\u5149\u306e\u300c\u66f2\u304c\u308a\u89d2\u300d<\/a><\/li>\n<\/ul>\n\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3092\u4eee\u5b9a\u3057\u306a\u3044\u4e0d\u5909\u7684\u306a\u89d2\u5ea6\u306e\u5b9a\u7fa9\u3067\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u3067\u306e\u66f2\u304c\u308a\u89d2\u3092\u3061\u3083\u3093\u3068\u8a08\u7b97\u3057\u3066\u3044\u308b\u3002<\/p>\n
\n\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3092\u4eee\u5b9a\u3057\u306a\u3044\u4e0d\u5909\u7684\u306a\u89d2\u5ea6\u306e\u5b9a\u7fa9<\/h3>\n
\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\u30b3\u30c8\u30e9\u30fc\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
AK \u3067\u306f Eq. (12) \u306e\u3042\u3068\u3059\u3050\u306b bending angle $\\alpha$ \u3092 Eq. (13) \u3067\u6c42\u3081\u3066\u3044\u308b\u304c\uff0c\u3053\u306e\u5c0e\u51fa\u306f\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u3092\u4eee\u5b9a\u3057\u305f\uff0c$r \\rightarrow \\infty$ \u3067\u306e\u632f\u308b\u821e\u3044\u3092\u4f7f\u3063\u3066\u3044\u308b\u3002\u3057\u304b\u3057\uff0cKottler \u306f\uff0c\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u304c\u306a\u304f\uff0c$r \\rightarrow \\infty$ \u3082\u3067\u304d\u306a\u3044\u306e\u3067\uff0c\u3053\u306e\u5b9a\u7fa9\u304c\u30ad\u30e2\u3068\u306a\u308b\u3002<\/p>\n
\u7a7a\u9593\u7684\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u89d2\u5ea6<\/h4>\n
\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
\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
\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
\n$$\\cos\\varPsi \\equiv \\boldsymbol{\\gamma}\\cdot\\boldsymbol{n}$$
\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
\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6<\/h4>\n
\u307e\u305a\uff0c\u30b3\u30c8\u30e9\u30fc\u6642\u7a7a\u4e2d\u306e\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u6210\u5206 \\(u^{\\mu}\\) \u306f
\n$$ u^{\\mu} = \\left(\\frac{1}{\\sqrt{1-\\frac{r_g}{r}- \\frac{\\Lambda}{3} r^2}}, 0, 0, 0\\right)$$
\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
\n$$P_{\\mu \\nu} = g_{\\mu\\nu} + u_{\\mu} u_{\\nu}$$<\/p>\n
\u52d5\u5f84\u65b9\u5411\u306e\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/h4>\n
\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} = g_{11} (n^1)^2 = 1\\) \u3088\u308a\u6700\u7d42\u7684\u306b
\n$$ n^{\\mu} =\\left( 0,\u00a0 – \\sqrt{1-\\frac{r_g}{r}- \\frac{\\Lambda}{3} r^2}, 0, 0\\right)$$<\/p>\n
\u5149\u306e\u9032\u3080\u5411\u304d\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/h4>\n
\\(\\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
\n$$ \\gamma_{\\mu} \\equiv\u00a0 \\frac{P_{\\mu\\nu} k^{\\nu}}{\\sqrt{P_{\\mu\\nu} k^{\\mu} k^{\\nu}}} = \\frac{P_{\\mu\\nu} k^{\\nu}}{\\omega}$$ \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\u30b3\u30c8\u30e9\u30fc\u6642\u7a7a\u3067\u306f<\/p>\n
\\begin{eqnarray}
\n\\omega = – \\boldsymbol{k}\\cdot\\boldsymbol{u}\u00a0 &=& -k_0\\, u^0 \\\\
\n&=& \\frac{\\omega_c}{\\sqrt{1-\\frac{r_g}{r}- \\frac{\\Lambda}{3} r^2}}
\n\\end{eqnarray}<\/p>\n
\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
\\begin{eqnarray}
\n\\gamma_1 &=& \\frac{1}{\\omega} P_{1\\mu} k^{\\mu} \\\\
\n&=& \\frac{1}{\\omega} g_{11} k^1 \\\\
\n&=& \\frac{1}{\\omega} g_{11} \\frac{dr}{dv} \\\\
\n&=& \\frac{1}{\\omega} g_{11} \\frac{d\\phi}{dv}\\frac{dr}{d\\phi}\u00a0 \\\\
\n&=& \\frac{\\sqrt{1-\\frac{r_g}{r}- \\frac{\\Lambda}{3} r^2}}{\\omega_c} \\ \\frac{1}{1-\\frac{r_g}{r}- \\frac{\\Lambda}{3} r^2} \\ \\frac{\\ell}{r^2} \\frac{dr}{d\\phi} \\\\
\n&=& \\frac{\\ell}{\\omega_c} \\frac{-1}{\\sqrt{1-\\frac{r_g}{r}- \\frac{\\Lambda}{3} r^2}} \\frac{d}{d\\phi}\\left(\\frac{1}{r}\\right)\\\\
\n&=& -\\frac{b}{\\sqrt{1-\\frac{r_g}{r}- \\frac{\\Lambda}{3} r^2}} \\frac{d}{d\\phi}\\left(\\frac{1}{r}\\right)
\n\\end{eqnarray}
\n\u3068\u306a\u308b\u3002<\/p>\n
\u3053\u3053\u3067 \\(\\displaystyle b \\equiv \\frac{\\ell}{\\omega_c} \\) \u306f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306e\u969b\u306e\u885d\u7a81\u30d0\u30e9\u30e1\u30fc\u30bf\u3067\u3042\u308a\uff0c\u30b3\u30c8\u30e9\u30fc\u6642\u7a7a\u306e\u5834\u5408\u306e<\/p>\n
$$\\frac{1}{B} \\equiv \\sqrt{\\left(\\frac{\\omega_c}{\\ell} \\right)^2 + \\frac{\\Lambda}{3}} = \\sqrt{\\left(\\frac{1}{b}\\right)^2 + \\frac{\\Lambda}{3} }$$<\/p>\n
\u3067\u5b9a\u7fa9\u3055\u308c\u308b \\(B\\) \u3068\u306f\u7570\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\uff08AK \u306e Eq. (11)\uff09<\/p>\n
\u5185\u7a4d\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u89d2\u5ea6<\/h4>\n
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
\n$$ \\cos\\varPsi \\equiv \\boldsymbol{\\gamma}\\cdot\\boldsymbol{n} = \\gamma_1 n^1 = \\frac{b}{B} \\frac{d}{d\\phi}\\left(\\frac{B}{r}\\right)$$<\/p>\n
\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
\u3053\u3053\u306b\uff0c\\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u89e3
\n$$\\frac{1}{r} = \\frac{\\sin\\phi}{B} + \\frac{r_g}{B^2} \\left(1 – \\frac{1}{2} \\sin^2\\phi\\right)$$ \u3092\u3044\u308c\u308b\u3068<\/p>\n
$$\\cos\\varPsi = \\frac{b}{B}\\cos\\phi \\left( 1 – \\frac{r_g}{B} \\sin\\phi\\right)$$ \u3068\u306a\u308b\u3002\u3053\u308c\u304b\u3089 \\(\\sin\\varPsi\\) \u3092\u6c42\u3081\u308b\u3068\uff0c\\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067<\/p>\n
\\begin{eqnarray}
\n\\sin\\varPsi &=& \\sqrt{1 – \\cos^2\\varPsi} \\\\
\n&\\simeq& \\sqrt{1 – \\left( 1+ \\frac{\\Lambda}{3} b^2\\right) \\cos^2\\phi \\left( 1 – 2 \\frac{r_g}{B} \\sin\\phi \\right) } \\end{eqnarray}<\/p>\n
\u3053\u308c\u3060\u3068\uff0c$\\phi = 0$ \u306e\u3068\u304d\u30eb\u30fc\u30c8\u306e\u4e2d\u304c\u8ca0\u306b\u306a\u3063\u3066\u3057\u307e\u3046\uff01\u3064\u307e\u308a\uff0cKottler \u3067\u306f $\\phi = 0$ \u306e $x$ \u8ef8\u3092\u6a2a\u5207\u3089\u306a\u3044\u3053\u3068\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u304c… \uff1f<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
\u30bf\u30a4\u30c8\u30eb\u305d\u306e\u307e\u307e\u306e\u8a71\u3002\u8352\u6728\u7530\u541b\u3078\u306e\u79c1\u4fe1\u3002Arakida & Kasai 2012 (AK) \u3067\u306f\uff0cKottler \u3067\u306e\u66f2\u304c\u308a\u89d2\u306e\u8a08\u7b97\u306e\u8a73\u7d30\u3092\u7aef\u6298\u3063\u305f\u3051\u3069\uff0c\u3061\u3083\u3093\u3068\u3084\u308b\u3068\u3069\u3046\u306a\u308b\u304b\uff0c\u3068\u3044\u3046\u8a71\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[20],"tags":[],"class_list":["post-3914","post","type-post","status-publish","format-standard","hentry","category-rel-cosmo","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3914","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"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=3914"}],"version-history":[{"count":7,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3914\/revisions"}],"predecessor-version":[{"id":3921,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3914\/revisions\/3921"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=3914"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=3914"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=3914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}