{"id":1612,"date":"2022-01-31T16:38:05","date_gmt":"2022-01-31T07:38:05","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1612"},"modified":"2024-07-29T17:30:40","modified_gmt":"2024-07-29T08:30:40","slug":"%e5%85%89%e5%ba%a6%e8%b7%9d%e9%9b%a2","status":"publish","type":"page","link":"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%e5%ae%87%e5%ae%99%e8%ab%96\/%e5%85%89%e5%ba%a6%e8%b7%9d%e9%9b%a2\/","title":{"rendered":"\u5149\u5ea6\u8ddd\u96e2"},"content":{"rendered":"

\u898b\u304b\u3051\u306e\u5149\u5ea6\u304c\u8ddd\u96e2\u306e2\u4e57\u306b\u53cd\u6bd4\u4f8b\u3059\u308b\u3053\u3068\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u5149\u5ea6\u8ddd\u96e2<\/strong><\/span>\u306b\u3064\u3044\u3066\uff0c\u4e00\u822c\u7684\u306a\u5b9a\u7fa9\u304b\u3089\u8aac\u304d\u8d77\u3053\u3059\u3002<\/p>\n

<\/p>\n

\u5fc5\u8981\u306a\u5f0f\u306e\u304a\u3055\u3089\u3044<\/h3>\n

\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c<\/h4>\n

\u3053\u3053<\/a>\u3067\u307e\u3068\u3081\u3066\u3044\u308b\u3088\u3046\u306b\uff0c4\u5143\u30d9\u30af\u30c8\u30eb $k^{\\mu}$ \u3067\u8868\u3055\u308c\u308b\u300c\u5358\u8272\u5149\u300d\u306e\u5149\u6e90\u5929\u4f53\u304b\u3089\u653e\u51fa\u3055\u308c\u308b\uff0c\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb\u306f<\/p>\n

$$T^{\\mu\\nu} = A^2 k^{\\mu}k^{\\nu}$$
\n$T^{\\mu\\nu}_{\\ \\ \\ ;\\nu}=0$ \u3088\u308a
\n$$A_{,\\nu} k^{\\nu}\u00a0 + \\frac{1}{2} A k^{\\nu}_{\\ \\ ;\\nu} = 0$$<\/p>\n

\u3061\u306a\u307f\u306b\uff0c\u30cc\u30eb\u6e2c\u5730\u7dda\u306e\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf $v$ \u3092\u4f7f\u3046\u3068\uff0c
\n$$A_{,\\nu} k^{\\nu} = \\frac{dx^{\\nu}}{dv} \\frac{\\partial}{\\partial x^{\\nu} }A = \\frac{d A}{dv}$$\u3067\u3042\u308b\u306e\u3067
\n$$\\frac{d A}{dv} +\u00a0 \\theta A\u00a0 = 0$$\u3068\u66f8\u3044\u305f\u65b9\u304c\u898b\u901a\u3057\u304c\u826f\u3044\u304b\u3082\u77e5\u308c\u306a\u3044\u3002$\\theta$ \u306f\u5149\u7dda\u675f\u306e expansion\u3002<\/p>\n

$k^{\\mu}$ \u306f\u3046\u305a\u306a\u3057\u306e\u30cc\u30eb\u6e2c\u5730\u7dda\u3067\u3042\u308b\u3002<\/p>\n

$$
\nk_{\\mu, \\nu} – k_{\\nu, \\mu} = k_{\\mu; \\nu} – k_{\\nu; \\mu} =0
\n$$
\n$$ k_{\\mu} k^{\\mu} = 0, \\quad k^{\\mu}_{\\ \\ ;\\nu} k^{\\nu} = 0$$<\/p>\n

\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6\u306b\u57fa\u3065\u3044\u305f \\(k^{\\mu}\\) \u306e $3+1$ \u5206\u89e3<\/h4>\n

$$k^{\\mu} = \\frac{dx^{\\mu}}{dv} = \\omega ( u^{\\mu} + \\gamma^{\\mu})$$
\n\u3053\u3053\u3067\uff0c$\\omega$ \u306f4\u5143\u901f\u5ea6 $u^{\\mu}$ \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u3067\u3042\u308a
\n$$\\omega \\equiv – k_{\\mu} u^{\\mu}$$<\/p>\n

\u307e\u305f\uff0c$\\gamma^{\\mu}$ \u306f $u^{\\mu}$ \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c
\n$$u_{\\mu} u^{\\mu} = -1, \\quad u_{\\mu} \\gamma^{\\mu} = 0, \\quad \\gamma_{\\mu} \\gamma^{\\mu} = 1$$<\/p>\n

\u5149\u6e90\u5929\u4f53\u304b\u3089\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u6d41\u675f<\/h3>\n

4\u5143\u901f\u5ea6 $u^{\\mu}$ \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\uff0c\u5149\u6e90\u5929\u4f53\u304b\u3089\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u6d41\u675f $f^{\\mu}$ \u306f<\/p>\n

\\begin{eqnarray}
\nf^{\\mu} &=& \\left(g^{\\mu\\alpha} + u^{\\mu} u^{\\alpha} \\right) T_{\\alpha\\beta} u^{\\beta}\\\\
\n&=& A^2 \\omega^2 \\gamma^{\\mu} \\\\
\n&\\equiv& f \\gamma^{\\mu}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067 $f = A^2 \\omega^2$ \u306f $\\gamma^{\\mu}$ \u306b\u5782\u76f4\u306a\u9762\u306b\u5bfe\u3059\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u6d41\u675f\uff08\u5358\u4f4d\u9762\u7a4d\u5358\u4f4d\u6642\u9593\u3042\u305f\u308a\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u6d41\u308c\uff09\u3067\u3042\u308a\uff0c\u7d76\u5bfe\u5149\u5ea6 $L$ \u306e\u5149\u6e90\u5929\u4f53\u306b\u5bfe\u3057\u3066\uff0c4\u5143\u901f\u5ea6 $u^{\\mu}$ \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u898b\u304b\u3051\u306e\u5149\u5ea6\u306b\u76f8\u5f53\u3059\u308b\u3002<\/p>\n

\u5149\u5ea6\u8ddd\u96e2\u306e\u5b9a\u7fa9<\/h3>\n

\u7d76\u5bfe\u5149\u5ea6 $L$ \u3068\u306f\u3059\u306a\u308f\u3061\uff0c\u5149\u6e90\u3092\u56f2\u3080\u5168\u8868\u9762\u7a4d\u304b\u3089\u653e\u51fa\u3055\u308c\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u3053\u3068\u3002\u898b\u304b\u3051\u306e\u660e\u308b\u3055 $f$ \u3068\u306f\uff0c\u5358\u4f4d\u9762\u7a4d\u3042\u305f\u308a\u306b\u53d7\u3051\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u3002$f$ \u306b\u5168\u8868\u9762\u7a4d\u3092\u304b\u3051\u308b\u3068 $L$ \u306b\u306a\u308b\u3002\u305d\u3053\u3067\uff0c\u5149\u6e90\u307e\u3067\u306e\uff08\u5149\u5ea6\uff09\u8ddd\u96e2\u3092 $d_L$\u3068\u3059\u308b\u3068\uff0c\u534a\u5f84 $d_L$ \u306e\u7403\u9762\u306e\u5168\u8868\u9762\u7a4d\u306f $4 \\pi \\left( d_L\\right)^2$ \u3067\u3042\u308b\u304b\u3089<\/p>\n

\\begin{eqnarray}
\nL &=& f \\times 4\u00a0 \\pi \\left( d_L\\right)^2\\\\
\n\\therefore\\ \\ f &=& \\frac{L}{4 \\pi\u00a0 \\left( d_L\\right)^2}
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u3088\u3046\u306b\uff0c\u898b\u304b\u3051\u306e\u5149\u5ea6\u304c\u8ddd\u96e2\u306e2\u4e57\u306b\u53cd\u6bd4\u4f8b\u3059\u308b\u3053\u3068\u304b\u3089\uff0c<\/p>\n

$$ f \\equiv \\frac{L}{4\\pi \\left(d_L\\right)^2}$$<\/p>\n

\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u8ddd\u96e2 $d_L$ \u304c\u5149\u5ea6\u8ddd\u96e2<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n

\"\"<\/p>\n

\u30a8\u30cd\u30eb\u30ae\u30fc\u6d41\u675f $f$ \u306e\u5f0f\u3092\u5165\u308c\u308b\u3068<\/p>\n

\\begin{eqnarray}
\nA^2 \\omega^2 &=& \\frac{L}{4\\pi \\left(d_L\\right)^2} \\\\
\n\\therefore \\ \\\u00a0 A \\omega d_L &=& \\sqrt{\\frac{L}{4\\pi}} = \\mbox{const.}\\\\
\n\\frac{d}{dv} \\left(A \\omega d_L \\right) &=& 0 \\\\
\n\\therefore \\ \\ \\frac{d}{dv}\\left( \\omega d_L\\right) &=& -\\frac{1}{A} \\frac{dA}{dv} \\left( \\omega d_L\\right) \\\\
\n&=& \\theta \\left( \\omega d_L\\right)
\n\\end{eqnarray}<\/p>\n

\u5149\u6e90\u306b\u304a\u3051\u308b\u632f\u52d5\u6570\u3092 $\\omega_e$ \u3068\u3059\u308b\u3068\uff0c\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570 $\\omega$ \u306f\u8d64\u65b9\u504f\u79fb $z$ \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002
\n$$\\omega = \\frac{\\omega_e}{1+z}$$
\n\u3053\u308c\u3092\u4f7f\u3046\u3068\uff0c\u5149\u5ea6\u8ddd\u96e2\u3092\u6c7a\u3081\u308b\u5f0f\u306f<\/p>\n

$$\\frac{d}{dv}\\left( \\frac{d_L}{1+z}\\right) = \\theta\\left( \\frac{d_L}{1+z}\\right)$$<\/p>\n

\u5149\u5ea6\u8ddd\u96e2\u306e\u89b3\u6e2c\uff08\u89d2\u5f84\u8ddd\u96e2\u3068\u6bd4\u8f03\u3057\u3066\uff09<\/h4>\n

\u4ee5\u4e0b\u3067\u7d39\u4ecb\u3059\u308b\u304c\uff0c\u5149\u5ea6\u8ddd\u96e2\u3068\u89d2\u5f84\u8ddd\u96e2\u306f\u72ec\u7acb\u306a\u8ddd\u96e2\u516c\u5f0f\u3067\u306f\u306a\u304f\uff0c\u4efb\u610f\u306e\u6642\u7a7a\u306b\u304a\u3044\u3066<\/p>\n

$$d_L(z) = (1+z)^2 \\, d_A(z)$$<\/p>\n

\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002\u3053\u308c\u3092 reciprocity theorem \u3068\u547c\u3093\u3060\u308a\u3059\u308b\u3002\u4ee5\u4e0b\u306e\u6587\u732e\u306b\u8a3c\u660e\u304c\u3042\u308b\u3002<\/p>\n