{"id":1757,"date":"2022-02-05T14:07:17","date_gmt":"2022-02-05T05:07:17","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1757"},"modified":"2023-12-01T12:49:05","modified_gmt":"2023-12-01T03:49:05","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae-31-%e5%ae%9a%e5%bc%8f%e5%8c%96","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\/%e8%86%a8%e5%bc%b5%e5%ae%87%e5%ae%99%e3%81%ae%e8%a8%88%e9%87%8f%e3%81%ae%e5%b0%8e%e5%87%ba%e3%81%a8%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae-31-%e5%ae%9a%e5%bc%8f%e5%8c%96\/","title":{"rendered":"\u88dc\u8db3\uff1a\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e 3+1 \u5b9a\u5f0f\u5316"},"content":{"rendered":"

<\/p>\n

4\u6b21\u5143\u6642\u7a7a\u306e\u8a08\u91cf\u306e $3+1$ \u5206\u89e3<\/h3>\n

\u7c21\u5358\u306e\u305f\u3081\u306b $g_{0i} = 0$ \u306e\u5834\u5408\u3092\u8003\u3048\uff0c\u7dda\u7d20\u3092<\/p>\n

$$ds^2 = – N^2 dt^2 + g_{ij} dx^i dx^j$$<\/p>\n

\u3068\u66f8\u304d\uff0c\u672c\u6765\u306f4\u6b21\u5143\u6642\u7a7a\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb $g_{\\mu\\nu}$ \u306e $ij$ \u6210\u5206\u3067\u3042\u308b $g_{ij}$ \u30923\u6b21\u5143\u7a7a\u9593\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u3068\u307f\u306a\u3059\u3002\u307e\u305f\uff0c<\/p>\n

$$K_{ij} \\equiv \\frac{1}{2N} \\dot{g}_{ij}, \\quad K^i_{\\ \\ j} = g^{ik} K_{kj} = \\frac{1}{2N} g^{ik} \\dot{g}_{kj}$$<\/p>\n

\u3068\u5b9a\u7fa9\u3059\u308b\u3002$\\displaystyle \\dot{{\\color{white}{g}}} \\equiv \\frac{\\partial}{\\partial t}$<\/p>\n

${}^{(3)}\\!\\varGamma^i_{\\ \\ \\ j k}, \\ {}^{(3)}\\!R^i_{\\ \\ j}$ \u306f $g_{ij}$ \u3067\u8868\u3055\u308c\u308b3\u6b21\u5143\u7a7a\u9593\u306e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3068\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u3092\u8868\u3059\u3002<\/p>\n

\u307e\u305f\uff0c${\\ }_{|i}$ \u306f3\u6b21\u5143\u7a7a\u9593\u306e\u5171\u5909\u5fae\u5206\u3092\u8868\u3059\u3002<\/p>\n

\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/h3>\n

\\begin{equation}\\label{eq:chr2}
\n\\varGamma^{\\lambda}_{\\ \\ \\mu\\nu} = \\frac{1}{2} g^{\\lambda\\sigma}
\n\\left(g_{\\sigma\\mu,\\nu} + g_{\\sigma\\nu,\\mu} – g_{\\mu\\nu,\\sigma}
\n\\right)
\n\\end{equation}<\/p>\n

\\begin{eqnarray}
\n\\varGamma^0_{\\ \\ \\ 00} &=& \\frac{1}{N} \\dot{N}\\\\
\n\\varGamma^0_{\\ \\ \\ 0i} &=& \\frac{1}{N} N_{|i}\\\\
\n\\varGamma^0_{\\ \\ \\ ij} &=& \\frac{1}{N} K_{ij}\\\\
\n\\varGamma^i_{\\ \\ \\ 00} &=& N N^{|i}\\\\
\n\\varGamma^i_{\\ \\ \\ 0j} &=& N K^i_{\\ \\ j}\\\\
\n\\varGamma^i_{\\ \\ \\ jk} &=& {}^{(3)}\\!\\varGamma^i_{\\ \\ \\ jk}
\n\\end{eqnarray}<\/p>\n

\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u30fb\u30ea\u30c3\u30c1\u30b9\u30ab\u30e9\u30fc<\/h3>\n

\\begin{eqnarray}
\nR_{\\mu\\rho} = R^{\\nu}_{\\ \\ \\mu\\nu\\rho} &=& \\varGamma^{\\nu}_{\\ \\ \\mu\\rho,\\nu} –
\n\\varGamma^{\\nu}_{\\ \\ \\mu\\nu,\\rho} \\nonumber + \\varGamma^{\\nu}_{\\ \\ \\lambda\\nu}\\varGamma^{\\lambda}_{\\ \\ \\mu\\rho}
\n– \\varGamma^{\\nu}_{\\ \\ \\lambda\\rho}\\varGamma^{\\lambda}_{\\ \\ \\mu\\nu}.
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\nR_{00} &=& N N^{|i}_{\\ \\ |i} – N \\dot{K}^i_{\\ \\ i} – N^2 K^i_{\\ \\ j} K^j_{\\ \\ i} \\\\
\nR_{0i} &=& N \\left(K^j_{\\ \\ i|j} – K^j_{\\ \\ j|i} \\right)\\\\
\nR_{ij} &=& \\frac{1}{N} \\dot{K}_{ij} – 2 K_{ik} K^k_{\\ \\ j} – \\frac{1}{N} N_{|i j}+ K^k_{\\ \\ k} K_{ij} + {}^{(3)}\\!R_{ij}\\\\
\n\\end{eqnarray}<\/p>\n

$$R = \\frac{2}{N} \\dot{K}^i_{\\ \\ i} + K^i_{\\ \\ j} K^j_{\\ \\ i} – \\frac{2}{N} N^{|i}_{\\ \\ |i} + {}^{(3)}\\!R^i_{\\ \\ i}$$<\/p>\n

\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/h3>\n

$$G_{00} = \\frac{N^2}{2} \\left( \\left(K^i_{\\ \\ i} \\right)^2 – K^i_{\\ \\ j} K^j_{\\ \\ i} + {}^{(3)}\\!R^i_{\\ \\ i}\\right) = 8\\pi G T_{00}$$<\/p>\n

$$G_{0i} = N \\left(K^j_{\\ \\ i|j} – K^j_{\\ \\ j|i} \\right) = 8\\pi G T_{0i}$$<\/p>\n

$ij$ \u6210\u5206\u306b\u3064\u3044\u3066\u306f\uff0ctrace-reversed \u306a\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e \\(ij\\) \u6210\u5206 \u3092\u3068\u308b\u3053\u3068\u306b\u3057\u3066<\/p>\n

$$R^i_{\\ \\ j} = \\frac{1}{N} \\dot{K}^i_{\\ \\ j} + K^k_{\\ \\ k} K^i_{\\ \\ j} – \\frac{1}{N} N^{|i}_{\\ \\ |j} + {}^{(3)}\\!R^i_{\\ \\ j} = 8\\pi G\\left(T^i_{\\ \\ j} – \\frac{1}{2} \\delta^i_{\\ \\ j} T^{\\mu}_{\\ \\ \\mu}\\right)$$<\/p>\n

\u3053\u3046\u3084\u3063\u3066\u66f8\u3044\u3066\u307f\u308b\u3068\uff0c$00$ \u6210\u5206\u3068 $0i$ \u6210\u5206\u306f $K^i_{\\ \\ j}$ \u3059\u306a\u308f\u3061\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6642\u9593\u306b\u95a2\u3059\u308b1\u968e\u5fae\u5206\u307e\u3067\u3057\u304b\u542b\u307e\u306a\u3044\u306e\u3067\uff0c\u4f55\u304b\u3057\u3089\u306e\u62d8\u675f\u6761\u4ef6\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u306b\u5bfe\u3057\u3066\uff0c$ij$ \u6210\u5206\u306f $K^i_{\\ \\ j}$ \u306e\u6642\u9593\u5fae\u5206\u3059\u306a\u308f\u3061\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6642\u9593\u306b\u95a2\u3059\u308b2\u968e\u5fae\u5206\u3092\u542b\u3080\u306e\u3067\uff0c\u306a\u3093\u3068\u306a\u304f\u904b\u52d5\u65b9\u7a0b\u5f0f\u3068\u3044\u3046\u304b\uff0c\u767a\u5c55\u65b9\u7a0b\u5f0f\u3068\u3044\u3046\u304b\u305d\u3093\u306a\u6642\u9593\u767a\u5c55\u3092\u8a18\u8ff0\u3059\u308b\u65b9\u7a0b\u5f0f\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u3088\u304f\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

trace-reversed \u306a\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3068\u306f<\/h4>\n

$$R_{\\mu\\nu} – \\frac{1}{2} g_{\\mu\\nu} R = 8\\pi G T_{\\mu\\nu}$$<\/p>\n

\u4e21\u8fba\u3092$g^{\\mu\\nu}$ \u3067\u7e2e\u7d04\u3059\u308b\u3068
\n$$R – 2 R = 8 \\pi G T^{\\mu}_{\\ \\ \\mu}, \\quad\\therefore\\ \\ R = – 8 \\pi G T^{\\mu}_{\\ \\ \\mu}$$<\/p>\n

\\begin{eqnarray}
\n\\therefore\\ \\ R_{\\mu\\nu} &=& 8\\pi G T_{\\mu\\nu} + \\frac{1}{2} g_{\\mu\\nu} R \\\\
\n&=& 8\\pi G T_{\\mu\\nu} + \\frac{1}{2} g_{\\mu\\nu} \\left(- 8 \\pi G T^{\\lambda}_{\\ \\ \\lambda} \\right) \\\\
\n&=& 8\\pi G\\left( T_{\\mu\\nu} – \\frac{1}{2} g_{\\mu\\nu} T^{\\lambda}_{\\ \\ \\lambda}\\right)
\n\\end{eqnarray}<\/p>\n

\u5de6\u8fba\u306b\u3042\u3063\u305f $R^{\\mu}_{\\ \\ \\nu}$ \u306e\u30c8\u30ec\u30fc\u30b9\u3067\u3042\u308b $R = R^{\\mu}_{\\ \\ \\mu}$ \u304c\u53f3\u8fba\u306e $T^{\\mu}_{\\ \\ \\nu}$ \u306e\u30c8\u30ec\u30fc\u30b9\u3067\u3042\u308b $T^{\\mu}_{\\ \\ \\mu}$\u00a0 \u3068\u306a\u3063\u305f\u306e\u3067\u300c\u30c8\u30ec\u30fc\u30b9\u53cd\u8ee2\u300d\u306a\u306e\u304b\u306a\u3041\u3002\u305d\u308c\u3068\u3082\uff0c$R^{\\mu}_{\\ \\ \\nu}$ \u306e\u30c8\u30ec\u30fc\u30b9\u3068$T^{\\mu}_{\\ \\ \\nu}$ \u306e\u30c8\u30ec\u30fc\u30b9\u3068\u306f\u8ca0\u53f7\u304c\u53cd\u8ee2\u3057\u3066\u3044\u308b\u306e\u3067\u300c\u30c8\u30ec\u30fc\u30b9\u53cd\u8ee2\u300d\u306a\u306e\u304b\u306a\u3041\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":1412,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1757","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t