{"id":513,"date":"2022-01-08T13:45:39","date_gmt":"2022-01-08T04:45:39","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=513"},"modified":"2022-06-21T16:54:27","modified_gmt":"2022-06-21T07:54:27","slug":"%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e6%b8%ac%e5%9c%b0%e7%b7%9a%e6%96%b9%e7%a8%8b%e5%bc%8f","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%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad\/%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e6%b8%ac%e5%9c%b0%e7%b7%9a%e6%96%b9%e7%a8%8b%e5%bc%8f\/","title":{"rendered":"\u91cd\u529b\u5834\u4e2d\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f"},"content":{"rendered":"

\u91cd\u529b\u5834\u4e2d\u3067\u3042\u3063\u3066\u3082\uff0c\u5149\u304c\u4f1d\u64ad\u3059\u308b\u7d4c\u8def\u306f\u30cc\u30eb\u6e2c\u5730\u7dda<\/strong><\/span>\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3053\u308c\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u73fe\u3059\u308b\u3002<\/p>\n

\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u307e\u305a\uff0c\u5149\u306e\u7d4c\u8def\u3092\u8868\u3059\u4e16\u754c\u7dda<\/strong><\/span>\u00a0 \\(x^{\\mu}(v)\\) \u3092\u8003\u3048\u308b\u3002\u3053\u3053\u3067 \\(v\\) \u306f\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span>\u3067\u3042\u308b\u3002<\/p>\n

\u3053\u306e\u4e16\u754c\u7dda<\/strong><\/span>\u306e\u63a5\u30d9\u30af\u30c8\u30eb\u304c\uff0c\u5149\u306e\u4f1d\u64ad\u3092\u8868\u30594\u5143\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c
\n$$ \\boldsymbol{k} = k^{\\mu} \\boldsymbol{e}_{\\mu} \\equiv \\frac{dx^{\\mu}}{dv} \\boldsymbol{e}_{\\mu}$$
\n\u3068\u66f8\u304f\u3002\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u306f \\(\\boldsymbol{k} \\) \u306e\u7a7a\u9593\u6210\u5206\u304c3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u306e\uff08\u89d2\uff09\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\\(\\boldsymbol{k} \\) \u30924\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3068\u547c\u3093\u3060\u308a\u3059\u308b\u3002<\/p>\n

\u30cc\u30eb\u6e2c\u5730\u7dda<\/h3>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306f\uff0c
\n\\begin{eqnarray}\\frac{d\\boldsymbol{k}}{dv} =
\n\\frac{d}{dv} \\left( k^{\\mu} \\boldsymbol{e}_{\\mu}\\right)
\n&=& \\frac{dk^{\\mu}}{dv} \\boldsymbol{e}_{\\mu} + k^{\\mu} \\, \\boldsymbol{e}_{\\mu , \\nu} \\, k^{\\nu}\\\\
\n&=& \\left(\\frac{dk^{\\lambda}}{dv} + \\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu} k^{\\mu} k^{\\nu} \\right) \\boldsymbol{e}_{\\lambda} = 0
\n\\end{eqnarray}
\n\u3068\u306a\u308b\u3002<\/p>\n

\u5171\u5909\u6210\u5206\u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/span><\/h3>\n

\u307e\u305f\uff0c\u5b9f\u969b\u306e\u8a08\u7b97\u306e\u969b\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\uff0c\u4fdd\u5b58\u91cf\u304c\u308f\u304b\u308a\u3084\u3059\u3044\u3088\u3046\u306b\u5909\u5f62\u3057\u305f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u4f7f\u3046\u3053\u3068\u304c\u4fbf\u5229\u306a\u5834\u5408\u304c\u3042\u308b\u3002<\/p>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u57fa\u672c\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{e}_{\\nu} \\) \u3068\u306e\u5185\u7a4d\u3092\u3068\u3063\u3066<\/p>\n

\\begin{eqnarray}
\n0 = \\boldsymbol{e}_{\\nu}\\cdot \\frac{d\\boldsymbol{k}}{dv} &=&
\n\\frac{d}{dv} \\left( \\boldsymbol{k}\\cdot\\boldsymbol{e}_{\\nu} \\right) –
\n\\boldsymbol{k}\\cdot\\boldsymbol{e}_{\\nu, \\lambda}\\frac{dx^{\\lambda}}{dv} \\\\
\n&=& \\frac{d}{dv} \\left( k^{\\mu} \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu} \\right) – \\boldsymbol{e}_{\\mu}\\cdot \\boldsymbol{e}_{\\lambda, \\nu} k^{\\mu} k^{\\lambda} \\\\
\n&=& \\frac{d}{dv} \\left( g_{\\nu\\mu} k^{\\mu}\u00a0 \\right) – \\frac{1}{2} g_{\\mu\\lambda, \\nu} k^{\\mu} k^{\\lambda}
\n\\end{eqnarray}<\/p>\n

\\( k_{\\nu} \\equiv g_{\\nu\\mu} k^{\\mu} \\) \u3068\u3059\u308b\u3068\uff0c\u300c\u5171\u5909\u6210\u5206<\/strong><\/span>\u300d \\(k_{\\nu} \\) \uff08\u6b74\u53f2\u7684\u306b\u306f\u4e0a\u4ed8\u6dfb\u5b57\u6210\u5206<\/strong><\/span> \u00a0\\(k^{\\mu} \\)\u00a0 \u3092\u53cd\u5909\u6210\u5206<\/strong><\/span>\u3068\u304b\u53cd\u5909\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u306a\u3069\u3068\u547c\u3073\uff0c\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206<\/strong><\/span> \\(k_{\\nu} \\) \u3092\u5171\u5909\u6210\u5206<\/strong><\/span>\u3068\u304b\u5171\u5909\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u306a\u3069\u3068\u547c\u3093\u3067\u3044\u305f\uff09\u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306f\uff08\u6dfb\u5b57\u3092\u66f8\u304d\u63db\u3048\u3066\uff09<\/p>\n

$$\\frac{d k_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}$$ \u3068\u306a\u308b\u3002<\/p>\n

\u3053\u306e\u3053\u3068\u304b\u3089\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u00a0 \\(g_{\\alpha\\beta} \\) \u304c\u3042\u308b\u7279\u5b9a\u306e\u5ea7\u6a19 \\(x^{\\mu} \\) \u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u5834\u5408\u306f\uff0c\u305f\u3060\u3061\u306b
\n$$\\displaystyle g_{\\alpha\\beta, \\mu} = 0\\quad\\Rightarrow\\quad \\frac{d k_{\\mu}}{dv} = 0 \\quad\\Rightarrow\\quad k_{\\mu} = \\mbox{const.} $$
\n\u3068\u306a\u308a \\(k_{\\mu} \\) \u6210\u5206\u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u3053\u306e\u300c\u5171\u5909\u6210\u5206\u300d\u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\uff0c\u89e3\u6790\u529b\u5b66\u306b\u304a\u3051\u308b\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u65b9\u7a0b\u5f0f\u306b\u76f8\u5f53\u3057\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb \\(g_{\\alpha\\beta}\\)\uff0c\u3057\u305f\u304c\u3063\u3066\u30e9\u30b0\u30e9\u30f3\u30b8\u30a2\u30f3\u304c\u7279\u5b9a\u306e\u5ea7\u6a19 \\(x^{\\mu}\\) \u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u5834\u5408\u306f\uff0c\\(x^{\\mu}\\) \u306f\u5faa\u74b0\u5ea7\u6a19\u3068\u547c\u3070\u308c\uff0c\u3053\u306e\u5faa\u74b0\u5ea7\u6a19\u306b\u5171\u5f79\u306a\u4e00\u822c\u5316\u904b\u52d5\u91cf\u3067\u3042\u308b \\(k_{\\mu} \\) \u304c\u4fdd\u5b58\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u8868\u3057\u3066\u3044\u308b\u3002\uff08\u89e3\u6790\u529b\u5b66\u3092\u5b66\u3093\u3060\u5b66\u751f\u3055\u3093\u306b\u306f\uff0c\u3053\u306e\u3088\u3046\u306a\u8aac\u660e\u304c\u30a6\u30b1\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u3046\u3061\u306e\u5b66\u79d1\u306f\u7269\u7406\u5b66\u79d1\u3067\u306f\u306a\u3044\u306e\u3067\u89e3\u6790\u529b\u5b66\u306e\u6388\u696d\u304c\u306a\u304f\uff0c\u30e9\u30b0\u30e9\u30f3\u30b8\u30a2\u30f3\u3042\u308a\u304d\u306e\u8aac\u660e\u306f\u5c01\u5370\u3002\uff09<\/p>\n

\u307e\u305f\uff0c\u91cd\u529b\u5834\u4e2d\u306b\u304a\u3044\u3066\u3082\uff0c\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb\u304c\u30cc\u30eb\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u306f\u5909\u308f\u3089\u306a\u3044\u3002<\/p>\n

$$ \\boldsymbol{k}\\cdot \\boldsymbol{k} = g_{\\mu\\nu} k^{\\mu} k^{\\nu} = 0$$<\/p>\n

\uff08\u8a3c\u660e\u306f\uff0c\\( \\boldsymbol{k}\\cdot \\boldsymbol{k} \\) \u306f\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u30b9\u30ab\u30e9\u30fc\u3067\u3042\u308b\u3053\u3068\uff0c\u304a\u3088\u3073\u4efb\u610f\u306e\u6642\u7a7a\u4e2d\u306e1\u70b9\u3067\u5c40\u6240\u6163\u6027\u7cfb\u3092\u3068\u308b\u3053\u3068\u304c\u3067\u304d\uff0c\u305d\u3053\u3067\u306f\u7279\u6b8a\u76f8\u5bfe\u8ad6\u3068\u540c\u3058\u8a08\u7b97\u304c\u3067\u304d\u308b\u3053\u3068\u304b\u3089\u660e\u3089\u304b\u3067\u3042\u308d\u3046\u3002\uff09<\/p>\n

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