{"id":1056,"date":"2024-04-08T14:00:34","date_gmt":"2024-04-08T05:00:34","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1056"},"modified":"2024-06-04T14:08:34","modified_gmt":"2024-06-04T05:08:34","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e5%bc%b1%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e7%b2%92%e5%ad%90%e3%81%ae%e8%bb%8c%e9%81%93%e3%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3%e6%b3%95%ef%bc%88%e3%82%88%e3%82%8d","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%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%bc%b1%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e7%b2%92%e5%ad%90%e3%81%ae%e8%bb%8c%e9%81%93%e3%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e5%bc%b1%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e7%b2%92%e5%ad%90%e3%81%ae%e8%bb%8c%e9%81%93%e3%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3%e6%b3%95%ef%bc%88%e3%82%88%e3%82%8d\/","title":{"rendered":"\u5f31\u91cd\u529b\u5834\u4e2d\u306e\u7c92\u5b50\u306e\u8ecc\u9053\u306e\u3088\u308d\u3057\u304f\u306a\u3044\u8fd1\u4f3c\u89e3\u6cd5\u4f8b"},"content":{"rendered":"

\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u8ecc\u9053\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\u7c92\u5b50\u306e\u8ecc\u9053\u306e\u3044\u305f\u308b\u3068\u3053\u308d\u3067\u91cd\u529b\u5834\u304c\u5f31\u3044\u3068\u3044\u3046\u8fd1\u4f3c\u306e\u3082\u3068\uff0c\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u8fd1\u4f3c\u7684\u306b\u89e3\u304f\u306e\u3060\u304c\uff0c\u5149\u306e\u7d4c\u8def\u306e\u5834\u5408\u3068\u540c\u69d8\u306a\u6442\u52d5\u6cd5\u306b\u3088\u308a\u8fd1\u4f3c\u7684\u306b\u89e3\u304f\u3068\uff0c\u529b\u5b66\u306b\u304a\u3051\u308b\u300c\u5f37\u5236\u632f\u52d5\u306b\u3088\u308b\u5171\u9cf4<\/strong><\/span>\u300d\u306e\u3088\u3046\u306a\uff0c\u5358\u8abf\u5897\u52a0\u3059\u308b\u632f\u5e45\u3092\u6301\u3064\u9805\u304c\u51fa\u3066\u304f\u308b\u5834\u5408\u304c\u3042\u308b<\/strong><\/span>\u3002\u3053\u308c\u306f\u3042\u307e\u308a\u3088\u308d\u3057\u304f\u306a\u3044\u3002\u3053\u306e\u3078\u3093\u306e\u4e8b\u60c5\u306b\u3064\u3044\u3066\u88dc\u8db3\u3057\u3066\u304a\u304f\u3002<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e<\/span>\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f<\/span><\/h3>\n

$\\displaystyle s \\equiv \\frac{1}{r}$ \u3068\u3059\u308b\u3068\uff0c\u300c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u7c92\u5b50\uff08\u89b3\u6e2c\u8005\uff09\u306e\u904b\u52d5<\/a>\u300d\u306e\u30da\u30fc\u30b8\u306b\u307e\u3068\u3081\u305f\u3088\u3046\u306b\uff08\u9069\u5b9c\u79fb\u9805\u3057\u3066\uff09<\/p>\n

\\begin{eqnarray}
\n\\left( \\frac{ds}{d\\phi} \\right)^2\u00a0 +s^2 -\\frac{2GM}{\\ell^2} s\u00a0\u00a0 -r_g\\, s^3&=& \\frac{\\epsilon^2 c^2 -c^2}{\\ell^2}
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u304c\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u3053\u3053\u3067\uff0c$\\epsilon$ \u304a\u3088\u3073 $\\ell$ \u306f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u969b\u306b\u5f97\u3089\u308c\u305f\u904b\u52d5\u306e\u5b9a\u6570\u3067\u3042\u308a\uff0c<\/p>\n

$$ \\frac{d x^0}{d\\tau} = \\frac{c\\, dt}{d\\tau} = \\frac{\\epsilon\\, c}{1 -\\frac{r_g}{r}}, \\quad \\frac{d x^3}{d\\tau} = \\frac{d\\phi}{d\\tau} = \\frac{\\ell}{r^2}$$<\/p>\n

\u904b\u52d5\u304c\u6709\u754c\u3067\u3042\u308c\u3070\uff0c<\/p>\n

$$ \\frac{1}{a(1+e)} \\equiv \\frac{1}{r_{\\rm max}} \\leq s \\leq \\frac{1}{r_{\\rm min}} \\equiv \\frac{1}{a(1 -e)}$$<\/p>\n

\u6975\u5024\u3092\u3068\u308b\u70b9\u3067 $\\displaystyle \\frac{ds}{d\\phi} =0$ \u3067\u3042\u308b\u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\left( \\frac{1}{a (1+e)}\\right)^2 -2 \\frac{GM}{\\ell^2} \\left( \\frac{1}{a (1+e)}\\right) -r_g\\, \\left( \\frac{1}{a (1+e)}\\right)^3 &=& \\frac{\\epsilon^2 c^2 -c^2}{\\ell^2} \\\\
\n\\left( \\frac{1}{a (1-e)}\\right)^2 -2 \\frac{GM}{\\ell^2} \\left( \\frac{1}{a (1-e)}\\right) -r_g\\, \\left( \\frac{1}{a (1-e)}\\right)^3 &=& \\frac{\\epsilon^2 c^2 -c^2}{\\ell^2}
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\uff08\u7dda\u5f62\u8fd1\u4f3c\u3067\u306f\u306a\u304f\uff0c\u53b3\u5bc6\u306b\u89e3\u3044\u3066\u3044\u308b\u306e\u3067\u3042\u308b\u304c\uff0c\u7d50\u679c\u306f $r_g$ \u306e1\u6b21\u307e\u3067\u306e\u5f62\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3082\u8208\u5473\u6df1\u3044\u3002\uff09<\/p>\n

\\begin{eqnarray}
\n\\frac{GM}{\\ell^2} &=& \\frac{1}{a (1 -e^2)} -\\frac{(3 + e^2) r_g}{2 a^2 (1 -e^2)^2} \\\\
\n\\frac{\\epsilon^2 c^2 -c^2}{\\ell^2} &=& -\\frac{1}{a^2 (1 -e^2)} + \\frac{2 r_g}{a^3 (1 -e^2)^2}
\n\\end{eqnarray}<\/p>\n

$a, \\, e$ \u306f\u6955\u5186\u8ecc\u9053\u306e\u5834\u5408\u306b\u306f\u8ecc\u9053\u9577\u534a\u5f84\uff0c\u96e2\u5fc3\u7387\u3068\u547c\u3070\u308c\u308b\u304c\uff0c\u3053\u3053\u3067\u306f $r_{\\rm max}, \\, r_{\\rm min}$ \u304b\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u6c7a\u3081\u3089\u308c\u308b\u5b9a\u6570\u3067\u3042\u308b\u3053\u3068\u3060\u3051\u3092\u899a\u3048\u3066\u304a\u304f\u3002<\/p>\n

\\begin{eqnarray}
\na &\\equiv& \\frac{1}{2} (r_{\\rm max} + r_{\\rm min}) \\\\
\ne &\\equiv& \\frac{r_{\\rm max} -r_{\\rm min}}{r_{\\rm max} + r_{\\rm min}}
\n\\end{eqnarray}<\/p>\n

\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u30922\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066\u89e3\u304f<\/span><\/h3>\n

\u3053\u306e\u307e\u307e\u3067\u3082\u8fd1\u4f3c\u7684\u306b\u89e3\u304f\u3053\u3068\u306f\u53ef\u80fd\u3067\u3042\u308b\u304c\uff0c\u4e21\u8fba\u3092 \\(\\phi\\) \u3067\u5fae\u5206\u3057\u3066<\/p>\n

\\begin{eqnarray}
\n2\\frac{ds}{d\\phi} \\frac{d^2s}{d\\phi^2}\u00a0 &=& -2 \\left(s -\\frac{GM}{\\ell^2} \\right) \\frac{ds}{d\\phi} + 3 r_g s^2 \\frac{ds}{d\\phi}\\\\
\n\\therefore\\ \\\u00a0 \\frac{d^2s}{d\\phi^2}\u00a0 &=& -\\left(s -\\frac{GM}{\\ell^2} \\right) + \\frac{3}{2}r_g s^2 \\\\
\n&=& -\\left(s -\\frac{1}{a (1 -e^2)} +\\frac{(3 + e^2) r_g}{2 a^2 (1 -e^2)^2} \\right) + \\frac{3}{2}r_g s^2
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\uff0c\u3053\u3061\u3089\u3092\u8fd1\u4f3c\u7684\u306b\u89e3\u304f\u65b9\u6cd5\u3092\u7d39\u4ecb\u3059\u308b\u3002\u306a\u3093\u3067\u3082\u30461\u968e\u5fae\u5206\u3057\u3066\uff0c\u3053\u306e2\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u3057\u305f\u304b\u3068\u3044\u3046\u3068\uff0c\u307f\u3066\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u5168\u4f53\u3092\u898b\u308f\u305f\u3059\u3068\u61d0\u304b\u3057\u3044\u5358\u632f\u52d5\u306e\u65b9\u7a0b\u5f0f\u306b\u88dc\u6b63\u9805\u304c\u3064\u3044\u305f\u5f62\u306b\u306a\u3063\u3066\u3044\u3066\uff0c\u3072\u3087\u3063\u3068\u3057\u305f\u3089\u89e3\u304d\u3084\u3059\u3044\u304b\u3082\u2026 \u3068\u601d\u308f\u308c\u308b\u304b\u3089\u3067\u3042\u308b\u3002<\/p>\n

\u7a4d\u5206\u5b9a\u6570\u3092\u6c7a\u3081\u308b\u521d\u671f\u6761\u4ef6<\/span><\/h4>\n

\u305f\u3060\u3057\uff0c\u3053\u306e\u307e\u307e\u3060\u3068\u672c\u67651\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3060\u304b\u3089\u89e3\u306f\u7a4d\u5206\u5b9a\u6570\u30921\u500b\u6301\u3064\u306f\u305a\u304c\uff0c\u3082\u30461\u968e\u5fae\u5206\u3057\u30662\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u306a\u3063\u305f\u306e\u3067\u89e3\u306f\u7a4d\u5206\u5b9a\u6570\u30922\u500b\u6301\u3064\u3053\u3068\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u3002<\/p>\n

\u5fae\u5206\u306e\u968e\u6570\u3092\u4eba\u70ba\u7684\u306b\u4e0a\u3052\u305f\u3053\u3068\u3067\u73fe\u308c\u308b\u3053\u3068\u306b\u306a\u3063\u3066\u3057\u307e\u3063\u305f\u4f59\u5206\u306e\u7a4d\u5206\u5b9a\u6570\u3082\u6c7a\u3081\u308b\u305f\u3081\u306e\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066\uff0c\u4ee5\u4e0b\u3092\u63a1\u7528\u3059\u308b\u3002<\/p>\n