{"id":139,"date":"2022-01-05T11:06:39","date_gmt":"2022-01-05T02:06:39","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=139"},"modified":"2024-10-03T13:59:37","modified_gmt":"2024-10-03T04:59:37","slug":"%e8%a6%b3%e6%b8%ac%e8%80%85%e3%81%ae4%e5%85%83%e9%80%9f%e5%ba%a6","status":"publish","type":"page","link":"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\/%e8%a6%b3%e6%b8%ac%e8%80%85%e3%81%ae4%e5%85%83%e9%80%9f%e5%ba%a6\/","title":{"rendered":"\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6"},"content":{"rendered":"<p><!--more--><\/p>\n<h3 id=\"yui_3_17_2_1_1641348390478_1059\">4\u5143\u901f\u5ea6<\/h3>\n<p>\u6163\u6027\u7cfb \\(S\\) \uff08\u305d\u306e\u5ea7\u6a19\u3092\\(x^{\\mu} =\u00a0 (c t, x, y, z)\\) \u3068\u3059\u308b\uff09\u306b\u304a\u3044\u3066\uff0c2\u4eba\u306e\u89b3\u6e2c\u8005 \\(A, B\\) \u3092\u8003\u3048\u308b\u3002\u89b3\u6e2c\u8005 \\(A\\) \u306f\u9759\u6b62\u3057\u3066\u3044\u308b\u57fa\u6e96\u89b3\u6e2c\u8005\u3067\u3042\u308a\uff0c\u89b3\u6e2c\u8005 \\(B\\) \u306f \\(A\\) \u306b\u5bfe\u3057\u3066\\(+x\\) \u65b9\u5411\u306b\u901f\u3055 \\(V\\) \u3067\u904b\u52d5\u3057\u3066\u3044\u308b\u3002<\/p>\n<p>\\begin{eqnarray} ds^2 &amp;\\equiv&amp; &#8211; c^2 d\\tau^2 \\\\ &amp;=&amp; \\eta_{\\mu\\nu} \\,dx^{\\mu} dx^{\\nu} \\\\<br \/>\n&amp;=&amp; -c^2 dt^2 + dx^2 + dy^2 + dz^2 \\end{eqnarray}<\/p>\n<p>\u306e\u3088\u3046\u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7dda\u7d20<\/strong><\/span> \\(ds^2\\) \u304b\u3089<span style=\"color: #ff0000;\"><strong><span style=\"font-family: helvetica, arial, sans-serif;\">\u56fa\u6709\u6642\u9593<\/span><\/strong><\/span>\u00a0 \\(\\tau\\) \uff08\u306e\u5fae\u5c0f\u5909\u4f4d \\(d\\tau\\)\uff09\u3092\u5b9a\u7fa9\u3057\uff0c<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>4\u5143\u901f\u5ea6<\/strong><\/span>\u306e\u6210\u5206\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3002\\(\\tau\\) \u306f\u4e16\u754c\u7dda\u3092\u30d1\u30e9\u30e1\u30c8\u30e9\u30a4\u30ba\u3059\u308b<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><b>\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf<\/b><\/span>\u306a\u306e\u3060\u304c\uff0c\u305d\u306e\u8a71\u306f\u307e\u305f\u3042\u3068\u3067\u3002\u307e\u305f\uff0c\u4ee5\u5f8c\u7c21\u5358\u306e\u305f\u3081\u306b\u5149\u901f \\(c\\) \u3092 \\(1\\) \u306b\u3059\u308b\u3053\u3068\u304c\u3057\u3070\u3057\u3070\u3042\u308b\u304c\uff0c\u305d\u306e\u5834\u5408\u306f\u306a\u308b\u3079\u304f\\(\\color{green} (\\because c \\Rightarrow 1)\\) \u306a\u3069\u3068\u6ce8\u8a18\u3059\u308b\u3088\u3046\u306b\u3059\u308b\u3002<\/p>\n<ul>\n<li>\u89b3\u6e2c\u8005 \\(A\\) \u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4e16\u754c\u7dda<\/strong><\/span>\u3092 \\(x^{\\mu}(\\tau)\\), \u4e16\u754c\u7dda\u306e\u63a5\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u901f\u5ea6<\/strong><\/span>\u306e\u6210\u5206\u3092<br \/>\n$$ u^{\\mu} = \\frac{1}{c} \\frac{\\ dx^{\\mu}}{d\\tau} {\\color{green}\\Rightarrow} \\frac{\\ dx^{\\mu}}{d\\tau} \\quad {\\color{green} (\\because c \\Rightarrow 1)}$$<\/li>\n<li>\u89b3\u6e2c\u8005 \\(B\\) \u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4e16\u754c\u7dda<\/strong><\/span>\u3092 \\(\\bar{x}^{\\mu}(\\tau)\\), \u4e16\u754c\u7dda\u306e\u63a5\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u901f\u5ea6<\/strong><\/span>\u306e\u6210\u5206\u3092<br \/>\n$$\\bar{u}^{\\mu} = \\frac{1}{c} \\frac{\\ d\\bar{x}^{\\mu}}{d\\tau} {\\color{green}\\Rightarrow} \\frac{\\ d\\bar{x}^{\\mu}}{d\\tau} \\quad {\\color{green} (\\because c \\Rightarrow 1)}$$<\/li>\n<\/ul>\n<p>\\(\\bar{x}^{\\mu}\\) \u306f\u5225\u306e\u6163\u6027\u7cfb\u3067\u306e\u5ea7\u6a19\u3067\u306f\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n<p>\u6163\u6027\u7cfb\u306f\u3042\u304f\u307e\u3067 \\(S\\) \u306e\u307f\u3002\u5168\u3066\u306e\u5ea7\u6a19\u306f \\( x^{\\mu} = (c t,x,y,z) \\) \u3067\u306e\u5024\u3067\u3042\u308b\u3002\u672c\u6765\u306f\u89b3\u6e2c\u8005 \\(A\\) \u306e\u4e16\u754c\u7dda\u3092 \\(x_A^{\\mu}(\\tau)\\)\uff0c\u89b3\u6e2c\u8005\\(B\\) \u306e\u4e16\u754c\u7dda\u3092 \\(x_B^{\\mu}(\\tau)\\) \u3068\u66f8\u304d\u305f\u3044\u3068\u3053\u308d\u3060\u304c\uff0c\u305d\u3046\u3059\u308b\u30684\u5143\u901f\u5ea6\u306e\u6210\u5206 \\( u_B^{\\mu} \\) \u306e\u4e0b\u6dfb\u5b57\u30d0\u30fc\u30b8\u30e7\u30f3\u304c \\(u_{B\\mu}\\) \u306a\u3069\u3068\u306a\u3063\u3066\u76ee\u304c\u30c1\u30ab\u30c1\u30ab\u3057\u3084\u3059\u3044\u306e\u3067\uff0c\u82e6\u6e0b\u306e\u6c7a\u65ad\u3067\uff0c\\(u_B^{\\mu} \\rightarrow \\bar{u}^{\\mu}\\) \u3068\u8868\u8a18\u3059\u308b\u3053\u3068\u306b\u3057\u305f\u3002\u3054\u4e86\u627f\u304f\u3060\u3055\u3044\u3002<\/p>\n<p>\u307e\u305f\uff0c\u5225\u30bb\u30af\u30b7\u30e7\u30f3\uff08\u300c<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\/4%e6%ac%a1%e5%85%83%e6%99%82%e7%a9%ba%e3%81%ae%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%83%bb%e7%b7%9a%e7%b4%a0%e3%83%bb%e8%a8%88%e9%87%8f%e3%83%86%e3%83%b3%e3%82%bd%e3%83%ab\/\">4\u6b21\u5143\u6642\u7a7a\u306e\u30d9\u30af\u30c8\u30eb\u30fb\u7dda\u7d20\u30fb\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/a>\u300d\u306e\u9805\uff09\u3067\u8aac\u660e\u3059\u308b\u3088\u3046\u306b\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(\\boldsymbol{u}\\) \u306e\u6b63\u5f0f\u8868\u8a18\u3092<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6210\u5206<\/strong><\/span> \\(u^{\\mu}\\) \u3068<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u57fa\u672c\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(\\boldsymbol{e}_{\\mu}\\) \u3092\u4f7f\u3063\u3066\u00a0 \\( \\boldsymbol{u} = u^{\\mu} \\boldsymbol{e}_{\\mu}\\) \u306a\u3069\u3068\u66f8\u304f\u3053\u3068\u306b\u306a\u308b\u3002\u307e\u305f\uff0c4\u5143\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5185\u7a4d<\/strong><\/span>\u306f \\(\\eta_{\\mu\\nu}\\) \u3067\u7e2e\u7d04\u3092\u3068\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<p>\u306a\u306e\u3067<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3068\u3044\u3063\u305f\u3089 \\(\\boldsymbol{u}\\) \u3067\u3042\u308a\uff0c\\(u^{\\mu}\\) \u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206<\/strong><\/span>\uff0c\u3068\u3067\u304d\u308b\u304b\u304e\u308a\u8a00\u3044\u5206\u3051\u308b\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n<h3>\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6<\/h3>\n<p>\u4eca\uff0c\u89b3\u6e2c\u8005 \\(A\\) \u306f \\(S\\) \u7cfb\u3067\u306f\u9759\u6b62\u3057\u3066\u3044\u308b\u306e\u3067\uff0c<br \/>\n\\begin{eqnarray}<br \/>\n\\frac{dx^i}{dt} &amp;=&amp; \\left( \\frac{dx}{dt}, \\frac{dy}{dt}, \\frac{dz}{dt} \\right) = (0, 0, 0)\\\\<br \/>\n&amp;=&amp; c \\frac{dx^i}{dx^0} = c \\frac{u^i}{u^0} \\\\<br \/>\n\\therefore\\ \\ u^i &amp;=&amp; (u^1, u^2, u^3) = (0, 0, 0)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u307e\u305f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u901f\u5ea6<\/strong><\/span> \\(\\boldsymbol{u}\\)\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u300c\u5927\u304d\u3055\u300d\u306e2\u4e57\uff08\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\uff09\u304c \\(-1\\) \u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\boldsymbol{u}\\cdot\\boldsymbol{u} &amp;=&amp;\u00a0 \\eta_{\\mu\\nu}\\, u^{\\mu} u^{\\nu} = -\\left(u^0\\right)^2 \\\\<br \/>\n&amp;=&amp; \\eta_{\\mu\\nu} \\frac{\\ dx^{\\mu}}{c d\\tau}\u00a0 \\frac{\\ dx^{\\nu}}{c d\\tau} \\\\<br \/>\n&amp;=&amp; \\frac{ds^2}{c^2 d\\tau^2} \\\\<br \/>\n&amp;=&amp; -1<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u306e\u3053\u3068\u304b\u3089\u6700\u7d42\u7684\u306b future-directed \\(u^0 &gt; 0\\) \u3092\u4eee\u5b9a\u3057\u3066<\/p>\n<p>$$ u^{\\mu} = (1, 0, 0, 0) $$<\/p>\n<h3>\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6<\/h3>\n<p>\u4e00\u65b9\uff0c\u89b3\u6e2c\u8005 \\(B\\) \u306f\u901f\u3055\\(V\\) \u3067 \\(+x\\) \u65b9\u5411\u306b\u904b\u52d5\u3057\u3066\u3044\u308b\u306e\u3067\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{d\\bar{x}^i}{d\\bar{t}} &amp;=&amp; \\left(\\frac{d\\bar{x}}{d\\bar{t}}, \\frac{d\\bar{y}}{d\\bar{t}}, \\frac{d\\bar{z}}{d\\bar{t}}\\right) = (V, 0, 0) \\\\<br \/>\n&amp;=&amp;c \\frac{\\bar{u}^i}{\\bar{u}^0} \\\\<br \/>\n\\therefore\\ \\ \\bar{u}^i &amp;=&amp; \\left( \\bar{u}^1, \\bar{u}^2, \\bar{u}^3\\right) = \\left( \\bar{u}^0 \\frac{V}{c}, 0, 0\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u898f\u683c\u5316\u6761\u4ef6 \\(\\bar{\\boldsymbol{u}}\\cdot\\bar{\\boldsymbol{u}} = \\eta_{\\mu\\nu}\\, \\bar{u}^{\\mu} \\bar{u}^{\\nu} = -\\left(\\bar{u}^0\\right)^2 + \\left(\\bar{u}^1\\right)^2 = -1 \\) \u304b\u3089\u6700\u7d42\u7684\u306b<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\bar{u}^{\\mu} &amp;=&amp;<br \/>\n\\left(\\frac{1}{\\sqrt{1-\\left(\\frac{V}{c}\\right)^2}}, \\frac{\\frac{V}{c}}{\\sqrt{1-\\left(\\frac{V}{c}\\right)^2}}, 0, 0\\right) \\\\<br \/>\n&amp;{\\color{green}\\Rightarrow}&amp;<br \/>\n\\left(\\frac{1}{\\sqrt{1-V^2}}, \\frac{V}{\\sqrt{1-V^2}}, 0, 0\\right)\\quad{\\color{green} (\\because c \\Rightarrow 1)}<br \/>\n\\end{eqnarray}<\/p>\n<p>&nbsp;<\/p>\n<h3>2\u4eba\u306e\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6\u306e\u9593\u306e\u95a2\u4fc2\u5f0f<\/h3>\n<p>\u89b3\u6e2c\u8005 \\(A\\) \u304b\u3089\u307f\u305f\uff0c\u89b3\u6e2c\u8005 \\(B\\) \u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7a7a\u9593\u7684<\/strong><\/span>\u306a\u904b\u52d5\u65b9\u5411\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3092 \\(\\boldsymbol{e}\\)\uff0c\u305d\u306e\u6210\u5206\u3092 \\(e^{\\mu}\\) \u3068\u3059\u308b\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2413 size-large\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela01-640x302.png\" alt=\"\" width=\"640\" height=\"302\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela01-640x302.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela01-300x141.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela01-1536x724.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela01-750x354.png 750w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela01.png 1540w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\\(\\boldsymbol{e}\\) \u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7a7a\u9593\u7684<\/strong><\/span>\u3067\u3042\u308b\u3068\u306f\uff0c\u89b3\u6e2c\u8005 \\(A\\) \u306e4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306b\u76f4\u4ea4\u3059\u308b\u3053\u3068\u3067\u3042\u308a\uff0c<\/p>\n<p>$$\\boldsymbol{u}\\cdot\\boldsymbol{e} = \\eta_{\\mu\\nu} u^{\\mu} e^{\\nu} = 0$$<\/p>\n<p>\u307e\u305f\uff0c\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u304b\u3089\uff0c<br \/>\n$$\\boldsymbol{e}\\cdot\\boldsymbol{e} = \\eta_{\\mu\\nu}e^{\\mu}e^{\\nu} = 1$$<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c\\(e^{\\mu}\\) \u306f\uff0c\\(u^{\\mu} = (1, 0, 0, 0)\\) \u3067\u3042\u308b \\(S\\) \u7cfb\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<br \/>\n$$ e^{\\mu} = (0, 1, 0, 0)$$<\/p>\n<p>\u6700\u7d42\u7684\u306b\uff0c\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6 \\(\\bar{u}^{\\mu} \\) \u3092\uff0c\u89b3\u6e2c\u8005 \\(A\\) \u306e4\u5143\u901f\u5ea6 \\(u^{\\mu} \\) \u3068\uff0c\\(A\\) \u306b\u5bfe\u3059\u308b \\(B\\) \u306e\u904b\u52d5\u65b9\u5411\u3092\u8868\u3059 \\(e^{\\mu}\\) \u3092\u4f7f\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\bar{u}^{\\mu} &amp;=&amp;<br \/>\n\\left(\\frac{1}{\\sqrt{1-\\left(\\frac{V}{c}\\right)^2}}, \\frac{\\frac{V}{c}}{\\sqrt{1-\\left(\\frac{V}{c}\\right)^2}}, 0, 0\\right) \\\\<br \/>\n&amp;=&amp; \\frac{1}{\\sqrt{1-\\left(\\frac{V}{c}\\right)^2}} (1, 0, 0, 0) + \\frac{\\frac{V}{c}}{\\sqrt{1-\\left(\\frac{V}{c}\\right)^2}} (0, 1, 0, 0) \\\\<br \/>\n&amp;{\\color{green}\\Rightarrow}&amp; \\frac{1}{\\sqrt{1-V^2}} u^{\\mu} +\\frac{V}{\\sqrt{1-V^2}} e^{\\mu} \\quad{\\color{green} (\\because c \\Rightarrow 1)}\\\\<br \/>\n\\therefore \\ \\ \\bar{\\boldsymbol{u}} &amp;=&amp; \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{u} +\\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{e}\\quad{\\color{green} (c = 1)}<br \/>\n\\end{eqnarray}<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2414\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela02-640x596.png\" alt=\"\" width=\"480\" height=\"447\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela02-640x596.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela02-300x279.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela02-750x698.png 750w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/rela02.png 1132w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3>4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h3>\n<p>\u4ee5\u4e0b\uff0c${\\color{green} c = 1}$.<\/p>\n<p>\u4e00\u65e6\u30d9\u30af\u30c8\u30eb\u5f0f\u3067\u8868\u3055\u308c\u305f\u4e0a\u306e\u95a2\u4fc2\u5f0f\u306f\uff0c\u5ea7\u6a19\u7cfb\u306e\u3068\u308a\u304b\u305f\u306b\u3088\u3089\u305a\u306b\u6210\u308a\u7acb\u3064\u3002\u5f93\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u8868\u73fe\u304c\u4e00\u822c\u306b\u6210\u308a\u7acb\u3064\u3002\u3053\u308c\u3092\uff08\u30ac\u30ea\u30ec\u30a4\u5909\u63db\u306b\u57fa\u3065\u304f3\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u3068\u306e\u985e\u4f3c\u6027\u304b\u3089\uff09<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/strong><\/span>\u3068\u547c\u3076\u3053\u3068\u306b\u3057\u3088\u3046\u3002<span style=\"color: #999999;\">\uff08\u4f55\u304b\uff0c\u3082\u3046\u5c11\u3057\u3044\u3044\u540d\u524d\u304c\u3042\u308c\u3070\u3044\u3044\u306e\u3060\u304c&#8230; \uff09<\/span><br \/>\n$$ \\bar{\\boldsymbol{u}} = \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{u} +\\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{e}$$<\/p>\n<h4>\u53c2\u8003\uff1a\u30ac\u30ea\u30ec\u30a4\u5909\u63db\u306b\u57fa\u3065\u304f3\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h4>\n<p>\u53c2\u8003\u307e\u3067\u306b\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66<\/strong><\/span>\u306b\u304a\u3044\u3066\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30ac\u30ea\u30ec\u30a4\u5909\u63db\u306b\u57fa\u3065\u304f\u901f\u5ea6\u306e\u5408\u6210\u5247<\/strong><\/span>\u3092\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n<p>4\u5143\u30d9\u30af\u30c8\u30eb\u3068\u533a\u5225\u3059\u308b\u305f\u3081\u306b\uff0c3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306f \\(\\vec{v}\\) \u306e\u3088\u3046\u306b \\(\\vec{\\ }\\) \u3092\u3064\u3051\u3066\u66f8\u304f\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n<p>\u901f\u5ea6 \\(\\vec{v}\\) \u306e\u7269\u4f53\u306b\u5bfe\u3057\u3066\u76f8\u5bfe\u7684\u306b\u901f\u5ea6 \\(\\vec{V}\\) \uff08\u305d\u306e\u5927\u304d\u3055\u3092 \\(V\\)\uff0c\u5411\u304d\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3092 \\(\\vec{e}\\) \u3068\u3059\u308b\u3068 \\(\\vec{V} = V \\vec{e}\\)\uff09\u3067\u904b\u52d5\u3059\u308b\u7269\u4f53\u306e\u901f\u5ea6 \\(\\vec{v}&#8217;\\) \u306f<\/p>\n<p>$$\\vec{v}&#8217; = \\vec{v} + \\vec{V} = \\vec{v} + V \\vec{e}$$<br \/>\n\u3053\u308c\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30ac\u30ea\u30ec\u30a4\u5909\u63db\u306b\u57fa\u3065\u304f3\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/strong><\/span>\u3067\u3042\u308b\u3002<\/p>\n<p>&nbsp;<\/p>\n<h3 id=\"yui_3_17_2_1_1641349024343_29\">4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u306e\u9006\u5909\u63db<\/h3>\n<p>4\u5143\u901f\u5ea6 \\(\\boldsymbol{u} \\) \u306e\u89b3\u6e2c\u8005 \\(A\\) \u306b\u5bfe\u3057\u3066\uff0c\\( \\boldsymbol{u}\\) \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{e}\\) \u306e\u65b9\u5411\u306b3\u6b21\u5143\u7684\u306a\u76f8\u5bfe\u901f\u5ea6\u306e\u5927\u304d\u3055 \\(V\\) \u3067\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6 \\(\\boldsymbol{u} \\) \u306f<\/p>\n<p>$$ \\bar{\\boldsymbol{u}} = \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{u} +\\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{e}$$<\/p>\n<p>\u3068\u66f8\u3051\uff0c\u3053\u308c\u3092<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/strong><\/span>\u3068\u547c\u3076\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n<p id=\"yui_3_17_2_1_1641352539305_989\">\u4e00\u65b9\uff0c4\u5143\u901f\u5ea6 \\(\\bar{\\boldsymbol{u}}\\) \u306e\u89b3\u6e2c\u8005 \\(B\\) \u304b\u3089\u307f\u308b\u3068\uff0c4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u89b3\u6e2c\u8005 \\(A\\) \u306f\uff0c\\(\\bar{\\boldsymbol{u}}\\) \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\bar{\\boldsymbol{e}}\\) \u306e\u30de\u30a4\u30ca\u30b9\u65b9\u5411\u306b3\u6b21\u5143\u7684\u306a\u76f8\u5bfe\u901f\u5ea6\u306e\u5927\u304d\u3055 \\(V\\) \u3067\u904b\u52d5\u3057\u3066\u3044\u308b\u3088\u3046\u306b\u307f\u3048\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c<br \/>\n$$ \\boldsymbol{u} = \\frac{1}{\\sqrt{1-V^2}} \\bar{\\boldsymbol{u}} -\\frac{V}{\\sqrt{1-V^2}}\\bar{\\boldsymbol{e}}$$<br \/>\n\u3068\u66f8\u3051\u308b\u306f\u305a\u3067\u3042\u308b\u3002\u3053\u3053\u3067\uff0c\\(\\bar{\\boldsymbol{e}}\\) \u306f\uff0c\\(\\bar{\\boldsymbol{u}} \\) \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068\u304b\u3089<br \/>\n\\begin{eqnarray}<br \/>\n\\bar{\\boldsymbol{u}}\\cdot\\bar{\\boldsymbol{e}} &amp;=&amp;\u00a0 \\eta_{\\mu\\nu} \\bar{u}^{\\mu}\\bar{e}^{\\nu} = 0\\\\<br \/>\n\\bar{\\boldsymbol{e}}\\cdot\\bar{\\boldsymbol{e}} &amp;=&amp;\\eta_{\\mu\\nu} \\bar{e}^{\\mu}\\bar{e}^{\\nu} = 1<br \/>\n\\end{eqnarray}<\/p>\n<p>\u4e0a\u8a18\u306e \\(\\bar{\\boldsymbol{u}}= \\cdots\\) \u306e\u5f0f\u3092 \\(\\boldsymbol{u}= \\cdots\\) \u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066 \\(\\bar{\\boldsymbol{e}}\\) \u306b\u3064\u3044\u3066\u89e3\u304f\u3068\uff0c\u4ee5\u4e0b\u306e\u5f0f\u304c\u5c0e\u304b\u308c\u308b\u3002<br \/>\n$$\\bar{\\boldsymbol{e}}= \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{e} + \\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{u}$$<\/p>\n<p>\u540c\u69d8\u306b\u4ee5\u4e0b\u3082\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<p>$$\\boldsymbol{e}= \\frac{1}{\\sqrt{1-V^2}} \\bar{\\boldsymbol{e}} &#8211; \\frac{V}{\\sqrt{1-V^2}} \\bar{\\boldsymbol{u}}$$<\/p>\n<h3>\u307e\u3068\u3081<\/h3>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u89b3\u6e2c\u8005 \\(A\\) \u306e<\/strong><strong>4\u5143\u901f\u5ea6<\/strong><\/span>: \\(\\boldsymbol{u}\\)<\/p>\n<p>\\(\\boldsymbol{u}\\) \u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u76f4\u4ea4\u3059\u308b<\/strong><strong>\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/strong><\/span>: \\(\\boldsymbol{e}\\)<br \/>\n\\begin{eqnarray}<br \/>\n{\\boldsymbol{u}}\\cdot{\\boldsymbol{e}} &amp;=&amp;\u00a0\u00a0 0\\\\<br \/>\n{\\boldsymbol{e}}\\cdot{\\boldsymbol{e}} &amp;=&amp; 1<br \/>\n\\end{eqnarray}<\/p>\n<p>\u89b3\u6e2c\u8005 \\(A\\) \u306b\u5bfe\u3057\u3066\uff0c\\(\\boldsymbol{e}\\) \u65b9\u5411\u306b\u901f\u3055 \\(V\\) \u3067\u904b\u52d5\u3059\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u89b3\u6e2c\u8005 \\(B\\) \u306e4\u5143\u901f\u5ea6<\/strong><\/span>: \\(\\bar{\\boldsymbol{u}}\\)<\/p>\n<p>\\(\\bar{\\boldsymbol{u}}\\) \u306b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/strong><\/span>: \\(\\bar{\\boldsymbol{e}}\\)<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\bar{\\boldsymbol{u}}\\cdot\\bar{\\boldsymbol{e}} &amp;=&amp;\u00a0\u00a0 0\\\\<br \/>\n\\bar{\\boldsymbol{e}}\\cdot\\bar{\\boldsymbol{e}} &amp;=&amp;\u00a0 1<br \/>\n\\end{eqnarray}<\/p>\n<h4>4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h4>\n<p>\\begin{eqnarray}<br \/>\n\\bar{\\boldsymbol{u}} &amp;=&amp; \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{u} + \\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{e}\\\\<br \/>\n\\bar{\\boldsymbol{e}} &amp;=&amp; \\frac{1}{\\sqrt{1-V^2}} \\boldsymbol{e} + \\frac{V}{\\sqrt{1-V^2}} \\boldsymbol{u}<br \/>\n\\end{eqnarray}<\/p>\n<h4>4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u306e\u9006\u5909\u63db<\/h4>\n<p>\\begin{eqnarray}<br \/>\n\\boldsymbol{u} &amp;=&amp; \\frac{1}{\\sqrt{1-V^2}} \\bar{\\boldsymbol{u}} &#8211; \\frac{V}{\\sqrt{1-V^2}} \\bar{\\boldsymbol{e}}\\\\<br \/>\n\\boldsymbol{e} &amp;=&amp; \\frac{1}{\\sqrt{1-V^2}} \\bar{\\boldsymbol{e}} &#8211; \\frac{V}{\\sqrt{1-V^2}} \\bar{\\boldsymbol{u}}<br \/>\n\\end{eqnarray}<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4ee5\u4e0a\u306e\u95a2\u4fc2\u306f\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306e\u307f\u306a\u3089\u305a\u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u3082\u6210\u308a\u7acb\u3064\u3002<\/strong><\/span>\u3053\u308c\u3089\u306f\u5358\u306b\uff0c\u6642\u7a7a\u306e\u4efb\u610f\u306e1\u70b9\u306b\u304a\u3051\u308b4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u8db3\u3057\u7b97\u5f15\u304d\u7b97\u3067\u3042\u308b\u3002\u91cd\u529b\u304c\u5b58\u5728\u3059\u308b\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u72b6\u6cc1\u4e0b\u306b\u304a\u3044\u3066\u3082\uff0c\u6642\u7a7a\u306e\u4efb\u610f\u306e1\u70b9\u3067\uff0c\u89b3\u6e2c\u8005 \\(A\\) \u306e\u77ac\u9593\u7684\u5171\u52d5\u5ea7\u6a19\u7cfb\u3068\u3044\u3046\u5c40\u6240\u6163\u6027\u7cfb\u3092\u3068\u308b\u3053\u3068\u304c\u3067\u304d\uff0c\u305d\u3053\u3067\u306f\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306b\u304a\u3051\u308b\u8a08\u7b97\u304c\u6210\u308a\u7acb\u3064\u304b\u3089\u3067\u3042\u308b\u3002<\/p>\n<h3>\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50<\/h3>\n<p>\u6700\u5f8c\u306b\u4e00\u8a00\u3002<\/p>\n<p>4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u304a\u3088\u3073\u305d\u306e\u9006\u5909\u63db\u304b\u3089\uff0c\u305f\u3060\u3061\u306b\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>$$- \\boldsymbol{u} \\cdot\\bar{\\boldsymbol{u}} = \\boldsymbol{e}\\cdot \\bar{\\boldsymbol{e}} = \\frac{1}{\\sqrt{1-V^2}}$$<\/p>\n<p>2\u3064\u306e4\u5143\u901f\u5ea6\u540c\u58eb\uff0c\u3042\u308b\u3044\u306f\u305d\u308c\u3089\u306b\u76f4\u4ea4\u3059\u308b2\u3064\u306e\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u3068\u3044\u3063\u305f\uff0c4\u5143\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u304c<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50<\/strong><\/span> \\( \\displaystyle \\frac{1}{\\sqrt{1-V^2}}\\) \u3092\u4e0e\u3048\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50\u306f\uff084\u5143\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u304b\u3089\u3064\u304f\u3089\u308c\u308b\uff094\u6b21\u5143\u30b9\u30ab\u30e9\u30fc\u91cf\u3067\u3042\u308a\uff0c\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u91cf\u3067\u3042\u308b\uff01<\/strong><\/span>\u3068\u3044\u3046\u306e\u304c\u3053\u3053\u3067\u306e\u7acb\u5834\u3067\u3042\u308b\u3002<\/p>\n<p><span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u6642\u9593\u304c\u9045\u308c\u305f\u308a\uff0c\u68d2\u304c\u7e2e\u3093\u305f\u308a\u3059\u308b\u306e\u3082\uff0c\u3059\u3079\u3066\u3053\u306e\u4e0d\u5909\u91cf\u3067\u3042\u308b\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50\u306e\u5f71\u97ff\u3067\u3042\u308b\uff01<\/strong><\/span><\/p>\n<h3>\u53c2\u8003\u6587\u732e<\/h3>\n<p>G. F. R. Ellis &#8211; Relativistic Cosmology, in &#8220;General Relativity and Cosmology&#8221; ed. <a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/3560\/\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>B. K. Sachs<\/strong><\/span><\/a> (Academic Press, New York, 1971) \u306e P.147 \u306e (6.11a) \u5f0f\uff1a<\/p>\n<p>$$u_a = \\cosh\\beta\\,u_a + \\sinh\\beta\\, e_a, \\quad\u00a0 e_a e^a = 1, \\quad e^a u_a = 0 $$<\/p>\n<p>\u3068\u3042\u308a\uff0c\u305d\u306e\u4e0b\u306b $V = \\tanh \\beta$ (\u539f\u6587\u3067\u306f $V = \\mbox{tgh}\\\u00a0 \\beta$) \u3068\u66f8\u3044\u3066\u3042\u308b\u3002\u3053\u308c\u3092\u81a8\u3089\u307e\u305b\u3066\u4e01\u5be7\u306b\u66f8\u3044\u3066\u307f\u308b\u3068\uff0c\u4e0a\u8a18\u306e\u3088\u3046\u306a\u8aac\u660e\u306b\u306a\u308b\u304b\u3068\u3002Ellis \u306e\u3053\u306e\u5f0f\u306f\uff0c\u8aa4\u690d\u306a\u306e\u304b\u5de6\u8fba\u3068\u53f3\u8fba\u306b\u540c\u3058 $u_a$ \u304c\u3042\u3063\u3066\uff0c\u3053\u308c\u306a\u3093\u306e\u3053\u3068\uff1f\u3068\u601d\u3063\u3066\u81ea\u5206\u304c\u7d0d\u5f97\u3067\u304d\u308b\u3088\u3046\u306b\u304b\u307f\u304f\u3060\u3044\u3066\u307f\u305f\u306e\u304c\u3053\u306e\u30da\u30fc\u30b8\u306e\u5185\u5bb9\u3002<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":71,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-139","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\/139","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=139"}],"version-history":[{"count":38,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/139\/revisions"}],"predecessor-version":[{"id":9453,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/139\/revisions\/9453"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/71"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=139"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}