{"id":487,"date":"2022-01-07T17:45:00","date_gmt":"2022-01-07T08:45:00","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=487"},"modified":"2024-10-08T11:32:28","modified_gmt":"2024-10-08T02:32:28","slug":"%e7%89%b9%e6%ae%8a%e7%9b%b8%e5%af%be%e8%ab%96%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad","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\/%e7%89%b9%e6%ae%8a%e7%9b%b8%e5%af%be%e8%ab%96%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad\/","title":{"rendered":"\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306b\u304a\u3051\u308b\u5149\u306e\u4f1d\u64ad"},"content":{"rendered":"<p><!--more-->\u7279\u6b8a\u76f8\u5bfe\u8ad6\u3067\u306f\uff0c\u5149\u306f\u771f\u7a7a\u4e2d\u3092\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u306a\u3044\u901f\u5ea6\u3067\u3042\u308b\u5149\u901f \\(c\\) \u3067\u76f4\u9032\u3059\u308b\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\uff0c\u5149\u306e\u7d4c\u8def\u306f<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30cc\u30eb\u6e2c\u5730\u7dda<\/strong><\/span>\u3067\u4e0e\u3048\u3089\u308c\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<h3>\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb<\/h3>\n<p>\u307e\u305a\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u6642\u7a7a<\/strong><\/span>\u4e2d\u306e\u5149\u306e\u7d4c\u8def\u3092\u8868\u3059<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u4e16\u754c\u7dda<\/strong><\/span>\u00a0 \\(x^{\\mu}(v)\\) \u3092\u8003\u3048\u308b\u3002\u3053\u3053\u3067 \\(v\\) \u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span>\u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u306e\u4e16\u754c\u7dda\u306e\u63a5\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{k} \\) \u304c\uff0c\u5149\u306e\u4f1d\u64ad\u3092\u8868\u30594\u5143\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c\u305d\u306e\u6210\u5206 \\(k^{\\mu}\\) \u3068\u57fa\u672c\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{e}_{\\mu}\\) \u3092\u4f7f\u3063\u3066<br \/>\n$$ \\boldsymbol{k} = k^{\\mu} \\boldsymbol{e}_{\\mu} \\equiv \\frac{dx^{\\mu}}{dv} \\boldsymbol{e}_{\\mu}$$<br \/>\n\u3068\u66f8\u304f\u3002<\/p>\n<h3>\u30cc\u30eb\u6e2c\u5730\u7dda<\/h3>\n<p>\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u307e\u3063\u3059\u3050\u306a\u7dda<\/strong><\/span>\u300d\u3068\u306f\uff0c\u4e16\u754c\u7dda\u306b\u305d\u3063\u3066\u63a5\u30d9\u30af\u30c8\u30eb\u304c\u4e00\u5b9a\u3067\u3042\u308b\u3088\u3046\u306a\u7dda\u306e\u3053\u3068\u3067\u3042\u308a\uff0c\u5149\u306b\u5bfe\u3057\u3066\u306f<br \/>\n$$\\frac{d\\boldsymbol{k}}{dv} = \\boldsymbol{0}$$<br \/>\n\u3068\u306a\u308b\u3002<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7279\u6b8a\u76f8\u5bfe\u8ad6<\/strong><\/span>\u306b\u9650\u3089\u305a<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4e00\u822c\u76f8\u5bfe\u8ad6<\/strong><\/span>\u306b\u304a\u3044\u3066\u3082\uff0c\u3053\u306e\u3088\u3046\u306b\u4e16\u754c\u7dda\u306b\u305d\u3063\u3066\u63a5\u30d9\u30af\u30c8\u30eb\u304c\u4e00\u5b9a\u3067\u3042\u308b\u7dda\u3092<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u6e2c\u5730\u7dda<\/strong><\/span>\u3068\u547c\u3076\u3002\u63a5\u30d9\u30af\u30c8\u30eb\u3092\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u5fae\u5206\u3057\u3066\u30bc\u30ed\u3068\u3044\u3046\u3053\u306e\u5f0f\u3092\uff08\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf \\(v\\) \u3067\u30d1\u30e9\u30e1\u30c8\u30e9\u30a4\u30ba\u3055\u308c\u305f\uff09<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u547c\u3076\u3002<\/p>\n<p>\u307e\u305f\uff0c\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb \\( \\boldsymbol{k} \\) \u306f\u305d\u306e\u300c\u5927\u304d\u3055\u300d\u306e2\u4e57\uff0c\u3064\u307e\u308a\uff08\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\uff09\u304c\u30bc\u30ed\u3068\u306a\u308b<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30cc\u30eb\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3067\u3042\u308a\uff0c\u3053\u306e\u3053\u3068\u306f\u5149\u901f\u4e0d\u5909\u6027\u304b\u3089\u304d\u3066\u3044\u308b\u306e\u3067\u3042\u3063\u305f\u3002\uff08<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e7%9b%b8%e5%af%be%e8%ab%96%e3%81%ae%e7%90%86%e8%a7%a3\/%e5%85%89%e3%81%ae4%e5%85%83%e3%83%99%e3%82%af%e3%83%88%e3%83%ab\/\">\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb<\/a>\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002\uff09<\/p>\n<p>$$ \\boldsymbol{k}\\cdot \\boldsymbol{k} = \\eta_{\\mu\\nu} k^{\\mu} k^{\\nu} = 0$$<\/p>\n<h3>\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306b\u304a\u3051\u308b\u5149\u306e\u632f\u52d5\u6570\u30fb\u6ce2\u6570\u306e\u4e00\u5b9a\u6027<\/h3>\n<p>\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\uff0c<br \/>\n\\begin{eqnarray}\\frac{d\\boldsymbol{k}}{dv} =<br \/>\n\\frac{d}{dv} \\left( k^{\\mu} \\boldsymbol{e}_{\\mu}\\right)<br \/>\n&amp;=&amp; \\frac{dk^{\\mu}}{dv} \\boldsymbol{e}_{\\mu} + k^{\\mu}\\frac{d\\boldsymbol{e}_{\\mu}}{dv} \\\\<br \/>\n&amp;=&amp; \\frac{dk^{\\mu}}{dv} \\boldsymbol{e}_{\\mu} + k^{\\mu}\\frac{\\partial\\boldsymbol{e}_{\\mu}}{\\partial x^{\\nu}} \\frac{dx^{\\nu}}{dv}\\\\<br \/>\n&amp;=&amp; \\frac{dk^{\\mu}}{dv} \\boldsymbol{e}_{\\mu} + k^{\\mu} \\, \\boldsymbol{e}_{\\mu , \\nu} \\, k^{\\nu} = 0<br \/>\n\\end{eqnarray}<br \/>\n\u3068\u306a\u308b\u3002 \\( x^{\\nu}\\) \u306b\u95a2\u3059\u308b\u504f\u5fae\u5206\u3092 \\( \\boldsymbol{e}_{\\mu , \\nu}\\) \u3068\u66f8\u304f\u306e\u306f\u7c21\u7565\u7684\u8868\u8a18\u6cd5\u3067\u3042\u308b\u3002<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e6%99%82%e7%a9%ba%e3%81%ae%e8%a1%a8%e3%81%97%e6%96%b9\/%e6%b8%ac%e5%9c%b0%e7%b7%9a%e3%81%a8%e6%8e%a5%e7%b6%9a%e4%bf%82%e6%95%b0%e3%83%bb%e3%82%af%e3%83%aa%e3%82%b9%e3%83%88%e3%83%83%e3%83%95%e3%82%a7%e3%83%ab%e8%a8%98%e5%8f%b7\/#i-3\">\u504f\u5fae\u5206\u306e\u7c21\u7565\u8868\u8a18\u6cd5<\/a>\u306e\u9805\u3092\u53c2\u7167\u3002<\/p>\n<p>\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u306f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u8a08\u91cf<\/strong><\/span>\u3092\u4f7f\u3063\u3066\uff0c<br \/>\n$$\\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu} = \\eta_{\\mu\\nu} = \\mbox{diag}(-1, 1, 1, 1)$$ \u3067\u3042\u308b\u304b\u3089\uff0c\u5ea7\u6a19\u57fa\u5e95 \\(\\boldsymbol{e}_{\\mu}\\) \u306f\u5ea7\u6a19\u4f9d\u5b58\u6027\u3092\u6301\u305f\u305a\uff0c\u4e00\u5b9a\u3067\u3042\u308a<br \/>\n$$\\boldsymbol{e}_{\\mu , \\nu} = \\boldsymbol{0}$$<br \/>\n\u7d50\u5c40\uff0c<br \/>\n$$\\frac{dk^{\\mu}}{dv} = 0, \\ \\ \\mbox{i.e.}, \\ \\ k^{\\mu} = \\mbox{const.}$$<br \/>\n\u304c\u5c0e\u304b\u308c\u308b\u3002\u4e00\u822c\u7684\u306b\u306f\uff0c\u30d9\u30af\u30c8\u30eb\u3092\u5fae\u5206\u3059\u308b\u969b\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306e\u5fae\u5206\u3060\u3051\u3067\u306a\u304f\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u5fae\u5206\u3082\u8003\u616e\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u306e\u3060\u304c\uff0c\u3053\u306e\u5834\u5408\u306e\u3088\u3046\u306b\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066<strong>\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u5ea7\u6a19\u7cfb<\/strong>\u306e\u5834\u5408\u306b\u306f\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\uff08\u5ea7\u6a19\u57fa\u5e95\uff09\u306f\u4e00\u5b9a\u3067\u3042\u308b\u306e\u3067\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306e\u5fae\u5206\u306e\u307f\u3092\u8003\u3048\u308c\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<p>4\u5143\u901f\u5ea6\u304c \\(\\boldsymbol{u} = u^{\\mu} \\boldsymbol{e}_{\\mu}\\), \\(u^{\\mu} = (1, 0, 0, 0)\\) \u3067\u3042\u308b\u89b3\u6e2c\u8005\uff08\u3053\u306e\u5834\u5408\u306f\u7279\u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9759\u6b62<\/strong><\/span>\u89b3\u6e2c\u8005\u3067\u3042\u308b\uff09\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570 \\(\\omega\\) \u306f<\/p>\n<p>$$\\omega \\equiv &#8211; \\boldsymbol{k}\\cdot\\boldsymbol{u} = &#8211; \\eta_{\\mu\\nu}k^{\\mu} u^{\\nu} = k^0 = \\mbox{const.}$$<br \/>\n\u3068\u306a\u308a\uff0c\u632f\u52d5\u6570\u00a0 \\(\\omega\\) \u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>\u9759\u6b62\u89b3\u6e2c\u8005\u306b\u9650\u3089\u305a\uff0c\u4e00\u822c\u306b4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u3067\u3042\u3089\u308f\u3055\u308c\u308b\u89b3\u6e2c\u8005\u304c4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb $\\boldsymbol{k}$ \u3067\u3042\u3089\u308f\u3055\u308c\u308b\u5149\u3092\u89b3\u6e2c\u3057\u305f\u5834\u5408\u306b\u6e2c\u5b9a\u3055\u308c\u308b\u5149\u306e\u632f\u52d5\u6570\u3092<br \/>\n$$\\omega = &#8211; \\boldsymbol{k}\\cdot\\boldsymbol{u}$$<br \/>\n\u3068\u66f8\u304f\u306e\u306f\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u72b6\u6cc1\u3067\u3082\u6709\u52b9\u3067\u3042\u308b\u3002\u3053\u306e\u5f0f\u306b\u3088\u3063\u3066\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u3088\u3089\u305a\u306b\u7279\u6b8a\u76f8\u5bfe\u8ad6\u7684\u52b9\u679c\u3067\u3042\u308b\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u3092\u7406\u89e3\u3067\u304d\u308b\u3060\u3051\u3067\u306a\u304f\uff0c\u91cd\u529b\u304c\u5b58\u5728\u3057\u3066\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u304c\u4f7f\u3048\u306a\u3044\u5834\u5408\u306b\u3082\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306a\u632f\u52d5\u6570\u306e\u5909\u5316\u3067\u3042\u308b\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u304c\u7406\u89e3\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n<p>\u307e\u305f\uff0c\u3053\u306e\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u9032\u884c\u65b9\u5411\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\gamma^{\\mu}\\) \u306f\uff0c\\(\\boldsymbol{u} \\) \u306b\u76f4\u4ea4\u3059\u308b3\u6b21\u5143\u8d85\u66f2\u9762\u3078\u5c04\u5f71\u3059\u308b \\(P^{\\mu}_{\\ \\ \\nu} = \\delta^{\\mu}_{\\ \\ \\nu} + u^{\\mu} u_{\\nu}\\) \u3092\u4f7f\u3063\u3066<\/p>\n<p>$$\\gamma^{\\mu} \\equiv \\frac{P^{\\mu}_{\\ \\ \\nu} k^{\\nu}}{\\omega} = (0, \\vec{\\gamma})$$<br \/>\n$$\\vec{\\gamma} = \\left( \\frac{k^1}{\\omega}, \\frac{k^2}{\\omega}, \\frac{k^3}{\\omega}\\right) = \\mbox{const.}$$<br \/>\n\u3068\u306a\u308a\uff0c$\\vec{\\gamma}$ 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