{"id":664,"date":"2024-04-09T15:00:43","date_gmt":"2024-04-09T06:00:43","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=664"},"modified":"2024-11-05T11:15:01","modified_gmt":"2024-11-05T02:15:01","slug":"%e5%bc%b1%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e5%85%89%e3%81%ae%e7%b5%8c%e8%b7%af%e3%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad\/%e5%bc%b1%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e5%85%89%e3%81%ae%e7%b5%8c%e8%b7%af%e3%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3\/","title":{"rendered":"\u5f31\u91cd\u529b\u5834\u4e2d\u306e\u5149\u306e\u7d4c\u8def\u306e\u8fd1\u4f3c\u89e3"},"content":{"rendered":"<p><!--more-->\u5149\u306e\u7d4c\u8def\u3092\u6c7a\u3081\u308b\u5f0f\u306f\uff0c\u4e00\u822c\u306b\u306f\u89e3\u6790\u7684\u306a\u53b3\u5bc6\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u3002\u3053\u3053\u3067\u306f\uff0c\u5149\u306e\u7d4c\u8def\u306e\u3044\u305f\u308b\u3068\u3053\u308d\u3067\u91cd\u529b\u5834\u304c\u5f31\u3044\u3068\u3044\u3046\u8fd1\u4f3c\u306e\u3082\u3068\uff0c\u5149\u306e\u7d4c\u8def\u3092\u6442\u52d5\u6cd5\u306b\u3088\u308a\u8fd1\u4f3c\u7684\u306b\u89e3\u304f\u3002<\/p>\n<h3>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/h3>\n<p>$$ ds^2 = -\\left(1-\\frac{r_g}{r}\\right) c^2 dt^2 + \\frac{dr^2} {1-\\frac{r_g}{r}} + r^2(d\\theta^2 + \\sin^2\\theta d\\varphi^2)$$<br \/>\n\u3053\u3053\u3067 \\(\\displaystyle r_g \\equiv \\frac{2 G M}{c^2} \\) \u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u91cd\u529b\u534a\u5f84<\/strong><\/span>\uff08\u307e\u305f\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84<\/strong><\/span>\uff09\u3002\u4ee5\u5f8c\u306f\u7279\u306b\u65ad\u3089\u306a\u3044\u9650\u308a\uff0c\\(c = 1\\) \u3068\u3059\u308b\u3002<\/p>\n<h3>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u5149\u306e\u7d4c\u8def\u3092\u6c7a\u3081\u308b\u5f0f<\/h3>\n<p id=\"yui_3_17_2_1_1641871600658_4584\">$$ \\frac{1}{r} \\equiv u$$<br \/>\n\u3068\u5909\u6570\u5909\u63db\u3057\u3066\u3084\u308b\u3068\uff0c<\/p>\n<p id=\"yui_3_17_2_1_1641871600658_4586\">\\begin{eqnarray}<br id=\"yui_3_17_2_1_1641871600658_4587\" \/>\\left(\\frac{du}{d\\phi}\\right)^2 <br id=\"yui_3_17_2_1_1641871600658_4588\" \/>&amp;=&amp; \\left(\\frac{\\omega_c}{\\ell}\\right)^2 -u^2 + r_g \\,u^3<br id=\"yui_3_17_2_1_1641871600658_4589\" \/>\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u304c\u5149\u306e\u7d4c\u8def\u3092\u6c7a\u3081\u308b\u5f0f\u3067\u3042\u3063\u305f\u3002\uff08\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%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad\/%e3%82%b7%e3%83%a5%e3%83%90%e3%83%ab%e3%83%84%e3%82%b7%e3%83%ab%e3%83%88%e6%99%82%e7%a9%ba%e4%b8%ad%e3%81%ae%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad\/\">\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u5149\u306e\u4f1d\u64ad<\/a>\u300d\u3092\u53c2\u7167\u3002\uff09\u3053\u3053\u3067\uff0c\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{k}\\) \u306e\u6210\u5206 \\(k^{\\mu}\\) \u306b\u3064\u3044\u3066<br \/>\n$$ k^0 = \\frac{\\omega_c}{1 -\\frac{r_g}{r}}, \\quad k^2 = 0\u00a0 \\ \\left( \\theta =\u00a0 \\frac{\\pi}{2}\\right), \\quad<br \/>\nk^3 =\u00a0 \\frac{d\\phi}{dv} = \\frac{\\ell}{r^2}$$<\/p>\n<p>\uff08\\(\\ell = 0\\) \u306e\u5834\u5408\u306f \\(\\phi\\) \u304c\u4e00\u5b9a\u3068\u306a\u308b\u7d4c\u8def\u3067\u3042\u308a\uff0c\\(r=0\\) \u3068\u3044\u3046\u5927\u5909\u306a\u3068\u3053\u308d\u3092\u901a\u308b\u3053\u3068\u306b\u306a\u308a\u305d\u3046\u306a\u306e\u3067\u9664\u5916\u3057\u3066\u3044\u308b\u3002\u3064\u307e\u308a\uff0c$\\ell \\neq 0$\uff09<\/p>\n<p>$r$ \u304c\u6700\u5c0f\u5024 $b$ \u3092\u3082\u3064\u5834\u5408\u306b\u306f\uff0c$\\displaystyle r = b$ \u3059\u306a\u308f\u3061 $\\displaystyle u = \\frac{1}{b}$ \u3067 $\\displaystyle \\frac{du}{d\\phi} = 0$ \u3060\u304b\u3089<\/p>\n<p>\\begin{eqnarray}<br \/>\n0 &amp;=&amp; \\left(\\frac{\\omega_c}{\\ell}\\right)^2 -\\left(\\frac{1}{b}\\right)^2 + r_g \\,\\left(\\frac{1}{b}\\right)^3 \\\\<br \/>\n\\therefore\\ \\ \\left(\\frac{\\omega_c}{\\ell}\\right)^2 &amp;=&amp; \\left(\\frac{1}{b}\\right)^2 -r_g \\,\\left(\\frac{1}{b}\\right)^3<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c\u6700\u8fd1\u63a5\u8ddd\u96e2 $b$ \u3092\u4f7f\u3063\u3066\u3042\u3089\u305f\u3081\u3066\u66f8\u304f\u3068<\/p>\n<p>$$\\left(\\frac{du}{d\\phi}\\right)^2 <br id=\"yui_3_17_2_1_1641871600658_4588\" \/>= \\frac{1}{b^2} -u^2 + r_g \\left(\\,u^3\u00a0 -\\frac{1}{b^3}\\right)$$<\/p>\n<p>\u3053\u306e\u5f0f\u3092\u8fd1\u4f3c\u7684\u306b\u89e3\u304f\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<h3>\\(r_g\\) \u306e\u30bc\u30ed\u6b21\u89e3<\/h3>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5149\u306e\u7d4c\u8def\u306e\u3044\u305f\u308b\u3068\u3053\u308d\u3067\u91cd\u529b\u5834\u304c\u5f31\u3044\uff0c\u3059\u306a\u308f\u3061\u5149\u306e\u7d4c\u8def \\(r\\) \u306f\u91cd\u529b\u534a\u5f84 $r_g$ \u306e\u5341\u5206\u5916\u5074\u3067\u3042\u308b<\/strong><\/span>\u3068\u3044\u3046\u72b6\u6cc1\u3067\u306f\uff0c \\(\\displaystyle 0 &lt; \\frac{r_g}{r} = r_g u \\ll 1\\) \u3068\u3057\u3066\u3088\u3044\u3002\u307e\u305f\uff0c\u6700\u8fd1\u63a5\u8ddd\u96e2 \\(b\\) \u3082\u91cd\u529b\u534a\u5f84\u306e\u5341\u5206\u5916\u5074\u3067\u3042\u308b\u3068\u3057\u3066 \\(\\displaystyle 0 &lt; \\frac{r_g}{b}\u00a0 \\ll 1\\) \u3002<\/p>\n<p>\u305d\u3053\u3067 $r_g$ \u306e\u9805\u3092\u7121\u8996\u3057\u305f\u5834\u5408\u306e\u89e3\u3092 $u_0$ \u3068\u304a\u304f\u3068<\/p>\n<p>$$\\left(\\frac{du_0}{d\\phi}\\right)^2 <br id=\"yui_3_17_2_1_1641871600658_4588\" \/>= \\frac{1}{b^2} -u_0^2$$<\/p>\n<p>\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f2\u5e74\u751f\u306e\u6642\u306e\u6388\u696d\u300c<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f\/%e7%b0%a1%e5%8d%98%e3%81%aa1%e9%9a%8e%e9%9d%9e%e7%b7%9a%e5%bd%a2%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e4%be%8b\/\">\u7c21\u5358\u306a1\u968e\u975e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/a>\u300d\u3067\u3084\u3063\u3066\u307e\u3059\u3002\u521d\u671f\u6761\u4ef6\u3092 $\\displaystyle \\phi = \\frac{\\pi}{2}$ \u306e\u3068\u304d $\\displaystyle u_0 = \\frac{1}{b}$ \u3068\u3059\u308b\u3068<\/p>\n<p>$$ u_0 = \\frac{\\sin\\phi}{b}$$<\/p>\n<h3>\\(r_g\\) \u306e1\u6b21\u89e3<\/h3>\n<p>$r_g$ \u306e1\u6b21\u306e\u52b9\u679c\u3092\u53d6\u308a\u5165\u308c\u305f\u89e3\u3092<\/p>\n<p>$$ u = u_0 + \\frac{r_g}{b} \\,u_1(\\phi) = \\frac{\\sin\\phi}{b}+ \\frac{r_g}{b}\\,u_1(\\phi)$$<\/p>\n<p>\u3068\u304a\u3044\u3066\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3057\uff0c\\(r_g\\) \u306e1\u6b21\u306e\u9805\u3092\u53d6\u308a\u51fa\u3059\u3068\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\left(\\frac{\\cos\\phi}{b} + \\frac{r_g}{b} \\frac{du_1}{d\\phi} \\right)^2 &amp;=&amp;<br \/>\n\\frac{1}{b^2} -\\left( \\frac{\\sin\\phi}{b} + \\frac{r_g}{b} u_1\\right)^2<br \/>\n+ r_g \\left( u_0^3 -\\frac{1}{b^3}\\right) \\\\<br \/>\n2 \\frac{\\cos\\phi}{b}\\, \\frac{r_g}{b}\\, \\frac{d u_1}{d\\phi} &amp;=&amp;<br \/>\n-2 \\frac{\\sin\\phi}{b} \\, \\frac{r_g}{b}\\, u_1 + r_g \\left(\\frac{\\sin^3\\phi}{b^3} -\\frac{1}{b^3}\\right) \\\\<br \/>\n\\therefore\\ \\ \\cos\\phi\\cdot \\frac{d u_1}{d\\phi} + \\sin\\phi\\cdot u_1 &amp;=&amp; \\frac{\\sin^3\\phi -1}{2 b}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u89e3\u3051\u308b\u30021\u5e74\u751f\u306e\u6388\u696d\u306e\u300c<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/sin-%f0%9d%91%a5-cos-%f0%9d%91%a5-%e3%81%ae%e6%9c%89%e7%90%86%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/#_4_displaystyle_int_fracsin3_x_-1cos2_x_dx\">\u7df4\u7fd2\u554f\u984c 4.<\/a>\u300d\u3082\u53c2\u7167\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\cos\\phi\\cdot \\frac{d u_1}{d\\phi} + \\sin\\phi\\cdot u_1 &amp;=&amp; \\frac{\\sin^3\\phi -1}{2 b} \\\\<br \/>\n\\cos^2\\phi \\cdot \\frac{d}{d\\phi} \\left( \\frac{u_1}{\\cos\\phi}\\right) &amp;=&amp; \\frac{\\sin^3\\phi -1}{2 b} \\\\<br \/>\n\\therefore\\ \\ u_1 &amp;=&amp; \\frac{\\cos\\phi}{2 b} \\int \\frac{\\sin^3\\phi -1}{\\cos^2\\phi }\\, d\\phi \\\\<br \/>\n&amp;=&amp; \\frac{1}{2b} \\left( 2 -\\sin\\phi -\\sin^2\\phi\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c$r_g$ \u306e1\u6b21\u307e\u3067\u306e\u89e3\u306f<\/p>\n<p>$$u = u_0 + \\frac{r_g}{b} u_1 = \\frac{\\sin\\phi}{b} + \\frac{r_g}{2 b^2} \\left( 2 -\\sin\\phi -\\sin^2\\phi\\right)$$<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":83,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-664","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\/664","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=664"}],"version-history":[{"count":26,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/664\/revisions"}],"predecessor-version":[{"id":9491,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/664\/revisions\/9491"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/83"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=664"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}