{"id":8268,"date":"2024-04-04T13:16:50","date_gmt":"2024-04-04T04:16:50","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=8268"},"modified":"2024-05-17T13:11:46","modified_gmt":"2024-05-17T04:11:46","slug":"%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%82%92%e8%a7%a3%e3%81%ae%e5%bd%a2%e6%b1%ba%e3%82%81%e6%89%93%e3%81%a1%e3%81%a7%e8%a7%a3%e3%81%8f","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\/%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%82%92%e8%a7%a3%e3%81%ae%e5%bd%a2%e6%b1%ba%e3%82%81%e6%89%93%e3%81%a1%e3%81%a7%e8%a7%a3%e3%81%8f\/","title":{"rendered":"\u5f31\u91cd\u529b\u5834\u4e2d\u306e\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u89e3\u306e\u5f62\u6c7a\u3081\u6253\u3061\u3067\u89e3\u304f\u4f8b"},"content":{"rendered":"

\u5185\u5c71\u9f8d\u96c4\u8457\u300c\u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6\u300d\uff08\u88f3\u83ef\u623f\uff09<\/a>\u7b49\uff0c\u591a\u304f\u306e\u30c6\u30ad\u30b9\u30c8\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u89e3\u6cd5\u4f8b\u3068\u3057\u3066\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\uff0c\u89e3\u306e\u5f62\u3092<\/p>\n

$$s \\equiv \\frac{1}{r} \\equiv \\frac{1 + e \\cos \\left(\\gamma\\, \\phi\\right)}{L}$$<\/p>\n

\u306e\u3088\u3046\u306b\u6c7a\u3081\u6253\u3061\u306b\u3057\u3066\u89e3\u304f\u65b9\u6cd5\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n

\u306a\u304a\uff0c\u5185\u5c71\u672c\u3067\u306f\u8868\u8a18\u3092<\/p>\n

$$u \\equiv \\frac{1}{r} = \\frac{1}{l} \\left\\{ 1 + e \\cos (\\eta\\, \\varphi)\\right\\}$$<\/p>\n

\u3068\u3057\u3066\u308b\u304c\uff0c\u672c\u30b5\u30a4\u30c8\u306e\u5909\u6570\u4f7f\u7528\u4f8b\u306b\u9451\u307f\uff0c\u5909\u6570\u3092\u5c11\u3057\u5909\u3048\u3066<\/p>\n

$$s \\equiv \\frac{1}{r} \\equiv \\frac{1 + e \\cos \\left(\\gamma\\, \\phi\\right)}{L}$$<\/p>\n

\u3068\u3057\u3066\u3044\u308b\u3002\u5b9a\u6570 $L$ \u306b\u3064\u3044\u3066\u306f\uff0c\u904b\u52d5\u304c\u6709\u754c\u306a\u675f\u7e1b\u904b\u52d5\u3067\u3042\u308c\u3070\uff0c<\/p>\n

$$ r_{\\rm min} \\equiv \\frac{L}{1 + e} \\leq r \\leq \\frac{L}{1 – e} \\equiv r_{\\rm max}$$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\nr_{\\rm min} &\\equiv& a ( 1 -e) \\\\
\nr_{\\rm max} &\\equiv& a ( 1 + e) \\\\
\n\\therefore\\ \\ a &=& \\frac{1}{2} (r_{\\rm min} + r_{\\rm max})
\n\\end{eqnarray}<\/p>\n

\u3067\u5c0e\u5165\u3055\u308c\u308b\u5909\u6570 $a$ \u3092\u4f7f\u3046\u3068<\/p>\n

$$L = a (1 -e^2)$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3002$\\gamma = 1$ \u306e\u3068\u304d\uff0c\u8ecc\u9053\u306f\u6955\u5186\u3068\u306a\u308a\uff0c$a$ \u306f\u8ecc\u9053\u9577\u534a\u5f84\uff0c$e$ \u306f\u96e2\u5fc3\u7387\u3068\u547c\u3070\u308c\u308b\u3088\u3046\u306b\u306a\u308b\u3002$\\gamma \\neq 1$ \u3067\u3042\u308c\u3070\u4e00\u822c\u306b\u306f\u9589\u3058\u305f\u6955\u5186\u8ecc\u9053\u3068\u306a\u3089\u306a\u3044\u305f\u3081\uff0c\u53b3\u5bc6\u306b\u306f $a, e$ \u3092\u8ecc\u9053\u9577\u534a\u5f84\u3084\u96e2\u5fc3\u7387\u3068\u547c\u3076\u3053\u3068\u306f\u3067\u304d\u306a\u3044\uff08\u306e\u3067\u3042\u308b\u304c\uff0c\u4ee5\u4e0b\u306e\u8a08\u7b97\u306e\u3088\u3046\u306b $r_g$ \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067\u306f $r_g$ \u306e\u304b\u304b\u3063\u305f\u9805\u306e\u306a\u304b\u306e $a$ \u3084 $e$ \u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u306e $a$ \u3084 $e$ \u3068\u3057\u3066\u3088\u3044\u306e\u3067\uff0c\u3053\u306e\u8fd1\u4f3c\u306e\u7bc4\u56f2\u3067\u306f $a, e$ \u3092\u8ecc\u9053\u9577\u534a\u5f84\u3084\u96e2\u5fc3\u7387\u3068\u547c\u3093\u3067\u3082\u3088\u3044\u3053\u3068\u306b\u306a\u308b\uff09\u3002<\/p>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/h3>\n

\u3055\u3066\uff0c$\\displaystyle s \\equiv \\frac{1}{r}$ \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 + \\frac{c^2 -\\epsilon^2 c^2}{\\ell^2} &=& r_g\\, s^3 \\tag{A}
\n\\end{eqnarray}<\/p>\n

\u3042\u3089\u305f\u3081\u3066 $s$ \u3092\u66f8\u304f\u3068\uff0c<\/p>\n

$$s = \\frac{1}{r} = \\frac{1 + e \\cos \\left(\\gamma\\, \\phi\\right)}{a (1 -e^2)} \\tag{B}$$<\/p>\n

\u3067\u3042\u308a\uff0c\u3053\u306e\u3088\u3046\u306b\u89e3\u306e\u5f62\u3092\u6c7a\u3081\u6253\u3061\u306b\u3057\u3066\u89e3\u3044\u3066\u307f\u308b\u3002<\/p>\n

\u305d\u306e\u524d\u306b\uff0c$\\displaystyle s_{\\rm min} = \\frac{1}{r_{\\rm max}} = \\frac{1}{a (1+e)}$ \u3068$\\displaystyle s_{\\rm max} = \\frac{1}{r_{\\rm min}} = \\frac{1}{a (1 -e)}$ \u3067\u6975\u5024\u3092\u3068\u308b\u306e\u3067\u3053\u306e\u3068\u304d $\\displaystyle \\frac{ds}{d\\phi} = 0$ \u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u6975\u5024\u3092\u3068\u308b2\u70b9\u306b\u304a\u3044\u3066\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f<\/p>\n

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

$\\mbox{(I)}$ \u304a\u3088\u3073 $\\mbox{(II)}$ \u5f0f\u3092\u9023\u7acb\u65b9\u7a0b\u5f0f\u3068\u3059\u308c\u3070\uff0c$\\mbox{(I)} -\\mbox{(II)}$ \u304b\u3089<\/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\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c$\\mbox{(I)}\\times a (1 + e) -\\mbox{(II)}\\times a (1 -e)$ \u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\frac{c^2 -\\epsilon^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

\u3068\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u3042\u3089\u305f\u3081\u3066\u5f62\u6c7a\u3081\u6253\u3061\u306e\u89e3 (B) \u5f0f\u3092\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f (A) \u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c\u5de6\u8fba\u306f<\/p>\n

\\begin{eqnarray}
\n\\mbox{\u5de6\u8fba} &=& \\left(- \\frac{e \\gamma \\sin (\\gamma\\,\\phi)}{a (1 -e^2)}\\right)^2 \\\\
\n&& + \\left\\{\\frac{1 + e \\cos \\left(\\gamma\\, \\phi\\right)}{a (1 -e^2)} \\right\\}^2 \\\\
\n&& -2 \\left\\{\\frac{1}{a (1 -e^2)} -\\frac{(3 + e^2) r_g}{2 a^2 (1 -e^2)^2} \\right\\} \\frac{1 + e \\cos \\left(\\gamma\\, \\phi\\right)}{a (1 -e^2)} \\\\
\n&& + \\left\\{\\frac{1}{a^2 (1 -e^2)} -\\frac{2 r_g}{a^3 (1 -e^2)^2} \\right\\} \\\\
\n&=& \\frac{e^2 \\gamma^2}{a^2 (1 -e^2)^2} \\left\\{ 1 -\\cos^2 \\left(\\gamma\\, \\phi\\right)\\right\\} \\\\
\n&& + \\frac{1}{a^2 (1 -e^2)^2} \\left\\{1 + 2 e \\cos(\\gamma\\,\\phi) + e^2 \\cos^2 (\\gamma\\,\\phi) \\right\\} \\\\
\n&& -\\left\\{\\frac{2}{a^2 (1 -e^2)^2} -\\frac{(3 + e^2) r_g}{a^3 (1 -e^2)^3} \\right\\}\\left\\{ 1 + e \\cos (\\gamma\\,\\phi)\\right\\} \\\\
\n&&+ \\left\\{\\frac{1}{a^2 (1 -e^2)} -\\frac{2 r_g}{a^3 (1 -e^2)^2} \\right\\}
\n\\end{eqnarray}<\/p>\n

\u53f3\u8fba\u306f<\/p>\n

\\begin{eqnarray}
\n\\mbox{\u53f3\u8fba} &=& \\frac{r_g}{a^3 (1 -e^2)^3} \\left\\{1 + 3 e \\cos (\\gamma\\,\\phi) + 3 e^2 \\cos^2 (\\gamma\\,\\phi) + e^3 \\cos^3 (\\gamma\\,\\phi) \\right\\}
\n\\end{eqnarray}<\/p>\n

\u4e21\u8fba\u306e $\\cos (\\gamma\\,\\phi)$ \u306e\u5404\u3079\u304d\u306e\u4fc2\u6570\u3092\u6bd4\u8f03\u3059\u308b\u3068\uff0c$\\cos (\\gamma\\,\\phi)$ \u306e\u30bc\u30ed\u6b21\uff0c\u3064\u307e\u308a\u5b9a\u6570\u9805\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{e^2 \\gamma^2}{a^2 (1 -e^2)^2} + \\frac{1}{a^2 (1 -e^2)^2}&& \\\\
\n-\\left\\{\\frac{2}{a^2 (1 -e^2)^2} -\\frac{(3 + e^2) r_g}{a^3 (1 -e^2)^3} \\right\\}&&\\\\
\n+ \\frac{1}{a^2 (1 -e^2)} -\\frac{2 r_g}{a^3 (1 -e^2)^2} &&\\\\
\n&=&\u00a0 \\frac{r_g}{a^3 (1 -e^2)^3} \\\\
\n\\therefore\\ \\ \\frac{e^2 (\\gamma^2 -1)}{a^2 (1 -e^2)^2} &=& -\\frac{3 e^2\\, r_g}{a^3 (1 -e^2)^3} \\\\
\n\\therefore\\ \\ \\gamma^2 &=& 1 -\\frac{3 r_g}{a (1 -e^2)}\\tag{1}
\n\\end{eqnarray}<\/p>\n

$\\cos (\\gamma\\,\\phi)$ \u306e1\u6b21\u306e\u9805\u306e\u4fc2\u6570\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{2 e}{a^2 (1 -e^2)^2} -e \\left\\{\\frac{2}{a^2 (1 -e^2)^2} -\\frac{(3 + e^2) \\,r_g}{a^3 (1 -e^2)^3} \\right\\}&=& \\frac{3 e\\, r_g}{a^3 (1 -e^2)^3}\u00a0 \\\\
\n\\therefore\\ \\ \\frac{ e^3\\, r_g}{a^3 (1 -e^2)^3} &=& 0 \\tag{2}
\n\\end{eqnarray}<\/p>\n

$\\cos (\\gamma\\,\\phi)$ \u306e2\u6b21\u306e\u9805\u306e\u4fc2\u6570\u306f<\/p>\n

\\begin{eqnarray}
\n-\\frac{e^2 \\gamma^2}{a^2 (1 -e^2)^2} + \\frac{e^2}{a^2 (1 -e^2)^2} &=& \\frac{3 e^2 r_g}{a^3 (1 -e^2)^3}\u00a0 \\\\
\n\\therefore\\ \\ \\gamma^2 &=& 1 -\\frac{3 r_g}{a (1 -e^2)}\\tag{3}
\n\\end{eqnarray}<\/p>\n

$\\cos (\\gamma\\,\\phi)$ \u306e3\u6b21\u306e\u9805\u306e\u4fc2\u6570\u306f<\/p>\n

\\begin{eqnarray}
\n0 &=& \\frac{e^3\\, r_g}{a^3 (1 -e^2)^3}\u00a0 \\tag{4}
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u307e\u307e\u3067\u306f\u5de6\u8fba\u306b $\\cos^3(\\gamma\\,\\phi)$ \u306e\u9805\u304c\u306a\u3044\u305f\u3081\u306b\u4e21\u8fba\u304c\u3064\u308a\u3042\u308f\u306a\u304f\u306a\u308b\u305f\u3081\uff0c\uff08\u82e6\u3057\u7d1b\u308c\u306b\uff09\u300c$e^3\\, r_g$ \u306e\u9805\u306f\u5c0f\u3055\u3044\u306e\u3067\u7121\u8996\u3059\u308b<\/strong><\/span>\u300d\u3068\u5ba3\u8a00\u3059\u308b\u3002\u8fd1\u70b9\u79fb\u52d5\u3092\u8868\u3059 $\\gamma$ \u3092\u6c42\u3081\u308b\u3060\u3051\u306a\u3089\u3053\u308c\u3067\u3082\u3088\u3044\u304c\uff0c\u3053\u306e\u7dbb\u3073\u306f\uff0c\u3064\u307e\u308a\u306f $r_g$ \u306e1\u6b21\u307e\u3067\u306e\u5b8c\u5168\u306a\u7dda\u5f62\u8fd1\u4f3c\u89e3\u306f\uff0c\u4e0a\u8a18\u306e\u3088\u3046\u306b\u6c7a\u3081\u6253\u3061\u3057\u305f\u5f62\u3067\u306f\u5b8c\u5168\u306b\u306f\u8868\u305b\u306a\u3044\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u308b\u3002\uff08\u3067\u306f\u3069\u306e\u3088\u3046\u306a\u5f62\u306b\u306a\u308b\u306e\u304b\u306f\u5225\u9014\u300c\u5f31\u91cd\u529b\u5834\u4e2d\u306e\u7c92\u5b50\u306e\u8ecc\u9053\u306e\u8fd1\u4f3c\u89e3\u306e\u5225\u89e3\u6cd5<\/a>\u300d\u306b\u307e\u3068\u3081\u3066\u3044\u307e\u3057\u305f\u3002\uff09<\/p>\n

\u300c$e^3\\, r_g$ \u306e\u9805\u306f\u5c0f\u3055\u3044\u306e\u3067\u7121\u8996\u3059\u308b\uff08\u30bc\u30ed\u3068\u307f\u306a\u3059\uff09<\/strong><\/span>\u300d\u3068\uff0c(4) \u5f0f\u304a\u3088\u3073 (2) \u5f0f\u306f $0 = 0$ \u306e\u6052\u7b49\u5f0f\u306b\u306a\u308b\u3002(1) \u5f0f\u3068 (3) \u5f0f\u306f\u540c\u3058\u5f0f\u3067\u3042\u308a\uff0c\u4ee5\u4e0b\u3092\u4e0e\u3048\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\gamma^2 &=&\u00a0 1 -\\frac{3 r_g}{a (1 -e^2)} \\\\
\n\\therefore\\ \\ \\gamma &=& \\sqrt{1 -\\frac{3 r_g}{a (1 -e^2)} } \\simeq 1 -\\frac{3 r_g}{2 a (1 -e^2)}
\n\\end{eqnarray}<\/p>\n

\u8fd1\u70b9\u79fb\u52d5\u89d2 $\\varDelta$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\gamma \\times (2 \\pi + \\varDelta) &=& 2 \\pi \\\\
\n\\therefore\\ \\ \\varDelta &=& \\frac{2 \\pi}{\\gamma}\u00a0 -2 \\pi \\\\
\n&\\simeq& \\frac{3 \\pi r_g}{a (1-e^2)} = \\frac{6 \\pi G M}{ c^2 a (1 -e^2)}
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5185\u5c71\u9f8d\u96c4\u8457\u300c\u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6\u300d\uff08\u88f3\u83ef\u623f\uff09\u7b49\uff0c\u591a\u304f\u306e\u30c6\u30ad\u30b9\u30c8\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u89e3\u6cd5\u4f8b\u3068\u3057\u3066\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\uff0c\u89e3\u306e\u5f62\u3092<\/p>\n

$$s \\equiv \\frac{1}{r} \\equiv \\frac{1 + e \\cos \\left(\\gamma\\, \\phi\\right)}{L}$$<\/p>\n

\u306e\u3088\u3046\u306b\u6c7a\u3081\u6253\u3061\u306b\u3057\u3066\u89e3\u304f\u65b9\u6cd5\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>

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