{"id":2970,"date":"2022-05-09T12:28:09","date_gmt":"2022-05-09T03:28:09","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=2970"},"modified":"2023-03-14T16:30:31","modified_gmt":"2023-03-14T07:30:31","slug":"maxima-%e3%81%a73%e6%ac%a1%e5%85%83%e7%90%83%e5%af%be%e7%a7%b0%e7%a9%ba%e9%96%93%e3%81%ae%e3%82%af%e3%83%aa%e3%82%b9%e3%83%88%e3%83%83%e3%83%95%e3%82%a7%e3%83%ab%e3%81%8a%e3%82%88%e3%81%b3%e3%83%aa","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/2970\/","title":{"rendered":"Maxima \u30673\u6b21\u5143\u7403\u5bfe\u79f0\u7a7a\u9593\u306e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u304a\u3088\u3073\u30ea\u30c3\u30c1\u3092\u6c42\u3081\u308b"},"content":{"rendered":"

<\/p>\n

\n
<\/div>\n
\n
\n

\u7403\u5bfe\u79f0\u7a7a\u9593\u306e\u8a08\u91cf<\/h2>\n

Maxima \u306e ctensor \u3092\u4f7f\u3063\u30663\u6b21\u5143\u7403\u5bfe\u79f0\u7a7a\u9593\u306e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u304a\u3088\u3073\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u3092\u6c42\u3081\u308b\u3002<\/p>\n

\u7403\u5bfe\u79f0\u306a3\u6b21\u5143\u7a7a\u9593\u306e\u8a08\u91cf\u306f\u4e00\u822c\u7684\u306b<\/p>\n

$$g_{ij} dx^i dx^j = A^2(r, t) dr^2 + B^2(r,t) (d\\theta^2 + \\sin^2\\theta d\\phi^2)$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3060\u308d\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
<\/div>\n
\n
\n

$\\dot{B} = 0$ \u306e\u5834\u5408\u306e\u7403\u5bfe\u79f0\u7a7a\u9593\u306e\u8a08\u91cf<\/h2>\n

$\\dot{B} = 0$ \u3068\u3057\u3066\uff0c$ B(t, r) \\Rightarrow B(r) \\equiv r$ \u3068\u3059\u308b\u3002<\/p>\n

$$g_{ij} dx^i dx^j = A^2(r, t) dr^2 + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
load<\/span>(<\/span>ctensor<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30e1\u30c8\u30ea\u30c3\u30af\u304c\u5bfe\u89d2\u7684\u306a\u306e\u3067\uff0c\u5165\u529b\u306e\u7c21\u4fbf\u6027\u306e\u305f\u3081\u306b load(\"diag\")$<\/code> \u3057\u3066 diag()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
load<\/span>(<\/span>\"diag\"<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
init_ctensor<\/span>()<\/span>$\r\n\r\n\/* \u504f\u5fae\u5206\u8868\u793a\u306e\u7c21\u4fbf\u6027\u306e\u305f\u3081\u306b *\/<\/span>\r\nderivabbrev<\/span>:<\/span> true<\/span>$\r\n\r\n\/* \u6b21\u5143\u3002\u30c7\u30d5\u30a9\u30eb\u30c8\u3067 4 *\/<\/span>\r\ndim<\/span>:<\/span> 3$\r\n\r\n\/* \u5ea7\u6a19\u3092\u30ea\u30b9\u30c8\u3067 *\/<\/span>\r\nct_coords<\/span>:<\/span>[<\/span>r<\/span>, theta<\/span>, phi<\/span>]<\/span>;\r\n\r\n\/* g_{i j} *\/<\/span>\r\nlg<\/span>:<\/span>diag<\/span>([<\/span>A<\/span>**<\/span>2<\/span>, r<\/span>**<\/span>2<\/span>, r<\/span>**<\/span>2*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2])<\/span>;\r\ndepends<\/span>(<\/span>A<\/span>, [<\/span>t<\/span>, r<\/span>])<\/span>$\r\n\r\n\r\n\/* g^{i j} \u3092\u8a08\u7b97\u3055\u305b\u3066\u304a\u304f *\/<\/span>\r\ncmetric<\/span>()<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\left[ r , \\vartheta , \\varphi \\right] \\]<\/div>\n<\/div>\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}\\begin{pmatrix}A^2 & 0 & 0 \\\\ 0 & r^2 & 0 \\\\ 0 & 0 & r^2\\,\\sin ^2\\vartheta \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/h3>\n
mcs[i,j,k]<\/code> = $\\varGamma^k_{\\ \\ ij}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
christof<\/span>(<\/span>mcs<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{10}$}{\\it mcs}_{1,1,1}=\\frac{A_{r}}{A}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{11}$}{\\it mcs}_{1,2,2}=\\frac{1}{r}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{12}$}{\\it mcs}_{1,3,3}=\\frac{1}{r}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{13}$}{\\it mcs}_{2,2,1}=-\\frac{r}{A^2}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{14}$}{\\it mcs}_{2,3,3}=\\frac{\\cos \\vartheta}{\\sin \\vartheta}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{15}$}{\\it mcs}_{3,3,1}=-\\frac{r\\,\\sin ^2\\vartheta}{A^2}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{16}$}{\\it mcs}_{3,3,2}=-\\cos \\vartheta\\,\\sin \\vartheta\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb<\/h3>\n
uric[i,j]<\/code> $\\equiv R_i^{\\ \\ j} = R^j_{\\ \\ i}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
uricci<\/span>(<\/span>true<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{17}$}{\\it uric}_{1,1}=\\frac{2\\,A_{r}}{A^3\\,r}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{18}$}{\\it uric}_{2,2}=\\frac{A_{r}\\,r+A^3-A}{A^3\\,r^2}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\\[\\tag{${\\it \\%t}_{19}$}{\\it uric}_{3,3}=\\frac{A_{r}\\,r+A^3-A}{A^3\\,r^2}\\]<\/div>\n<\/div>\n
\n
<\/div>\n
\n
<\/pre>\n<\/div>\n<\/div>\n
\n
Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{19}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30ea\u30c3\u30c1\u30b9\u30ab\u30e9\u30fc<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
scurvature<\/span>()<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{20}$}\\frac{4\\,A_{r}\\,r+2\\,A^3-2\\,A}{A^3\\,r^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"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-2970","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\/2970","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\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2970"}],"version-history":[{"count":2,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/2970\/revisions"}],"predecessor-version":[{"id":2972,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/2970\/revisions\/2972"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2970"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=2970"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=2970"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}