\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005<\/a>\u300d\u3067\u8a08\u7b97\u3057\u305f\u3088\u3046\u306b\uff0c<\/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
4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h4>\n
\u9759\u6b62\u89b3\u6e2c\u8005\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\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
\u3067\u3042\u308b\u304b\u3089\uff0c4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u306f<\/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{\\sqrt{1 – \\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}
\n\\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}}}
\n\\left(0, 0, 0, \\frac{1}{r}\\right)
\n\\end{eqnarray}<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n
\\begin{eqnarray}
\nV &=& \\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{r_g}{r}}}
\n\\end{eqnarray}<\/p>\n
\u3068\u306a\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n
\u307e\u305f\uff0c\u904b\u52d5\u89b3\u6e2c\u8005 \\(\\bar{u}^{\\mu}\\) \u304b\u3089\u307f\u305f\u904b\u52d5\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\bar{e}^{\\mu}\\) \u306f\uff0c<\/p>\n
$$\\bar{u}_{\\mu}\\bar{e}^{\\mu} = 0, \\quad \\bar{e}_{\\mu}\\bar{e}^{\\mu} = 1$$<\/p>\n
\u3067\u3042\u308a\uff0c\u5177\u4f53\u7684\u306a\u6210\u5206\u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\bar{e}^{\\mu} &=& \\frac{1}{\\sqrt{1 – V^2}} e^{\\mu} + \\frac{V}{\\sqrt{1 – V^2}} u^{\\mu} \\\\
\n&=& \\frac{\\sqrt{1 – \\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}
\n\\left(0, 0, 0, \\frac{1}{r} \\right) +
\n\\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}\u00a0 \\left(\\frac{1}{\\sqrt{1 – \\frac{r_g}{r}}}, 0, 0, 0 \\right) \\\\
\n&\\equiv& \\left(\\bar{e}^0, 0, 0, \\bar{e}^3 \\right)
\n\\end{eqnarray}<\/p>\n
4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e $3+1$ \u5206\u89e3<\/h3>\n
$$k_{\\mu} = \\bar{\\omega} \\left(\\bar{u}_{\\mu} + \\bar{\\gamma}_{\\mu} \\right) $$<\/p>\n
\u3053\u3053\u3067\uff0c$\\bar{\\omega} \\equiv – k_{\\mu} \\bar{u}^{\\mu}$ \u306f\u904b\u52d5\u89b3\u6e2c\u8005 \\(\\bar{u}^{\\mu}\\) \u304c\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570\u3067\u3042\u308a\uff0c$\\bar{\\gamma}_{\\mu}$ \u306f\u904b\u52d5\u89b3\u6e2c\u8005 \\(\\bar{u}^{\\mu}\\) \u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u9032\u884c\u65b9\u5411\u3092\u8868\u3059\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c<\/p>\n
$$\\bar{\\gamma}_{\\mu} \\bar{u}^{\\mu} = 0, \\quad \\bar{\\gamma}_{\\mu} \\bar{\\gamma}^{\\mu} = 1$$<\/p>\n
\u7279\u306b\uff0c\u904b\u52d5\u89b3\u6e2c\u8005\u306b\u3068\u3063\u3066\uff0c\u9032\u884c\u65b9\u5411\u3068\u5149\u306e\u5165\u5c04\u65b9\u5411\u304c\u76f4\u4ea4\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f<\/p>\n
$$\\bar{\\gamma}_{\\mu} \\bar{e}^{\\mu} = 0$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u307e\u305f\u306f\uff0c$\\bar{u}_{\\mu} \\bar{e}^{\\mu} = 0$ \u3067\u3042\u308b\u3053\u3068\u3092\u601d\u3044\u8d77\u3053\u305b\u3070<\/p>\n
\\begin{eqnarray}
\nk_{\\mu} \\bar{e}^{\\mu} &=& k_0 \\bar{e}^0 + k_3 \\bar{e}^3 \\\\
\n&=& -\\omega_c \\bar{e}^0\u00a0 + \\ell\\, \\bar{e}^3= 0 \\\\ \\ \\\\
\n\\therefore\\ \\ \\ell &=& \\omega_c \\frac{\\bar{e}^0}{ \\bar{e}^3} \\\\
\n&=& \\omega_c r \\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{1 – \\frac{r_g}{r}}
\n\\end{eqnarray}<\/p>\n
\u3068\u3057\u3066\u3082\u3088\u3044\u3002\u3053\u306e\u3068\u304d\u306e \\(\\bar{\\omega}\\) \u306f<\/p>\n
\\begin{eqnarray}
\n\\bar{\\omega} &=& – k_{\\mu} \\bar{u}^{\\mu} \\\\
\n&=& – k_0 \\bar{u}^0 – k_3 \\bar{u}^3 \\\\
\n&=& \\omega_c \\bar{u}^0 – \\ell \\bar{u}^3 \\\\
\n&=& \\omega_c \\frac{1}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}} – \\omega_c r \\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{1 – \\frac{r_g}{r}} \\frac{1}{r}\\frac{\\sqrt{\\frac{1}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}\\\\
\n&=& \\frac{\\omega_c}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}
\n\\left(1\u00a0 – \\frac{\\frac{1}{2}\\frac{r_g}{r}}{1 – \\frac{r_g}{r}} \\right) \\\\
\n&=& \\frac{\\omega_c}{\\sqrt{1 – \\frac{r_g}{r}}} \\frac{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{r_g}{r}}} \\\\
\n&=& \\omega \\frac{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{r_g}{r}}}
\n\\end{eqnarray}<\/p>\n
\u91cd\u529b\u5834\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306b\u3088\u308b\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c<\/h3>\n
\u4f4d\u7f6e $r$ \u3067\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570 \\(\\omega\\) \u3068\uff0c\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5165\u5c04\u89d2 $\\frac{\\pi}{2}$ \u306e\u5149\u306e\u632f\u52d5\u6570 \\(\\bar{\\omega}\\) \u3068\u306e\u6bd4\u306f\uff0c<\/p>\n
\\begin{eqnarray}
\n\\frac{\\bar{\\omega}}{\\omega}
\n&=&\u00a0 \\frac{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}}{\\sqrt{1 – \\frac{r_g}{r}}} < 1
\n\\end{eqnarray}<\/p>\n
\u3068\u306a\u308b\u3002\u3053\u306e\u52b9\u679c\u306f\u89b3\u6e2c\u8005\u304c\u904b\u52d5\u3059\u308b\u5834\u5408\u306e\u91cd\u529b\u5834\u4e2d\u306e\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c<\/strong><\/span>\u3067\u3042\u308b\u3002<\/p>\n\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f\u3068\u306e\u6574\u5408\u6027<\/h3>\n
\u3055\u3066\uff0c\u672c\u30b5\u30a4\u30c8\u3067\u306f\u5225\u30da\u30fc\u30b8\u300c\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c<\/a>\u300d\u306b\u304a\u3044\u3066\uff0c\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f\u306f\uff0c\uff08\u7279\u6b8a\u76f8\u5bfe\u8ad6\u3067\u306e\u307f\u6709\u52b9\u306a\uff09\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u308f\u305a\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f<\/h4>\n
\u7279\u306b\uff0c\u9759\u6b62\u5149\u6e90\u304b\u3089\u306e\u632f\u52d5\u6570\u3092 \\(\\omega\\)\uff0c\u904b\u52d5\u3059\u308b\u5149\u6e90\u304b\u3089\u306e\u632f\u52d5\u6570\u3092 \\(\\bar{\\omega}\\) \u3068\u8868\u8a18\u3059\u308b\u3068\uff0c\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f<\/p>\n
$$ \\bar{\\omega} =\u00a0 \\omega \\sqrt{1 – V^2} $$<\/p>\n
\u306f\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u7684\u72b6\u6cc1\u306e\u307f\u306a\u3089\u305a\uff0c\u305d\u306e\u5c0e\u51fa\u65b9\u6cd5\u306e\u4e00\u822c\u6027\u304b\u3089\uff0c\u91cd\u529b\u304c\u3042\u308b\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306a\u72b6\u6cc1\u306b\u304a\u3044\u3066\u3082\u540c\u69d8\u306b\u6709\u52b9\u3067\u3042\u308b\u3002<\/p>\n
\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u306e3\u6b21\u5143\u7684\u901f\u3055 $V$<\/h4>\n
\u52d5\u5f84\u5ea7\u6a19 $r$ \u306e\u9759\u6b62\u5149\u6e90\u304b\u3089\u307f\u308b\u3068\uff0c\u76ee\u306e\u524d\u3092\u901a\u904e\u3059\u308b\u77ac\u9593\u306e\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e\u901f\u3055 $V$ \u306f\uff0c<\/p>\n
\\begin{eqnarray}
\nV &=& \\frac{ \\sqrt{\\frac{1}{2} \\frac{r_g}{r} } }{ \\sqrt{1-\\frac{r_g}{r}} }
\n\\end{eqnarray}<\/p>\n
\u3068\u306a\u308b\u3002<\/p>\n
\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066…<\/h4>\n
\u4e0a\u8a18\u306e $V$ \u3092\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\frac{\\bar{\\omega}}{ \\omega} &=&\u00a0 \\sqrt{1-V^2} \\\\
\n&=& \\sqrt{1 – \\frac{ {\\frac{1}{2} \\frac{r_g}{r} } }{ {1-\\frac{r_g}{r}} }}
\n&=& \\sqrt{\\frac{ 1-{\\frac{3}{2} \\frac{r_g}{r} } }{ {1-\\frac{r_g}{r}} }}\\end{eqnarray}<\/p>\n
\u3068\u306a\u308a\uff0c\uff08\u5f53\u7136\u306e\u7d50\u679c\u3067\u306f\u3042\u308b\u304c\uff09\u4e0a\u8a18\u3067\u6c42\u3081\u305f $\\displaystyle \\frac{\\bar{\\omega}}{\\omega}$ \u3068\u4e00\u81f4\u3057\u3066\u3044\u308b\u3002<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":3059,"menu_order":25,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-3473","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\/3473","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\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=3473"}],"version-history":[{"count":15,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3473\/revisions"}],"predecessor-version":[{"id":3489,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3473\/revisions\/3489"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3059"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=3473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}