{"id":137,"date":"2022-01-05T11:05:03","date_gmt":"2022-01-05T02:05:03","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=137"},"modified":"2024-10-03T12:43:54","modified_gmt":"2024-10-03T03:43:54","slug":"%e5%85%89%e3%81%ae4%e5%85%83%e3%83%99%e3%82%af%e3%83%88%e3%83%ab","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\/%e5%85%89%e3%81%ae4%e5%85%83%e3%83%99%e3%82%af%e3%83%88%e3%83%ab\/","title":{"rendered":"\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb"},"content":{"rendered":"<p>\u5149\u306e\u8af8\u91cf\u3092\u3042\u3089\u308f\u30594\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306b\u3064\u3044\u3066\u3002<\/p>\n<p><!--more--><\/p>\n<h3>\u30d9\u30af\u30c8\u30eb\u306e\u8868\u8a18<\/h3>\n<p>3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306f \\(\\vec{k}\\) \u3067\u8868\u3057\uff0c<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\/%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e3%81%ae%e8%a6%8f%e7%b4%84%e3%81%a8%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e5%86%85%e7%a9%8d\/\">\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u306e\u898f\u7d04<\/a>\u3092\u4f7f\u3063\u3066<br \/>\n\\(\\vec{k} =\u00a0 k^i \\,\\vec{e}_i, \\ \\ \u00a0 k^i = (k_x, k_y, k_z) \\ \\ \\mbox{\u3042\u308b\u3044\u306f} \\ \\ \\vec{k} = (k_x, k_y, k_z) \\)<\/p>\n<p>4\u5143\u30d9\u30af\u30c8\u30eb\u306f \\(\\boldsymbol{k}\\) \u3067\u8868\u3057\uff0c<br \/>\n\\(\\boldsymbol{k}= k^{\\mu} \\,\\boldsymbol{e}_{\\mu}, \\ k^{\\mu} =\u00a0 (k^0, k^i) = (k^0, \\vec{k}) \\)<\/p>\n<h3>\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d<\/h3>\n<p>3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f<br \/>\n$$\\vec{k}\\cdot\\vec{k} = k_x^2 + k_y^2 + k_z^2$$4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f\uff08\u7279\u6b8a\u76f8\u5bfe\u8ad6\u3067\u306f\uff09<br \/>\n\\begin{eqnarray}<br \/>\n\\boldsymbol{k}\\cdot\\boldsymbol{k} &amp;=&amp; \\left(k^{\\mu} \\boldsymbol{e}_{\\mu}\\right) \\cdot \\left(k^{\\nu} \\boldsymbol{e}_{\\nu}\\right) \\\\<br \/>\n&amp;=&amp; \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu}\\,\u00a0 k^{\\mu} k^{\\nu} \\\\<br \/>\n&amp;=&amp; \\eta_{\\mu\\nu} \\, k^{\\mu} k^{\\nu} \\\\<br \/>\n&amp;=&amp; &#8211; \\left(k^0\\right)^2 + \\left(k^1\\right)^2 + \\left(k^2\\right)^2 + \\left(k^3\\right)^2<br \/>\n\\end{eqnarray}<\/p>\n<h3 id=\"yui_3_17_2_1_1641348286986_1033\">\u5149\u306e\u8af8\u91cf<\/h3>\n<p>\u5149\uff08\u4e00\u822c\u306b\u96fb\u78c1\u6ce2\uff09\u306e\u8af8\u91cf\u306e\u307e\u3068\u3081<\/p>\n<ul>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6ce2\u9577<\/strong><\/span>\uff1a\\(\\lambda\\)\u3000\u5358\u4f4d \\(\\mbox{m}\\) \uff08\u30e1\u30fc\u30c8\u30eb\uff09\u3002\u6ce2\uff11\u500b\u5206\u306e\u9577\u3055\u3092\u8868\u3059\u3002<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5468\u671f<\/strong><\/span>\uff1a\\(T\\)\u3000\u5358\u4f4d \\(\\mbox{s}\\)\uff08\u79d2\uff09\u3002\u4e00\u56de\u632f\u52d5\u3059\u308b\u306e\u306b\u304b\u304b\u308b\u6642\u9593\u3092\u8868\u3059\u3002<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6ce2\u306e\u901f\u3055<\/strong><\/span>\uff1a\u3000\u4e00\u822c\u306b \\( \\lambda \/ T \\) \u304c\u6ce2\u306e\u901f\u3055\u3002\u3053\u3053\u3067\u306f\u5149\u3060\u304b\u3089\u5149\u901f\u3092 \\(c\\) \u3068\u3057\u3066<br \/>\n$$ \\frac{\\lambda}{T} = c$$<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u632f\u52d5\u6570<\/strong><\/span>\uff1a\\(\\nu\\)\u3000\u5358\u4f4d \\(\\mbox{Hz} = \\mbox{s}^{-1}\\)\uff08\u30d8\u30eb\u30c4\u307e\u305f\u306f\u6bce\u79d2\uff09\u3002\u5468\u671f \\(T\\) \u306e\u9006\u6570\u3067\u4e0e\u3048\u3089\u308c\uff0c\u5358\u4f4d\u6642\u9593\u306b\u4f55\u56de\u632f\u52d5\u3059\u308b\u304b\u3092\u8868\u3059\u3002\u5468\u6ce2\u6570\u3068\u3082\u3002$$ \\nu = \\frac{1}{ T}, \\quad\\therefore \\ \\ \\frac{\\lambda}{T}\\\u00a0 = \\lambda\\,\\nu = c$$<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u89d2\u632f\u52d5\u6570<\/strong><\/span>\uff1a\\(\\omega\\)\u3000\u5358\u4f4d \\( \\mbox{rad}\/\\mbox{s} \\)\uff08\u30e9\u30b8\u30a2\u30f3\u6bce\u79d2\uff09\u3002\u632f\u52d5\u6570\u306b \\(2\\pi\\) \u30e9\u30b8\u30a2\u30f3\u3092\u4e57\u3058\u305f\u3082\u306e\u3067\uff0c\u5358\u4f4d\u6642\u9593\u306b\u4f55\u30e9\u30b8\u30a2\u30f3\u4f4d\u76f8\u304c\u5909\u5316\u3059\u308b\u304b\u3092\u8868\u3059\u3002\u696d\u754c\u306b\u3088\u3063\u3066\u306f\uff0c\u3057\u3070\u3057\u3070\u300c\u89d2\u300d\u3092\u7701\u7565\u3057\u3066\u5358\u306b\u300c\u632f\u52d5\u6570\u300d\u3068\u8a00\u3046\u50be\u5411\u304c\u3042\u308b\u3002\uff08\u79c1\u306e\u307e\u308f\u308a\u3060\u3051\uff1f\u304b\u3082\u3002\uff09<br \/>\n$$ \\omega = 2\\pi \\nu$$<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6ce2\u6570<\/strong><\/span>\uff1a\\(\\tilde{\\nu}\\)\u3000\u5358\u4f4d \\( \\mbox{m}^{-1} \\)\uff08\u6bce\u30e1\u30fc\u30c8\u30eb\uff09\u3002\u6ce2\u9577\u306e\u9006\u6570 \\( 1\/\\lambda \\) \u3067\u4e0e\u3048\u3089\u308c\uff0c\u5358\u4f4d\u9577\u3055\u3042\u305f\u308a\u306e\u6ce2\u306e\u500b\u6570\u3092\u8868\u3059\u3002$$\\tilde{\\nu} = \\frac{1}{\\lambda}$$<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u89d2\u6ce2\u6570<\/strong><\/span>\uff1a\\(k\\)\u3000\u5358\u4f4d \\( \\mbox{rad}\/\\mbox{m} \\)\uff08\u30e9\u30b8\u30a2\u30f3\u6bce\u30e1\u30fc\u30c8\u30eb\uff09\u3002\u6ce2\u6570\u306b \\(2\\pi\\) \u30e9\u30b8\u30a2\u30f3\u3092\u4e57\u3058\u305f\u3082\u306e\u3067\uff0c\u5358\u4f4d\u9577\u3055\u3042\u305f\u308a\u4f55\u30e9\u30b8\u30a2\u30f3\u4f4d\u76f8\u304c\u5909\u5316\u3059\u308b\u304b\u3092\u8868\u3059\u3002\u696d\u754c\u306b\u3088\u3063\u3066\u306f\uff0c\u3057\u3070\u3057\u3070\u300c\u89d2\u300d\u3092\u7701\u7565\u3057\u3066\u5358\u306b\u300c\u6ce2\u6570\u300d\u3068\u8a00\u3046\u50be\u5411\u304c\u3042\u308b\u3002\uff08\u79c1\u306e\u307e\u308f\u308a\u3060\u3051\uff1f\u304b\u3082\u3002\uff09<br \/>\n$$ k = 2\\pi \\tilde{\\nu} = \\frac{2\\pi}{\\lambda} $$<\/li>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u89d2\u6ce2\u6570\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\uff1a\\(\\vec{k}\\)\u3000\u30d9\u30af\u30c8\u30eb\u306e\u5411\u304d\u304c\u5149\u306e\u4f1d\u64ad\u3059\u308b\u5411\u304d\u3092\u8868\u3057\uff0c\u305d\u306e\u5927\u304d\u3055\u306f\u300c\u89d2\u6ce2\u6570\u300d\\(k\\) \u3092\u8868\u3059\u3002\u3057\u3070\u3057\u3070\u300c\u89d2\u300d\u3092\u7701\u7565\u3057\u3066\u5358\u306b\u300c\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u300d\u3068\u8a00\u3046\u50be\u5411\u304c\u3042\u308b\u3002<br \/>\n\uff08\u79c1\u306e\u307e\u308f\u308a\u3060\u3051\uff1f\u304b\u3082\u3002\uff09<br \/>\n$$ \\sqrt{\\vec{k} \\cdot\\vec{k}} = k = \\frac{2 \\pi}{\\lambda}$$<\/li>\n<\/ul>\n<p>\u8af8\u91cf\u306e\u9593\u306e\u95a2\u4fc2\u3092\u307e\u3068\u3081\u308b\u3068\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\lambda\\,\\nu &amp;=&amp; c\\\\<br \/>\n\\omega &amp;=&amp; 2\\pi\\nu = \\frac{2\\pi}{T} \\\\<br \/>\nk &amp;=&amp; \\frac{2\\pi}{\\lambda} = \\frac{\\omega}{c}<br \/>\n\\end{eqnarray}<\/p>\n<h3>\u5149\u306e\u4f1d\u64ad\u3092\u8868\u30594\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb<\/h3>\n<p id=\"yui_3_17_2_1_1641359028187_1243\">\u5149\u304c\u4f1d\u64ad\u3059\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4e16\u754c\u7dda<\/strong><\/span>\u3092 \\(x^{\\mu}(v)\\)\uff0c\u305d\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u63a5\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u306e\u6210\u5206\u3092 \\(\\displaystyle k^{\\mu}\\equiv \\frac{dx^{\\mu}}{dv}\\) \u3068\u3059\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>\u3002<\/p>\n<p>\u5149\u306f\u5149\u901f \\(c\\)\u00a0 \u3067\u4f1d\u64ad\u3059\u308b\u3053\u3068\u304b\u3089\uff0c\\(k^{\\mu}\\) \u3092\u6210\u5206\u3068\u3059\u308b\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{k}\\) \u306f\u30cc\u30eb\u30d9\u30af\u30c8\u30eb<br \/>\n$$\\boldsymbol{k} \\cdot \\boldsymbol{k} = \\eta_{\\mu\\nu} k^{\\mu} k^{\\nu} = k_{\\mu} k^{\\mu} = 0 $$<br \/>\n\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u4ee5\u4e0b\u306f\u305d\u306e\u8aac\u660e\u3002<\/p>\n<h4 id=\"yui_3_17_2_1_1641359028187_1264\">\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb\u304c\u30cc\u30eb\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068<\/h4>\n<p>\u6642\u523b \\(t\\) \u306b\u70b9 \\(P (x^i)\\) \u304b\u3089\u51fa\u305f\u5149\u304c\uff0c\u6642\u523b \\(t+dt\\) \u306b\u70b9 \\(Q (x^i + dx^i) \\) \u306b\u5230\u9054\u3059\u308b\u3002\u5149\u901f\u306f \\(c\\)\u00a0 \u3067\u3042\u308b\u304b\u3089\uff0c<br \/>\n$$\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2 + \\left(\\frac{dz}{dt}\\right)^2 =<br \/>\n\\delta_{ij} \\frac{dx^i}{dt} \\frac{dx^j}{dt} = c^2 $$<br \/>\n$$\\therefore -c^2 \\,dt^2 + \\delta_{ij} dx^i dx^j = 0$$<br \/>\n$$\\eta_{\\mu\\nu} dx^{\\mu} dx^{\\nu} = 0$$<br \/>\n$$\\eta_{\\mu\\nu} \\frac{dx^{\\mu}}{dv} \\frac{dx^{\\nu}}{dv} = 0$$<br \/>\n$$\\therefore\\ \\ \u00a0 \\boldsymbol{k}\\cdot\\boldsymbol{k} = \\eta_{\\mu\\nu} k^{\\mu} k^{\\nu} = k_{\\mu} k^{\\mu} = 0$$<br \/>\n\u3053\u3053\u3067\uff0c\\( k_{\\mu} = \\eta_{\\mu\\nu} k^{\\nu}\\) \u3068\u3057\u305f\u3002<\/p>\n<p>\\(\\boldsymbol{k}\\) \u306e\u3088\u3046\u306b\uff0c<\/p>\n<ul>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u304c\u30bc\u30ed\u3068\u306a\u308b\u3088\u3046\u306a\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3092<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30cc\u30eb\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3068\u3044\u3046\u3002<\/li>\n<\/ul>\n<p>\u3061\u306a\u307f\u306b\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>MTW \u6d41\u306e sign convention<\/strong><\/span> \\((-, +, +, +) \\) \u3067\u306f<\/p>\n<ul>\n<li><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u304c\u8ca0\u306e\u5024\u306b\u306a\u308b\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3092<span style=\"color: #ff0000;\"><strong><span style=\"font-family: helvetica, arial, sans-serif;\">\u6642\u9593\u7684\u30d9\u30af\u30c8\u30eb<\/span><\/strong><\/span>\uff0c<\/li>\n<li><strong><span style=\"font-family: helvetica, arial, sans-serif;\">\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u304c\u6b63\u306e\u5024\u306b\u306a\u308b\u30d9\u30af\u30c8\u30eb<\/span><\/strong>\u3092<span style=\"color: #ff0000;\"><strong><span style=\"font-family: helvetica, arial, sans-serif;\">\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb<\/span><\/strong><\/span>\u3068\u3044\u3046\u3002<\/li>\n<\/ul>\n<h3>\u5149\u306e4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206<\/h3>\n<p>\u7279\u6b8a\u76f8\u5bfe\u8ad6\u7684\u72b6\u6cc1\u3067\u306f\uff0c\u9759\u6b62\u3057\u3066\u3044\u308b\u89b3\u6e2c\u8005\u306b\u3068\u3063\u3066\u306f\uff0c \\(k^0\\) \u304c\u5149\u306e\u89d2\u632f\u52d5\u6570 \\(\\omega\\) \u3092\u5149\u901f \\(c\\) \u3067\u308f\u3063\u305f\u3082\u306e\uff0c \\(k^i\\) \u304c\u5149\u306e\u6ce2\u6570\u30d9\u30af\u30c8\u30eb \\(\\vec{k}\\) \u3092\u8868\u3059\u3002<\/p>\n<p>\u3055\u3059\u304c\u306b\u9762\u5012\u306b\u306a\u3063\u3066\u304d\u305f\u306e\u3067\uff0c\u4ee5\u5f8c\u306f \\(c = 1\\) \u3068\u3059\u308b\u3002<\/p>\n<p>$$ k^{\\mu} = \\left(k^0,k^i \\right) = \\left(\\frac{\\omega}{c}, \\ \\vec{k} \\right) \\Rightarrow \\left(\\omega, \\ \\vec{k} \\right)\\quad (\\because c = 1)$$<\/p>\n<p>\u89d2\u632f\u52d5\u6570 \\(\\omega\\) \u306e\u5358\u4f4d\u306f \\(\\mbox{rad}\/\\mbox{s}\\)\uff08\u30e9\u30b8\u30a2\u30f3\u6bce\u79d2\uff09\u3002<br \/>\n\u6ce2\u6570\u30d9\u30af\u30c8\u30eb \\(\\vec{k}\\) \u306e\u5927\u304d\u3055\u3067\u3042\u308b\u6ce2\u6570 \\(k \\equiv \\sqrt{\\vec{k}\\cdot\\vec{k}}\\) \u306e\u5358\u4f4d\u306f rad\/m\uff08\u30e9\u30b8\u30a2\u30f3\u6bce\u30e1\u30fc\u30c8\u30eb\uff09\u3002<br \/>\n\u632f\u52d5\u6570 \\(\\nu\\)\uff0c\u5468\u671f \\(T\\)\uff0c\u6ce2\u9577 \\(\\lambda\\) \u3068\u306e\u95a2\u4fc2\u306f\uff0c<br \/>\n\\begin{eqnarray}<br \/>\n\\omega &amp;=&amp; 2\\pi\\nu = \\frac{2\\pi}{T}\\\\ \\ \\\\<br \/>\nk &amp;=&amp; \\frac{2\\pi}{\\lambda} = \\frac{\\omega}{c} \\Rightarrow \\omega \\quad (\\because c = 1)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c\u6ce2\u6570\u30d9\u30af\u30c8\u30eb \\(\\vec{k}\\) \u3092\u305d\u306e\u5927\u304d\u3055 \\(k\\) \u3068\u5411\u304d\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\vec{\\gamma}, \\ \\ \\vec{\\gamma}\\cdot\\vec{\\gamma}=1\\) \u3067\u8868\u3059\u3068<br \/>\n$$ k^{\\mu} = (k^0, \\vec{k}) = (\\omega, \\omega \\vec{\\gamma}) = \\omega (u^{\\mu} + \\gamma^{\\mu})$$<\/p>\n<p>\u3053\u3053\u3067\uff0c<br \/>\n$$u^{\\mu} \\equiv (1, 0, 0, 0)$$<br \/>\n\u306f\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6\uff0c<br \/>\n$$\\gamma^{\\mu} = (0, \\ \\vec{\\gamma})$$<br \/>\n\u306f \\(u^{\\mu}\\) \u306b\u76f4\u4ea4\u3059\u308b\uff08\u3064\u307e\u308a \\(\\eta_{\\mu\\nu} u^{\\mu} k^{\\nu} = 0\\)\uff09\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\uff08\u3064\u307e\u308a \\( \\eta_{\\mu\\nu} \\gamma^{\\mu} \\gamma^{\\nu} = 1 &gt; 0\\)\uff09\u3067\u3042\u308a\uff0c\u5149\u306e\u4f1d\u64ad\u3059\u308b\u5411\u304d\u3092\u3042\u3089\u308f\u3059\u3002<\/p>\n<h3>\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6\u306b\u57fa\u3065\u3044\u305f \\( \\boldsymbol{k}\\) \u306e \\(3+1\\) \u5206\u89e3<\/h3>\n<p>\u4e0a\u8a18\u306e\u3088\u3046\u306b\u3057\u3066\u5c0e\u3044\u305f4\u5143\u30d9\u30af\u30c8\u30eb\u3067\u8868\u3057\u305f\u95a2\u4fc2\u306f\uff0c\u89b3\u6e2c\u8005\u306e\u9759\u6b62\u5ea7\u6a19\u7cfb\u3060\u3051\u3067\u306a\u304f\uff0c\u4e00\u822c\u7684\u306b\u6210\u308a\u7acb\u3064\u3002<\/p>\n<p>\u3064\u307e\u308a\uff0c\u5149\u306e\u4f1d\u64ad\u3092\u8868\u30594\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{k} \\) \u306f\uff0c\u6642\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u3068\uff0c\\(\\boldsymbol{u}\\) \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{\\gamma}\\) \u3092\u4f7f\u3063\u3066\uff0c\u4e00\u822c\u7684\u306b\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u304b\u3051\u308b\u3002<br \/>\n$$ \\boldsymbol{k}\u00a0 = \\omega (\\boldsymbol{u} + \\boldsymbol{\\gamma})$$<br \/>\n$$\\boldsymbol{u} \\cdot\\boldsymbol{u} =-1, \\ \\boldsymbol{u} \\cdot\\boldsymbol{\\gamma} = 0, \\ \\boldsymbol{\\gamma} \\cdot\\boldsymbol{\\gamma} =1$$<br \/>\n\u3053\u3053\u3067 \\(\\omega\\) \u306f4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u89b3\u6e2c\u8005\u304c\u6e2c\u5b9a\u3059\u308b\u5149\u306e\u89d2\u632f\u52d5\u6570\u3067\u3042\u308a\uff0c<br \/>\n$$\\omega = &#8211; \\boldsymbol{k} \\cdot \\boldsymbol{u}$$<br \/>\n\u306a\u304a\uff0c\u4ee5\u4e0b\u3067\u306f\u7c21\u5358\u306e\u305f\u3081\u306b \\(\\omega\\) \u3092\u5358\u306b\uff08\u300c\u89d2\u300d\u632f\u52d5\u6570\u306e\u300c\u89d2\u300d\u3092\u7701\u7565\u3057\u3066\uff09\u300c\u632f\u52d5\u6570\u300d\u3068\u547c\u3076\u3002<\/p>\n<h3>\\( k^{\\mu} = \\omega (u^{\\mu} + \\gamma^{\\mu})\\) \u306e\u5225\u306e\u5c0e\u51fa\u6cd5<\/h3>\n<p>\\( k^{\\mu} = \\omega (u^{\\mu} + \\gamma^{\\mu})\\) \u3068\u3044\u3046\u95a2\u4fc2\u306f\uff0c\u4f55\u3082 \\(u^{\\mu} = (1, 0, 0, 0)\\) \u3068\u306a\u308b\u3088\u3046\u306a\u89b3\u6e2c\u8005\u306e\u9759\u6b62\u5ea7\u6a19\u7cfb\u3092\u6301\u3061\u51fa\u3055\u306a\u304f\u3066\u3082\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u4e00\u822c\u7684\u306b\u6c42\u3081\u3089\u308c\u308b\u3002<\/p>\n<p>\u5149\u306e\u4f1d\u64ad\u3092\u8868\u30594\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb \\(k^{\\mu}\\) \u3092\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 \\(u^{\\mu}\\) \u306b\u5e73\u884c\u306a\u6210\u5206\u3068\u5782\u76f4\u306a\u6210\u5206\u306b\u5206\u89e3\u3059\u308b\u3002<br \/>\n\\begin{eqnarray} k^{\\mu} &amp;=&amp; \\delta^{\\mu}_{\\ \\ \\nu} k^{\\nu} \\\\<br \/>\n&amp;=&amp; \\left\\{ -u^{\\mu} u_{\\nu} + \\left( \\delta^{\\mu}_{\\ \\ \\nu} + u^{\\mu} u_{\\nu}\\right) \\right\\}k^{\\nu} \\\\<br \/>\n&amp;=&amp; (-k_{\\nu} u^{\\nu}) u^{\\mu} + P^{\\mu}_{\\ \\ \\, \\nu} \\,k^{\\nu} \\\\<br \/>\n&amp;=&amp; \\omega \\, u^{\\mu} + k^{\\mu}_{\\perp}<br \/>\n\\end{eqnarray}<br \/>\n\u3053\u3053\u3067\uff0c\\(\\omega \\equiv -k_{\\nu} u^{\\nu} \\) \u306f\u89b3\u6e2c\u8005 \\(u^{\\mu}\\) \u304c\u6e2c\u5b9a\u3059\u308b\u632f\u52d5\u6570\uff0c \\(k^{\\mu}_{\\perp}\\) \u306f \\(u^{\\mu}\\) \u306b\u5782\u76f4\u306a\u7a7a\u9593\u7684\u6210\u5206\u3067\u3042\u308a\uff0c\u305d\u306e\u5927\u304d\u3055\u306f\uff0c<br \/>\n\\begin{eqnarray}<br \/>\n\\sqrt{g_{\\mu\\nu} k^{\\mu}_{\\perp} k^{\\nu}_{\\perp}} &amp;=&amp; \\sqrt{P_{\\mu\\nu} k^{\\mu} k^{\\nu}} \\\\<br \/>\n&amp;=&amp; \\sqrt{(g_{\\mu\\nu} + u_{\\mu} u_{\\nu}) k^{\\mu} k^{\\nu}}\\\\<br \/>\n&amp;=&amp; \\sqrt{0 + (u_{\\mu} k^{\\mu})^2} = \\omega<br \/>\n\\end{eqnarray}<br \/>\n\u3057\u305f\u304c\u3063\u3066\uff0c\\(k^{\\mu}_{\\perp}\\) \u306f\u305d\u306e\u5927\u304d\u3055 \\(\\omega\\) \u3068\uff0c\u5411\u304d\u3092\u8868\u3059\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\gamma^{\\mu}\\) \u3092\u4f7f\u3063\u3066\uff0c<br \/>\n$$ k^{\\mu}_{\\perp} = \\omega\\, \\gamma^{\\mu}, \\quad \\gamma_{\\mu} u^{\\mu} = 0, \\ \\ \\gamma_{\\mu} \\gamma^{\\mu} = 1$$<br \/>\n\u3068\u66f8\u3051\u308b\u3002\u6700\u7d42\u7684\u306b\uff0c<br \/>\n$$ k^{\\mu} = \\omega\\, u^{\\mu} + k^{\\mu}_{\\perp} = \\omega (u^{\\mu} + \\gamma^{\\mu})$$<br \/>\n4\u5143\u30d9\u30af\u30c8\u30eb\u8868\u8a18\u3067\u306f\uff0c<br \/>\n$$ \\boldsymbol{k}\u00a0 = \\omega (\\boldsymbol{u} + \\boldsymbol{\\gamma})$$<\/p>\n<h3>\u8ffd\u8a18<\/h3>\n<p>\u3053\u306e\u30b5\u30a4\u30c8\u3067\u306f\u5f53\u521d\uff0c$\\boldsymbol{k} = k^{\\mu} \\,\\boldsymbol{e}_{\\mu}$ \u3092\u300c\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb\u300d\u3068\u547c\u3093\u3067\u3044\u305f\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u305d\u306e\u7a7a\u9593\u6210\u5206\u304c3\u6b21\u5143\u306e\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306b\u76f8\u5f53\u3059\u308b\u3053\u3068\u304b\u3089\uff0c\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5149\u306e4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u300d\u3068\u547c\u3093\u3060\u65b9\u304c\uff0c\u3088\u308a\u306f\u3063\u304d\u308a\u3068\u308f\u304b\u308b\u304b\u3068\u3002<\/p>\n<p>\u306a\u304a\uff0c\u4e00\u90e8\u306e\u6559\u79d1\u66f8\u3067\u306f\uff0c\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u3092\u8aac\u660e\u3059\u308b\u969b\u306b\uff0c\u300c\u5149\u5b50\u300d\u306e4\u5143\u904b\u52d5\u91cf $\\boldsymbol{P}$ \u306b\u5bfe\u3059\u308b\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u3046\u5834\u5408\u304c\u3042\u308b\u3002\u91cf\u5b50\u8ad6\u7684\u77e5\u898b\u306b\u3088\u308a\uff0c\u300c\u5149\u5b50\u300d\u306e4\u5143\u904b\u52d5\u91cf $\\boldsymbol{P}$ \u3068\u300c\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb\u300d$\\boldsymbol{k} $ \u306e\u9593\u306b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6bd4\u4f8b\u95a2\u4fc2\u304c\u3042\u308b\u3002<\/p>\n<p>$$\\boldsymbol{P} = \\hbar\\, \\boldsymbol{k} $$<\/p>\n<p>\u3053\u3053\u3067 $\\displaystyle \\hbar\\equiv \\frac{h}{2\\pi}$ \u306f\u300c\u30a8\u30a4\u30c1\u30fb\u30d0\u30fc\u300d\u3068\u8aad\u307f\uff0c\u300c\u63db\u7b97\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\u300d\u306a\u3069\u3068\u547c\u3070\u308c\u308b\u3002$h$ \u304c\u300c\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\u300d\u305d\u306e\u3082\u306e\u3002<\/p>\n<h3>\u5149\u5b50\u3068 $h$ \u306b\u307e\u3064\u308f\u308b\u4f59\u8ac7<\/h3>\n<p>\u300c\u5149\u5b50\u300d\u3067\u601d\u3044\u51fa\u3059\u306e\u306f20\u4e16\u7d00\u306e\u6614\u306e\u9ad8\u6821\u6642\u4ee3\u3002\u9ad83\u306e\u540c\u3058\u30af\u30e9\u30b9\u306b\u306f\uff0c\u7269\u7406\u9078\u629e\u30af\u30e9\u30b9\u3060\u3051\u3042\u3063\u3066\u300c\u5149\u5b50\u300d\u3055\u3093\u3082\u300c\u967d\u5b50\u300d\u3055\u3093\u3082\u3044\u305f\u3002\uff08\u3055\u3059\u304c\u306b\u96fb\u5b50\u3055\u3093\u3068\u304b\u4e2d\u6027\u5b50\u3055\u3093\u3068\u306f\u3044\u306a\u304b\u3063\u305f\u304c\u3002\uff09<\/p>\n<p>\u300c\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\u300d\u3067\u601d\u3044\u51fa\u3059\u306e\u306f\uff0cD\u8ad6\u5be9\u67fb\u6642\u306e\u767a\u8868\u4f1a\u3002\u6307\u5c0e\u6559\u54e1\u3092\u542b\u3080\u5168\u3066\u306e\u7269\u7406\u5b66\u5c02\u653b\u6559\u6388\u306e\u524d\u3067\u767a\u8868\u3057\u305f\u3068\u304d\u306e\u8cea\u554f\uff1a\u300c\u541b\uff0c\u305d\u306e $h$ \u306f\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\u304b\u306d\uff1f\u300d\u305d\u306e\u5fc3\u306f&#8230;<\/p>\n<p>\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u3092 $H_0 = 100\\, h \\,\\mbox{km\/s\/Mpc}$ \u3068\u66f8\u3044\u305f\u3068\u304d\u306e $h$ \uff08\u898f\u683c\u5316\u3057\u305f\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\uff09\u3067\u3057\u305f\uff0c\u3068\u3044\u3046\u30aa\u30c1\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5149\u306e\u8af8\u91cf\u3092\u3042\u3089\u308f\u30594\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306b\u3064\u3044\u3066\u3002<\/p><p><a class=\"more-link btn\" 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\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":71,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-137","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\/137","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=137"}],"version-history":[{"count":29,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/137\/revisions"}],"predecessor-version":[{"id":9449,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/137\/revisions\/9449"}],"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=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}