{"id":738,"date":"2022-01-12T14:10:53","date_gmt":"2022-01-12T05:10:53","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=738"},"modified":"2024-04-09T14:05:11","modified_gmt":"2024-04-09T05:05:11","slug":"%e5%85%89%e3%81%ae%e3%80%8c%e6%9b%b2%e3%81%8c%e3%82%8a%e8%a7%92%e3%80%8d","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\/%e5%85%89%e3%81%ae%e3%80%8c%e6%9b%b2%e3%81%8c%e3%82%8a%e8%a7%92%e3%80%8d\/","title":{"rendered":"\u5149\u306e\u300c\u66f2\u304c\u308a\u89d2\u300d"},"content":{"rendered":"<p><!--more--><\/p>\n<p>\u4f55\u5ea6\u3082\u8a00\u3046\u304c\uff0c\u5149\u306e\u7d4c\u8def\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6e2c\u5730\u7dda<\/strong><\/span>\u3067\u3042\u308a\uff0c\u63a5\u30d9\u30af\u30c8\u30eb\u304c\u7d4c\u8def\u306b\u6cbf\u3063\u3066\u4e00\u5b9a\u3067\u3042\u308a\u7d9a\u3051\u308b\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u307e\u3063\u3059\u3050\u306a<\/strong><\/span>\u300d\u7dda\u3067\u3042\u308b\u3002<br \/>\n$$\\frac{d\\boldsymbol{k}}{dv} = \\boldsymbol{0}$$<\/p>\n<p>\u3067\u306f\u306a\u305c\uff0c\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u91cd\u529b\u306b\u3088\u3063\u3066\u5149\u3055\u3048\u3082\u66f2\u304c\u308b<\/strong><\/span>\u300d\u306a\u3069\u3068\u8a00\u3046\u306e\u304b\uff0c\u306b\u3064\u3044\u3066\u88dc\u8db3\u3059\u308b\u3002<\/p>\n<h3>\\(r_g \\) \u306e\u30bc\u30ed\u6b21\u89e3\u304c\u76f4\u7dda\u3067\u3042\u308b\u3053\u3068<\/h3>\n<p>\\(r_g\\) \u306e\u30bc\u30ed\u6b21\u89e3\u306f<br \/>\n$$u =\u00a0 \\frac{1}{r} = \\frac{\\sin\\phi}{b}$$<br \/>\n\\(r \\rightarrow \\infty\\) \u3067\u306e\u6f38\u8fd1\u7684\u306a\u632f\u308b\u821e\u3044\u3092\u307f\u308b\u3068\uff0c\\(\\displaystyle \\frac{1}{r} \\rightarrow 0 \\) \u3068\u306a\u308b\u3053\u3068\u304b\u3089 \\(\\sin\\phi = 0\\) \u3088\u308a \\(\\phi = 0, \\ \\pi\\)\u3002<\/p>\n<p>\u3064\u307e\u308a\u5149\u306f \\(\\phi = 0\\) \u306e\u7121\u9650\u9060\u304b\u3089\u3084\u3063\u3066\u304d\u3066\uff0c\\(\\phi = \\pi\\) \u306e\u7121\u9650\u9060\u3078\u53bb\u3063\u3066\u3044\u304f\u3002\u89d2\u5ea6\u5dee\u306f \\(\\pi &#8211; 0 = \\pi\\)\uff0c\u3064\u307e\u308a\u6f38\u8fd1\u7684\u7121\u9650\u9060\u3067\u306e\u632f\u308b\u821e\u3044\u304b\u3089\u7d4c\u8def\u306f\u76f4\u7dda\u3067\u3042\u308b\uff0c\u3068\u7d50\u8ad6\u3065\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<p>\u3042\u308b\u3044\u306f\uff0c\u6975\u5ea7\u6a19\u304b\u3089\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3078\u5909\u63db\u3059\u308b\u3068\uff0c<br \/>\n$$ y = r \\sin\\theta \\sin\\phi = \\frac{b}{\\sin\\phi} \\sin\\frac{\\pi}{2} \\sin\\phi = b = \\mbox{const.}$$<br \/>\n\u3064\u307e\u308a\uff0c\\( y = b \\) \u3068\u3044\u3046 \\(x\\) \u8ef8\u306b\u5e73\u884c\u306a\u76f4\u7dda\u3092\u8868\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>\\(b\\) \u306f\u3053\u306e\u76f4\u7dda\u3068\u5ea7\u6a19\u539f\u70b9\uff08\u306b\u3053\u308c\u304b\u3089\u7f6e\u304b\u308c\u308b\u3067\u3042\u308d\u3046\u5929\u4f53\uff09\u3068\u306e\u6700\u77ed\u8ddd\u96e2\u3092\u8868\u3057\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u885d\u7a81\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u3002\u539f\u70b9\u306e\u3069\u308c\u3060\u3051\u8fd1\u304f\u3092\u304b\u3059\u3081\u3066\u3044\u304f\u306e\u304b\u3068\u3044\u3046\uff0c\u3059\u308c\u3059\u308c\u5ea6\u5408\u3044\u3092\u3042\u3089\u308f\u3059\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-742\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f1-640x449.png\" alt=\"\" width=\"480\" height=\"337\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f1-640x449.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f1-300x211.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f1-1536x1078.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f1-2048x1437.png 2048w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f1-750x526.png 750w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<h3>\\(r_g\\) \u306e1\u6b21\u306e\u52b9\u679c\u3092\u53d6\u308a\u5165\u308c\u305f\u89e3\u306e\u6f38\u8fd1\u7684\u632f\u308b\u821e\u3044<\/h3>\n<p>\\(r_g\\) \u306e1\u6b21\u306e\u52b9\u679c\u3092\u53d6\u308a\u5165\u308c\u305f\u89e3<br \/>\n$$ \\frac{1}{r} = \\frac{\\sin\\phi}{b} + \\frac{r_g}{2 b^2} \\left(2 -\\sin\\phi -\\sin^2\\phi\\right)$$ \u306b\u3064\u3044\u3066\uff0c\\( r \\rightarrow \\infty\\) \u3068\u306a\u308b\u306e\u306f<br \/>\n$$ \\sin\\phi + \\frac{r_g}{2 b} \\left(2 -\\sin\\phi -\\sin^2\\phi\\right) = 0 $$ \u306e\u3068\u304d\u3002<\/p>\n<p>$r_g$ \u306e\u30bc\u30ed\u6b21\u89e3\u306f $\\sin\\phi = 0$ \u3067\u3042\u308b\u304b\u3089\uff0c$\\sin\\phi$ \u306f $O(r_g)$ \u3067\u3042\u308a\uff0c$r_g$ \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067<\/p>\n<p>$$ \\sin\\phi \\simeq &#8211; \\frac{r_g}{b} $$<\/p>\n<p>\u306e\u3088\u3046\u306b\u89e3\u3051\u308b\u3002<\/p>\n<p>\\( |\\phi| \\ll 1 \\) \u306b\u3064\u3044\u3066\u306f \\(\\sin\\phi \\simeq \\phi \\) \u3067\u3042\u308b\u304b\u3089\uff0c\\(\\displaystyle \\frac{r_g}{b} \\ll 1 \\) \u3067\u3042\u308c\u3070\uff0c\u3064\u307e\u308a\u5149\u306e\u6f38\u8fd1\u7684\u7121\u9650\u9060\u3067\u306e\u632f\u308b\u821e\u3044\u3092\u307f\u308b\u3068 \\(\\displaystyle \\phi = -\\frac{r_g}{b} \\) \u3060\u3051\u50be\u3044\u305f\u6f38\u8fd1\u7dda\u306b\u4e26\u884c\u306a\u7121\u9650\u9060\u304b\u3089\u3084\u3063\u3066\u304d\u3066\uff0c\\(\\displaystyle \\phi = \\pi + \\frac{r_g}{b} \\) \u3060\u3051\u50be\u3044\u305f\u6f38\u8fd1\u7dda\u306b\u4e26\u884c\u306a\u7121\u9650\u9060\u306b\u53bb\u3063\u3066\u3044\u304f\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-746\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f2-640x449.png\" alt=\"\" width=\"480\" height=\"337\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f2-640x449.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f2-300x211.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f2-1536x1078.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f2-2048x1437.png 2048w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f2-750x526.png 750w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<h3>\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3092\u4eee\u5b9a\u3057\u305f\u5149\u306e\u300c\u66f2\u304c\u308a\u89d2\u300d\u306e\u5b9a\u7fa9<\/h3>\n<p>\u7121\u9650\u9060\u3067\u306e\u6f38\u8fd1\u7dda\u306e\u89d2\u5ea6\u5dee\u306f<br \/>\n$$ \\pi + \\frac{r_g}{b} &#8211; \\left( -\\frac{r_g}{b}\\right) = \\pi + 2\\frac{r_g}{b}$$<br \/>\n\u3064\u307e\u308a\uff0c\u76f4\u7dda\u304b\u3089 \\(\\displaystyle \\alpha \\equiv \\frac{2 r_g}{b}\\) \u3060\u3051\u66f2\u304c\u3063\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u3053\u306e \\(\\alpha\\) \u3092\u5149\u306e\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u66f2\u304c\u308a\u89d2<\/strong><\/span>\u300d\u3068\u3044\u3046\u3002<\/p>\n<p>\\(r \\rightarrow \\infty \\) \u3067\u306f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306f\u307b\u307c\u5e73\u5766<br \/>\n$$ds^2 \\simeq -c^2 dt^2 + dr^2 + r^2 (d\\theta^2 + \\sin^2\\theta\\,d\\phi^2)$$\u3068\u306a\u308b\u306e\u3067\uff0c\u3053\u306e\u3088\u3046\u306a<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6f38\u8fd1\u7684\u7121\u9650\u9060<\/strong><\/span>\u306b\u304a\u3051\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6f38\u8fd1\u7684\u5e73\u5766\u6642\u7a7a<\/strong><\/span>\u3067\u306f\u89d2\u5ea6\u5ea7\u6a19 \\(\\phi\\) \u304c\u5b9f\u969b\u306e\u89d2\u5ea6\u3092\u3042\u3089\u308f\u3057\u3066\u3044\u308b\u3068\u3057\u3066\u554f\u984c\u306a\u3044\u3002<\/p>\n<p>\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u4e16\u306b\u6570\u591a\u3042\u308b\u6559\u79d1\u66f8\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u8aac\u660e\u3067\u306f\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3092\u4eee\u5b9a\u3057\uff0c\u5149\u306e\u7d4c\u8def\u306e\u6f38\u8fd1\u7684\u7121\u9650\u9060\u3067\u306e\u6f38\u8fd1\u7dda\u306e\u50be\u304d\u3092\u3082\u3063\u3066\u5149\u306e\u7d4c\u8def\u306e\u89d2\u5ea6\u3068\u3057\uff0c\u6f38\u8fd1\u7dda\u306e\u50be\u304d\u306e\u5dee\u3092\u3082\u3063\u3066\u5149\u306e\u300c\u66f2\u304c\u308a\u89d2\u300d\u3068\u5b9a\u7fa9\u3057\u3066\u3044\u308b\u3002<\/strong><\/span><\/p>\n<p>&nbsp;<\/p>\n<h3>\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3092\u4eee\u5b9a\u3057\u306a\u3044\u4e0d\u5909\u7684\u306a\u89d2\u5ea6\u306e\u5b9a\u7fa9<\/h3>\n<p>\u4e00\u822c\u306e\u6642\u7a7a\u5185\u306e\u4efb\u610f\u306e\u70b9\u306b\u304a\u3051\u308b3\u6b21\u5143\u7684\u306a\u89d2\u5ea6\u306e\u4e0d\u5909\u306a\u5b9a\u7fa9\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u308b\u3002\u4ee5\u4e0b\u3067\u306f\uff0c\u5177\u4f53\u7684\u306a\u8a08\u7b97\u306e\u969b\u306b\u306f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u7b49\u3092\u4f7f\u3063\u3066\u3044\u308b\u304c\uff0c\u5b9a\u7fa9\u81ea\u4f53\u306f\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u7684\u306a\u5b9a\u7fa9\u3067\u3042\u308a\uff0c\u4efb\u610f\u306e\u6642\u7a7a\u3067\u4f7f\u3048\u308b\u3002<\/p>\n<h4>\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u89d2\u5ea6<\/h4>\n<p>\u4e00\u822c\u306b\uff0c\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3092 \\(\\boldsymbol{n}, \\ \\boldsymbol{\\gamma}\\) \u3068\u3059\u308b\u3068\uff0c<br \/>\n$$\\boldsymbol{n}\\cdot\\boldsymbol{u} = 0, \\ \\boldsymbol{\\gamma}\\cdot\\boldsymbol{u} = 0, \\ \\boldsymbol{n}\\cdot\\boldsymbol{n} = 1, \\ \\boldsymbol{\\gamma}\\cdot\\boldsymbol{\\gamma} = 1$$<\/p>\n<p>\u3053\u306e2\u3064\u306e\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{\\gamma}, \\ \\boldsymbol{n}\\) \u306e\u306a\u3059\u89d2\u3092 \\(\\varPsi\\) \u306f\uff0c\u4ee5\u4e0b\u306e\u5f0f\u304b\u3089\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<br \/>\n$$\\cos\\varPsi \\equiv \\boldsymbol{\\gamma}\\cdot\\boldsymbol{n}$$<br \/>\n4\u5143\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u305f \\(\\cos\\varPsi\\) \u3057\u305f\u304c\u3063\u3066 \\(\\varPsi\\) \u306f4\u6b21\u5143\u30b9\u30ab\u30e9\u30fc\u3067\u3042\u308a\uff0c\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u91cf\u3067\u3042\u308b\u3002<\/p>\n<h4>\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6<\/h4>\n<p>\u307e\u305a\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u6210\u5206 \\(u^{\\mu}\\) \u306f<br \/>\n$$ u^{\\mu} = \\left(\\frac{1}{\\sqrt{1-\\frac{r_g}{r}}}, 0, 0, 0\\right)$$<br \/>\n\\(\\boldsymbol{u}\\) \u306b\u76f4\u4ea4\u3059\u308b\u540c\u6642\u76843\u6b21\u5143\u7a7a\u9593\u3078\u306e\u5c04\u5f71\u6f14\u7b97\u5b50 \\(P^{\\mu}_{\\ \\ \\ \\nu}\\) \u306f<br \/>\n$$P_{\\mu \\nu} = g_{\\mu\\nu} + u_{\\mu} u_{\\nu}$$<\/p>\n<h4>\u52d5\u5f84\u65b9\u5411\u306e\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/h4>\n<p>\u3055\u3066\uff0c\\(\\boldsymbol{n}\\) \u3092\u3053\u306e3\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u52d5\u5f84\u65b9\u5411\uff08\u4e2d\u5fc3\u5411\u304d\uff09\u306e\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\uff0c\u305d\u306e\u6210\u5206\u3092 \\(n^{\\mu}\\) \u3068\u3059\u308b\u3068\uff0c\u7a7a\u9593\u6210\u5206\u306f\u52d5\u5f84\u6210\u5206\u306e\u307f\u3067\u3042\u308b\u3053\u3068\u304b\u3089 \\(n^2 = n^3 = 0\\)\u3002\u307e\u305f\uff0c\\(\\boldsymbol{u}\\) \u306b\u76f4\u4ea4\u3059\u308b\u3053\u3068 \\( \\boldsymbol{u}\\cdot\\boldsymbol{n} = u_{\\mu} n^{\\mu} = u_0 n^0 = 0 \\) \u3088\u308a \\(n^0 = 0\\)\uff0c\u3055\u3089\u306b\u898f\u683c\u5316\u6761\u4ef6\u6761\u4ef6 \\(\\boldsymbol{n}\\cdot\\boldsymbol{n} = 1\\) \u3088\u308a\u6700\u7d42\u7684\u306b<br \/>\n$$ n^{\\mu} =\\left( 0,\u00a0 &#8211; \\sqrt{1-\\frac{r_g}{r}}, 0, 0\\right)$$<\/p>\n<h4>\u5149\u306e\u9032\u3080\u5411\u304d\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/h4>\n<p>\\(\\boldsymbol{\\gamma}\\) \u3092 \\(\\boldsymbol{k}\\) \u3067\u8868\u3055\u308c\u308b\u5149\u306e\u9032\u3080\u5411\u304d\u3092\u8868\u3059\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3068\u3059\u308b\u3068\uff0c\u305d\u306e\u6210\u5206\u306f<br \/>\n$$ \\gamma_{\\mu} \\equiv\u00a0 \\frac{P_{\\mu\\nu} k^{\\nu}}{\\omega}$$ \u3068\u306a\u308b\u3002\u3053\u3053\u3067 \\(\\omega \\) \u306f4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u3092\u3042\u3089\u308f\u3057\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u3067\u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\omega = &#8211; \\boldsymbol{k}\\cdot\\boldsymbol{u}\u00a0 &amp;=&amp; -k_0\\, u^0 \\\\<br \/>\n&amp;=&amp; \\frac{\\omega_c}{\\sqrt{1-\\frac{r_g}{r}}}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u304b\u3089 \\(\\boldsymbol{\\gamma}\\) \u306e\u6210\u5206 \\(\\gamma_{\\mu} = g_{\\mu\\nu} \\gamma^{\\nu}\\) \u306e\u3046\u3061\uff0c\u5b9f\u969b\u306b\u8a08\u7b97\u306b\u4f7f\u3046 \\(\\gamma_1\\) \u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\gamma_1 &amp;=&amp; \\frac{1}{\\omega} P_{1\\mu} k^{\\mu} \\\\<br \/>\n&amp;=&amp; \\frac{1}{\\omega} g_{11} k^1 \\\\<br \/>\n&amp;=&amp; \\frac{1}{\\omega} g_{11} \\frac{dr}{d\\phi} \\frac{d\\phi}{dv} \\\\<br \/>\n&amp;=&amp; \\frac{\\sqrt{1-\\frac{r_g}{r}}}{\\omega_c} \\frac{1}{1-\\frac{r_g}{r}} \\frac{\\ell}{r^2} \\frac{dr}{d\\phi} \\\\<br \/>\n&amp;=&amp; \\frac{\\ell}{\\omega_c} \\frac{-1}{\\sqrt{1-\\frac{r_g}{r}}} \\frac{d}{d\\phi}\\left(\\frac{1}{r}\\right)<br \/>\n\\end{eqnarray}<br \/>\n\u3068\u306a\u308b\u3002<\/p>\n<h4>\u5185\u7a4d\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u89d2\u5ea6<\/h4>\n<p>2\u3064\u306e\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{n}\\) \u3068 \\(\\boldsymbol{\\gamma}\\) \u306e\u306a\u3059\u89d2 \\(\\varPsi\\) \u306f\uff0c\u5149\u306e\u4f1d\u64ad\u65b9\u5411\u3068\u52d5\u5f84\u65b9\u5411\u306e\u3068\u306e\u306a\u3059\u89d2\u3067\u3042\u308a\uff0c\u305d\u306e \\(\\cos\\) \u306f\u4ee5\u4e0b\u306e\u5f0f\u304b\u3089\u6c42\u3081\u3089\u308c\u308b\u3002<br \/>\n$$ \\cos\\varPsi \\equiv \\boldsymbol{\\gamma}\\cdot\\boldsymbol{n} = \\gamma_1 n^1 =\\frac{\\ell}{\\omega_c} \\frac{d}{d\\phi}\\left(\\frac{1}{r}\\right)$$<\/p>\n<p>\u3053\u306e\u89d2\u5ea6 \\(\\varPsi\\) \u304c\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306e\u8d64\u9053\u9762\u4e0a\uff08\\(\\theta = \\pi\/2\\)\uff09\u306e\u4efb\u610f\u306e\u70b9 \\(\\displaystyle \\left( r(\\phi),\u00a0 \\phi\\right)\\) \u306b\u304a\u3051\u308b\u5149\u304c\u52d5\u5f84\u65b9\u5411\u3068\u306a\u3059\u89d2\u3067\u3042\u308b\u30024\u5143\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3067\u5b9a\u7fa9\u3055\u308c\u308b \\(\\varPsi\\) \u306f4\u5143\u30b9\u30ab\u30e9\u30fc\u3067\u3042\u308a\uff0c\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u91cf\u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u3053\u306b\uff0c\\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u89e3<br \/>\n$$\\frac{1}{r} = \\frac{\\sin\\phi}{b} + \\frac{r_g}{2 b^2} \\left(2 -\\sin\\phi -\\sin^2\\phi\\right)$$ \u3092\u3044\u308c\u308b\u3068<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\cos\\varPsi &amp;=&amp; \\left( \\frac{1}{b^2} -\\frac{r_g}{b^3}\\right)^{-\\frac{1}{2}} \\,<br \/>\n\\frac{\\cos\\phi}{b}\\, \\left\\{1 -\\frac{r_g}{2 b} (1 + 2 \\sin\\phi) \\right\\} \\\\<br \/>\n&amp;\\simeq&amp; \\cos\\phi\\, \\left( 1 -\\frac{r_g}{b} \\sin\\phi\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308b\u3002\u3053\u308c\u304b\u3089 \\(\\sin\\varPsi\\) \u3092\u6c42\u3081\u308b\u3068\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\sin\\varPsi &amp;=&amp; \\sqrt{1 &#8211; \\cos^2\\varPsi} \\\\<br \/>\n&amp;\\simeq&amp; \\sqrt{1 &#8211; \\cos^2\\phi \\left( 1 &#8211; 2 \\frac{r_g}{b} \\sin\\phi \\right) } \\\\<br \/>\n&amp;=&amp; \\sin\\phi \\sqrt{ 1 + 2\\frac{r_g}{b} \\frac{\\cos^2\\phi}{\\sin\\phi} }\\\\<br \/>\n&amp;\\simeq&amp; \\sin\\phi + \\frac{r_g}{b} \\cos^2\\phi<br \/>\n\\end{eqnarray}<\/p>\n<h4>\u5149\u304c $x$ \u8ef8\u3068\u4ea4\u308f\u308b\u89d2\u5ea6\u3068\u300c\u66f2\u304c\u308a\u89d2\u300d<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-749\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hikari-magari-640x222.png\" alt=\"\" width=\"480\" height=\"166\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hikari-magari-640x222.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hikari-magari-300x104.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hikari-magari-1536x532.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hikari-magari-750x260.png 750w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hikari-magari.png 1579w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<p>\u5149\u306f \\(\\phi = 0\\) \u3067 \\(x\\) \u8ef8\u3068\u4ea4\u308f\u308b\u3002\u305d\u306e\u3068\u304d\u306e\u89d2\u5ea6 \\(\\varPsi\\) \u3092 \\(\\varPsi_0\\) \u3068\u3059\u308b\u3068\uff0c<br \/>\n$$\\sin\\varPsi_0 = \\sin 0 + \\frac{r_g}{b} \\cos^2 0 = \\frac{r_g}{b}$$<br \/>\n\\(|\\varPsi_0 | \\ll 1 \\) \u3067\u3042\u308b\u304b\u3089<br \/>\n$$\\varPsi_0 \\simeq \\frac{r_g}{b}$$ \\(\\phi = \\pi\\) \u3067 \\(x\\) \u8ef8\u3068\u4ea4\u308f\u308b\u3068\u304d\u306e\u89d2\u5ea6\u3082\u540c\u3058\u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u3053\u307e\u3067\u306e\u89d2\u5ea6\u306f\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3067\u306e \\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u7cbe\u5ea6\u3067\u6c42\u3081\u305f\uff08\u304f\u3069\u3044\u3088\u3046\u3060\u304c\uff09\u73fe\u5b9f\u306e\u66f2\u304c\u3063\u305f\u6642\u7a7a\u3067\u306e\u89d2\u5ea6\u3067\u3042\u308b\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-751\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f3-640x419.png\" alt=\"\" width=\"480\" height=\"314\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f3-640x419.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f3-300x196.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f3-1536x1004.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f3-2048x1339.png 2048w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/f3-750x490.png 750w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<p>\u6700\u5f8c\u306b\uff0c\u4e0a\u56f3\u304b\u3089<strong>\u66f2\u304c\u308a\u89d2<\/strong> \\(\\alpha\\) \u306f\uff0c\u300c<strong>\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u306e\u9802\u5916\u89d2\u306f\u5e95\u89d2\u306e2\u500d<\/strong>\u300d\u3068\u3044\u3046\uff08<strong>\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u5e7e\u4f55\u5b66\u3067\u300c\u306e\u307f\u300d\u6210\u308a\u7acb\u3064<\/strong>\uff09\u95a2\u4fc2\u3092\u4f7f\u3048\u3070\uff0c<br \/>\n$$ \\alpha = 2 \\varPsi_0 = \\frac{2 r_g}{b}$$<br \/>\n\u3068\u306a\u308b\u3002\\(\\varPsi_0\\) \u81ea\u4f53\u306e\u8a08\u7b97\u306e\u969b\u306b\u306f\uff0c\\( r \\rightarrow \\infty\\) \u3067\u306e\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3092\u4eee\u5b9a\u3057\u3066\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3002\u3060\u304c\uff0c\u66f2\u304c\u308a\u89d2 \\(\\alpha\\) \u306e\u5b9a\u7fa9\u306e\u969b\u306b\u306f\uff0c\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3069\u3053\u308d\u304b\uff0c\u5149\u304c\u4f1d\u64ad\u3059\u308b\u7a7a\u9593\u5168\u4f53\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u5e7e\u4f55\u5b66\u306e\u4eee\u5b9a\u3092\u7528\u3044\u3066\u3044\u308b\u3053\u3068\u306b\u3082\u7559\u610f\u3055\u308c\u305f\u3044\u3002<\/p>\n<h3>\u300c\u307e\u3063\u3059\u3050\u306a\u300d\u7dda\u306f\u66f2\u304c\u308b\u306e\u304b\uff1f<\/h3>\n<p>\u300c\u307e\u3063\u3059\u3050\u306a\u300d\u7dda\u306f\u66f2\u304c\u308b\u306e\u304b\uff1f\u3068\u3044\u3046\u306a\u3093\u3060\u304b\u7985\u554f\u7b54\uff08\u9069\u5207\u306a\u4f8b\u3048\u304b\u308f\u304b\u3089\u306a\u3044\u304c\uff09\u307f\u305f\u3044\u306a\u554f\u984c\u306b\u5bfe\u3057\u3066\u306f\uff0c\u8aad\u8005\u81ea\u8eab\u304c\u7d0d\u5f97\u3067\u304d\u308b\u3088\u3046\u306a\u89e3\u7b54\u3092\u81ea\u5206\u3067\u63a2\u3059\u306e\u304c\u4e00\u756a\u3060\u3068\u601d\u3046\u304c\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u4f8b\u3048\u8a71\u306f\u3069\u3046\u3067\u3057\u3087\u3046\u304b\u306d\u3047\u3002<\/p>\n<p>\u73fe\u5b9f\u4e16\u754c\u3067\u306f\u5149\u306f\u66f2\u304c\u3063\u305f\u7a7a\u9593\uff08\u4e0b\u56f3\u3067\u306f Schwarzschild \u3068\u66f8\u3044\u305f\u307b\u3046\uff09\u3092\u3072\u305f\u3059\u3089\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306b\u3057\u305f\u304c\u3063\u3066\u300c\u307e\u3063\u3059\u3050\u300d\u4f1d\u64ad\u3057\u3066\u3044\u304f\u3002\u3053\u308c\u3092\uff08Flat\u00a0 \u3068\u66f8\u3044\u305f\u307b\u3046\u306e\uff09\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u5c04\u5f71\u3057\u3066\u307f\u308b\u3068\uff0c\u305d\u306e\u7d4c\u8def\u306f\u307e\u3063\u3059\u3050\u306b\u306f\u898b\u3048\u306a\u3044\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-753\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig1-640x480.png\" alt=\"\" width=\"480\" height=\"360\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig1-640x480.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig1-300x225.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig1-1536x1152.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig1-2048x1536.png 2048w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig1-750x563.png 750w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<p>\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u5e7e\u4f55\u5b66\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a\u5e73\u5766\u306a\u7a7a\u9593\u306b\u5c04\u5f71\u3057\u305f\u7d4c\u8def\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u300c\u66f2\u304c\u308a\u89d2\u300d \\(\\alpha\\) \u3060\u3051\u66f2\u304c\u3063\u3066\u3044\u308b\u3068\u89e3\u91c8\u3055\u308c\u308b&#8230; \u3068\u3053\u3093\u306a\u8003\u3048\u65b9\u3082\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-752\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig2-640x480.png\" alt=\"\" width=\"480\" height=\"360\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig2-640x480.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig2-300x225.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig2-1536x1152.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig2-2048x1536.png 2048w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig2-750x563.png 750w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<p>\u53c2\u8003\uff1a<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1110.6735\">Effect of the cosmological constant on the bending of light and the cosmological lens equation by Arakida &amp; kasai<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":83,"menu_order":6,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-738","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\/738","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=738"}],"version-history":[{"count":20,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/738\/revisions"}],"predecessor-version":[{"id":8359,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/738\/revisions\/8359"}],"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=738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}