{"id":1875,"date":"2022-02-07T17:55:02","date_gmt":"2022-02-07T08:55:02","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=1875"},"modified":"2023-03-14T16:37:45","modified_gmt":"2023-03-14T07:37:45","slug":"einsteinpy","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/1875\/","title":{"rendered":"EinsteinPy \u306e\u4f7f\u7528\u4f8b\uff1a\u7403\u5bfe\u79f0\u771f\u7a7a\u975e\u9759\u7684\u30e1\u30c8\u30ea\u30c3\u30af"},"content":{"rendered":"
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from<\/span> sympy<\/span> import<\/span> *<\/span>\r\nfrom<\/span> einsteinpy.symbolic<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/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\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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Python, EinsteinPy \u3067\u5206\u6570\u3092\u6271\u3046\u3068\u304d\u306f\u6ce8\u610f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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2<\/span>\/<\/span>3<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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0.6666666666666666<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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Rational<\/span>(<\/span>2<\/span>,<\/span> 3<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
$\\displaystyle \\frac{2}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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tau<\/span>,<\/span> R<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span> =<\/span> symbols<\/span>(<\/span>'tau, R, theta, phi'<\/span>)<\/span>\r\nf<\/span> =<\/span> (<\/span>Rational<\/span>(<\/span>3<\/span>,<\/span>2<\/span>)<\/span>*<\/span>(<\/span>R<\/span>-<\/span>tau<\/span>))<\/span>**<\/span>(<\/span>Rational<\/span>(<\/span>2<\/span>,<\/span>3<\/span>))<\/span>\r\n\r\nMetric<\/span> =<\/span> diag<\/span>(<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>\/<\/span>f<\/span>,<\/span> f<\/span>**<\/span>2<\/span>,<\/span> f<\/span>**<\/span>2<\/span>*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>)<\/span>.<\/span>tolist<\/span>()<\/span>\r\ng<\/span> =<\/span> MetricTensor<\/span>(<\/span>Metric<\/span>,<\/span> [<\/span>tau<\/span>,<\/span> R<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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g<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & \\frac{1}{\\left(\\frac{3 R}{2} – \\frac{3 \\tau}{2}\\right)^{\\frac{2}{3}}} & 0 & 0\\\\0 & 0 & \\left(\\frac{3 R}{2} – \\frac{3 \\tau}{2}\\right)^{\\frac{4}{3}} & 0\\\\0 & 0 & 0 & \\left(\\frac{3 R}{2} – \\frac{3 \\tau}{2}\\right)^{\\frac{4}{3}} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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ric<\/span> =<\/span> RicciTensor<\/span>.<\/span>from_metric<\/span>(<\/span>g<\/span>)<\/span>\r\nprint<\/span>(<\/span>ric<\/span>.<\/span>config<\/span>)<\/span>\r\nric<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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ll\r\n<\/pre>\n<\/div>\n<\/div>\n
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$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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ein<\/span> =<\/span> EinsteinTensor<\/span>.<\/span>from_metric<\/span>(<\/span>g<\/span>)<\/span>\r\nprint<\/span>(<\/span>ein<\/span>.<\/span>config<\/span>)<\/span>\r\nein<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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ll\r\n<\/pre>\n<\/div>\n<\/div>\n
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$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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… \u3068\u3044\u3046\u3053\u3068\u3067\uff0cRicci \u30c6\u30f3\u30bd\u30eb\u3082\uff0c\u305d\u3057\u3066 Einstein \u30c6\u30f3\u30bd\u30eb\u3082\u5168\u3066\u306e\u6210\u5206\u304c\u30bc\u30ed\u306b\u306a\u308b\u306e\u3067\uff0c\u771f\u7a7a\u89e3\u3067\u3042\u308b\u3002<\/p>\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 $\\Omega_{\\rm m} = \\Omega_{\\Lambda} = 0$ \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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In\u00a0[8]:<\/div>\n
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t<\/span>,<\/span> r<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span> =<\/span> symbols<\/span>(<\/span>'t, r, theta, phi'<\/span>)<\/span>\r\n\r\nMetric<\/span> =<\/span> diag<\/span>(<\/span>-<\/span>1<\/span>,<\/span> t<\/span>**<\/span>2<\/span>\/<\/span>(<\/span>1<\/span>+<\/span>r<\/span>**<\/span>2<\/span>),<\/span> t<\/span>**<\/span>2<\/span>*<\/span>r<\/span>**<\/span>2<\/span>,<\/span> t<\/span>**<\/span>2<\/span>*<\/span>r<\/span>**<\/span>2<\/span>*<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>)<\/span>.<\/span>tolist<\/span>()<\/span>\r\ng<\/span> =<\/span> MetricTensor<\/span>(<\/span>Metric<\/span>,<\/span> [<\/span>t<\/span>,<\/span> r<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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g<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & \\frac{t^{2}}{r^{2} + 1} & 0 & 0\\\\0 & 0 & r^{2} t^{2} & 0\\\\0 & 0 & 0 & r^{2} t^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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ein<\/span> =<\/span> EinsteinTensor<\/span>.<\/span>from_metric<\/span>(<\/span>g<\/span>)<\/span>\r\nprint<\/span>(<\/span>ein<\/span>.<\/span>config<\/span>)<\/span>\r\nein<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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ll\r\n<\/pre>\n<\/div>\n<\/div>\n
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$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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… \u3068\u3044\u3046\u3053\u3068\u3067\uff0cEinstein \u30c6\u30f3\u30bd\u30eb\u306e\u5168\u3066\u306e\u6210\u5206\u304c\u30bc\u30ed\u306b\u306a\u308b\u306e\u3067\uff0c\u771f\u7a7a\u89e3\u3067\u3042\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
In\u00a0[1]: from sympy import * from einsteinpy.symbolic import *<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","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":[17],"tags":[],"class_list":["post-1875","post","type-post","status-publish","format-standard","hentry","category-einsteinpy","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