\u89b3\u6e2c\u8005\u304c\u904b\u52d5\u3057\u3066\u3044\u308b\u5834\u5408\u306b\u306f\uff0c\\( \\Delta t_1 = \\Delta t_2\\) \u304c\u6210\u7acb\u3059\u308b\u6839\u62e0\u304c\u5d29\u308c\u308b\u305f\u3081\uff0c\u3053\u306e\u3088\u3046\u306a\u6a19\u6e96\u7684\u306a\u8aac\u660e\u81ea\u4f53\u304c\u6709\u52b9\u3067\u306f\u306a\u3044\uff0c\u3068\u8003\u3048\u308b\u3002<\/p>\n<p dir=\"ltr\">\u5ea7\u6a19\u6642\u9593\u304c \\(\\Delta t\\) \u3060\u3051\u7d4c\u904e\u3059\u308b\u9593\u306b\uff0c\\(r = r_1\\) \u306e\u9759\u6b62\u89b3\u6e2c\u8005\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u56fa\u6709\u6642\u9593<\/strong><\/span>\u306f<\/p>\n<p dir=\"ltr\">$$ds^2 = &#8211; d\\tau^2 = &#8211; \\left( 1 &#8211; \\frac{r_g}{r}\\right) dt^2$$<\/p>\n<p dir=\"ltr\">\u3088\u308a\uff0c$$\\Delta \\tau_1 = \\sqrt{1 &#8211; \\frac{r_g}{r_1}} \\Delta t_1 = \\sqrt{1 &#8211; \\frac{r_g}{r_1}} \\Delta t$$\u3060\u3051\u7d4c\u904e\u3057\uff0c<\/p>\n<p dir=\"ltr\">\\(r = r_2\\) \u306e\u9759\u6b62\u89b3\u6e2c\u8005\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u56fa\u6709\u6642\u9593<\/strong><\/span>\u306f$$\\Delta \\tau_2 = \\sqrt{1 &#8211; \\frac{r_g}{r_2}} \\Delta t_2 =\\sqrt{1 &#8211; \\frac{r_g}{r_2}} \\Delta t$$\u3060\u3051\u7d4c\u904e\u3059\u308b\u3002<\/p>\n<p dir=\"ltr\">\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u306f\uff0c\u7570\u306a\u308b\u5834\u6240\u306b\u304a\u3051\u308b\u5ea7\u6a19\u6642\u9593\u9593\u9694\u304c\u7b49\u3057\u3044 \\( \\Delta t_1 = \\Delta t_2 \\) \u3068\u3044\u3046\u4e8b\u5b9f\u3042\u308a\u304d\u304b\u3089\u306f\u3058\u307e\u3063\u3066\uff0c\u305d\u308c\u305e\u308c\u306e\u56fa\u6709\u6642\u9593\u9593\u9694\u304c\u4e0a\u8a18\u306e\u3088\u3046\u306b\u306a\u308b\u3068\u3059\u308b\u8aac\u660e\u3082\u591a\u304f\u898b\u304b\u3051\u3089\u308c\u308b\u304c\uff0c\\( \\Delta t_1 = \\Delta t_2\\) \u306f\uff08\u308f\u305f\u304f\u3057\u7684\u306b\u306f\u305d\u3093\u306a\u306b\u81ea\u660e\u306a\u3053\u3068\u3067\u306f\u306a\u304b\u3063\u305f\u306e\u3067\uff09<span style=\"font-family: helvetica, arial, sans-serif;\"><strong><span style=\"color: #ff0000;\">\u5149\u6e90\u304a\u3088\u3073\u89b3\u6e2c\u8005\u304c\u9759\u6b62\u3057\u3066\u3044\u308b\u3068\u3044\u3046\u4eee\u5b9a\u306e\u3082\u3068\u3067\u5c0e\u304b\u308c\u308b\u306e\u3060<\/span>\u3068\u3044\u3046\u3053\u3068\u3092\u5f37\u8abf\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3068\u8003\u3048\u308b\u3002<\/strong><\/span><\/p>\n<p dir=\"ltr\">\u3055\u3066\uff0c\u3053\u3053\u304c\u5927\u4e8b\u306a\u30dd\u30a4\u30f3\u30c8\u3060\u304c\uff0c<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u306f\uff0c\u6ce2\u306e\u500b\u6570\u3092<span style=\"color: #0000ff;\">\uff08\u5ea7\u6a19\u6642\u9593\u9593\u9694 \\(\\color{blue}{dt}\\) \u3067\u5272\u3063\u305f\u3082\u306e\u3067\u306f\u306a\u304f\u3066\uff09<\/span>\u56fa\u6709\u6642\u9593\u9593\u9694 \\(\\color{red}{d\\tau}\\) \u3067\u5272\u3063\u305f\u3082\u306e\u3067\u3042\u308b\uff01\u3068\u3059\u308b\u3002<\/strong><\/span><\/p>\n<p dir=\"ltr\">\\(r = r_1\\) \u306e\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570 \\(\\nu_1\\) \u306f<\/p>\n<p dir=\"ltr\">$$\\nu_1 = \\frac{n}{\\Delta \\tau_1}$$<\/p>\n<p dir=\"ltr\">\u3053\u306e\u540c\u3058\u5149\u3092 \\(r = r_2\\) \u306e\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u3068\uff0c\u305d\u306e\u632f\u52d5\u6570 \\(\\nu_2\\) \u306f<\/p>\n<p dir=\"ltr\">$$\\nu_2 = \\frac{n}{\\Delta \\tau_2}$$<\/p>\n<p dir=\"ltr\">$$\\therefore\\ \\ \\frac{\\nu_2}{\\nu_1} = \\frac{\\Delta \\tau_1}{\\Delta \\tau_2} = \\frac{\\sqrt{1 &#8211; \\frac{r_g}{r_1}} \\Delta t}{\\sqrt{1 &#8211; \\frac{r_g}{r_2}} \\Delta t} = \\frac{\\sqrt{1 &#8211; \\frac{r_g}{r_1}} }{\\sqrt{1 &#8211; \\frac{r_g}{r_2}}} &lt; 1\\ \\ (\\because r_1 &lt; r_2)$$<\/p>\n<p dir=\"ltr\">\u7279\u306b \\(r_2 \\gg r_1\\) \u306e\u5834\u5408\u306f\uff0c\\(r_2 \\rightarrow \\infty\\) \u3068\u3057\u3066<\/p>\n<p dir=\"ltr\">$$\\frac{\\nu_2}{\\nu_1} \\simeq \\sqrt{1 &#8211; \\frac{r_g}{r_1}}$$<\/p>\n<p dir=\"ltr\">\u3053\u308c\u304c\u5185\u5c71\u672c\u306e (37.5) \u5f0f\u3002<\/p>\n<p dir=\"ltr\">\u3053\u3053\u304b\u3089\u304c\uff0c\u3053\u306e\u672c\u306e\u304a\u304b\u3057\u306a\u70b9\u3002\u3053\u306e (37.5) \u5f0f\u3092\u6ce2\u9577\u3092\u7528\u3044\u3066\u66f8\u304d\u63db\u3048\u308b\u306e\u3060\u304c\uff0c\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u307e\u305a\uff0c\u5149\u306e \\(r\\) \u65b9\u5411\u306e\u901f\u3055\u3092 \\(v\\) \u3068\u3059\u308c\u3070<\/strong><\/span><\/p>\n<p dir=\"ltr\">$${v = \\sqrt{g_{11}} \\frac{dr}{\\color{blue}{dt}}}$$<\/p>\n<p dir=\"ltr\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u3068\u306a\u308b<\/strong><\/span>\u300d\u65e8\u306e\u8a18\u8ff0\u304c\u3042\u308b\u3002\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<p dir=\"ltr\">$$d\\ell^2 = g_{11} dr^2, \\quad \\therefore\\ \\ d\\ell = \\sqrt{g_{11}} dr$$\u307e\u305f\uff0c\u30cc\u30eb\u3067\u3042\u308b\u304b\u3089$$g_{00} dt^2 + g_{11} dr^2 = 0, \\quad \\therefore\\ \\ \\sqrt{g_{11}} {dr} = \\sqrt{-g_{00}} dt$$<\/p>\n<p dir=\"ltr\">\u3068\u306a\u308b\u306e\u306f\u826f\u3044\u3068\u3057\u3066\u3082<\/p>\n<p dir=\"ltr\">$$v = \\frac{d\\ell}{dt} = \\sqrt{g_{11}} \\frac{dr}{\\color{blue}{dt}}$$<\/p>\n<p dir=\"ltr\">\u306e\u3088\u3046\u306b\u5ea7\u6a19\u6642\u9593 \\(\\color{blue}{dt}\\) \u3067\u5272\u308b\u306e\u306f\u3044\u304b\u304c\u306a\u3082\u306e\u304b\u3002<\/p>\n<p dir=\"ltr\">\u632f\u52d5\u6570\u306e\u5b9a\u7fa9\u306e\u3068\u3053\u308d\u3067\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong><span style=\"color: #0000ff;\">\u5ea7\u6a19\u6642\u9593\u304c \\(\\color{blue}{dt}\\) \u7d4c\u904e\u3059\u308b\u3068\u304d\uff0c<\/span>\u9759\u6b62\u89b3\u6e2c\u8005\u306b\u3068\u3063\u3066\u306f<span style=\"color: #ff0000;\">\u56fa\u6709\u6642\u9593 \\(\\color{red}{d\\tau}\\) \u304c\u7d4c\u904e\u3059\u308b<\/span>\u6642\u9593\u3067\u3042\u308b<\/strong><\/span>\u3068\u3044\u3046\u4eee\u5b9a\u3092\u4f7f\u3063\u3066\u3044\u308b\u306e\u3067\uff0c\u3053\u3053\u3067\u3082\u89b3\u6e2c\u8005\u304c\u6e2c\u5b9a\u3059\u308b\u901f\u3055\u306f\u7a7a\u9593\u7684\u9577\u3055 \\(d\\ell\\) \u3092\u56fa\u6709\u6642\u9593 \\(\\color{red}{d\\tau}\\) \u3067\u5272\u308b\u3079\u304d\u304b\u3068\u601d\u3046\u3002<\/p>\n<p dir=\"ltr\">\u3053\u306e\u5b9a\u7fa9\u3092\u63a1\u7528\u3059\u308b\u3068\uff0c<\/p>\n<p dir=\"ltr\">$$v = \\frac{d\\ell}{\\color{red}{d\\tau}} = \\sqrt{g_{11}} \\frac{dr}{\\color{red}{d\\tau}} = \\sqrt{-g_{00}} \\frac{dt}{\\sqrt{1-\\frac{r_g}{r}}dt} = 1 = c$$<\/p>\n<p dir=\"ltr\">\uff08\u3053\u3053\u3067\u306f \\(c=1\\) \u3068\u3057\u3066\u3044\u305f\u3053\u3068\u306b\u6ce8\u610f\u3002\uff09<\/p>\n<p dir=\"ltr\">\u3064\u307e\u308a\uff0c\u91cd\u529b\u5834\u4e2d\u3067\u3082\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u901f\u3055\u306f \\(v = c\\) \u3067\u4e00\u5b9a\u3068\u306a\u308b\u3002<\/p>\n<p dir=\"ltr\">\u3053\u308c\u3092\u4f7f\u3046\u3068\uff0c\u6ce2\u9577 \\(\\lambda\\) \u3068\u632f\u52d5\u6570 \\(\\nu\\) \u306e\u95a2\u4fc2\u306f<\/p>\n<p dir=\"ltr\">$$\\lambda \\nu = c, \\quad \\therefore\\ \\ \\lambda \\propto \\frac{1}{\\nu}$$<\/p>\n<p dir=\"ltr\">\u3057\u305f\u304c\u3063\u3066<\/p>\n<p dir=\"ltr\">$$\\frac{\\lambda_2}{\\lambda_1} = \\frac{\\nu_1}{\\nu_2} = \\frac{\\sqrt{1 &#8211; \\frac{r_g}{r_2}} }{\\sqrt{1 &#8211; \\frac{r_g}{r_1}}} &gt; 1$$<\/p>\n<p dir=\"ltr\">\u7279\u306b \\(r_2 \\gg r_1\\) \u306e\u5834\u5408\u306f\uff0c\\(r_2 \\rightarrow \\infty\\) \u3068\u3057\u3066<\/p>\n<p dir=\"ltr\">$$\\frac{\\lambda_2}{\\lambda_1} \\simeq \\frac{1}{\\sqrt{1 &#8211; \\frac{r_g}{r_1}}} = \\left( 1 &#8211; \\frac{r_g}{r_1}\\right)^{\\color{red}{-\\frac{1}{2}}}$$<\/p>\n<p>\u3068\u306a\u308b\u3079\u304d\u3067\u3042\u308b\u3002\u5185\u5c71\u672c (37.7) \u5f0f\u306f\u3053\u306e\u3088\u3046\u306b\u8a02\u6b63\u3055\u308c\u308b\u3079\u304d\u3067\u3042\u308b\u3068\u8003\u3048\u308b\u3002<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u306f\uff0c\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u306f\u3069\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u304b\u3002\u4f8b\u3048\u3070\u300c\u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6\u300d\uff08\u5185\u5c71\u9f8d\u96c4\u8457\uff0c\u88f3\u83ef\u623f\uff09\u306e\u00a737. \u3092\u53c2\u8003\u306b\uff0cnotation \u3092\u82e5\u5e72\u5909\u66f4\u3057\u3066\u304a\u3055\u3089\u3044\u3059\u308b\u3002\u307e\u305f\uff0c\u3053\u306e\u30c6\u30ad\u30b9\u30c8\u4e2d\u306e\u9593\u9055\u3044\u306b\u3064\u3044\u3066\u3082\u6307\u6458\u3057\u3066\u304a\u304f\u3002<\/p><p><a class=\"more-link btn\" href=\"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\/%e9%87%8d%e5%8a%9b%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\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":3059,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-3072","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" 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