{"id":1971,"date":"2022-02-10T18:10:20","date_gmt":"2022-02-10T09:10:20","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=1971"},"modified":"2023-03-14T16:37:06","modified_gmt":"2023-03-14T07:37:06","slug":"maxima-%e3%81%ae-ctensor-%e3%81%a7-kottler-%e8%a8%88%e9%87%8f%e3%81%ae%e3%83%81%e3%82%a7%e3%83%83%e3%82%af","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/1971\/","title":{"rendered":"Maxima \u306e ctensor \u3067 Kottler \u8a08\u91cf\u306e\u30c1\u30a7\u30c3\u30af"},"content":{"rendered":"

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Kottler spacetime<\/h3>\n

\u4ee5\u4e0b\u306e\u8a08\u91cf\u304c $G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = 0$ \u306e\u89e3\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u304b\u3081\u308b\u3002<\/p>\n

$$ f(r) = 1 – \\frac{r_g}{r} – \\frac{\\Lambda}{3} r^2$$$$ds^2 = -f(r) dt^2 + \\frac{dr^2}{f(r)}
\n+ r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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load<\/span>(<\/span>ctensor<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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f<\/span>:<\/span> 1<\/span> -<\/span> rg<\/span>\/<\/span>r<\/span> -<\/span> Lambda<\/span>\/<\/span>3<\/span> *<\/span> r<\/span>**<\/span>2<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{2}$}-\\frac{{\\it rg}}{r}-\\frac{\\Lambda\\,r^2}{3}+1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Maxima \u306f 1 \u59cb\u307e\u308a\u306a\u306e\u3067\uff0c\u7b2c\u30bc\u30ed\u6210\u5206\u3092\u7b2c4\u6210\u5206\u3068\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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csetup();<\/code> \u3092\u4f7f\u308f\u305a\u306b\uff0c\u3084\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n
\u30e1\u30c8\u30ea\u30c3\u30af\u304c\u5bfe\u89d2\u7684\u306a\u306e\u3067\uff0c\u5165\u529b\u306e\u7c21\u4fbf\u6027\u306e\u305f\u3081\u306b load(\"diag\")$<\/code> \u3057\u3066 diag()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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load<\/span>(<\/span>\"diag\"<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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init_ctensor<\/span>()<\/span>$\r\n\r\n\/* \u504f\u5fae\u5206\u8868\u793a\u306e\u7c21\u4fbf\u6027\u306e\u305f\u3081\u306b *\/<\/span>\r\nderivabbrev<\/span>:<\/span>true<\/span>$\r\n\/* \u6b21\u5143\u3002\u30c7\u30d5\u30a9\u30eb\u30c8\u3067 4 *\/<\/span>\r\ndim<\/span>:<\/span>4$\r\n\/* \u5ea7\u6a19\u7cfb\u3092\u30ea\u30b9\u30c8\u3067 *\/<\/span>\r\nct_coords<\/span>:<\/span>[<\/span>r<\/span>, theta<\/span>, phi<\/span>, t<\/span>]<\/span>;\r\n\/* g_{\\mu\\nu} *\/<\/span>\r\nlg<\/span>:<\/span>diag<\/span>([<\/span>1\/<\/span>f<\/span>, r<\/span>**<\/span>2<\/span>, r<\/span>**<\/span>2*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>, -<\/span>f<\/span>])<\/span>;\r\n\/* g^{\\mu\\nu} \u3092\u8a08\u7b97\u3055\u305b\u3066\u304a\u304f *\/<\/span>\r\ncmetric<\/span>()<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}\\left[ r , \\vartheta , \\varphi , t \\right] \\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{8}$}\\begin{pmatrix}\\frac{1}{-\\frac{{\\it rg}}{r}-\\frac{\\Lambda\\,r^2}{3}+1} & 0 & 0 & 0 \\\\ 0 & r^2 & 0 & 0 \\\\ 0 & 0 & r^2\\,\\sin ^2\\vartheta & 0 \\\\ 0 & 0 & 0 & \\frac{{\\it rg}}{r}+\\frac{\\Lambda\\,r^2}{3}-1 \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{9}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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einstein<\/span>(<\/span>true<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%t}_{10}$}{\\it ein}_{1,1}=-\\Lambda\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%t}_{11}$}{\\it ein}_{2,2}=-\\Lambda\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%t}_{12}$}{\\it ein}_{3,3}=-\\Lambda\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%t}_{13}$}{\\it ein}_{4,4}=-\\Lambda\\]<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = 0$ \u3088\u308a\uff0c\u78ba\u304b\u306b
\n$$G^{\\mu}_{\\ \\ \\nu} = – \\Lambda \\delta^{\\mu}_{\\ \\ \\nu}$$
\n\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[16],"tags":[],"class_list":["post-1971","post","type-post","status-publish","format-standard","hentry","category-ctensor","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\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