{"id":2008,"date":"2022-02-19T13:13:30","date_gmt":"2022-02-19T04:13:30","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2008"},"modified":"2024-07-29T16:15:11","modified_gmt":"2024-07-29T07:15:11","slug":"%e5%b9%be%e4%bd%95%e5%85%89%e5%ad%a6%e8%bf%91%e4%bc%bc%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%85%89%e7%b7%9a%e6%9d%9f%e3%81%a8%e5%85%89%e5%ad%a6%e3%82%b9%e3%82%ab%e3%83%a9%e3%83%bc","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%b9%be%e4%bd%95%e5%85%89%e5%ad%a6%e8%bf%91%e4%bc%bc%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%85%89%e7%b7%9a%e6%9d%9f%e3%81%a8%e5%85%89%e5%ad%a6%e3%82%b9%e3%82%ab%e3%83%a9%e3%83%bc\/","title":{"rendered":"\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c\u306b\u304a\u3051\u308b\u5149\u7dda\u675f\u3068\u5149\u5b66\u30b9\u30ab\u30e9\u30fc"},"content":{"rendered":"

\u591a\u6570\u306e\u5149\u7dda\u306e\u675f\uff08\u591a\u6570\u306e\u8fd1\u63a5\u30cc\u30eb\u6e2c\u5730\u7dda<\/strong><\/span>\u304c\u675f\u306b\u306a\u3063\u3066\u308b\u30a4\u30e1\u30fc\u30b8\uff09\u3092\u5149\u7dda\u675f<\/strong><\/span>\u3068\u3044\u3046\u3002\u3053\u306e\u5149\u7dda\u675f\u306e\u65ad\u9762\uff082\u6b21\u5143\u9762\uff09\u306e\u5f62\u306e\u5909\u5316\u3092\u6c7a\u3081\u308b\u306e\u304c expansion $\\theta$ \u3084 shear $\\sigma$ \u306a\u3069\u306e\u5149\u5b66\u30b9\u30ab\u30e9\u30fc<\/strong><\/span>\u3067\u3042\u308b\u3002\u306a\u3093\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f0f\u304c\u5fc5\u8981\u306b\u306a\u308b\u304b\u3068\u3044\u3046\u3068\uff0c\u89d2\u5f84\u8ddd\u96e2\u3084\u5149\u5ea6\u8ddd\u96e2\u306a\u3069\u306e\u5b87\u5b99\u8ad6\u7684\u8ddd\u96e2\u306f\uff0c\u5149\u7dda\u675f\u306e\u65ad\u9762\u306e\u5909\u5316\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u304b\u3089\u3067\u3042\u308b\u3002<\/p>\n

\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c\u306e\u304a\u3055\u3089\u3044<\/h3>\n

\u3053\u3053\u306e\u307e\u3068\u3081<\/a>\u3092\u518d\u63b2\u3002\u5149\u306e\u4f1d\u64ad\u3092\u3042\u3089\u308f\u3059\u4e16\u754c\u7dda $x^{\\mu}(v)$\uff0c\u3053\u308c\u304c\u5149\u7dda\u3002\u3053\u3053\u3067\uff0c$v$ \u306f\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u3002<\/p>\n

\u5149\u7dda\u306e\u63a5\u30d9\u30af\u30c8\u30eb $\\displaystyle k^{\\mu}= \\frac{dx^{\\mu}}{dv}$ \u306f\u6e26\u7121\u3057\u30cc\u30eb\u6e2c\u5730\u7dda\u3067\u3042\u3063\u305f\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

\u8fd1\u63a5\u3057\u305f2\u672c\u306e\u5149\u7dda\u306e\u504f\u5dee\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u5149\u7dda\u675f\u306e\u57fa\u6e96\u3068\u306a\u308b\uff08\u30bb\u30f3\u30bf\u30fc\u306b\u3042\u308b\u3068\u304b\u4ee3\u8868\u9078\u624b\u3067\u3042\u308b\u3068\u304b\uff09\u5149\u7dda $x^{\\mu}(v)$\u00a0 \u3068\u305d\u308c\u306b\u8fd1\u63a5\u3059\u308b\u4efb\u610f\u306e\u3082\u30461\u672c\u306e\u5149\u7dda $\\tilde{x}^{\\mu}(v)$ \u3092\u8003\u3048\u308b\u3002\u300c\u8fd1\u63a5\u300d\u3057\u3066\u3044\u308b\u306e\u3067<\/p>\n

$$\\tilde{x}^{\\mu}(v) = x^{\\mu}(v) + \\epsilon \\xi^{\\mu}, \\quad |\\epsilon| \\ll 1$$<\/p>\n

\uff08\u3053\u3053\u304b\u3089\u306e\u8ad6\u7406\u5c55\u958b\u306f\u5225\u30da\u30fc\u30b8<\/a>\u306e\u504f\u5dee\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408\u3068\u540c\u3058\u3002\uff09<\/p>\n

\u305d\u308c\u305e\u308c\u306e\u63a5\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
\nk^{\\mu} &=& \\frac{dx^{\\mu}}{dv^{\\ }} \\\\
\n\\tilde{k}^{\\mu} &=& \\frac{d\\tilde{x}^{\\mu}}{dv^{\\ }} = \\frac{dx^{\\mu}}{dv^{\\ }} + \\epsilon \\frac{d\\xi^{\\mu}}{dv^{\\ }} \\\\
\n&=& k^{\\mu}(x) + \\epsilon \\frac{d\\xi^{\\mu}}{dv^{\\ }} \\tag{1}
\n\\end{eqnarray}<\/p>\n

\u4e00\u65b9\u3067<\/p>\n

\\begin{eqnarray}
\n\\tilde{k}^{\\mu} = k^{\\mu}\\left(\\tilde{x}\\right) &=& k^{\\mu}(x + \\epsilon \\xi) \\\\
\n&\\simeq& k^{\\mu}(x) + \\epsilon k^{\\mu}_{\\ \\ , \\nu} \\xi^{\\nu} \\tag{2}
\n\\end{eqnarray}<\/p>\n

$(1)$ \u5f0f\u3068 $(2)$ \u5f0f\u306e $\\epsilon$ \u306e1\u6b21\u306e\u9805\u3092\u7b49\u3057\u3044\u3068\u304a\u304f\u3068<\/p>\n

\\begin{eqnarray}
\n\\frac{d\\xi^{\\mu}}{dv^{\\ }} &=& k^{\\mu}_{\\ \\ , \\nu} \\xi^{\\nu} \\\\
\n\\therefore\\ \\ \\xi^{\\mu}_{\\ \\ ,\\nu} k^{\\nu} &=& k^{\\mu}_{\\ \\ , \\nu} \\xi^{\\nu}\u00a0 \\\\
\n\\xi^{\\mu}_{\\ \\ ,\\nu} k^{\\nu}\u00a0 + \\varGamma^{\\mu}_{\\ \\ \\ \\nu\\lambda} k^{\\nu} \\xi^{\\lambda} &=& k^{\\mu}_{\\ \\ , \\nu} \\xi^{\\nu} + \\varGamma^{\\mu}_{\\ \\ \\ \\nu\\lambda} k^{\\nu} \\xi^{\\lambda} \\\\
\n\\therefore\\ \\\u00a0 \\xi^{\\mu}_{\\ \\ ;\\nu} k^{\\nu} &=& k^{\\mu}_{\\ \\ ; \\nu} \\xi^{\\nu} \\\\
\n\\mbox{\u307e\u305f\u306f} \\ \\ \\xi_{\\mu ;\\nu} k^{\\nu} \\equiv \\frac{D\\xi_{\\mu}}{Dv} &=& k_{\\mu ; \\nu} \\xi^{\\nu}
\n\\end{eqnarray}<\/p>\n

\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n

\u504f\u5dee\u30d9\u30af\u30c8\u30eb\u306e\u76f4\u4ea4\u6027<\/h3>\n

$k^{\\mu}$ \u3068\u504f\u5dee\u30d9\u30af\u30c8\u30eb\uff08\u9023\u7d50\u30d9\u30af\u30c8\u30eb\uff09$\\xi^{\\mu}$ \u306e\u5185\u7a4d\u306f\u5149\u7dda\u306b\u6cbf\u3063\u3066\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u793a\u3055\u308c\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\frac{d}{dv} \\left(k_{\\mu} \\xi^{\\mu} \\right) &=& \\left(k_{\\mu} \\xi^{\\mu} \\right)_{,\\nu} k^{\\nu}\\\\
\n&=& \\left(k_{\\mu} \\xi^{\\mu} \\right)_{;\\nu} k^{\\nu}\\\\
\n&=& k_{\\mu; \\nu} k^{\\nu} \\, \\xi^{\\mu} + k_{\\mu}\\, \\xi^{\\mu}_{\\ \\ ;\\nu} k^{\\nu} \\\\
\n&=& k_{\\mu} k^{\\mu}_{\\ \\ ; \\nu} \\xi^{\\nu} \\qquad\\qquad(\\because k_{\\mu; \\nu} k^{\\nu} =0)\\\\
\n&=& \\frac{1}{2} \\left( k_{\\mu} k^{\\mu} \\right)_{;\\nu}\\xi^{\\nu}\\\\
\n&=& 0 \\qquad\\qquad\\quad(\\because k_{\\mu} k^{\\mu} = 0)\\\\
\n\\therefore\\ \\ k_{\\mu} \\xi^{\\mu} &=& \\mbox{const.}
\n\\end{eqnarray}<\/p>\n

\u5f93\u3063\u3066\uff0c\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $k_{\\mu} \\xi^{\\mu} = 0$ \u3068\u3059\u308c\u3070\uff0c\u305a\u30fc\u3063\u3068 $k_{\\mu} \\xi^{\\mu} = 0$ \u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u304c\u30bc\u30ed $k_{\\mu} \\xi^{\\mu} = 0$ \u3068\u3044\u3046\u3053\u3068\u306f\uff0c$\\xi^{\\mu}$ \u306f $k^{\\mu}$ \u306b\u300c\u76f4\u4ea4\u3059\u308b<\/strong><\/span>\u300d\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u901a\u5e38\u306a\u3089\u3070\uff0c$\\xi^{\\mu}$ \u306f $k^{\\mu}$ \u306b\u5e73\u884c\u306a\u6210\u5206\u306f\u6301\u305f\u306a\u3044\uff01\u3068\u8a00\u3044\u305f\u3044\u3068\u3053\u308d\u3060\u304c\uff0c$k^{\\mu}$ \u304c\u30cc\u30eb\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u72b6\u6cc1\u304c\u751f\u3058\u3066\u3057\u307e\u3046\u3002<\/p>\n

\u307e\u305a\uff0c $k_{\\mu} \\xi^{\\mu} = 0$ \u3068\u3057\u3066 $k^{\\mu}$ \u306b\u76f4\u4ea4\u3059\u308b<\/strong><\/span> $\\xi^{\\mu}$ \u306b $k^{\\mu}$ \u306b\u5e73\u884c\u306a\u4efb\u610f\u306e\u6210\u5206<\/strong><\/span>\u3092\u52a0\u3048\u305f $\\tilde{\\xi}^{\\mu}$ \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n

$$\\tilde{\\xi}^{\\mu} \\equiv \\xi^{\\mu} + a k^{\\mu}$$<\/p>\n

\u3053\u3053\u3067 $a$ \u306f\u4efb\u610f\u306e\u5b9a\u6570\uff08\u95a2\u6570\u3067\u3042\u3063\u3066\u3082\u53ef\uff09\u3067\u3042\u308b\u3002\u3068\u3053\u308d\u304c<\/p>\n

$$k_{\\mu} \\tilde{\\xi}^{\\mu} = k_{\\mu} \\xi^{\\mu} + a k_{\\mu} k^{\\mu} = 0$$<\/p>\n

\u3068\u306a\u308a\uff0c$\\tilde{\\xi}^{\\mu}$ \u3082\u307e\u305f $k^{\\mu}$ \u306b\u300c\u76f4\u4ea4\u3059\u308b<\/strong><\/span>\u300d\u3053\u3068\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u3002\u3053\u308c\u3067\u306f\uff0c$\\xi^{\\mu}$ \u306f\u3044\u3063\u305f\u3044\u4f55\u306b\u76f4\u4ea4\u3057\uff0c\u4f55\u306b\u5e73\u884c\u306a\u306e\u304b\uff0c\u8a33\u304c\u308f\u304b\u3089\u306a\u304f\u306a\u3063\u3066\u3057\u307e\u3046\u3002<\/p>\n

4\u5143\u901f\u5ea6\u306e\u5c0e\u5165\u306b\u3088\u308b $2+1+1$ \u5206\u89e3<\/h3>\n

 <\/p>\n

\u305d\u3053\u3067\uff0c\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6\u3067\u3042\u308b\u6642\u9593\u7684\u30d9\u30af\u30c8\u30eb $u^{\\mu}$ \u3092\u5c0e\u5165\u3057\uff0c\u307e\u305a\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb $k^{\\mu}$ \u3092 $3+1$ \u5206\u89e3\u3059\u308b\u3002\uff08\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306e\u9805<\/a>\u3068\u540c\u3058\u3002\uff09<\/p>\n

$$k^{\\mu} = \\omega \\left( u^{\\mu} + \\gamma^{\\mu} \\right)$$<\/p>\n

\u3053\u3053\u3067 $\\omega \\equiv -k_{\\mu} u^{\\mu}$ \u306f4\u5143\u901f\u5ea6 $u^{\\mu}$ \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u3067\u3042\u308a\uff0c$\\gamma^{\\mu}$ \u306f $u^{\\mu}$ \u306b\u76f4\u4ea4\u3059\u308b\u7a7a\u9593\u7684\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c\u89b3\u6e2c\u8005\u306b\u3068\u3063\u3066\u306f\u5149\u304c\u3084\u3063\u3066\u304f\u308b\u8996\u7dda\u65b9\u5411\u3092\u8868\u3059\u30d9\u30af\u30c8\u30eb\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n

$$u_{\\mu} u^{\\mu} = -1, \\quad u_{\\mu} \\gamma^{\\mu} = 0, \\quad \\gamma_{\\mu}\\gamma^{\\mu} =1$$<\/p>\n

${\\color{red}{2}}+{\\color{blue}{1}}+{\\color{green}{1}}$ \u5206\u89e3\u3068\u306f\uff0c<\/p>\n