{"id":1021,"date":"2022-01-15T20:42:14","date_gmt":"2022-01-15T11:42:14","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1021"},"modified":"2022-06-24T14:21:59","modified_gmt":"2022-06-24T05:21:59","slug":"%e5%8b%95%e5%be%84%e6%96%b9%e5%90%91%e3%81%ae%e8%87%aa%e7%94%b1%e8%90%bd%e4%b8%8b%e9%81%8b%e5%8b%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%8b%95%e5%be%84%e6%96%b9%e5%90%91%e3%81%ae%e8%87%aa%e7%94%b1%e8%90%bd%e4%b8%8b%e9%81%8b%e5%8b%95\/","title":{"rendered":"\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005"},"content":{"rendered":"

<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6\u306b\u3064\u3044\u3066\u3002<\/p>\n

\u3053\u306e\u30da\u30fc\u30b8\u3067\u306f \\(c = 1\\) \u3068\u3059\u308b\u3002<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/h3>\n

\u7dda\u7d20<\/strong>\u306f\uff0c<\/p>\n

$$ ds^2 = -\\left(1-\\frac{r_g}{r}\\right)\u00a0 dt^2 + \\frac{dr^2} {1-\\frac{r_g}{r}} + r^2(d\\theta^2 + \\sin^2\\theta d\\phi^2)$$<\/p>\n

\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u30bc\u30ed\u3067\u306a\u3044\u7a0e\u5206\u306f<\/p>\n

\\begin{eqnarray}
\ng_{00} &=& -\\left(1-\\frac{r_g}{r}\\right) \\\\
\ng_{11} &=& \\frac{1} {1-\\frac{r_g}{r}}\\\\
\ng_{22} &=& r^2\\\\
\ng_{33} &=& r^2 \\sin^2\\theta
\n\\end{eqnarray}<\/p>\n

\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6<\/h3>\n

\\(r = r_i\\) \u306e\u3068\u304d \\(\\bar{u}^1 = 0\\) \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6<\/strong><\/span>\u306e\u3082\u3068\uff0c\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b<\/strong><\/span>\uff08\\(\\bar{u}^1 < 0\\)\uff09\u3057\u3066\u3044\u304f\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6<\/strong><\/span>\u306e\u6210\u5206\u3092$$\\bar{u}^{\\mu} = (\\bar{u}^0, \\bar{u}^1, 0, 0)$$\u3068\u3059\u308b\u3002<\/p>\n

\\(\\bar{u}_{\\mu} \\equiv g_{\\mu\\nu} \\bar{u}^{\\nu}\\) \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d\\bar{u}_{\\mu}}{d\\tau} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} \\bar{u}^{\\alpha}\\bar{u}^{\\beta}$$<\/p>\n

\u304b\u3089\uff0c\\(g_{\\alpha\\beta}\\) \u306f \\(x^0 = t\\) \u3092\u542b\u307e\u306a\u3044\u306e\u3067 \\(g_{\\alpha\\beta, 0} = 0\\) \u3068\u306a\u308a\uff0c<\/p>\n

$$\\frac{d\\bar{u}_{0}}{d\\tau}= 0, \\quad\\therefore\\ \\ \\bar{u}_{0} = \\mbox{const.} \\equiv – \\epsilon, \\quad \\bar{u}^0 = g^{00} \\bar{u}_{0} = \\frac{\\epsilon}{1-\\frac{r_g}{r}}$$<\/p>\n

\u307e\u305f\uff0c4\u5143\u901f\u5ea6\u306e\u898f\u683c\u5316\u6761\u4ef6 \\(g_{\\mu\\nu} \\bar{u}^{\\mu} \\bar{u}^{\\nu} = -1\\) \u304b\u3089<\/p>\n

$$g_{00} \\left(\\bar{u}^0\\right)^2 + g_{11} \\left(\\bar{u}^1\\right)^2 \u00a0 = -1, \\quad \\therefore\\ \\\u00a0 \\left(\\bar{u}^1\\right)^2 = \\epsilon^2 –\u00a0 \\left( 1 – \\frac{r_g}{r}\\right)$$<\/p>\n

\\(\\epsilon\\) \u306f \\(r = r_i\\) \u306e\u3068\u304d \\(\\bar{u}^1 = 0\\) \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u304b\u3089<\/p>\n

$$0 = \\epsilon^2 –\u00a0 \\left( 1 – \\frac{r_g}{r_i}\\right), \\quad \\therefore\\ \\ \\epsilon =\u00a0 \\sqrt{1 – \\frac{r_g}{r_i}}$$<\/p>\n

\u6700\u7d42\u7684\u306b<\/p>\n

$$\\bar{u}^1 = –\u00a0 \\sqrt{\\left(1 – \\frac{r_g}{r_i}\\right) – \\left(1 – \\frac{r_g}{r}\\right)} = -\\sqrt{\\frac{r_g}{r} – \\frac{r_g}{r_i}}$$<\/p>\n

$$\\bar{u}^{\\mu} = \\left(\\frac{ \\sqrt{1 – \\frac{r_g}{r_i}}}{1 – \\frac{r_g}{r}},\u00a0 -\\sqrt{\\frac{r_g}{r} – \\frac{r_g}{r_i}}, 0, 0\\right) $$<\/p>\n

\u9759\u6b62\u89b3\u6e2c\u8005 \\(A\\) \u306e4\u5143\u901f\u5ea6<\/h3>\n

\u4e00\u65b9\uff0c\u52d5\u5f84\u5ea7\u6a19 \\(r\\) \u306e\u5730\u70b9\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u89b3\u6e2c\u8005 \\(A\\) \u306e4\u5143\u901f\u5ea6<\/strong><\/span>\u306e\u6210\u5206\u3092 \\(u^{\\mu}\\) \u3068\u3059\u308b\u3068
\n$$u^{\\mu} = \\left(u^0, 0, 0, 0 \\right) = \\left(\\frac{1}{\\sqrt{1 – \\frac{r_g}{r}}}, 0, 0, 0 \\right)$$<\/p>\n

\u904b\u52d5\u65b9\u5411\u306e\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u9759\u6b62\u89b3\u6e2c\u8005 \\(A\\) \u304b\u3089\u307f\u3066\uff0c\u52d5\u5f84\u65b9\u5411\uff08\u4e2d\u5fc3\u5411\u304d\uff09\u306b\u904b\u52d5\u3057\u3066\u3044\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e\u904b\u52d5\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3092 \\(e^{\\mu}\\) \u3068\u3059\u308b\u3068\uff0c\\(u_{\\mu} e^{\\mu} = 0, \\ e_{\\mu}e^{\\mu} = 1\\) \u3088\u308a\uff0c<\/p>\n

$$e^{\\mu} = \\left(0, e^{1}, 0, 0\\right) = \\left(0, – \\sqrt{1 – \\frac{r_g}{r}}, 0, 0\\right)$$<\/p>\n

4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h3>\n

\u9759\u6b62\u89b3\u6e2c\u8005 \\(A\\) \u304b\u3089\u307f\u308b\u3068\uff0c\u52d5\u5f84\u5ea7\u6a19 \\(r\\) \u306e\u5730\u70b9\u3092\u901a\u904e\u3059\u308b\u77ac\u9593\u306e\u89b3\u6e2c\u8005 \\(B\\) \u306f\uff0c\u52d5\u5f84\u65b9\u5411\u3078\u901f\u3055 \\(V\\) \u3067\u904b\u52d5\u3057\u3066\u3044\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306e\u3068\u304d\u306b\u7d39\u4ecb\u3057\u305f<\/a>4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/strong><\/span>\u304c\u3053\u3053\u3067\u3082\u6210\u308a\u7acb\u3061\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n

$$\\bar{u}^{\\mu} = \\frac{1}{\\sqrt{1 – V^2}} u^{\\mu} + \\frac{V}{\\sqrt{1 – V^2}} e^{\\mu} $$<\/p>\n

\u5177\u4f53\u7684\u306a\u6210\u5206\u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\bar{u}^{\\mu} &=& \\left(\\frac{ \\sqrt{1 – \\frac{r_g}{r_i}}}{1 – \\frac{r_g}{r}},\u00a0 -\\sqrt{\\frac{r_g}{r} – \\frac{r_g}{r_i}}, 0, 0\\right) \\\\
\n&=& \\frac{1}{\\sqrt{1 – V^2}} \\left(\\frac{1}{\\sqrt{1 – \\frac{r_g}{r}}}, 0, 0, 0 \\right) +
\n\\frac{V}{\\sqrt{1 – V^2}} \\left(0, – \\sqrt{1 – \\frac{r_g}{r}}, 0, 0\\right) \\\\
\n&=& \\frac{ \\sqrt{1-\\frac{r_g}{r_i}} }{ \\sqrt{1-\\frac{r_g}{r}} } \\left(\\frac{1}{\\sqrt{1 – \\frac{r_g}{r}}}, 0, 0, 0 \\right) +
\n\\frac{ \\sqrt{ \\frac{r_g}{r} – \\frac{r_g}{r_i} } }{ \\sqrt{1-\\frac{r_g}{r}} } \\left(0, – \\sqrt{1 – \\frac{r_g}{r}}, 0, 0\\right)
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{1}{\\sqrt{1 – V^2}} &=& \\frac{ \\sqrt{1-\\frac{r_g}{r_i}} }{ \\sqrt{1-\\frac{r_g}{r}} } \\\\
\n\\frac{V}{\\sqrt{1 – V^2}} &=& \\frac{ \\sqrt{ \\frac{r_g}{r} – \\frac{r_g}{r_i} } }{ \\sqrt{1-\\frac{r_g}{r}} } \\\\
\n\\therefore\\ \\ V &=& \\frac{ \\sqrt{ \\frac{r_g}{r} – \\frac{r_g}{r_i} } }{ \\sqrt{1-\\frac{r_g}{r_i}} }
\n\\end{eqnarray}<\/p>\n

\u3068 \\(V\\) \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u3053\u308c\u304c\u9759\u6b62\u89b3\u6e2c\u8005 \\(A\\) \u304c\u6e2c\u5b9a\u3059\u308b\uff0c\u76ee\u306e\u524d\u3092\u901a\u904e\u3059\u308b\u77ac\u9593\u306e\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e3\u5143\u7684\u901f\u3055\u3067\u3042\u308b\uff01\u3068\u601d\u3044\u307e\u3059\u304c\uff0c\u3044\u304b\u304c\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n

\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8<\/h3>\n

\u3061\u306a\u307f\u306b\uff0c\u81ea\u7531\u843d\u4e0b\u3059\u308b\u901f\u3055\u3092\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u304b\u3089\u6c42\u3081\u3066\u307f\u308b\u3068\uff0c\u529b\u5b66\u7684\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\u5247\u304b\u3089<\/p>\n

$$\\frac{1}{2} m v^2 – \\frac{GMm}{r} = \\mbox{const.}$$<\/p>\n

\\(r = r_i\\) \u3067 \\(v = 0\\) \u3068\u3059\u308b\u306a\u3089<\/p>\n

\\begin{eqnarray}
\nv^2 – \\frac{2GM}{r} &=& – \\frac{2GM}{r_i} \\\\
\n\\therefore\\ \\ v &=& \\sqrt{ \\frac{r_g}{r} – \\frac{r_g}{r_i} }\\quad (r_g \\equiv 2GM)
\n\\end{eqnarray}<\/p>\n

\u3068\u8a55\u4fa1\u3055\u308c\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u3067\u8a55\u4fa1\u3057\u305f \\(v\\) \u306f\u4e0a\u8a18\u3067\u6c42\u3081\u305f \\(V\\) \u3068\u5206\u6bcd\u306e\u90e8\u5206\u3060\u3051\u7570\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n

\u305f\u3060\u3057\uff0c\u521d\u671f\u4f4d\u7f6e\u304c\u5341\u5206\u9060\u65b9\u3067\u3042\u308b \\( r_i \\gg r\\) \u3068\u3059\u308b\u3068\uff0c\\(r_i \\rightarrow \\infty\\) \u3068\u3057\u3066\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u3067\u8a55\u4fa1\u3057\u305f \\(v\\) \u306f<\/p>\n

$$v \\simeq \\sqrt{ \\frac{r_g}{r} }$$<\/p>\n

\u4e0a\u8a18\u3067\u6c42\u3081\u305f \\(V\\)\u3082<\/p>\n

$$V \\simeq \\sqrt{ \\frac{r_g}{r} }$$<\/p>\n

\u3068\u306a\u308a\uff0c\\( v \\simeq V\\) \u3068\u3057\u3066\u3088\u3044\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":85,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1021","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