{"id":1873,"date":"2022-02-07T17:54:52","date_gmt":"2022-02-07T08:54:52","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=1873"},"modified":"2023-03-14T16:37:59","modified_gmt":"2023-03-14T07:37:59","slug":"maxima-%e3%81%ae-ctensor","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/1873\/","title":{"rendered":"Maxima \u306e ctensor \u306e\u4f7f\u7528\u4f8b\uff1a\u7403\u5bfe\u79f0\u771f\u7a7a\u975e\u9759\u7684\u30e1\u30c8\u30ea\u30c3\u30af"},"content":{"rendered":"

<\/p>\n

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\u7403\u5bfe\u79f0\u3060\u3051\u3069\u6642\u9593\u4f9d\u5b58\u6027\u304c\u3042\u308b\u30e1\u30c8\u30ea\u30c3\u30af\uff1a\u305d\u306e1<\/h3>\n

\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u00a7102. \u7403\u72b6\u7269\u4f53\u306e\u91cd\u529b\u5d29\u58ca\u306e\u9805\u306b\u8f09\u3063\u3066\u3044\u308b\u3002<\/p>\n

$$ds^2 = -d\\tau^2 + \\frac{dR^2}{\\left(\\frac{3}{2}(R-\\tau)\\right)^{2\/3}}
\n+ \\left(\\frac{3}{2}(R-\\tau)\\right)^{4\/3} (d\\theta^2 + \\sin^2\\theta d\\phi^2)$$<\/p>\n

\u3053\u306e\u30e1\u30c8\u30ea\u30c3\u30af\u304c\u771f\u7a7a\u306e\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e\u89e3\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u304b\u3081\u308b\u3002<\/p>\n

\uff08\u305f\u3060\u3057\uff0c\u3053\u306e\u89e3\u306f\u5ea7\u6a19\u5909\u63db\u3067\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u30e1\u30c8\u30ea\u30c3\u30af\u306b\u306a\u308b\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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load<\/span>(<\/span>ctensor<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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f<\/span>:<\/span> (<\/span>3\/<\/span>2*<\/span>(<\/span>R<\/span>-<\/span>tau<\/span>))<\/span>**<\/span>(<\/span>2\/<\/span>3<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\frac{3^{\\frac{2}{3}}\\,\\left(R-\\tau\\right)^{\\frac{2}{3}}}{2^{\\frac{2}{3}}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Maxima \u306f 1 \u59cb\u307e\u308a\u306a\u306e\u3067\uff0c\u7b2c\u30bc\u30ed\u6210\u5206\u3092\u7b2c4\u6210\u5206\u3068\u3059\u308b\u3002<\/p>\n
csetup();<\/code> \u3067\u5bfe\u8a71\u7684\u306b\u5165\u529b\u3067\u304d\u308b\u306e\u3067\u308f\u304b\u308a\u3084\u3059\u3044\u3002\u3057\u304b\u3057\uff0c\u5165\u529b\u3057\u9593\u9055\u3048\u308b\u3068\u521d\u3081\u304b\u3089\u3084\u308a\u76f4\u3057\u306a\u306e\u304c\u3061\u3087\u3063\u3068\u5927\u5909\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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init_ctensor<\/span>()<\/span>$\r\ncsetup<\/span>()<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Enter the dimension of the coordinate system:<\/div>\n<\/div>\n
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4;\r\n<\/pre>\n<\/div>\n<\/div>\n
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Do you wish to change the coordinate names?<\/div>\n<\/div>\n
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y;\r\n<\/pre>\n<\/div>\n<\/div>\n
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Enter a list containing the names of the coordinates in order<\/div>\n<\/div>\n
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[R, theta, phi, tau];\r\n<\/pre>\n<\/div>\n<\/div>\n
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Do you want to
\n1. Enter a new metric?
\n2. Enter a metric from a file?
\n3. Approximate a metric with a Taylor series?<\/div>\n<\/div>\n
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<\/div>\n
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1;\r\n\r\nIs the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General\r\n\r\nIs the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General\r\n\u001bXAnswer 1, 2, 3 or 4 : \u001b\\Answer 1, 2, 3 or 4 : 1;\r\n\r\n\u001bXRow 1 Column 1: \u001b\\Row 1 Column 1: 1\/f;\r\n\r\n\u001bXRow 2 Column 2: \u001b\\Row 2 Column 2: f**2;\r\n\r\n\u001bXRow 3 Column 3: \u001b\\Row 3 Column 3: f**2 * sin(theta)**2;\r\n\r\n\u001bXRow 4 Column 4: \u001b\\Row 4 Column 4: -1;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Enter functional dependencies with DEPENDS or ‘N’ if none<\/div>\n<\/div>\n
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Matrix entered.\r\nN;\r\n<\/pre>\n<\/div>\n<\/div>\n
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Do you wish to see the metric?<\/div>\n<\/div>\n
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y;\r\n<\/pre>\n<\/div>\n<\/div>\n
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\\[\\begin{pmatrix}\\frac{2^{\\frac{2}{3}}}{3^{\\frac{2}{3}}\\,\\left(R-\\tau\\right)^{\\frac{2}{3}}} & 0 & 0 & 0 \\\\ 0 & \\frac{3^{\\frac{4}{3}}\\,\\left(R-\\tau\\right)^{\\frac{4}{3}}}{2^{\\frac{4}{3}}} & 0 & 0 \\\\ 0 & 0 & \\frac{3^{\\frac{4}{3}}\\,\\left(R-\\tau\\right)^{\\frac{4}{3}}\\,\\sin ^2\\vartheta}{2^{\\frac{4}{3}}} & 0 \\\\ 0 & 0 & 0 & -1 \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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leinstein(true)<\/code> \u3067\uff0c$G_{\\mu\\nu}$ \u306e\u30bc\u30ed\u3067\u306a\u3044\u6210\u5206\u304c\u8868\u793a\u3055\u308c\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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leinstein<\/span>(<\/span>true<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u3067\u4f55\u3082\u8868\u793a\u3055\u308c\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5168\u3066\u306e\u6210\u5206\u304c\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n
\u5ff5\u306e\u305f\u3081\uff0c\u6210\u5206\u3092\u66f8\u304d\u51fa\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
lein[i,i]<\/code> = $G_{i i}$\uff0clein[4,4]<\/code> = $G_{00}$ \u306b\u76f8\u5f53\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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lein<\/span>[<\/span>1<\/span>,1]<\/span>;\r\nlein<\/span>[<\/span>2<\/span>,2]<\/span>;\r\nlein<\/span>[<\/span>3<\/span>,3]<\/span>;\r\nlein<\/span>[<\/span>4<\/span>,4]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{6}$}0\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{7}$}0\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{8}$}0\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{9}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7403\u5bfe\u79f0\u3060\u3051\u3069\u6642\u9593\u4f9d\u5b58\u6027\u304c\u3042\u308b\u30e1\u30c8\u30ea\u30c3\u30af\uff1a\u305d\u306e2<\/h3>\n
\\begin{eqnarray}
\nds^2 &=& -dt^2 + t^2 \\left(\\frac{dr^2}{1 + r^2} + r^2(d\\theta^2 + \\sin^2\\theta d\\phi^2) \\right)
\n\\end{eqnarray}<\/p>\n
\u3053\u306e\u30e1\u30c8\u30ea\u30c3\u30af\u304c\u771f\u7a7a\u306e\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e\u89e3\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u304b\u3081\u308b\u3002<\/p>\n
\uff08\u3053\u308c\u306f\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3067\uff0c$\\Omega_{\\rm m} = \\Omega_{\\Lambda} = 0$ \u3068\u3057\u305f\u3068\u304d\u306e\u89e3\u3067\uff0c\u30df\u30eb\u30f3\u5b87\u5b99\u3068\u547c\u3070\u308c\u3066\u3044\u308b\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u4eca\u5ea6\u306f 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\")$<\/code> \u3057\u3066 diag()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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load<\/span>(<\/span>\"diag\"<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/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$\r\n\/* \u5ea7\u6a19\u7cfb\u3092\u30ea\u30b9\u30c8\u3067 *\/<\/span>\r\nct_coords<\/span>:<\/span>[<\/span>r<\/span>, theta<\/span>, phi<\/span>, t<\/span>]<\/span>;\r\n\/* g_{\\mu\\nu} *\/<\/span>\r\nlg<\/span>:<\/span>diag<\/span>([<\/span>t<\/span>**<\/span>2\/<\/span>(<\/span>1+<\/span>r<\/span>**<\/span>2<\/span>)<\/span>, t<\/span>**<\/span>2*<\/span>r<\/span>**<\/span>2<\/span>, t<\/span>**<\/span>2*<\/span>r<\/span>**<\/span>2*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>, -<\/span>1])<\/span>;\r\n\/* g^{\\mu\\nu} \u3092\u8a08\u7b97\u3055\u305b\u3066\u304a\u304f *\/<\/span>\r\ncmetric<\/span>()<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}\\left[ r , \\vartheta , \\varphi , t \\right] \\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{15}$}\\begin{pmatrix}\\frac{t^2}{r^2+1} & 0 & 0 & 0 \\\\ 0 & r^2\\,t^2 & 0 & 0 \\\\ 0 & 0 & r^2\\,t^2\\,\\sin ^2\\vartheta & 0 \\\\ 0 & 0 & 0 & -1 \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{16}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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einstein<\/span>(<\/span>true<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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THIS SPACETIME IS EMPTY AND\/OR FLAT<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{17}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u3067\u4f55\u3082\u8868\u793a\u3055\u308c\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5168\u3066\u306e\u6210\u5206\u304c\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n
\u5ff5\u306e\u305f\u3081\uff0c\u6210\u5206\u3092\u66f8\u304d\u51fa\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
ein[i,j]<\/code> = $G^i_{\\ \\ j}$\uff0cein[4,4]<\/code> = $G^0_{\\ \\ 0}$ \u306b\u76f8\u5f53\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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ein<\/span>[<\/span>1<\/span>,1]<\/span>;\r\nein<\/span>[<\/span>2<\/span>,2]<\/span>;\r\nein<\/span>[<\/span>3<\/span>,3]<\/span>;\r\nein<\/span>[<\/span>4<\/span>,4]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{18}$}0\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{19}$}0\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{20}$}0\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{21}$}0\\]<\/div>\n<\/div>\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-1873","post","type-post","status-publish","format-standard","hentry","category-ctensor","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/1873","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=1873"}],"version-history":[{"count":5,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/1873\/revisions"}],"predecessor-version":[{"id":1948,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/1873\/revisions\/1948"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=1873"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=1873"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=1873"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}