{"id":5131,"date":"2023-01-25T12:02:11","date_gmt":"2023-01-25T03:02:11","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=5131"},"modified":"2025-02-04T12:34:37","modified_gmt":"2025-02-04T03:34:37","slug":"maxima-%e3%81%ae-ctensor-%e3%81%a7%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%82%92%e8%a7%a3%e3%81%84%e3%81%a6%e3%82%b7%e3%83%a5%e3%83%90","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%e6%99%82%e7%a9%ba%e3%81%ae%e8%a1%a8%e3%81%97%e6%96%b9\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e4%bb%a3%e6%95%b0%e3%82%b7%e3%82%b9%e3%83%86%e3%83%a0%e3%81%a7%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\/maxima-%e3%81%ae-ctensor-%e3%81%a7%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%82%92%e8%a7%a3%e3%81%84%e3%81%a6%e3%82%b7%e3%83%a5%e3%83%90\/","title":{"rendered":"Maxima \u306e ctensor \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u3092\u6c42\u3081\u308b"},"content":{"rendered":"
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Maxima \u306e ctensor \u3092\u4f7f\u3063\u3066\u7403\u5bfe\u79f0\u306a\u8a08\u91cf\u304b\u3089\u771f\u7a7a\u306e\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

$$G^{\\mu}_{\\ \\ \\nu} = 0$$<\/p>\n

\u3092\u89e3\u304d\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306e load<\/h3>\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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load<\/span>(<\/span>ctensor<\/span>)<\/span>$\r\nload<\/span>(<\/span>diag<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7403\u5bfe\u79f0\u306a\u8a08\u91cf<\/h3>\n
\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u306e\u8a18\u8ff0\u306b\u305d\u3063\u3066\uff0c\u7403\u5bfe\u79f0\u306a\u8a08\u91cf\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u304a\u304f\u3002\uff08$\\lambda$ \u306f\u4e88\u7d04\u8a9e\uff1f\u306a\u306e\u3067 $\\mu$ \u306b\u3057\u305f\u3002\uff09\u306a\u304a\uff0cMaxima \u306e\u30ea\u30b9\u30c8\u306f 1 \u59cb\u307e\u308a\u306a\u306e\u3067\uff0c\u7b2c\u30bc\u30ed\u6210\u5206\u3092\u7b2c4\u6210\u5206\u3068\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
$$ds^2 = e^{\\mu(t,r)} dr^2 + r^2 \\left(d\\theta^2 + \\sin^2\\theta\\,d\\phi^2 \\right) – e^{\\nu(t,r)} dt^2 $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/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\r\n\/* \u6b21\u5143\u3002\u30c7\u30d5\u30a9\u30eb\u30c8\u3067 4 *\/<\/span>\r\ndim<\/span>:<\/span>4$\r\n\r\n\/* \u5ea7\u6a19\u7cfb\u3092\u30ea\u30b9\u30c8\u3067 *\/<\/span>\r\nct_coords<\/span>:<\/span>[<\/span>r<\/span>, theta<\/span>, phi<\/span>, t<\/span>]<\/span>;\r\n\r\n\/* mu(r, t), nu(r, t) *\/<\/span>\r\ndepends<\/span>(<\/span>mu<\/span>, [<\/span>r<\/span>, t<\/span>])<\/span>$\r\ndepends<\/span>(<\/span>nu<\/span>, [<\/span>r<\/span>, t<\/span>])<\/span>$\r\n\r\n\/* g_{\\mu\\nu} *\/<\/span>\r\nlg<\/span>:<\/span>diag<\/span>([<\/span>exp<\/span>(<\/span>mu<\/span>)<\/span>, r<\/span>**<\/span>2<\/span>, r<\/span>**<\/span>2*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>, -<\/span>exp<\/span>(<\/span>nu<\/span>)])<\/span>;\r\n\r\n\/* g^{\\mu\\nu} \u3092\u8a08\u7b97\u3055\u305b\u3066\u304a\u304f *\/<\/span>\r\ncmetric<\/span>()<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\left[ r , \\vartheta , \\varphi , t \\right] \\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{9}$}\\begin{pmatrix}e^{\\mu} & 0 & 0 & 0 \\\\ 0 & r^2 & 0 & 0 \\\\ 0 & 0 & r^2\\,\\sin ^2\\vartheta & 0 \\\\ 0 & 0 & 0 & -e^{\\nu} \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb<\/h3>\n
$\\displaystyle G^{\\mu}_{\\ \\ \\nu} = R^{\\mu}_{\\ \\ \\nu} – \\frac{1}{2} R \\delta^{\\mu}_{\\ \\ \\nu} $ = ein<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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\/* \u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb\u306e\u8a08\u7b97\u3002<\/span>\r\n   false \u3067\u7d50\u679c\u3092\u975e\u8868\u793a\uff0ctrue \u306a\u3089\u30ce\u30f3\u30bc\u30ed\u6210\u5206\u3092\u8868\u793a *\/<\/span>\r\neinstein<\/span>(<\/span>false<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/h3>\n
$$G^{\\mu}_{\\ \\ \\nu} = 8\\pi G T^{\\mu}_{\\ \\ \\nu} – \\Lambda \\delta^{\\mu}_{\\ \\ \\nu} $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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EinEq<\/span>(<\/span>a<\/span>, b<\/span>)<\/span>:=<\/span> \r\n  (<\/span>expand<\/span>(<\/span>ein<\/span>[<\/span>a<\/span>,b<\/span>])<\/span> =<\/span>  0<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{12}$}{\\it EinEq}\\left(a , b\\right):={\\it expand}\\left({\\it ein}_{a,b}\\right)=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\mu$ \u306f\u6642\u9593\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3053\u3068<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 0}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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EinEq<\/span>(<\/span>1<\/span>, 4<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}-\\frac{\\mu_{t}\\,e^ {- \\nu }}{r}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 0} = 0$ \u3088\u308a<\/p>\n
$$\\mu_t = \\frac{\\partial \\mu}{\\partial t} = 0, \\quad \\therefore\\ \\ \\mu(t, r) \\Rightarrow \\mu(r)$$<\/p>\n
$\\mu(r)$ \u3068\u3057\u3066\uff0c\u3042\u3089\u305f\u3081\u3066\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb\u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/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\r\n\/* \u6b21\u5143\u3002\u30c7\u30d5\u30a9\u30eb\u30c8\u3067 4 *\/<\/span>\r\ndim<\/span>:<\/span>4$\r\n\r\n\/* \u5ea7\u6a19\u7cfb\u3092\u30ea\u30b9\u30c8\u3067 *\/<\/span>\r\nct_coords<\/span>:<\/span>[<\/span>r<\/span>, theta<\/span>, phi<\/span>, t<\/span>]<\/span>$\r\n\r\n\/* mu(r), nu(r, t) *\/<\/span>\r\ndepends<\/span>(<\/span>mu<\/span>, [<\/span>r<\/span>])<\/span>$\r\ndepends<\/span>(<\/span>nu<\/span>, [<\/span>r<\/span>, t<\/span>])<\/span>$\r\n\r\n\/* g_{\\mu\\nu} *\/<\/span>\r\nlg<\/span>:<\/span>diag<\/span>([<\/span>exp<\/span>(<\/span>mu<\/span>)<\/span>, r<\/span>**<\/span>2<\/span>, r<\/span>**<\/span>2*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>, -<\/span>exp<\/span>(<\/span>nu<\/span>)])<\/span>$\r\n\r\n\/* g^{\\mu\\nu} \u3092\u8a08\u7b97\u3055\u305b\u3066\u304a\u304f *\/<\/span>\r\ncmetric<\/span>()<\/span>$\r\n\/* \u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb\u306e\u8a08\u7b97\u3002<\/span>\r\n   false \u3067\u7d50\u679c\u3092\u975e\u8868\u793a\uff0ctrue \u306a\u3089\u30ce\u30f3\u30bc\u30ed\u6210\u5206\u3092\u8868\u793a *\/<\/span>\r\neinstein<\/span>(<\/span>false<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{0}_{\\ \\ 0} = 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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EinEq<\/span>(<\/span>4<\/span>, 4<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{23}$}-\\frac{e^ {- \\mu }\\,\\mu_{r}}{r}+\\frac{e^ {- \\mu }}{r^2}-\\frac{1}{r^2}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 1} = 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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EinEq<\/span>(<\/span>1<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{24}$}\\frac{e^ {- \\mu }\\,\\nu_{r}}{r}+\\frac{e^ {- \\mu }}{r^2}-\\frac{1}{r^2}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{2}_{\\ \\ 2} = 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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EinEq<\/span>(<\/span>2<\/span>, 2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{25}$}\\frac{e^ {- \\mu }\\,\\nu_{r}}{2\\,r}-\\frac{e^ {- \\mu }\\,\\mu_{r}}{2\\,r}+\\frac{e^ {- \\mu }\\,\\nu_{r\\,r}}{2}+\\frac{e^ {- \\mu }\\,\\left(\\nu_{r}\\right)^2}{4}-\\frac{e^ {- \\mu }\\,\\mu_{r}\\,\\nu_{r}}{4}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{3}_{\\ \\ 3} = 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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EinEq<\/span>(<\/span>3<\/span>, 3<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{26}$}\\frac{e^ {- \\mu }\\,\\nu_{r}}{2\\,r}-\\frac{e^ {- \\mu }\\,\\mu_{r}}{2\\,r}+\\frac{e^ {- \\mu }\\,\\nu_{r\\,r}}{2}+\\frac{e^ {- \\mu }\\,\\left(\\nu_{r}\\right)^2}{4}-\\frac{e^ {- \\mu }\\,\\mu_{r}\\,\\nu_{r}}{4}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{2}_{\\ \\ 2} = \\displaystyle G^{3}_{\\ \\ 3}$ \u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3002$\\displaystyle G^{2}_{\\ \\ 2} – \\displaystyle G^{3}_{\\ \\ 3}$ \u304c\u30bc\u30ed\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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ein<\/span>[<\/span>2<\/span>,2]<\/span> -<\/span> ein<\/span>[<\/span>3<\/span>,3]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{27}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\nu = – \\mu$ \u3068\u304a\u3051\u308b\u3053\u3068<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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EinEq<\/span>(<\/span>1<\/span>, 1<\/span>)<\/span> -<\/span> EinEq<\/span>(<\/span>4<\/span>, 4<\/span>)<\/span>;\r\n%<\/span> *<\/span> r<\/span> *<\/span> exp<\/span>(<\/span>mu<\/span>)<\/span>, expand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{28}$}\\frac{e^ {- \\mu }\\,\\nu_{r}}{r}+\\frac{e^ {- \\mu }\\,\\mu_{r}}{r}=0\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{29}$}\\nu_{r}+\\mu_{r}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$G^1_{\\ \\ 1} – G^2_{\\ \\ 2} = 0$ \u3088\u308a\uff0c\u4ee5\u4e0b\u306e\u5f0f\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n
$$\\nu_r + \\mu_r = \\frac{\\partial}{\\partial r}\\left(\\nu(t, r) + \\mu(r)\\right) = 0$$<\/p>\n
\u3053\u308c\u304b\u3089\uff0c$f(t)$ \u3092\u7a4d\u5206\u300c\u5b9a\u6570\u300d\u3068\u3057\u3066<\/p>\n
$$\\nu(t, r) = – \\mu(r) + f(t)$$<\/p>\n
\u3068\u306a\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\therefore\\ \\ e^{\\nu(t, r)} dt^2 &=& e^{- \\mu(r) + f(t)} dt^2 \\\\
\n&=& e^{- \\mu(r)} \\left( e^{\\frac{f(t)}{2}} dt\\right)^2
\n\\end{eqnarray}<\/p>\n
\u6642\u9593 $t$ \u306e\u307f\u306e\u4efb\u610f\u95a2\u6570 $f(t)$ \u306e\u81ea\u7531\u5ea6\u306f\uff0c$e^{\\frac{f(t)}{2}} dt \\Rightarrow dt’$ \u306a\u308b\u65b0\u3057\u3044\u6642\u9593\u5ea7\u6a19\u306e\u5b9a\u7fa9\u306b\u3088\u3063\u3066\u5438\u53ce\u3067\u304d\u308b\u306e\u3067\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f $f(t) = 0$ \u3059\u306a\u308f\u3061<\/p>\n
$$\\nu(t, r) = – \\mu(r)$$<\/p>\n
\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\u30d0\u30fc\u30b3\u30d5\u306e\u5b9a\u7406<\/h4>\n
\u3053\u3053\u307e\u3067\u306f\uff0c\u7403\u5bfe\u79f0\u771f\u7a7a\u89e3\u306f metric \u304c\u6642\u9593\u306b\u3088\u3089\u306a\u3044\uff0c\u3064\u307e\u308a\u9759\u7684\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u308f\u3051\u3067\uff0c\u30d0\u30fc\u30b3\u30d5\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u30d0\u30fc\u30b3\u30d5\u306e\u5b9a\u7406\u306e\u3082\u3046\u4e00\u3064\u306e\u5e30\u7d50\u3067\u3042\u308b\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u3067\u793a\u3059\u3088\u3046\u306b\u89e3\u304c $e^{-\\mu(r)} = \\displaystyle 1 – \\frac{r_g}{r}$ \u3068\u306a\u308b\u3053\u3068\u3067 $r \\rightarrow \\infty$ \u3067 $e^{-\\mu(r)} \\rightarrow 1$ \u3068\u306a\u308b\u3053\u3068\u304b\u3089\u308f\u304b\u308b\u3002<\/p>\n