{"id":3452,"date":"2023-04-18T00:00:19","date_gmt":"2023-04-17T15:00:19","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3452"},"modified":"2023-04-18T13:02:05","modified_gmt":"2023-04-18T04:02:05","slug":"%e3%82%b7%e3%83%a5%e3%83%90%e3%83%ab%e3%83%84%e3%82%b7%e3%83%ab%e3%83%88%e6%99%82%e7%a9%ba%e4%b8%ad%e3%82%92%e5%86%86%e9%81%8b%e5%8b%95%e3%81%99%e3%82%8b%e5%85%89%e6%ba%90%e3%81%ab%e3%82%88%e3%82%8b","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%81%ae%e7%b5%b1%e4%b8%80%e7%9a%84%e7%90%86%e8%a7%a3\/%e3%82%b7%e3%83%a5%e3%83%90%e3%83%ab%e3%83%84%e3%82%b7%e3%83%ab%e3%83%88%e6%99%82%e7%a9%ba%e4%b8%ad%e3%82%92%e5%86%86%e9%81%8b%e5%8b%95%e3%81%99%e3%82%8b%e5%85%89%e6%ba%90%e3%81%ab%e3%82%88%e3%82%8b\/","title":{"rendered":"\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u306b\u3088\u308b\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c"},"content":{"rendered":"

\u3053\u3053\u3067\u306f\uff0c\u8d64\u65b9\u504f\u79fb\u3092\u7d71\u4e00\u7684\u306b\u7406\u89e3\u3059\u308b\u65b9\u6cd5\u306b\u5f93\u3063\u3066\uff0c<\/p>\n

I. \u91cd\u529b\u5834\u4e2d\u306e\u5149\u306e\u4f1d\u64ad\u306f\uff0c\u30cc\u30eb\u6e2c\u5730\u7dda\u3067\u4e0e\u3048\u3089\u308c\u308b<\/p>\n

II. 4\u5143\u901f\u5ea6 $u^{\\mu}$ \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u306f $\\omega = – k_{\\mu} u^{\\mu}$ \u3067\u4e0e\u3048\u3089\u308c\u308b<\/p>\n

\u3068\u3044\u30462\u3064\u306e\u539f\u7406\u539f\u5247\u304b\u3089\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u306b\u3088\u308b\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f\u3092\u5c0e\u304f\u3002<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u52d5\u5f84\u65b9\u5411\u306b\u4f1d\u64ad\u3059\u308b\u5149<\/h3>\n

\u4eca\u56de\u306f\uff0c\u9759\u6b62\u5149\u6e90<\/strong><\/span>\u3068\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u5149\u6e90<\/strong><\/span>\u306e2\u3064\u304b\u3089\u5149\u304c\u653e\u51fa\u3055\u308c\u308b\u306e\u3067\uff0c\u305d\u308c\u3092\u533a\u5225\u3057\u3066\u8868\u8a18\u3059\u308b\u3002<\/p>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\u5225\u30da\u30fc\u30b8\u300c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u5149\u306e\u4f1d\u64ad<\/a>\u300d\u3067\u89e3\u3044\u3066\u3044\u308b\u304b\u3089\uff0c\u305d\u306e\u7d50\u679c\u3092\u4f7f\u3046\u3068\uff0c\u9759\u6b62\u5149\u6e90\u304b\u3089\u306e\u5149\u306e4\u5143\u6ce2\u52d5\u30d9\u30af\u30c8\u30eb $k_{\\mu} = g_{\\mu\\nu} \\,k^{\\nu}$ \u306b\u3064\u3044\u3066<\/p>\n

$$k_{\\mu} = \\left(k_0, k_1, 0, 0 \\right) = \\left(-\\omega_c, \\ \\pm \\frac{\\omega_c}{1 – \\frac{r_g}{r}}, \\ 0, \\ 0 \\right)$$
\n\u8907\u53f7 \\(\\pm\\) \u306f\uff0c\u52d5\u5f84\u65b9\u5411\u5916\u5411\u304d\u306b\u4f1d\u64ad\u3059\u308b\u5149\u306b\u5bfe\u3057\u3066\u30d7\u30e9\u30b9\uff0c\u5185\u5411\u304d\u306e\u5149\u306b\u5bfe\u3057\u3066\u30de\u30a4\u30ca\u30b9\u3092\u3068\u308b\u3002<\/p>\n

\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u5149\u6e90\u304b\u3089\u306e\u5149\u306e4\u5143\u6ce2\u52d5\u30d9\u30af\u30c8\u30eb\u306f\u4e0a\u3068\u533a\u5225\u3057\u3066 $k’_{\\mu}$ \u3068\u3057\uff0c<\/p>\n

$$k’_{\\mu} = \\left(k’_0, k’_1, 0, 0 \\right) = \\left(-\\omega’_c, \\ \\pm \\frac{\\omega’_c}{1 – \\frac{r_g}{r}}, \\ 0, \\ 0 \\right)$$<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u9759\u6b62\u89b3\u6e2c\u8005\u30fb\u9759\u6b62\u5149\u6e90<\/h3>\n

\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 $u^{\\mu}$ \u306f<\/p>\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

\u9759\u6b62\u5149\u6e90\u304b\u3089\u306e\u5149 \\(k_{\\mu}\\) \u3092\uff0c\u4f4d\u7f6e $r$ \u3067\u9759\u6b62\u3057\u3066\u3044\u308b\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3057\u305f\u3068\u304d\u306e\u632f\u52d5\u6570 \\(\\omega\\) \u306f<\/p>\n

$$\\omega = – k_{\\mu} u^{\\mu} = \\frac{\\omega_c}{\\sqrt{1 – \\frac{r_g}{r}}}$$<\/p>\n

\u7279\u306b\uff0c$r = r_1$ \u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u5149\u6e90\u306e\u632f\u52d5\u6570\u306f\uff0c\u5149\u6e90\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u5149\u6e90\u3068\u540c\u3058\u4f4d\u7f6e\u3067\u89b3\u6e2c\u3059\u308b\u306e\u3067\u3042\u308b\u304b\u3089<\/p>\n

$$\\omega_1 \\equiv\u00a0 \\omega(r_1) = \\frac{\\omega_c}{\\sqrt{1 – \\frac{r_g}{r_1}}}$$<\/p>\n

\u307e\u305f\uff0c\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u5149\u6e90\u304b\u3089\u306e\u5149 $k’_{\\mu}$ \u3092\uff0c\u4f4d\u7f6e $r$ \u3067\u9759\u6b62\u3057\u3066\u3044\u308b\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3057\u305f\u3068\u304d\u306e\u632f\u52d5\u6570 \\(\\omega’ \\) \u306f<\/p>\n

$$\\omega’ = – k’_{\\mu} u^{\\mu} = \\frac{\\omega’_c}{\\sqrt{1 – \\frac{r_g}{r}}}$$<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90<\/h3>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u306e4\u5143\u901f\u5ea6 $\\bar{u}^{\\mu}$ \u306f\uff0c\u5225\u30da\u30fc\u30b8\u300c\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

\u3053\u306e\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u304b\u3089 $r=r_1$ \u3067\u52d5\u5f84\u65b9\u5411\u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u306e\u632f\u52d5\u6570 \\(\\bar{\\omega}’_1\\) \u306f\uff0c\u5149\u6e90\u9759\u6b62\u89b3\u6e2c\u8005\uff08\u5149\u6e90\u3068\u540c\u30584\u5143\u901f\u5ea6\u3067\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\uff09\u304c\u5149\u6e90\u3068\u540c\u3058\u4f4d\u7f6e\u3067\u89b3\u6e2c\u3059\u308b\u306e\u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\n\\bar{\\omega}’_1 &=& – k’_{\\mu}(r_1)\\, \\bar{u}^{\\mu}(r_1) \\\\
\n&=& \\omega’_c \\frac{1}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r_1}}} \\
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\u306f\uff0c\u7c21\u5358\u306e\u305f\u3081\u306b \\(r = r_1\\) \u306b\u304a\u3051\u308b\u9759\u6b62\u5149\u6e90\u304b\u3089\u306e\u632f\u52d5\u6570\u3068\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u304b\u3089\u306e\u632f\u52d5\u6570\u306f\u7b49\u3057\u3044\u3068\u3059\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\omega_1 &\\equiv& \\bar{\\omega}’_1 \\\\
\n\\therefore\\ \\ \\frac{\\omega_c}{\\sqrt{1 – \\frac{r_g}{r_1}}} &=& \\frac{\\omega’_c}{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r}}} \\\\
\n\\therefore\\ \\ \\omega’_c &=& \\omega_c \\frac{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r_1}}}{\\sqrt{1 – \\frac{r_g}{r_1}}}
\n\\end{eqnarray}<\/p>\n

\u91cd\u529b\u5834\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u306b\u3088\u308b\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c<\/h3>\n

\"\"<\/p>\n

\u4f4d\u7f6e $r$ \u3067\u9759\u6b62\u89b3\u6e2c\u8005\u304c\uff0c\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u304b\u3089\u306e\u5149\u3092\u89b3\u6e2c\u3059\u308b\u5834\u5408\u306e\u632f\u52d5\u6570 \\(\\omega’\\) \u3068\uff0c$r = r_1$ \u306e\u9759\u6b62\u5149\u6e90\u304b\u3089\u306e\u5149\u3092\u89b3\u6e2c\u3059\u308b\u5834\u5408\u306e\u632f\u52d5\u6570 \\(\\omega\\) \u3068\u306e\u6bd4\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{\\omega’}{\\omega}
\n&=& \\frac{\\omega’_c}{\\omega_c} \\\\
\n&=&\u00a0 \\frac{\\sqrt{1 – \\frac{3}{2}\\frac{r_g}{r_1}}}{\\sqrt{1 – \\frac{r_g}{r_1}}} < 1
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u304b\u3089\u306e\u5149\u306e\u632f\u52d5\u6570 $\\omega’$ \u306e\u307b\u3046\u304c\uff0c\u9759\u6b62\u5149\u6e90\u304b\u3089\u306e\u5149\u306e\u632f\u52d5\u6570 $\\omega$ \u3088\u308a\u5c0f\u3055\u304f\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3053\u306e\u5834\u5408\uff0c\u5186\u904b\u52d5\u3059\u308b\u5149\u6e90\u306e\u9032\u884c\u65b9\u5411\u3068\uff0c\u5149\u306e\u4f1d\u64ad\u65b9\u5411\u3068\u306e\u306a\u3059\u89d2\u306f $\\displaystyle \\frac{\\pi}{2}$ \u3067\u3042\u308a\uff0c\u89b3\u6e2c\u8005\u65b9\u5411\u3078\u306e\u5149\u6e90\u306e\u901f\u5ea6\u6210\u5206\u304c\u30bc\u30ed\u3067\u3042\u308b\u306b\u3082\u304b\u304b\u308f\u3089\u305a\uff0c\u632f\u52d5\u6570\u304c\u5909\u5316\u3057\u3066\u304a\u308a\uff0c\u3053\u306e\u52b9\u679c\u306f\u91cd\u529b\u5834\u4e2d\u306e\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c<\/strong><\/span>\u3067\u3042\u308b\u3002<\/p>\n

\u306a\u304a\uff0c\u5ff5\u306e\u305f\u3081\u306b\u88dc\u8db3\u3059\u308b\u304c\uff0c\u4e0a\u56f3\u3067\u306f $r_1 < r$\uff0c\u3059\u306a\u308f\u3061\u5149\u304c\u5916\u5411\u304d\u306b\u4f1d\u64ad\u3059\u308b\u5834\u5408\u3092\u8003\u3048\u3066\u3044\u308b\u304c\uff0c\\(\\displaystyle \\frac{\\omega’}{\\omega}\\) \u306e\u7d50\u679c\u306f $r < r_1$\uff0c\u3059\u306a\u308f\u3061\u5149\u304c\u5185\u5411\u304d\u306b\u4f1d\u64ad\u3059\u308b\u5834\u5408\u306b\u3082\u540c\u69d8\u306b\u6210\u308a\u7acb\u3064\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 \\(\\omega’\\) \u3068\u8868\u8a18\u3059\u308b\u3068\uff0c\u6a2a\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f<\/p>\n

$$ \\omega’ =\u00a0 \\omega \\sqrt{1 – V^2} $$<\/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 $r=r_1$ \u306e\u5730\u70b9\u306e\u9759\u6b62\u5149\u6e90\u304b\u3089\u307f\u308b\u3068\uff0c\u76ee\u306e\u524d\u3092\u901a\u904e\u3059\u308b\u77ac\u9593\u306e\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u5149\u6e90\u306e\u901f\u3055 $V$ \u306f\uff0c\u300c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005<\/a>\u300d\u306b\u8a18\u8f09\u306e\u3088\u3046\u306b\uff0c4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/p>\n

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

\u3092\u4f7f\u3046\u3068\uff0c\u305d\u308c\u305e\u308c\u306e4\u5143\u901f\u5ea6\u306e\u5185\u7a4d\u304c\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50\u3092\u8868\u3059\u3053\u3068<\/p>\n

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

\u304b\u3089 $V$ \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\uff0c<\/p>\n

\\begin{eqnarray}
\nV &=& \\frac{ \\sqrt{\\frac{1}{2} \\frac{r_g}{r_1} } }{ \\sqrt{1-\\frac{r_g}{r_1}} }
\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{\\omega’}{ \\omega} &=&\u00a0 \\sqrt{1-V^2} \\\\
\n&=& \\sqrt{1 – \\frac{ {\\frac{1}{2} \\frac{r_g}{r_1} } }{ {1-\\frac{r_g}{r_1}} }}
\n&=& \\sqrt{\\frac{ 1-{\\frac{3}{2} \\frac{r_g}{r_1} } }{ {1-\\frac{r_g}{r_1}} }}\\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 $\\displaystyle \\frac{\\omega’}{\\omega}$ \u3068\u4e00\u81f4\u3057\u3066\u3044\u308b\u3002<\/p>\n

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