{"id":543,"date":"2022-01-08T16:13:51","date_gmt":"2022-01-08T07:13:51","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=543"},"modified":"2022-01-15T11:37:44","modified_gmt":"2022-01-15T02:37:44","slug":"%e9%87%8d%e5%8a%9b%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb","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%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb\/","title":{"rendered":"\u91cd\u529b\u8d64\u65b9\u504f\u79fb"},"content":{"rendered":"

<\/p>\n

\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6<\/h3>\n

\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6<\/strong><\/span> \\(\\boldsymbol{u}\\) \u306e\u6210\u5206 \\(u^{\\mu}\\) \u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7dda\u7d20<\/strong><\/span>\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u56fa\u6709\u6642\u9593<\/strong><\/span>\u00a0 \\(\\tau\\)
\n$$ds^2 = –\u00a0 d\\tau^2 = g_{\\mu\\nu} dx^{\\mu} dx^{\\nu}\\quad (c=1)$$
\n\u3092\u4f7f\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002
\n$$u^{\\mu} \\equiv \\frac{dx^{\\mu}}{d\\tau}$$<\/p>\n

\\(\\boldsymbol{u}\\) \u306f\u300c\u5927\u304d\u3055\u300d\u306e2\u4e57\uff08\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\uff09\u304c \\(-1\\) \u306b\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308b\u3002
\n\\begin{eqnarray}
\n\\boldsymbol{u}\\cdot\\boldsymbol{u} &=& g_{\\mu\\nu} u^{\\mu} u^{\\nu} \\\\
\n&=& g_{\\mu\\nu} \\frac{dx^{\\mu}}{d\\tau}\\frac{dx^{\\nu}}{d\\tau}\\\\
\n&=& \\frac{g_{\\mu\\nu} dx^{\\mu} dx^{\\nu}}{d\\tau^2} \\\\
\n&=& \\frac{-d\\tau^2}{d\\tau^2} \\\\
\n&=& -1
\n\\end{eqnarray}
\n\u3053\u306e\u5f0f\u3092\uff0c\u3068\u304d\u3069\u304d4\u5143\u901f\u5ea6\u306e\u898f\u683c\u5316\u6761\u4ef6<\/strong><\/span>\u3068\u8a00\u3063\u305f\u308a\u3059\u308b\u3002<\/p>\n

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

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/strong><\/span>\u4e2d\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u89b3\u6e2c\u8005<\/strong><\/span>\u306e4\u5143\u901f\u5ea6<\/strong><\/span> \\(u^{\\mu}\\) \u306f\uff08\u300c\u9759\u6b62\u300d\u3068\u306f\u7a7a\u9593\u5ea7\u6a19\u304c\u4e00\u5b9a\u3068\u3044\u3046\u3053\u3068\u3060\u304b\u3089\uff09
\n$$ u^{\\mu} = \\frac{dx^{\\mu}}{d\\tau} = (u^0, 0, 0, 0)$$
\n\u307e\u305f\uff0c\u898f\u683c\u5316\u6761\u4ef6\u304b\u3089
\n$$\\boldsymbol{u}\\cdot\\boldsymbol{u}= g_{00} \\left(u^0\\right)^2 = -1, \\quad\\therefore \\ u^0 = \\frac{1}{\\sqrt{-g_{00}}} = \\frac{1}{\\sqrt{1 – \\frac{r_g}{r}}}$$<\/p>\n

\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570<\/h3>\n

4\u5143\u901f\u5ea6<\/strong><\/span> \\(\\boldsymbol{u}\\) \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570<\/strong><\/span> \\(\\omega\\) \u306f\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u3082\uff0c\u307e\u305f\u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u3082\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002<\/p>\n

$$\\omega \\equiv – \\boldsymbol{k}\\cdot\\boldsymbol{u} = – k_{\\mu} u^{\\mu}$$
\n4\u5143\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u632f\u52d5\u6570\u306f\uff0c\u5f53\u7136\u306a\u304c\u3089\u5ea7\u6a19\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u30b9\u30ab\u30e9\u30fc\u91cf\u3067\u3042\u308b\u3002<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/strong><\/span>\u4e2d\u306e\u9759\u6b62\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u306f\uff0c<\/p>\n

$$\\omega = -k_0 u^0 = \\frac{\\omega_c}{\\sqrt{1 – \\frac{r_g}{r}}}$$
\n\u3068\u306a\u308a\uff0c\u540c\u3058\u5149\u6e90\u304b\u3089\u653e\u305f\u308c\u305f\u540c\u3058\u5149\uff0c\u3064\u307e\u308a\u540c\u3058 \\(\\boldsymbol{k}\\) \u3092\u89b3\u6e2c\u3057\u3066\u3044\u3066\u3082\uff0c\u632f\u52d5\u6570 \\(\\omega\\) \u306f\u89b3\u6e2c\u8005\u306e\u4f4d\u7f6e\u3092\u8868\u3059\u52d5\u5f84\u5ea7\u6a19 \\(r\\) \u306b\u4f9d\u5b58\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u5149\u306e\u91cd\u529b\u8d64\u65b9\u504f\u79fb<\/h3>\n

\u7279\u306b \\(r\u00a0 = r_1\\) \u306e\u89b3\u6e2c\u8005\u304c\u632f\u52d5\u6570\u3092 \\(\\omega_1\\) \u3068\u6e2c\u5b9a\u3057\u305f\u5149\u3092 \\(r = r_2 > r_1\\) \u306b\u3044\u308b\u5225\u306e\u89b3\u6e2c\u8005\u304c\u6e2c\u5b9a\u3059\u308b\u3068\u305d\u306e\u632f\u52d5\u6570 \\(\\omega_2\\) \u306f
\n$$\\frac{\\omega_2}{\\omega_1} = \\frac{\\sqrt{1 – \\frac{r_g}{r_1}}}{\\sqrt{1 – \\frac{r_g}{r_2}}} < 1,\\quad\\therefore\\ \\ \\omega_2 < \\omega_1$$
\n\u3068\u306a\u308b\u3002<\/p>\n

\u3053\u306e\u3088\u3046\u306b\uff0c\u91cd\u529b\u6e90\u306b\u8fd1\u3044\uff08\u300c\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u304c\u6df1\u3044\u300d\uff09\u3068\u3053\u308d\u304b\u3089\u653e\u51fa\u3055\u308c\u305f\u5149\u3092\u91cd\u529b\u6e90\u304b\u3089\u9060\u304f\u96e2\u308c\u305f\uff08\u300c\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u304c\u6d45\u3044\u300d\uff09\u5834\u6240\u3067\u89b3\u6e2c\u3059\u308b\u3068\uff0c\u632f\u52d5\u6570\u304c\u5c0f\u3055\u3044\u307b\u3046\u3078\u5909\u5316\u3057\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3002<\/p>\n

\u632f\u52d5\u6570 \\(\\omega\\) \u304c\u5c0f\u3055\u304f\u306a\u308c\u3070\u6ce2\u9577 \\(\\lambda\\) \u304c\u5927\u304d\u304f\u306a\u308b\uff08\u4f38\u3073\u308b\uff09\u3002<\/p>\n

$$\\lambda\u00a0 = \\frac{2\\pi c}{\\omega} \\propto \\frac{1}{\\omega} $$<\/p>\n

\u53ef\u8996\u5149\u3067\u306f\u6ce2\u9577\u306e\u9577\u3044\u5149\u306f\u8d64\u304f\u898b\u3048\u308b\u306e\u3067\uff0c\u305f\u3068\u3048\u53ef\u8996\u5149\u3067\u306a\u304f\u3066\u3082\uff0c\u91cd\u529b\u306b\u3088\u3063\u3066\u4e00\u822c\u306b\u5149\uff08\u96fb\u78c1\u6ce2\uff09\u306e\u6ce2\u9577\u304c\u4f38\u3073\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3053\u3068\u3092\u5149\u306e\u91cd\u529b\u8d64\u65b9\u504f\u79fb<\/strong><\/span>\u3068\u547c\u3093\u3067\u3044\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":83,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-543","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\/543","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\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=543"}],"version-history":[{"count":24,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/543\/revisions"}],"predecessor-version":[{"id":569,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/543\/revisions\/569"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/83"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}