{"id":8278,"date":"2024-04-04T13:10:08","date_gmt":"2024-04-04T04:10:08","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=8278"},"modified":"2024-06-08T10:51:14","modified_gmt":"2024-06-08T01:51:14","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%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3%e3%81%ae%e5%88%a5%e8%a7%a3%e6%b3%95","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%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3%e3%81%ae%e5%88%a5%e8%a7%a3%e6%b3%95\/","title":{"rendered":"\u5f31\u91cd\u529b\u5834\u4e2d\u306e\u7c92\u5b50\u306e\u8ecc\u9053\u306e\u8fd1\u4f3c\u89e3\u306e\u5225\u89e3\u6cd5"},"content":{"rendered":"

\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u30921\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306e\u307e\u307e\u3067\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>\n

<\/p>\n

\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f<\/h3>\n

$\\displaystyle s \\equiv \\frac{1}{r}$ \u3068\u3059\u308b\u3068\uff0c\u300c\u5f31\u91cd\u529b\u5834\u4e2d\u306e\u7c92\u5b50\u306e\u8ecc\u9053\u306e\u8fd1\u4f3c\u89e3\uff1a\u8fd1\u70b9\u79fb\u52d5<\/a>\u300d\u306e\u30da\u30fc\u30b8\u306b\u307e\u3068\u3081\u305f\u3088\u3046\u306b<\/p>\n

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

\u3042\u308b\u3044\u306f (1) \u5f0f\u306e\u4e21\u8fba\u3092 $\\gamma^2$ \u3067\u308f\u3063\u3066<\/p>\n

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

\u3053\u3053\u307e\u3067\u306f\u53b3\u5bc6\u306a\u5f0f\u3067\u3042\u308b\u3002\u3053\u306e\u307e\u307e\u3067\u306f\u89e3\u6790\u306b\u89e3\u3051\u306a\u3044\u3002\u7279\u306b\u53f3\u8fba\u306e\u7b2c3\u9805\uff083\u4e57\u306e\u9805\uff09\u304c\u66f2\u8005\u3067\u3042\u308b\u3002\u305d\u3053\u3067\uff0c\u4ee5\u4e0b\u3067\u306f\u4f55\u3089\u304b\u306e\u65b9\u6cd5\u3067\u8fd1\u4f3c\u7684\u306b\u89e3\u304f\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

$r_g$ \u306e\u30bc\u30ed\u6b21\u89e3<\/h4>\n

\u5929\u4f53\u306e\u8ecc\u9053\u306f\u91cd\u529b\u534a\u5f84 $r_g$ \u306e\u5341\u5206\u5916\u5074\u3067\u3042\u308b\u3068\u3044\u3046\u72b6\u6cc1\u3067\u306f\uff0c$\\displaystyle \\frac{r_g}{r} = r s \\ll 1$\uff0c\u3042\u308b\u3044\u306f\u540c\u3058\u3053\u3068\u3060\u304c $\\displaystyle \\frac{r_g}{a} \\ll 1$ \u3068\u3057\u3066\u3088\u3044\u3002\u4e0a\u306e (1) \u5f0f\u3067 $r_g$ \u304c\u304b\u304b\u3063\u3066\u3044\u308b\u9805\u3092\u7121\u8996\u3057\u305f\u5834\u5408\u306e\u89e3\u3092 $s_0$ \u3068\u66f8\u304f\u3068\uff0c\u3053\u306e\u8fd1\u4f3c\u3067\u306f $\\gamma=1$ \u3068\u3057\u3066\u3088\u3044\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\n\\left(\\frac{ds_0}{d\\phi} \\right)^2 + \\left(s_0 -\\frac{1}{a (1 -e^2)} \\right)^2 &=&
\n\\left(\\frac{e}{a (1 -e^2)} \\right)^2
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7c21\u5358\u306b\u89e3\u3051\u308b\u30022\u5e74\u751f\u306e\u6388\u696d\u300c\u7c21\u5358\u306a1\u968e\u975e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/a>\u300d\u3067\u3084\u3063\u3066\u307e\u3059\u3002\u3053\u3053\u3067\u306f\uff0c\u521d\u671f\u6761\u4ef6\u3092 $\\phi = 0$ \u3067 $s = 1\/r_{\\rm min} = 1\/a(1-e)$ \u3068\u3057\u3066…<\/p>\n

\\begin{eqnarray}
\ns_0 -\\frac{1}{a (1 -e^2)} &=& \\frac{e \\cos \\phi}{a (1 -e^2)} \\\\
\n\\therefore\\ \\ s_0 = \\frac{1}{r} &=& \\frac{1 + e \\cos \\phi}{a (1 -e^2)}
\n\\end{eqnarray}<\/p>\n

$O(e^3\\, r_g)$ \u306e\u9805\u3092\u7121\u8996\u3057\u305f\u5834\u5408\u306e\u89e3<\/h4>\n

(2) \u5f0f\u53f3\u8fba\u306e\u7b2c2\u9805\uff0c\u7b2c3\u9805\u306b $s_0$ \u3092\u5165\u308c\u3066\u8a55\u4fa1\u3057\u3066\u3084\u308b\u3068\u3069\u3061\u3089\u3082 $O({\\color{red}{e^3\\, r_g}})$ \u306e\u9805\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u6955\u5186\u8ecc\u9053\u306e\u5834\u5408\u306b\u96e2\u5fc3\u7387\u3068\u547c\u3070\u308c\u308b $e$ \u306e\u5024\u306f\u4e00\u822c\u306b $0 \\leq e < 1$ \u3067\u3042\u308b\u304b\u3089\uff0c $O({\\color{red}{e^3\\, r_g}})$ \u306e\u9805\u3092\u7121\u8996\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{1}{\\gamma^2}\\left(\\frac{ds}{d\\phi} \\right)^2 + \\left(s -\\frac{1}{a (1 -e^2)} \\right)^2 &=&
\n\\left(\\frac{e}{a (1 -e^2)} \\right)^2
\n\\end{eqnarray}<\/p>\n

\u306e\u5f62\u3092\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\uff0c2\u5e74\u751f\u306e\u3068\u304d\u306b\u3084\u3063\u305f\u300c\u7c21\u5358\u306a1\u968e\u975e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/a>\u300d\u306e\u30da\u30fc\u30b8\u3067\u7c21\u5358\u306b\u89e3\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3002\u521d\u671f\u6761\u4ef6\u3092 $\\phi = 0 $ \u3067 $s= 1\/a(1 -e)$ \u3068\u3057\u3066<\/p>\n

\\begin{eqnarray}
\ns -\\frac{1}{a (1 -e^2)}\u00a0 &=& \\frac{e \\cos(\\gamma \\phi )}{a (1 -e^2)} \\\\
\n\\therefore\\ \\ s &=& \\frac{1}{a (1 -e^2)} + \\frac{e \\cos(\\gamma \\phi )}{a (1 -e^2)} \\\\
\n&=& \\frac{1 + e \\cos(\\gamma \\phi )}{a (1 -e^2)}
\n\\end{eqnarray}<\/p>\n

$O(e^3\\, r_g)$ \u306e\u9805\u3092\u7121\u8996\u3057\u306a\u3044\u5834\u5408\u306e\u5f0f<\/h4>\n

(1) \u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u304a\u3088\u3073\u7b2c3\u9805\u306b\uff0c$O({\\color{red}{e^3\\, r_g}})$ \u306e\u9805\u3092\u7121\u8996\u3057\u305f\u5834\u5408\u306e\u89e3\u3092\u4ee3\u5165\u3057\u3066\u8a55\u4fa1\u3057\u3066\u3084\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\left(\\frac{1}{\\gamma}\\frac{ds}{d\\phi} \\right)^2 + \\left(s -\\frac{1}{a (1 -e^2)} \\right)^2 &\\simeq&
\n\\left(\\frac{e}{a (1 -e^2)} \\right)^2 \\\\
\n&& + \\frac{{\\color{red}{e^3\\, r_g}}}{\\gamma^2 a^3 (1 -e^2)^3} \\left(\\cos^3 \\gamma\\phi -\\cos\\gamma\\phi \\right) \\tag{2}
\n\\end{eqnarray}<\/p>\n

$O(e^3\\, r_g)$ \u306e\u9805\u3092\u7121\u8996\u3057\u306a\u3044\u5834\u5408\u306e\u89e3<\/h4>\n

$r_g$ \u306e1\u6b21\u307e\u3067\u306e\u7dda\u5f62\u8fd1\u4f3c\u89e3\u3092<\/p>\n

\\begin{eqnarray}
\ns -\\frac{1}{a (1 -e^2)}
\n&=& \\frac{e \\cos(\\gamma \\phi )}{a (1 -e^2)} + \\frac{e^2 \\,r_g}{a^2 (1 -e^2)^2}\u00a0 {\\color{blue}{s_1(\\phi)}}
\n\\end{eqnarray}<\/p>\n

\u3068\u304a\u3044\u3066 (2) \u5f0f\u306e\u5de6\u8fba\u306b\u4ee3\u5165\u3057\uff0c$r_g$ \u306e1\u6b21\u307e\u3067\u3068\u3063\u3066\u3084\u308b\u3068 ${\\color{blue}{s_1(\\phi)}}$ \u306b\u5bfe\u3059\u308b\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n-2 \\sin \\gamma \\phi\\, \\frac{1}{\\gamma}\\frac{d {\\color{blue}{s_1}}}{d\\phi} + 2 \\cos \\gamma \\phi \\,{\\color{blue}{s_1}} &=& \\frac{\\cos^3 \\gamma \\phi -\\cos\\gamma\\phi}{\\gamma^2}
\n\\end{eqnarray}<\/p>\n

\u5ff5\u306e\u305f\u3081\uff0c$\\varphi \\equiv \\gamma\\phi$ \u3068\u3059\u308b\u3068<\/p>\n

\\begin{eqnarray}
\n-2 \\sin \\varphi\\, \\frac{d {\\color{blue}{s_1}}}{d\\varphi} + 2 \\cos\u00a0 \\varphi \\,{\\color{blue}{s_1}}
\n&=& \\frac{\\cos^3\u00a0 \\varphi -\\cos \\varphi}{\\gamma^2}
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u30822\u5e74\u751f\u306e\u3068\u304d\u306b\u300c\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u4f8b\u984c<\/a>\u300d\u3067\u3084\u3063\u3066\u3044\u3066\u7b54\u3048\u306f…<\/p>\n

\\begin{eqnarray}
\n{\\color{blue}{s_1(\\varphi)}} &=& \\frac{\\sin^2 \\varphi}{2} = \\frac{\\sin^2 \\gamma\\phi}{2 \\gamma^2}
\n\\end{eqnarray}<\/p>\n

\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u6700\u7d42\u7684\u306a\u7b54\u3048\u306f<\/p>\n

\\begin{eqnarray}
\ns -\\frac{1}{a (1 -e^2)}
\n&=& \\frac{e \\cos(\\gamma \\phi )}{a (1 -e^2)} + \\frac{e^2 \\,r_g}{a^2 (1 -e^2)^2} \\frac{\\sin^2 \\gamma\\phi}{2\\gamma^2} \\\\
\n\\therefore\\ \\ s = \\frac{1}{r} &=& \\frac{1 +e \\cos(\\gamma \\phi )}{a (1 -e^2)} + \\frac{e^2 \\,r_g \\sin^2 \\gamma \\phi }{2 \\gamma^2 a^2 (1 -e^2)^2} \\\\
\n&\\simeq&
\n\\frac{1 + e\\cos (\\gamma \\phi)}{a(1-e^2)}
\n\\left\\{ 1 + \\frac{r_g e^2}{2 a (1-e^2)} \\frac{\\sin^2 (\\gamma \\phi)}{1+e\\cos(\\gamma\\phi)}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u306a\u305c $O(e^2\\, r_g)$ \u306e\u9805\u307e\u3067\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308b\u306e\u304b<\/h3>\n

\u8fd1\u70b9\u79fb\u52d5\u3092\u6c42\u3081\u308b\u3060\u3051\u306a\u3089\uff0c$O(e^2\\, r_g)$ \u306e\u9805\u3092\u7121\u8996\u3057\u305f\u5834\u5408\u306e\u89e3<\/p>\n

\\begin{eqnarray}
\ns &=& \\frac{1 + e \\cos(\\gamma \\phi )}{a (1 -e^2)}
\n\\end{eqnarray}<\/p>\n

\u3067\u5341\u5206\u3067\u3042\u308b\u3002$\\gamma \\neq 1$ \u3067\u3042\u308b\u305f\u3081\u306b\u8fd1\u70b9\u79fb\u52d5\u304c\u304a\u3053\u308a\uff0c\u305d\u306e\u8fd1\u70b9\u79fb\u52d5\u89d2 $\\varDelta$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\gamma (2 \\pi + \\varDelta) &=& 2 \\pi \\\\
\n\\therefore\\ \\ \\varDelta &=& \\frac{2 \\pi}{\\gamma} -2 \\pi \\\\
\n&\\simeq& \\frac{3 \\pi r_g}{a (1 -e^2)} = \\frac{6 \\pi GM}{c^2 \\, a (1 -e^2)} \\tag{A}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u304c\uff0c<\/p>\n

$$\\frac{3 \\pi r_g}{a (1 -e^2)}\u00a0 -\\frac{3 \\pi r_g}{a} = \\frac{3 e^2\\, r_g}{a (1 -e^2)} = O({\\color{red}{e^2\\, r_g}}) $$<\/p>\n

\u3067\u3042\u308b\u306e\u3067\uff0c$O({\\color{red}{e^2\\, r_g}})$ \u306e\u9805\u306f\u7121\u8996\u3059\u308b\u3068\u3044\u3046\u8fd1\u4f3c\u306e\u65b9\u91dd\u306b\u305f\u3066\u3070\uff0c\u8fd1\u70b9\u79fb\u52d5\u89d2\u306f<\/p>\n

\\begin{eqnarray}
\n\\varDelta
\n&\\simeq& \\frac{3 \\pi r_g}{a} = \\frac{6 \\pi GM}{c^2 \\, a} \\ \\ \\mbox{?} \\tag{B}
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\u3082\u304b\u307e\u308f\u306a\u3044\uff0c\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002(A) \u5f0f\u3067\u3082 (B) \u5f0f\u3067\u3082\u3069\u3061\u3089\u3067\u3082\u3044\u3044\uff0c\u3068\u3044\u3046\u306e\u3067\u306f\u306a\u304f\uff0c\u3069\u3063\u3061\u304b\uff0c(B) \u3067\u306f\u306a\u304f\u3066 (A) \u3067\u306a\u3051\u308c\u3070\u3060\u3081\u306a\u3093\u3060\uff0c\u3068\u306f\u3063\u304d\u308a\u3082\u306e\u3092\u8a00\u3046\u305f\u3081\u306b\u306f\uff0c$O(e^2\\, r_g)$ \u306e\u9805\u3082\u7121\u8996\u305b\u305a\u306b\uff0c$r_g$ \u306e1\u6b21\u307e\u3067\u306e\u5b8c\u5168\u306a\u7dda\u5f62\u8fd1\u4f3c\u89e3\u304c\u5fc5\u8981\u3060\uff01\u3068\u3044\u3046\u306e\u304c\u79c1\u306e\u8003\u3048\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u30921\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306e\u307e\u307e\u3067\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":1025,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-8278","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t