{"id":2937,"date":"2022-04-25T10:54:02","date_gmt":"2022-04-25T01:54:02","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=2937"},"modified":"2023-03-14T16:53:54","modified_gmt":"2023-03-14T07:53:54","slug":"%e3%83%ab%e3%83%a1%e3%83%bc%e3%83%88%e3%83%ab%e5%ba%a7%e6%a8%99%e3%81%a7%e3%81%82%e3%82%89%e3%82%8f%e3%81%97%e3%81%9f%e3%82%b7%e3%83%a5%e3%83%90%e3%83%ab%e3%83%84%e3%82%b7%e3%83%ab%e3%83%88%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/2937\/","title":{"rendered":"\u30eb\u30e1\u30fc\u30c8\u30eb\u5ea7\u6a19\u3067\u3042\u3089\u308f\u3057\u305f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3"},"content":{"rendered":"<p><!--more--><\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/1871\/#dotB_neq_0\">\u3053\u306e\u3078\u3093<\/a><\/strong><\/span>\u3067\u66f8\u3044\u305f\uff0c\u7403\u5bfe\u79f0\u771f\u7a7a\u89e3\u3060\u3051\u3069\u9759\u7684\u3067\u306f\u306a\u3055\u305d\u3046\u306a\u89e3\u306e\u4e00\u3064\u306b\u3064\u3044\u3066\u3002<\/p>\n<ul>\n<li>\u53c2\u8003\uff1a\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u00a7102 \u306b\u305d\u306e\u307e\u3093\u307e\u66f8\u3044\u3066\u3042\u308b\u3002<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>\u4ef6\u306e\u30e1\u30c8\u30ea\u30c3\u30af\u3092 \\(r_g = 1\\) \u3068\u305b\u305a\u306b\u66f8\u304f\u3068\uff0c<\/p>\n<p>$$ds^2 = -d\\tau^2 + \\frac{dR^2}{\\left\\{\\frac{3}{2 r_g} \\left(R &#8211; \\tau\\right) \\right\\}^{\\frac{2}{3}}}<br \/>\n+ \\left\\{\\frac{3}{2 r_g} \\left(R &#8211; \\tau\\right) \\right\\}^{\\frac{4}{3}} r_g^2 \\left( d\\theta^2 + \\sin^2\\theta d\\phi^2\\right)$$<\/p>\n<p>\u3053\u308c\u306f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3<\/p>\n<p>$$ds^2 = -\\left(1-\\frac{r_g}{r}\\right) dt^2 + \\frac{dr^2}{1-\\frac{r_g}{r}} + r^2 \\left(d\\theta^2 + \\sin^2\\theta d\\phi^2\\right)$$<\/p>\n<p>\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30eb\u30e1\u30fc\u30c8\u30eb\u5ea7\u6a19 \\(\\tau, R\\) \u3067\u3042\u3089\u308f\u3057\u305f\u30e1\u30c8\u30ea\u30c3\u30af\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\nd\\tau &amp;=&amp; dt + \\frac{\\sqrt{\\frac{r_g}{r}}}{1 &#8211; \\frac{r_g}{r}} dr \\tag{1} \\\\<br \/>\ndR &amp;=&amp; dt + \\frac{1}{\\left(1 &#8211; \\frac{r_g}{r}\\right) \\sqrt{\\frac{r_g}{r}}} dr \\tag{2}<br \/>\n\\end{eqnarray}<\/p>\n<p>\\(\\displaystyle (1) &#8211; (2)\\times\u00a0 \\frac{r_g}{r} \\) \u304b\u3089<\/p>\n<p>$$d\\tau &#8211; \\frac{r_g}{r} dR = \\left(1 &#8211; \\frac{r_g}{r}\\right) dt$$<\/p>\n<p>\\(\\displaystyle (2) &#8211; (1) \\) \u304b\u3089<\/p>\n<p>\\begin{eqnarray}<br \/>\ndR &#8211; d\\tau &amp;=&amp; \\frac{1}{\\left(1 &#8211; \\frac{r_g}{r}\\right) \\sqrt{\\frac{r_g}{r}}} dr &#8211; \\frac{\\sqrt{\\frac{r_g}{r}}}{1 &#8211; \\frac{r_g}{r}} dr\\\\<br \/>\n&amp;=&amp; \\frac{1 &#8211; \\frac{r_g}{r}}{\\left(1 &#8211; \\frac{r_g}{r}\\right) \\sqrt{\\frac{r_g}{r}}} dr \\\\<br \/>\n&amp;=&amp; \\left(\\frac{r}{r_g} \\right)^{\\frac{1}{2}} dr\\\\ \\ \\\\<br \/>\n\\therefore\\ \\ R &#8211; \\tau &amp;=&amp; \\int \\left(\\frac{r}{r_g} \\right)^{\\frac{1}{2}} dr = \\frac{2}{3} r_g \\left(\\frac{r}{r_g} \\right)^{\\frac{3}{2}} \\\\<br \/>\n\\therefore\\ \\ r &amp;=&amp; r_g \\left\\{\\frac{3}{2 r_g} \\left( R &#8211; \\tau\\right) \\right\\}^{\\frac{2}{3}}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u3089\u3092\u5143\u306e\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u306b\u3044\u308c\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n-\\left(1-\\frac{r_g}{r}\\right) dt^2 + \\frac{dr^2}{1-\\frac{r_g}{r}}<br \/>\n&amp;=&amp; -\\left(1-\\frac{r_g}{r}\\right) \\left(\\frac{d\\tau &#8211; dR}{1-\\frac{r_g}{r}} \\right)^2 \\\\<br \/>\n&amp;&amp;\\quad +<br \/>\n\\frac{1}{1-\\frac{r_g}{r}}\\left\\{\\sqrt{\\frac{r_g}{r}} (dR &#8211; d\\tau)\\right\\}^2 \\\\<br \/>\n&amp;=&amp; \\frac{1}{1-\\frac{r_g}{r}}\\left\\{ &#8211; \\left(\\frac{1}{1-\\frac{r_g}{r}}\\right) d\\tau^2 +\\left(\\frac{r_g}{r}\\right) \\left(\\frac{1}{1-\\frac{r_g}{r}}\\right) dR^2\\right\\} \\\\<br \/>\n&amp;=&amp; &#8211; d\\tau^2 + \\left(\\frac{r_g}{r}\\right) dR^2 \\\\<br \/>\n&amp;=&amp; &#8211; d\\tau^2 +\\frac{dR^2}{\\left\\{\\frac{3}{2 r_g} \\left(R &#8211; \\tau\\right) \\right\\}^{\\frac{2}{3}}}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u6700\u7d42\u7684\u306b<\/p>\n<p>\\begin{eqnarray}<br \/>\nds^2 &amp;=&amp; -\\left(1-\\frac{r_g}{r}\\right) dt^2 + \\frac{dr^2}{1-\\frac{r_g}{r}} + r^2 \\, d\\Omega^2 \\\\<br \/>\n&amp;=&amp; &#8211; d\\tau^2 +\\frac{dR^2}{\\left\\{\\frac{3}{2 r_g} \\left(R &#8211; \\tau\\right) \\right\\}^{\\frac{2}{3}}} + \\left\\{\\frac{3}{2 r_g} \\left( R &#8211; \\tau\\right) \\right\\}^{\\frac{4}{3}} r_g^2 \\, d\\Omega^2<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u3053\u3067\uff0c\\(d\\Omega^2 \\equiv d\\theta^2 + \\sin^2\\theta d\\phi^2\\)\u3002<\/p>\n<p>\u3061\u306a\u307f\u306b\uff0c\u30eb\u30e1\u30fc\u30c8\u30eb\u5ea7\u6a19 \\(\\tau, R\\) \u306f\u3069\u3046\u3044\u3046\u610f\u5473\u304c\u3042\u308b\u304b\u3068\u3044\u3046\u3068\uff0c\uff08\\(\\tau\\) \u306f\u56fa\u6709\u6642\u9593\u3068\u3057\u3066\uff09\u300c<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e3%83%86%e3%82%b9%e3%83%88%e7%b2%92%e5%ad%90%e3%81%ae%e9%81%8b%e5%8b%95\/%e5%8b%95%e5%be%84%e6%96%b9%e5%90%91%e3%81%ae%e8%87%aa%e7%94%b1%e8%90%bd%e4%b8%8b%e9%81%8b%e5%8b%95\/#_B_4\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005<\/strong><\/span><\/a>\u300d\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u304b\u3089<\/p>\n<p>$$\\left(\\frac{dr}{d\\tau}\\right)^2 = \\epsilon^2 &#8211; \\left(1 &#8211; \\frac{r_g}{r}\\right), \\quad u_0 =\\mbox{const.} \\equiv -\\epsilon$$<\/p>\n<p>\u3067\uff0c\\(\\epsilon = 1\\) \u3068\u304a\u3044\u305f\u5f0f\u304b\u3089<\/p>\n<p>$$\\left(\\frac{dr}{d\\tau}\\right)^2 =\\frac{r_g}{r}, \\quad \\therefore\\ \\ \\frac{dr}{d\\tau} = &#8211; \\sqrt{\\frac{r_g}{r}}$$<\/p>\n<p>\u304c\u5f97\u3089\u308c\u308b\u306e\u3067\uff0c\u3053\u308c\u3092\u30eb\u30e1\u30fc\u30c8\u30eb\u5ea7\u6a19\u306e\u5b9a\u7fa9\u5f0f\u3092\u4f7f\u3046\u3068<\/p>\n<p>\\begin{eqnarray}<br \/>\ndR &#8211; d\\tau &amp;=&amp; \\left( \\frac{r}{r_g}\\right)^{\\frac{1}{2}}\u00a0 dr\\\\<br \/>\n\\therefore\\ \\ \\frac{dR}{d\\tau} &#8211; 1 &amp;=&amp; \\left( \\frac{r}{r_g}\\right)^{\\frac{1}{2}} \\frac{dr}{d\\tau}<br \/>\n&amp;=&amp;\u00a0 -1 \\\\<br \/>\n\\therefore\\ \\ \\frac{dR}{d\\tau} &amp;=&amp; 0<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308a\uff0c$R$ \u306f\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u3059\u308b\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u5171\u52d5\u5ea7\u6a19\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n<p>\u30eb\u30e1\u30fc\u30c8\u30eb\u5ea7\u6a19 \\(x^{\\mu} = (\\tau, R, \\theta, \\phi)\\)\u3067\u8868\u3057\u305f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306b\u9650\u3089\u305a\uff0c\u540c\u671f\u5316\u3055\u308c\u305f\u5ea7\u6a19\u7cfb\u3067\u306f\uff0c4\u5143\u901f\u5ea6 \\(u^{\\mu} = (1, 0, 0, 0)\\) \u306e\u5171\u52d5\u30c6\u30b9\u30c8\u7c92\u5b50\u306f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\u3053\u3068\u306f\u7c21\u5358\u306b\u793a\u305b\u308b\u304c\uff0c\u3053\u3053\u3067\u3082\u78ba\u304b\u306b\u7a7a\u9593\u5ea7\u6a19 \\(R, \\theta, \\phi\\) \u304c\u4e00\u5b9a\u3067\u3042\u308b\u30c6\u30b9\u30c8\u7c92\u5b50\u306f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","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-2937","post","type-post","status-publish","format-standard","hentry","category-rel-cosmo","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\" content=\"\u3053\u306e\u3078\u3093\u3067\u66f8\u3044\u305f\uff0c\u7403\u5bfe\u79f0\u771f\u7a7a\u89e3\u3060\u3051\u3069\u9759\u7684\u3067\u306f\u306a\u3055\u305d\u3046\u306a\u89e3\u306e\u4e00\u3064\u306b\u3064\u3044\u3066\u3002 \u53c2\u8003\uff1a\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u00a7102 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