{"id":530,"date":"2022-01-08T14:25:13","date_gmt":"2022-01-08T05:25:13","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=530"},"modified":"2024-10-22T15:34:56","modified_gmt":"2024-10-22T06:34:56","slug":"%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","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\/%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\/","title":{"rendered":"\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u5149\u306e\u4f1d\u64ad"},"content":{"rendered":"<p><!--more--><\/p>\n<h3>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/h3>\n<p>\u66f2\u304c\u3063\u305f\u6642\u7a7a\u3092\u8a18\u8ff0\u3059\u308b\u306e\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u3067\u3042\u308a\uff0c\u305d\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306b\u3088\u3063\u3066\u6c7a\u307e\u308b\u3002<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306f\uff0c10\u500b\u306e\u72ec\u7acb\u306a\u6210\u5206\u3092\u6301\u3064<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span> \\(g_{\\mu\\nu}\\) \u3092\u672a\u77e5\u95a2\u6570\u3068\u3057\u305f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>10\u5143\u9023\u7acb\u975e\u7dda\u5f622\u968e\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306e\u5f62\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u5e73\u305f\u304f\u8a00\u3048\u3070\u89e3\u6790\u7684\u306b\u89e3\u304f\u306e\u304c\u3068\u3066\u3082\u96e3\u3057\u3044\u3002<\/p>\n<p>\u305d\u306e\u4e2d\u3067\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7403\u5bfe\u79f0<\/strong><\/span>\u304b\u3064<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u771f\u7a7a<\/strong><\/span> (\\(T^{\\mu\\nu} = 0\\)) \u3092\u4eee\u5b9a\u3057\u3066\u5f97\u3089\u308c\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306e\u6700\u3082\u7c21\u5358\u306a\u89e3\u304c\uff0c\u4ee5\u4e0b\u306e<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3<\/strong><\/span>\u3067\u3042\u308b\u3002\u5c0e\u51fa\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u3082\u53c2\u7167\uff1a<\/p>\n<ul>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e6%99%82%e7%a9%ba%e3%81%ae%e8%a1%a8%e3%81%97%e6%96%b9\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e4%bb%a3%e6%95%b0%e3%82%b7%e3%82%b9%e3%83%86%e3%83%a0%e3%81%a7%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9\/maxima-%e3%81%ae-ctensor-%e3%81%a7%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e8%a7%a3%e3%81%84%e3%81%a6%e3%82%b7%e3%83%a5%e3%83%90\/\">Maxima \u306e ctensor \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u3092\u6c42\u3081\u308b<\/a><\/li>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e6%99%82%e7%a9%ba%e3%81%ae%e8%a1%a8%e3%81%97%e6%96%b9\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e4%bb%a3%e6%95%b0%e3%82%b7%e3%82%b9%e3%83%86%e3%83%a0%e3%81%a7%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9\/5116-2\/\">EinsteinPy \u3068 SymPy \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u3092\u6c42\u3081\u308b<\/a><\/li>\n<\/ul>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u91cd\u529b\u6e90\u3067\u3042\u308b\u5929\u4f53\u306e\u5916\u90e8\u91cd\u529b\u5834<\/strong><\/span>\u306f\uff08\u53b3\u5bc6\u306b\u306f\u7403\u5bfe\u79f0\u3067\u306f\u306a\u3044\u3057\uff0c\u307e\u305f\u81ea\u8ee2\u3057\u3066\u3044\u305f\u308a\u3059\u308b\u304c&#8230; \uff09\u7403\u5bfe\u79f0\u771f\u7a7a\u306a<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3<\/strong><\/span>\u3067\u8a18\u8ff0\u3055\u308c\u3066\u308b\u3068\u3057\u3066\u3088\u3044\u305f\u3081\uff0c\u7c21\u5358\u304b\u3064\u6700\u3082\u5fdc\u7528\u3055\u308c\u3066\u3044\u308b\u89e3\u3067\u3042\u308b\u3002\u672c\u7a3f\u3067\u3082\uff0c\u7740\u76ee\u3057\u3066\u3044\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5929\u4f53\uff08\u592a\u967d\u3068\u304b\u5730\u7403\u3068\u304b\uff09\u306e\u5916\u90e8\u91cd\u529b\u5834<\/strong><\/span>\u304c\u3053\u306e\u89e3\u3067\u8a18\u8ff0\u3055\u308c\u308b\u3068\u3057\u3066\u8a71\u3092\u3059\u3059\u3081\u308b\u3002<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3<\/strong><\/span>\u3067\u8868\u3055\u308c\u308b\u6642\u7a7a\uff0c<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/strong><\/span>\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\uff0c\u7565\u3057\u3066<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf<\/strong><\/span>\u306e\u6210\u5206\u306f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7dda\u7d20<\/strong><\/span>\u306e\u4e2d\u306b\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3042\u3089\u308f\u308c\u308b\u3002<\/p>\n<p>$$ ds^2 = -\\left(1-\\frac{r_g}{r}\\right) c^2 dt^2 + \\frac{dr^2} {1-\\frac{r_g}{r}} + r^2(d\\theta^2 + \\sin^2\\theta d\\phi^2)$$ \u3053\u3053\u3067 \\(\\displaystyle r_g \\equiv \\frac{2 G M}{c^2} \\) \u306f<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84<\/strong><\/span>\u307e\u305f\u306f<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u91cd\u529b\u534a\u5f84<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u9577\u3055\u3067\u3042\u308a\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u52b9\u679c\u304c\u9855\u8457\u306b\u306a\u308b\u91cd\u529b\u6e90\u304b\u3089\u306e\u8ddd\u96e2\u3092\u8868\u3057\u3066\u3044\u308b\u3002\u4ee5\u5f8c\u306f\u7279\u306b\u65ad\u3089\u306a\u3044\u9650\u308a\uff0c\\(c = 1\\) \u3068\u3059\u308b\u3002<\/p>\n<p>\u4e0a\u8a18\u306e\u3088\u3046\u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7dda\u7d20<\/strong><\/span>\u3092\u66f8\u304f\u3053\u3068\u306b\u3088\u3063\u3066\uff0c\u5ea7\u6a19\u7cfb\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3068\u3063\u3066\u3044\u308b\u3053\u3068<br \/>\n$$x^{\\nu} = (x^0, x^1, x^2, x^3) = (t, r, \\theta, \\phi)$$<br \/>\n\u304a\u3088\u3073\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span> \\(g_{\\mu\\nu}\\) \u306e\u30bc\u30ed\u3067\u306a\u3044\u6210\u5206\u306f<br \/>\n\\begin{eqnarray}<br \/>\ng_{00} &amp;=&amp; -\\left(1-\\frac{r_g}{r}\\right) \\\\<br \/>\ng_{11} &amp;=&amp; \\frac{1} {1-\\frac{r_g}{r}}\\\\<br \/>\ng_{22} &amp;=&amp; r^2\\\\<br \/>\ng_{33} &amp;=&amp; r^2 \\sin^2\\theta<br \/>\n\\end{eqnarray}<br \/>\n\u3067\u3042\u308a\uff0c\u305d\u308c\u4ee5\u5916\u306e\u6210\u5206\u306f\u5168\u3066\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u4e00\u6319\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3093\u3060\u3063\u305f\u306d\u3002<\/p>\n<h3>\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/h3>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7403\u5bfe\u79f0\u771f\u7a7a<\/strong><\/span>\u306a<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/strong><\/span>\u306f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u304c\u6642\u9593\u5ea7\u6a19\u306b\u3088\u3089\u306a\u3044<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9759\u7684\u306a\u6642\u7a7a<\/strong><\/span>\u3067\u3082\u3042\u308b\u3002<\/p>\n<p>\u3053\u306e\u3088\u3046\u306a\u9ad8\u3044\u5bfe\u79f0\u6027\u3092\u5229\u7528\u3059\u308b\u305f\u3081\uff0c\u4fdd\u5b58\u91cf\u304c\u308f\u304b\u308a\u3084\u3059\u3044\u3088\u3046\u306b\u5909\u5f62\u3057\u305f\uff0c\u4ee5\u4e0b\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3092\u4f7f\u3046\u3002\u3053\u308c\u306f \\(\\nu = 0, 1, 2, 3\\) \u3067\u3042\u308b\u3053\u3068\u304b\u30894\u672c\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n<p>$$\\frac{d k_{\\nu}}{dv} = \\frac{1}{2} g_{\\lambda\\mu, \\nu} k^{\\lambda} k^{\\mu}$$<\/p>\n<p>\u3053\u306e\u5f0f\u304b\u3089\uff0c\u4e00\u822c\u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206 \\(g_{\\lambda\\mu} \\) \u304c \\(x^{\\nu} \\) \u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u5834\u5408\u306f\uff0c<br \/>\n$$ g_{\\lambda\\mu, \\nu} = 0 \\quad\\Rightarrow\\quad \\frac{d k_{\\nu}}{dv} = 0 \\quad\\Rightarrow\\quad\u00a0 k_{\\nu} = \\mbox{const.} $$\u3068\u306a\u308a \\(k_{\\nu} \\) \u6210\u5206\u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n<p>\u306a\u304a\uff0c\u5ff5\u306e\u305f\u3081\u306b\u8ffd\u8a18\u3059\u308b\u304c\uff0c\u4e0a\u8a18\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u305f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/strong><\/span>\u306b\u9650\u3089\u305a\uff0c\u4e00\u822c\u306e\u6642\u7a7a\u3067\u3082\u52ff\u8ad6\u4f7f\u3048\u308b\u3002<\/p>\n<h4>\\(k^0\\) \u306e\u89e3<\/h4>\n<p id=\"yui_3_17_2_1_1641617883829_1695\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf<\/strong><\/span>\u306e\u6210\u5206\u306f \\(x^0 = t\\) \u3092\u542b\u307e\u306a\u3044\u306e\u3067\uff0c\\(k_0\\) \u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3002\u3053\u306e\u91cf\u3092 \\(-\\omega_c\\) \u3068\u3059\u308b\u3068\uff0c<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_1717\">\\begin{eqnarray}<br id=\"yui_3_17_2_1_1641617883829_1718\" \/>k_0 = g_{0\\mu} k^{\\mu} &amp;=&amp; \\mbox{const.} \\equiv -\\omega_c \\\\<br id=\"yui_3_17_2_1_1641617883829_1719\" \/>\\therefore \\ \\ k^0 &amp;=&amp; \\frac{-\\omega_c}{g_{00}} = \\frac{\\omega_c}{1 &#8211; \\frac{r_g}{r}}<br id=\"yui_3_17_2_1_1641617883829_1720\" \/>\\end{eqnarray}<\/p>\n<h4>\\(k^3\\) \u306e\u89e3<\/h4>\n<p id=\"yui_3_17_2_1_1641617883829_1919\" dir=\"ltr\">\u307e\u305f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf<\/strong><\/span>\u306e\u6210\u5206\u306f \\( x^3 = \\phi\\) \u3082\u542b\u307e\u306a\u3044\u306e\u3067\uff0c\\(k_3\\) \u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3002\u3053\u306e\u91cf\u3092 \\(\\ell\\) \u3068\u3059\u308b\u3068\uff0c<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_1920\" dir=\"ltr\">\\begin{eqnarray}<br id=\"yui_3_17_2_1_1641617883829_1921\" \/>k_3 = g_{3\\mu} k^{\\mu} &amp;=&amp; \\mbox{const.} \\equiv \\ell \\\\<br id=\"yui_3_17_2_1_1641617883829_1922\" \/>\\therefore \\ \\ k^3 &amp;=&amp; \\frac{\\ell}{g_{33}} = \\frac{\\ell}{r^2 \\sin^2\\theta}<br id=\"yui_3_17_2_1_1641617883829_1923\" \/>\\end{eqnarray}<\/p>\n<h4>\\(k^2\\) \u306f\u521d\u671f\u6761\u4ef6\u304b\u3089<\/h4>\n<p dir=\"ltr\">\\( \\displaystyle k^2 = \\frac{d x^2}{dv} = \\frac{d\\theta}{dv} \\) \u306b\u3064\u3044\u3066\u306f\uff0c<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_2460\" dir=\"ltr\">$$ \\frac{dk_{2}}{dv} =\\frac{d}{dv} \\left( g_{22} \\frac{d\\theta}{dv}\\right)\u00a0 = \\frac{d}{dv} \\left( r^2 \\frac{d\\theta}{dv}\\right) = \\frac{1}{2} g_{33, 2} k^3 k^3$$ \u3088\u308a<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_2461\" dir=\"ltr\">$$ \\frac{d}{dv}\\frac{d\\theta}{dv} + \\frac{2}{r} \\frac{dr}{dv} \\frac{d\\theta}{dv} = \\frac{\\ell^2 \\cos\\theta}{r^4 \\sin^3\\theta}$$\u3092\u5f97\u308b\u3002<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_2462\" dir=\"ltr\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u521d\u671f\u6761\u4ef6<\/strong><\/span>\u3068\u3057\u3066\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf \\(v \\) \u304c\u3042\u308b\u5024 \\(v=0\\) \u306e\u3068\u304d\uff0c<br \/>\n$$\\displaystyle \\theta(0) = \\frac{\\pi}{2}, \\quad \\frac{d\\theta}{dv}\\Biggr|_0 = 0 $$<br \/>\n\u3068\u3059\u308b\u3068\uff0c\\(\\displaystyle \\frac{d}{dv}\\frac{d\\theta}{dv}\\Biggr|_{0} = 0 \\) \u3068\u306a\u308a\uff0c\u5e38\u306b \\( \\displaystyle \\frac{d\\theta}{dv} = 0 \\) \u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\uff0c\u3053\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u521d\u671f\u6761\u4ef6<\/strong><\/span>\u3092\u63a1\u7528\u3057\uff0c<br \/>\n$$ \\theta = \\frac{\\pi}{2}, \\quad k^2 = \\frac{d\\theta}{dv} = 0$$\u3068\u3059\u308b\u3002<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_2463\" dir=\"ltr\">\u3053\u306e\u3053\u3068\u306f<\/p>\n<p dir=\"ltr\" style=\"text-align: center;\"><span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong id=\"yui_3_17_2_1_1641617883829_2464\">\u7403\u5bfe\u79f0\u6027\u306b\u3088\u308a\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f<br \/>\n<\/strong><\/span><span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong id=\"yui_3_17_2_1_1641617883829_2464\">\u8d64\u9053\u9762\u4e0a \\( \\displaystyle \\theta = \\frac{\\pi}{2}\\) \u306b\u904b\u52d5\u3092\u5236\u9650\u3067\u304d\u308b<\/strong><\/span><\/p>\n<p dir=\"ltr\">\u3053\u3068\u3092\u521d\u671f\u5024\u554f\u984c\u3068\u3057\u3066\u793a\u3057\u305f\u3082\u306e\u3067\u3042\u308b\u3002<\/p>\n<h3>\\(k^1\\) \u306f\u30cc\u30eb\u6761\u4ef6\u304b\u3089<\/h3>\n<p dir=\"ltr\">\u3055\u3066\uff0c\u3053\u308c\u307e\u3067\u306e\u3068\u3053\u308d\uff0c\u308f\u304b\u3063\u305f\u306e\u306f<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3010\" dir=\"ltr\">$$ k^0 = \\frac{\\omega_c}{1 &#8211; \\frac{r_g}{r}}, \\quad k^2 = 0\u00a0 \\ \\left( \\theta =\u00a0 \\frac{\\pi}{2}\\right), \\quad<br id=\"yui_3_17_2_1_1641617883829_3129\" \/>k^3 = \\frac{\\ell}{r^2 \\sin^2\\theta} = \\frac{\\ell}{r^2}$$<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3130\" dir=\"ltr\">\u6b8b\u308a\u306e \\(\\displaystyle k^1 = \\frac{dr}{dv} \\) \u306b\u3064\u3044\u3066\u306f\uff0c\u30cc\u30eb\u6761\u4ef6\u3088\u308a<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3131\" dir=\"ltr\">\\begin{eqnarray}<br id=\"yui_3_17_2_1_1641617883829_3132\" \/>0 = \\boldsymbol{k}\\cdot\\boldsymbol{k} &amp;=&amp; g_{\\mu\\nu} k^{\\mu} k^{\\mu} \\\\<br id=\"yui_3_17_2_1_1641617883829_3133\" \/>&amp;=&amp; g_{00} \\left( k^0 \\right)^2 + g_{11} \\left(\\frac{dr}{dv}\\right)^2 <br id=\"yui_3_17_2_1_1641617883829_3134\" \/>+ g_{33} \\left(k^3\\right)^2 \\\\<br id=\"yui_3_17_2_1_1641617883829_3135\" \/>&amp;=&amp; -\\left( 1 &#8211; \\frac{r_g}{r}\\right)\\left(\\frac{\\omega_c}{1 &#8211; \\frac{r_g}{r}} \\right)^2<br id=\"yui_3_17_2_1_1641617883829_3136\" \/>+ \\frac{1}{1 &#8211; \\frac{r_g}{r}} \\left(\\frac{dr}{dv}\\right)^2<br id=\"yui_3_17_2_1_1641617883829_3137\" \/>+ r^2 \\left( \\frac{\\ell}{r^2}\\right)^2<br id=\"yui_3_17_2_1_1641617883829_3138\" \/>\\end{eqnarray}<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3139\" dir=\"ltr\">$$\\therefore \\ \\ \\left(\\frac{dr}{dv}\\right)^2 = \\omega_c^2 &#8211; \\left( 1 &#8211; \\frac{r_g}{r}\\right)\\frac{\\ell^2}{r^2}$$<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3140\" dir=\"ltr\">\\(r(v) \\) \u3068 \\(\\phi(v) \\) \u306f\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf \\(v\\) \u3092\u901a\u3057\u3066\u95a2\u4fc2\u3065\u3051\u3089\u308c\u3066\u3044\u308b\u306e\u3067\uff0c\\( \\displaystyle k^3 = \\frac{d\\phi}{dv} = \\frac{\\ell}{r^2} \\) \u3092\u4f7f\u3063\u3066<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3141\" dir=\"ltr\">$$ \\frac{dr}{dv} = \\frac{d\\phi}{dv} \\frac{dr}{d\\phi} = \\frac{\\ell}{r^2} \\frac{dr}{d\\phi}$$<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3143\" dir=\"ltr\">$$\\therefore \\ \\ \\left(\\frac{1}{r^2}\\frac{dr}{d\\phi}\\right)^2 = \\left(\\frac{\\omega_c}{\\ell}\\right)^2 &#8211; \\left( 1 &#8211; \\frac{r_g}{r}\\right)\\frac{1}{r^2}$$<\/p>\n<h3>\u5149\u306e\u7d4c\u8def\u3092\u6c7a\u3081\u308b\u5f0f<\/h3>\n<p>\u6c42\u3081\u305f\u3044\u5909\u6570 \\(r\\) \u304c\u5206\u6bcd\u306b\u3070\u3063\u304b\u308a\u3042\u3089\u308f\u308c\u308b\u306e\u3067\uff0c\u3044\u3063\u305d\u306e\u3053\u3068<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3146\">$$ \\frac{1}{r} \\equiv u$$<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3147\">\u3068\u5909\u6570\u5909\u63db\u3057\u3066\u3084\u308b\u3068\uff0c$$\u00a0 -\\frac{1}{r^2} \\frac{dr}{d\\phi} = \\frac{du}{d\\phi}$$\u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3148\">\\begin{eqnarray}<br id=\"yui_3_17_2_1_1641617883829_3149\" \/>\\left(\\frac{du}{d\\phi}\\right)^2 &amp;=&amp; \\left(\\frac{\\omega_c}{\\ell}\\right)^2 &#8211; (1 &#8211; r_g\\,u) u^2 \\\\<br id=\"yui_3_17_2_1_1641617883829_3150\" \/>&amp;=&amp; \\left(\\frac{\\omega_c}{\\ell}\\right)^2 &#8211; u^2 + r_g \\,u^3<br id=\"yui_3_17_2_1_1641617883829_3151\" \/>\\end{eqnarray}<\/p>\n<p id=\"yui_3_17_2_1_1641617883829_3060\">\u6700\u7d42\u7684\u306b\uff0c\u3053\u308c\u304c\u5149\u306e\u7d4c\u8def\u3092\u6c7a\u3081\u308b\u5f0f\u3067\u3042\u308b\u3002<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":83,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-530","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\/530","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=530"}],"version-history":[{"count":22,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/530\/revisions"}],"predecessor-version":[{"id":9484,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/530\/revisions\/9484"}],"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=530"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}