{"id":1023,"date":"2022-01-15T20:43:28","date_gmt":"2022-01-15T11:43:28","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1023"},"modified":"2024-07-01T11:13:50","modified_gmt":"2024-07-01T02:13:50","slug":"%e5%86%86%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%86%86%e9%81%8b%e5%8b%95\/","title":{"rendered":"\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005"},"content":{"rendered":"

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306e\u539f\u70b9\u306e\u307e\u308f\u308a\u3092\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6\u306b\u3064\u3044\u3066\u3002<\/p>\n

\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6<\/h3>\n

\u539f\u70b9\u306e\u307e\u308f\u308a\u3092\u534a\u5f84 \\(r = \\mbox{const.}\\) \u3067\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6<\/strong><\/span>\u306e\u6210\u5206\u3092\uff08\u9759\u6b62\u89b3\u6e2c\u8005\u306e \\(u^{\\mu}\\) \u3068\u533a\u5225\u3059\u305f\u3081 \\(\\bar{\u3000}\\) \u3092\u3064\u3051\u3066\uff09$$\\bar{u}^{\\mu} = (\\bar{u}^0, 0, 0, \\bar{u}^3) = \\left(\\frac{\\epsilon\\, c}{1-\\frac{r_g}{r}}, 0, 0, \\frac{\\ell}{r^2} \\right)$$\u3068\u3059\u308b\u3002<\/p>\n

\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f<\/a><\/p>\n

\\begin{eqnarray}
\n\\left( \\frac{dr}{d\\tau} \\right)^2 &=&\\epsilon^2 c^2 -c^2\u00a0 + \\frac{2 G M}{r}\u00a0 -\\frac{\\ell^2}{r^2}
\n+r_g\\frac{\\ell^2}{r^3}
\n\\end{eqnarray}<\/p>\n

\u3092\u4f7f\u3044\uff0c\\(\\displaystyle \\frac{dr}{d\\tau}= 0\\) \u3088\u308a<\/p>\n

$$\\epsilon^2 c^2 -c^2\u00a0 + \\frac{2 G M}{r}\u00a0 -\\frac{\\ell^2}{r^2}
\n+r_g\\frac{\\ell^2}{r^3} = 0$$<\/p>\n

\u3053\u308c\u3092\u3055\u3089\u306b \\(r\\) \u3067\u5fae\u5206\u3057\u3066 \\(r\\) \u3092\u304b\u3051\u308b\u3068\uff0c<\/p>\n

$$-\\frac{2 G M}{r}\u00a0 + 2 \\frac{\\ell^2}{r^2}
\n-3 r_g\\frac{\\ell^2}{r^3} = 0$$<\/p>\n

\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\\(\\displaystyle \\frac{\\ell^2}{r^2}\\) \u306b\u3064\u3044\u3066\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

$$ \\frac{\\ell^2}{r^2} = \\frac{\\frac{G M }{r}}{1 – \\frac{3}{2}\\frac{r_g}{r}} = \\frac{\\frac{1}{2}\\frac{r_g c^2 }{r}}{1 – \\frac{3}{2}\\frac{r_g}{r}}$$<\/p>\n

$$\\therefore\\ c^2 + \\frac{\\ell^2}{r^2} = \\frac{c^2 – \\frac{3}{2}\\frac{r_g c^2}{r}+\\frac{1}{2}\\frac{r_g c^2}{r}}{1 – \\frac{3}{2}\\frac{r_g}{r}} = c^2 \\frac{1 – \\frac{r_g}{r}}{1 – \\frac{3}{2}\\frac{r_g}{r}}$$<\/p>\n

$$\\therefore\\ \\epsilon^2 = \\frac{1}{c^2} \\left(1-\\frac{r_g}{r} \\right) \\left( c^2 + \\frac{\\ell^2}{r^2}\\right)
\n= \\frac{\\left(1 – \\frac{r_g}{r}\\right)^2}{1 – \\frac{3}{2}\\frac{r_g}{r}}$$<\/p>\n

\u6700\u7d42\u7684\u306b\uff0c\uff08\u3053\u308c\u3088\u308a \\(c = 1\\) \u3068\u3057\u3066\uff09<\/p>\n

$$\\bar{u}^{\\mu} = (\\bar{u}^0, 0, 0, \\bar{u}^3) = \\left(\\frac{1}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}},\u00a0 0, 0,
\n\\frac{1}{r}\\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}\\right) $$<\/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\\(\\theta = \\frac{\\pi}{2}\\) \u306e\u8d64\u9053\u9762\u4e0a\u3092\u5186\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\u306f<\/p>\n

$$e^{\\mu} = \\left(0, 0, 0, e^3\\right) = \\left(0, 0, 0, \\frac{1}{r}\\right)$$<\/p>\n

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

\u9759\u6b62\u89b3\u6e2c\u8005 \\(A\\)<\/strong> <\/span>\u304b\u3089\u307f\u308b\u3068\uff0c\u8d64\u9053\u9762\u4e0a\u3092\u534a\u5f84 \\(r = \\mbox{const.}\\) \u306e\u5186\u8ecc\u9053\u3092\u63cf\u3044\u3066\u76ee\u306e\u524d\u3092\u901a\u904e\u3059\u308b\u77ac\u9593\u306e\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\)<\/strong> <\/span>\u306f\uff0c\\(\\phi\\) \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\u306f\u305a\u3067\u3042\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{1}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}},\u00a0 0, 0,
\n\\frac{1}{r}\\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}\\right) \\\\
\n&=&
\n\\frac{1}{\\sqrt{1 – V^2}}\u00a0 \\left(\\frac{1}{\\sqrt{1 – \\frac{r_g}{r}}}, 0, 0, 0 \\right) +
\n\\frac{V}{\\sqrt{1 – V^2}} \\left(0, 0, 0, \\frac{1}{r}\\right) \\\\
\n&=&
\n\\frac{\\sqrt{1 – \\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}} \\left(\\frac{1}{\\sqrt{1 – \\frac{r_g}{r}}}, 0, 0, 0 \\right) +
\n\\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}\u00a0 \\left(0, 0, 0, \\frac{1}{r}\\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}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}} \\\\
\n\\frac{V}{\\sqrt{1 – V^2}} &=& \\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}\u00a0 \\\\
\n\\therefore\\ \\ V &=& \\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{r_g}{r}}}
\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\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e3\u6b21\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\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u306b\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u901f\u3055\u3068\u540c\u3058\u3088\u3046\u306b $\\displaystyle\u00a0 v \\equiv r \\frac{d\\phi}{dt} $ \u3067\u5b9a\u7fa9\u3055\u308c\u308b $v$ \u30923\u6b21\u5143\u7684\u901f\u3055\u3068\u3059\u308b\u3082\u306e\u304c\u3042\u308b\u304c\uff0c\u3053\u306e\u3088\u3046\u306a\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u3067\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068<\/p>\n

$$v \\equiv r \\frac{d\\phi}{dt} = \\frac{r \\bar{u}^3}{\\bar{u}^0}\u00a0 = \\sqrt{\\frac{1}{2}\\frac{r_g}{r}}$$<\/p>\n

\u3068\u306a\u308b\u3002\u307e\u305f\uff0c\u9060\u5fc3\u529b\u3068\u4e07\u6709\u5f15\u529b\u304c\u3064\u308a\u3042\u3046\u3068\u3044\u3046\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u985e\u63a8\u3067\u8a55\u4fa1\u3057\u3066\u3082\uff0c<\/p>\n

\\begin{eqnarray}
\nm \\frac{v^2}{r} &=& \\frac{GMm}{r^2} \\\\
\nv^2 &=& \\frac{GM}{r} \\\\
\n\\therefore\\ \\ v &=&\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}\\quad (r_g \\equiv 2 GM)
\n\\end{eqnarray}<\/p>\n

\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\u306f\u5206\u6bcd\u306e\u5206\u3060\u3051\u7570\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":85,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1023","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\/1023","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=1023"}],"version-history":[{"count":15,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1023\/revisions"}],"predecessor-version":[{"id":9043,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1023\/revisions\/9043"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/85"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=1023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}