{"id":5189,"date":"2024-07-16T11:50:28","date_gmt":"2024-07-16T02:50:28","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=5189"},"modified":"2025-02-04T10:39:32","modified_gmt":"2025-02-04T01:39:32","slug":"einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b","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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/","title":{"rendered":"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b"},"content":{"rendered":"<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f$$G^{\\mu}_{\\ \\ \\nu} + \\Lambda \\delta^{\\mu}_{\\ \\ \\nu} = 8\\pi G T^{\\mu}_{\\ \\ \\nu}$$\u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><!--more--><\/p>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30e9\u30a4\u30d6\u30e9\u30ea\u306e-import\">\u30e9\u30a4\u30d6\u30e9\u30ea\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[1]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"kn\">from<\/span> <span class=\"nn\">sympy.abc<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">sympy<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">einsteinpy.symbolic<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc\u8a08\u91cf\">\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc\u8a08\u91cf<\/h3>\n<p>Wikipedia \u306e\u8a18\u8ff0\u306b\u305d\u3063\u3066\uff0cFLRW \u8a08\u91cf\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u304a\u304f\u3002<\/p>\n<p>$$ds^2 = -dt^2 + a^2(t) \\left[\\frac{dr^2}{1-k r^2} + r^2 \\left(d\\theta^2 + \\sin^2\\theta\\,d\\phi^2 \\right)\\right] $$<\/p>\n<ul>\n<li><a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%83%95%E3%83%AA%E3%83%BC%E3%83%89%E3%83%9E%E3%83%B3%E3%83%BB%E3%83%AB%E3%83%A1%E3%83%BC%E3%83%88%E3%83%AB%E3%83%BB%E3%83%AD%E3%83%90%E3%83%BC%E3%83%88%E3%82%BD%E3%83%B3%E3%83%BB%E3%82%A6%E3%82%A9%E3%83%BC%E3%82%AB%E3%83%BC%E8%A8%88%E9%87%8F\">\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc\u8a08\u91cf &#8211; Wikipedia<\/a><\/li>\n<\/ul>\n<p>\u306a\u305c\u3053\u306e\u3088\u3046\u306a\u8a08\u91cf\u306b\u306a\u308b\u304b\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3082\u53c2\u7167\uff1a<\/p>\n<ul>\n<li><a href=\"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%e5%ae%87%e5%ae%99%e8%ab%96\/%e8%86%a8%e5%bc%b5%e5%ae%87%e5%ae%99%e3%81%ae%e8%a8%88%e9%87%8f%e3%81%ae%e5%b0%8e%e5%87%ba%e3%81%a8%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f\/\">\u81a8\u5f35\u5b87\u5b99\u306e\u8a08\u91cf\u306e\u5c0e\u51fa\u3068\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>a<\/code> $\\equiv a(t)$, <code>g<\/code> $\\equiv g_{\\mu\\nu}$ \u3068\u5b9a\u7fa9\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[2]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 a(t) \u306f\u6642\u9593\u306e\u307f\u306e\u95a2\u6570<\/span>\r\n<span class=\"n\">a<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a'<\/span><span class=\"p\">)(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># diag() \u3092\u4f7f\u3046\u3068\u304d\u306f<\/span>\r\n<span class=\"c1\"># .tolist() \u3092\u3064\u3051\u308b<\/span>\r\n<span class=\"n\">Metric<\/span> <span class=\"o\">=<\/span> <span class=\"n\">diag<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> \r\n              <span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">k<\/span><span class=\"o\">*<\/span><span class=\"n\">r<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">),<\/span> \r\n              <span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">r<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">,<\/span> \r\n              <span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">r<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span> <span class=\"o\">*<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">theta<\/span><span class=\"p\">)<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">tolist<\/span><span class=\"p\">()<\/span>\r\n\r\n<span class=\"n\">g<\/span> <span class=\"o\">=<\/span> <span class=\"n\">MetricTensor<\/span><span class=\"p\">(<\/span><span class=\"n\">Metric<\/span><span class=\"p\">,<\/span> <span class=\"p\">[<\/span><span class=\"n\">t<\/span><span class=\"p\">,<\/span> <span class=\"n\">r<\/span><span class=\"p\">,<\/span> <span class=\"n\">theta<\/span><span class=\"p\">,<\/span> <span class=\"n\">phi<\/span><span class=\"p\">])<\/span>\r\n\r\n<span class=\"c1\"># \u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u5168\u6210\u5206\u306e\u8868\u793a\u306f .tensor() \u3092\u3064\u3051\u308b<\/span>\r\n<span class=\"n\">g<\/span><span class=\"o\">.<\/span><span class=\"n\">tensor<\/span><span class=\"p\">()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[2]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\left[\\begin{matrix}-1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; \\frac{a^{2}{\\left(t \\right)}}{-k r^{2} + 1} &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; r^{2} a^{2}{\\left(t \\right)} &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; r^{2} a^{2}{\\left(t \\right)} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb\">\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb <code><span class=\"n\">ein<\/span><\/code><\/h3>\n<p><code>ein<\/code> $\\equiv \\displaystyle G^{\\mu}_{\\ \\ \\nu} = R^{\\mu}_{\\ \\ \\nu} -\\frac{1}{2} R \\delta^{\\mu}_{\\ \\ \\nu} $ \u3068\u5b9a\u7fa9\u3002\uff08<code>.change_config('ul')<\/code> \u3067\u4e0a\u4ed8\u4e0b\u4ed8\u306b\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[3]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ein<\/span> <span class=\"o\">=<\/span> <span class=\"n\">EinsteinTensor<\/span><span class=\"o\">.<\/span><span class=\"n\">from_metric<\/span><span class=\"p\">(<\/span><span class=\"n\">g<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">change_config<\/span><span class=\"p\">(<\/span><span class=\"s1\">'ul'<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5168\u6210\u5206\u3092\u8868\u793a\">\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5168\u6210\u5206\u3092\u8868\u793a<\/h4>\n<p><code>ein<\/code> $ = G^{\\mu}_{\\ \\ \\nu}$ \u30924\u884c4\u5217\u306e\u884c\u5217\u3068\u3057\u3066\u8868\u793a\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[4]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5168\u6210\u5206\u8868\u793a\u306f .tensor() \u3092\u3064\u3051\u3066...<\/span>\r\n<span class=\"n\">ein<\/span><span class=\"o\">.<\/span><span class=\"n\">tensor<\/span><span class=\"p\">()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[4]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\left[\\begin{matrix}\\frac{3 \\left(-k -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}\\right)}{a^{2}{\\left(t \\right)}} &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; \\frac{-k -2 a{\\left(t \\right)} \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)} -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; \\frac{-k -2 a{\\left(t \\right)} \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)} -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; \\frac{-k -2 a{\\left(t \\right)} \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)} -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$G^{\\mu}_{\\-\\-\\nu}$-\u306e\u30bc\u30ed\u3067\u306a\u3044\u6210\u5206\u3092\u500b\u5225\u306b\u8868\u793a\">$G^{\\mu}_{\\ \\ \\nu}$ \u306e\u30bc\u30ed\u3067\u306a\u3044\u6210\u5206\u3092\u500b\u5225\u306b\u8868\u793a<\/h4>\n<p>\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30c6\u30f3\u30bd\u30eb $G^{\\mu}_{\\ \\ \\nu}$ \u306e\u30bc\u30ed\u3067\u306a\u3044\u6210\u5206\u3092\u500b\u5225\u306b\u8868\u793a\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"ein[0,-0]-$=-G^{0}_{\\-\\-0}$\"><code>ein[0, 0]<\/code> $= G^{0}_{\\ \\ 0}$<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[5]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ein<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[5]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{3 \\left(-k -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}\\right)}{a^{2}{\\left(t \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"ein[1,-1]-$=-G^{1}_{\\-\\-1}$\"><code>ein[1, 1]<\/code> $= G^{1}_{\\ \\ 1}$<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[6]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ein<\/span><span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[6]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{-k -2 a{\\left(t \\right)} \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)} -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"ein[2,-2]-$=-G^{2}_{\\-\\-2}$\"><code>ein[2, 2]<\/code> $= G^{2}_{\\ \\ 2}$<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[7]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ein<\/span><span class=\"p\">[<\/span><span class=\"mi\">2<\/span><span class=\"p\">,<\/span><span class=\"mi\">2<\/span><span class=\"p\">]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[7]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{-k -2 a{\\left(t \\right)} \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)} -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"ein[3,-3]-$=-G^{3}_{\\-\\-3}$\"><code>ein[3, 3]<\/code> $= G^{3}_{\\ \\ 3}$<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[8]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ein<\/span><span class=\"p\">[<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span><span class=\"mi\">3<\/span><span class=\"p\">]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[8]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{-k -2 a{\\left(t \\right)} \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)} -\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u5ff5\u306e\u305f\u3081\uff0c$G^{2}_{\\ \\ 2}$ \u3068 $G^{3}_{\\ \\ 3}$ \u304c\u7b49\u3057\u3044\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[9]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ein<\/span><span class=\"p\">[<\/span><span class=\"mi\">2<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">]<\/span> <span class=\"o\">==<\/span> <span class=\"n\">ein<\/span><span class=\"p\">[<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[9]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u5b8c\u5168\u6d41\u4f53\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb\">\u5b8c\u5168\u6d41\u4f53\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb <code>T(mu, nu, P)<\/code><\/h3>\n<p>\\begin{eqnarray}<br \/>\nT^{\\mu}_{\\ \\ \\nu} &amp;=&amp; (\\rho + P) u^{\\mu} u_{\\nu} + P \\delta^{\\mu}_{\\ \\ \\nu} \\\\<br \/>\nu^{\\mu} &amp;=&amp; (1, 0, 0, 0) \\\\<br \/>\nu_{\\nu} &amp;=&amp; g_{\\nu\\mu} u^{\\mu} = (-1, 0, 0, 0)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u306e\u30c7\u30eb\u30bf\u306f\uff0c$\\delta^{\\mu}_{\\ \\ \\nu} = $ <code>KroneckerDelta(mu, nu)<\/code><\/p>\n<ul>\n<li><code>rho<\/code> $ \\equiv \\rho(t) $<\/li>\n<li><code>P<\/code> $\\equiv P(t) $<\/li>\n<li><code>uu<\/code> $ \\equiv u^{\\mu}$<\/li>\n<li><code>ud<\/code> $ \\equiv u_{\\nu}$<\/li>\n<li><code>T(mu, nu, P)<\/code> $ \\equiv T^{\\mu}_{\\ \\ \\nu}$<\/li>\n<\/ul>\n<p>\u3068\u5b9a\u7fa9\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[10]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6 \u03c1<\/span>\r\n<span class=\"n\">rho<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'rho'<\/span><span class=\"p\">)(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span>\r\n<span class=\"c1\"># \u5727\u529b P<\/span>\r\n<span class=\"n\">P<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'P'<\/span><span class=\"p\">)(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># u^{\\mu}<\/span>\r\n<span class=\"n\">uu<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n<span class=\"c1\"># u_{\\nu}<\/span>\r\n<span class=\"n\">ud<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n\r\n<span class=\"k\">def<\/span> <span class=\"nf\">T<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">nu<\/span><span class=\"p\">,<\/span> <span class=\"n\">P<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"p\">(<\/span><span class=\"n\">rho<\/span> <span class=\"o\">+<\/span> <span class=\"n\">P<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">uu<\/span><span class=\"p\">[<\/span><span class=\"n\">mu<\/span><span class=\"p\">]<\/span><span class=\"o\">*<\/span><span class=\"n\">ud<\/span><span class=\"p\">[<\/span><span class=\"n\">nu<\/span><span class=\"p\">]<\/span> <span class=\"o\">+<\/span> <span class=\"n\">P<\/span><span class=\"o\">*<\/span><span class=\"n\">KroneckerDelta<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">nu<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\">\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f <code>EinEq(mu, nu, P)<\/code><\/h3>\n<p>\\begin{eqnarray}<br \/>\nG^{\\mu}_{\\ \\ \\nu} +\\Lambda \\delta^{\\mu}_{\\ \\ \\nu} &amp;=&amp; 8\\pi G T^{\\mu}_{\\ \\ \\nu} \\\\<br \/>\n\\therefore\\ \\ G^{\\mu}_{\\ \\ \\nu} &amp;=&amp; 8\\pi G T^{\\mu}_{\\ \\ \\nu} -\\Lambda \\delta^{\\mu}_{\\ \\ \\nu}<br \/>\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[11]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u5b87\u5b99\u5b9a\u6570 \u039b \u3068 sympy.core.function.Lambda \u304c\u304b\u3061\u5408\u3046\u306e\u3067<\/span>\r\n<span class=\"c1\"># \u3042\u3089\u305f\u3081\u3066\u5909\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'Lambda'<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[11]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\Lambda$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>EinEq(mu, nu, P)<\/code>: $ G^{\\mu}_{\\ \\ \\nu} = 8\\pi G T^{\\mu}_{\\ \\ \\nu} -\\Lambda \\delta^{\\mu}_{\\ \\ \\nu}$ \u3068\u5b9a\u7fa9\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[12]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">nu<\/span><span class=\"p\">,<\/span> <span class=\"n\">P<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">expand<\/span><span class=\"p\">(<\/span><span class=\"n\">ein<\/span><span class=\"p\">[<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">nu<\/span><span class=\"p\">]),<\/span> \r\n              <span class=\"mi\">8<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"o\">*<\/span><span class=\"n\">G<\/span><span class=\"o\">*<\/span><span class=\"n\">T<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">nu<\/span><span class=\"p\">,<\/span> <span class=\"n\">P<\/span><span class=\"p\">)<\/span> <span class=\"o\">-<\/span> <span class=\"n\">Lambda<\/span><span class=\"o\">*<\/span><span class=\"n\">KroneckerDelta<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">nu<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u30c0\u30b9\u30c8\u7269\u8cea\u306e\u5834\u5408\">\u30c0\u30b9\u30c8\u7269\u8cea\u306e\u5834\u5408<\/h4>\n<p>$P = 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"EinEq(0,-0,-0):-$G^{0}_{\\-\\-0}-=-8\\pi-G-T^{0}_{\\-\\-0}---\\Lambda-\\delta^{0}_{\\-\\-0}-$\"><code>EinEq(0, 0, 0)<\/code>: $G^{0}_{\\ \\ 0} = 8\\pi G T^{0}_{\\ \\ 0} -\\Lambda \\delta^{0}_{\\ \\ 0} $<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[13]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[13]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -\\frac{3 k}{a^{2}{\\left(t \\right)}} -\\frac{3 \\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = -8 \\pi G \\rho{\\left(t \\right)} -\\Lambda$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>Eq(LHS, RHS)<\/code> \u3067\u8868\u3055\u308c\u308b\u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u306b\u5b9a\u6570 $-\\tfrac{1}{3}$ \u3092\u304b\u3051\u308b\u306b\u306f\uff0c<\/p>\n<p><code>Eq(LHS, RHS)*(-1\/3)<\/code> \u3067\u306f\u306a\u304f\u3066\uff0c<\/p>\n<p>\u5de6\u8fba <code>LHS<\/code>\uff0c\u53f3\u8fba <code>RHS<\/code> \u305d\u308c\u305e\u308c\u306b $-\\tfrac{1}{3}$ \u3092\u304b\u3051\u3066\u304b\u3089\u3042\u3089\u305f\u3081\u3066<\/p>\n<p><code>Eq(LHS*(-1\/3), RHS*(-1\/3))<\/code> \u3068\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>\u65b9\u7a0b\u5f0f <code>Eq()<\/code> \u306e\u5de6\u8fba\u306f <code>Eq().lhs<\/code>\uff0c\u53f3\u8fba\u306f <code>Eq().rhs<\/code> \u3068\u3057\u3066\u53c2\u7167\u3067\u304d\u307e\u3059\u3002\u4ee5\u4e0b\u3067\u306f\u306b $-\\tfrac{1}{3}$ \u3092\u304b\u3051\u307e\u3059\u3002<\/p>\n<p><code>EinEq(0,0,0) * (-Rational(1,3))<\/code> \u3059\u306a\u308f\u3061<\/p>\n<p>$\\displaystyle -\\frac{1}{3} G^{0}_{\\ \\ 0} = -\\frac{8 \\pi G}{3} + \\frac{\\Lambda}{3}$ \u3092\u8868\u793a\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[14]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">LHS<\/span> <span class=\"o\">=<\/span> <span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span>\r\n<span class=\"n\">RHS<\/span> <span class=\"o\">=<\/span> <span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span>\r\n\r\n<span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">LHS<\/span> <span class=\"o\">*<\/span> <span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">3<\/span><span class=\"p\">)),<\/span> \r\n   <span class=\"n\">RHS<\/span> <span class=\"o\">*<\/span> <span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">3<\/span><span class=\"p\">)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[14]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{k}{a^{2}{\\left(t \\right)}} + \\frac{\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = \\frac{8 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\">\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/h5>\n<p><code>Friedmann<\/code> $=$ <code>EinEq(0,0,0) * (-Rational(1,3))<\/code> \u3068\u304a\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[15]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e0a\u306e\u30bb\u30eb\u306e\u51fa\u529b\u7d50\u679c\u3092\u5909\u6570 Friemann \u306b\u4ee3\u5165\u3057\uff0c<\/span>\r\n<span class=\"c1\"># Friedmann \u65b9\u7a0b\u5f0f\u3068\u3059\u308b<\/span>\r\n<span class=\"n\">Friedmann<\/span> <span class=\"o\">=<\/span> <span class=\"n\">_<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"EinEq(1,-1,-0):-$G^{1}_{\\-\\-1}-=-8\\pi-G-T^{1}_{\\-\\-1}---\\Lambda-\\delta^{1}_{\\-\\-1}-$\"><code>EinEq(1, 1, 0)<\/code>: $G^{1}_{\\ \\ 1} = 8\\pi G T^{1}_{\\ \\ 1} -\\Lambda \\delta^{1}_{\\ \\ 1} $<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[16]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[16]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -\\frac{k}{a^{2}{\\left(t \\right)}} -\\frac{2 \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a{\\left(t \\right)}} -\\frac{\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = -\\Lambda$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>Friedmann<\/code> \u3059\u306a\u308f\u3061 <code>EinEq(0,0,0) * (-Rational(1,3))<\/code> \u3068 <code>EinEq(1,1,0)<\/code> \u306e\u4e21\u8fba\u3092\u8db3\u3057\u3066 $-2$ \u3067\u308f\u308a\uff0c$k$ \u3068 $\\displaystyle \\left(\\frac{\\dot{a}}{a}\\right)^2$ \u306e\u9805\u3092\u6d88\u53bb\u3057\uff0c$\\displaystyle \\frac{\\ddot{a}}{a}$ \u306e\u5f0f\u306b\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[17]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e0a\u306e\u5f0f\u3068 Friedmann \u65b9\u7a0b\u5f0f\u3092\u8db3\u3057\u3066 -2 \u3067\u5272\u308b<\/span>\r\n<span class=\"n\">Eq<\/span><span class=\"p\">((<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span> <span class=\"o\">+<\/span> <span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">2<\/span><span class=\"p\">)),<\/span> \r\n   <span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span> <span class=\"o\">+<\/span> <span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">2<\/span><span class=\"p\">)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[17]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{\\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a{\\left(t \\right)}} = -\\frac{4 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"$\\ddot{a}$-\u306e\u5f0f\">$\\ddot{a}$ \u306e\u5f0f<\/h5>\n<p>\u4e0a\u3067\u5f97\u3089\u308c\u305f $\\displaystyle \\frac{\\ddot{a}}{a}$ \u306e\u5f0f\u3092<br \/>\n<code>att<\/code> \u3068\u304a\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[18]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e0a\u306e\u30bb\u30eb\u306e\u51fa\u529b\u7d50\u679c\u3092\u5909\u6570 att \u306b\u4ee3\u5165\u3057\uff0c <\/span>\r\n<span class=\"c1\"># a \u306e2\u968e\u5fae\u5206\u306e\u65b9\u7a0b\u5f0f\u3068\u3059\u308b<\/span>\r\n<span class=\"n\">att<\/span> <span class=\"o\">=<\/span> <span class=\"n\">_<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u306e\u5f0f\">\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u306e\u5f0f<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Friedmann \u65b9\u7a0b\u5f0f <code>Friedmann<\/code> \u306f $\\displaystyle \\left(\\frac{\\dot{a}}{a} \\right)^2$ \u3092\u542b\u3080\u5f0f\u3067\u3042\u308a&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[19]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Friedmann<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[19]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{k}{a^{2}{\\left(t \\right)}} + \\frac{\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = \\frac{8 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>att<\/code> \u306f $\\displaystyle \\frac{\\ddot{a}}{a}$ \u306b\u3064\u3044\u3066\u306e\u65b9\u7a0b\u5f0f\u3067\u3042\u308a&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[20]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">att<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[20]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{\\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a{\\left(t \\right)}} = -\\frac{4 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>Friedmann<\/code> \u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u3092 $t$ \u3067\u5fae\u5206\u3059\u308b\u3068&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[21]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">,<\/span> <span class=\"n\">t<\/span><span class=\"p\">),<\/span> \r\n   <span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span><span class=\"p\">,<\/span> <span class=\"n\">t<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[21]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -\\frac{2 k \\frac{d}{d t} a{\\left(t \\right)}}{a^{3}{\\left(t \\right)}} + \\frac{2 \\frac{d}{d t} a{\\left(t \\right)} \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a^{2}{\\left(t \\right)}} -\\frac{2 \\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{3}}{a^{3}{\\left(t \\right)}} = \\frac{8 \\pi G \\frac{d}{d t} \\rho{\\left(t \\right)}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>Friedmann<\/code> \u5f0f\u3068\u305d\u306e\u5fae\u5206\uff0c\u305d\u3057\u3066 <code>att<\/code> \u5f0f\u3092\u4f7f\u3063\u3066\uff0c$\\rho$ \u304c\u6e80\u305f\u3059\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3002<\/p>\n<p>$\\displaystyle \\Bigl(\\displaystyle \\frac{d}{dt}$ (<code>Friedmann<\/code>) + $\\displaystyle 2 \\frac{\\dot{a}}{a} \\times$ (<code>Friedmann<\/code>) -$\\displaystyle 2 \\frac{\\dot{a}}{a} \\times$ (<code>att<\/code>) $\\displaystyle \\Bigr) \\times \\frac{3}{8 \\pi G}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[22]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">LHS<\/span> <span class=\"o\">=<\/span> <span class=\"p\">((<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">,<\/span> <span class=\"n\">t<\/span><span class=\"p\">)<\/span>  \r\n        <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span>  \r\n        <span class=\"o\">-<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">att<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span> <span class=\"mi\">3<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">8<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"o\">*<\/span><span class=\"n\">G<\/span><span class=\"p\">))<\/span><span class=\"o\">.<\/span><span class=\"n\">expand<\/span><span class=\"p\">()<\/span>\r\n\r\n<span class=\"n\">RHS<\/span> <span class=\"o\">=<\/span> <span class=\"p\">((<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span><span class=\"p\">,<\/span> <span class=\"n\">t<\/span><span class=\"p\">)<\/span>  \r\n        <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span>  \r\n        <span class=\"o\">-<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">att<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span> <span class=\"mi\">3<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">8<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"o\">*<\/span><span class=\"n\">G<\/span><span class=\"p\">))<\/span><span class=\"o\">.<\/span><span class=\"n\">expand<\/span><span class=\"p\">()<\/span>\r\n\r\n<span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">RHS<\/span><span class=\"p\">,<\/span> <span class=\"n\">LHS<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[22]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d}{d t} \\rho{\\left(t \\right)} + \\frac{3 \\rho{\\left(t \\right)} \\frac{d}{d t} a{\\left(t \\right)}}{a{\\left(t \\right)}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[23]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">eqrho<\/span> <span class=\"o\">=<\/span> <span class=\"n\">_<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[24]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">display<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">display<\/span><span class=\"p\">(<\/span><span class=\"n\">att<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">display<\/span><span class=\"p\">(<\/span><span class=\"n\">eqrho<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">$\\displaystyle \\frac{k}{a^{2}{\\left(t \\right)}} + \\frac{\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = \\frac{8 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">$\\displaystyle \\frac{\\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a{\\left(t \\right)}} = -\\frac{4 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">$\\displaystyle \\frac{d}{d t} \\rho{\\left(t \\right)} + \\frac{3 \\rho{\\left(t \\right)} \\frac{d}{d t} a{\\left(t \\right)}}{a{\\left(t \\right)}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u4ee5\u4e0b\u306e\u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u305f\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} &amp;=&amp; \\frac{8\\pi G}{3} \\rho + \\frac{\\Lambda}{3}\\\\<br \/>\n\\frac{\\ddot{a}}{a} &amp;=&amp; -\\frac{4\\pi G}{3} \\rho + \\frac{\\Lambda}{3} \\\\<br \/>\n\\dot{\\rho} + 3 \\frac{\\dot{a}}{a} \\rho &amp;=&amp; 0<br \/>\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u5727\u529b\u304c\u3042\u308b\u7269\u8cea\u306e\u5834\u5408\">\u5727\u529b\u304c\u3042\u308b\u7269\u8cea\u306e\u5834\u5408<\/h4>\n<p>$P \\neq 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"EinEq(0,-0,-P):-$G^{0}_{\\-\\-0}-=-8\\pi-G-T^{0}_{\\-\\-0}---\\Lambda-\\delta^{0}_{\\-\\-0}-$\"><code>EinEq(0, 0, P)<\/code>: $G^{0}_{\\ \\ 0} = 8\\pi G T^{0}_{\\ \\ 0} -\\Lambda \\delta^{0}_{\\ \\ 0} $<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[25]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"n\">P<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[25]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -\\frac{3 k}{a^{2}{\\left(t \\right)}} -\\frac{3 \\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = -8 \\pi G \\rho{\\left(t \\right)} -\\Lambda$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>EinEq(0,0,0) * (-Rational(1,3))<\/code> \u3059\u306a\u308f\u3061<\/p>\n<p>$\\displaystyle -\\frac{1}{3} G^{0}_{\\ \\ 0} = -\\frac{8 \\pi G}{3} + \\frac{\\Lambda}{3}$ \u3092\u8868\u793a\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[26]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e21\u8fba\u3092 -3 \u3067\u5272\u308b<\/span>\r\n<span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"n\">P<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span> <span class=\"o\">*<\/span> <span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">3<\/span><span class=\"p\">)),<\/span> \r\n   <span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span><span class=\"n\">P<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span> <span class=\"o\">*<\/span> <span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">3<\/span><span class=\"p\">)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[26]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{k}{a^{2}{\\left(t \\right)}} + \\frac{\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = \\frac{8 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\">\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/h5>\n<p><code>Friedmann<\/code> $=$ <code>EinEq(0,0,P) * (-Rational(1,3))<\/code> \u3068\u304a\u304d\uff0c<code>Friedmann<\/code> \u65b9\u7a0b\u5f0f\u3068\u3057\u3066\u53c2\u7167\u3059\u308b\u3002\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u306f $P=0$ \u306e\u5834\u5408\u3068\u540c\u3058\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[27]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e0a\u306e\u30bb\u30eb\u306e\u51fa\u529b\u7d50\u679c\u3092\u5909\u6570 Friemann \u306b\u4ee3\u5165\u3057\uff0c<\/span>\r\n<span class=\"c1\"># Friedmann \u65b9\u7a0b\u5f0f\u3068\u3059\u308b<\/span>\r\n<span class=\"n\">Friedmann<\/span> <span class=\"o\">=<\/span> <span class=\"n\">_<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"EinEq(1,-1,-P):-$G^{1}_{\\-\\-1}-=-8\\pi-G-T^{1}_{\\-\\-1}---\\Lambda-\\delta^{1}_{\\-\\-1}-$\"><code>EinEq(1, 1, P)<\/code>: $G^{1}_{\\ \\ 1} = 8\\pi G T^{1}_{\\ \\ 1} -\\Lambda \\delta^{1}_{\\ \\ 1} $<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[28]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">P<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[28]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -\\frac{k}{a^{2}{\\left(t \\right)}} -\\frac{2 \\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a{\\left(t \\right)}} -\\frac{\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = 8 \\pi G P{\\left(t \\right)} -\\Lambda$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3053\u306e\u5f0f\u3068 <code>Friedmann<\/code> \u306e\u4e21\u8fba\u3092\u8db3\u3057\u3066 $-2$ \u3067\u308f\u308a\uff0c$k$ \u3068 $\\displaystyle \\left(\\frac{\\dot{a}}{a}\\right)^2$ \u306e\u9805\u3092\u6d88\u53bb\u3057\uff0c$\\displaystyle \\frac{\\ddot{a}}{a}$ \u306e\u5f0f\u306b\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[29]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e0a\u306e\u5f0f\u3068 Friedmann \u65b9\u7a0b\u5f0f\u3092\u8db3\u3057\u3066 -2 \u3067\u5272\u308b<\/span>\r\n<span class=\"n\">Eq<\/span><span class=\"p\">((<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span> <span class=\"o\">+<\/span> <span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"n\">P<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">2<\/span><span class=\"p\">)),<\/span> \r\n   <span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span> <span class=\"o\">+<\/span> <span class=\"n\">EinEq<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"n\">P<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">Rational<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span><span class=\"mi\">2<\/span><span class=\"p\">)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[29]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{\\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a{\\left(t \\right)}} = -4 \\pi G P{\\left(t \\right)} -\\frac{4 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"$\\ddot{a}$-\u306e\u5f0f\">$\\ddot{a}$ \u306e\u5f0f<\/h5>\n<p>\u4e0a\u3067\u5f97\u3089\u308c\u305f $\\displaystyle \\frac{\\ddot{a}}{a}$ \u306e\u5f0f\u3092<br \/>\n<code>att<\/code> \u3068\u304a\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[30]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e0a\u306e\u30bb\u30eb\u306e\u51fa\u529b\u7d50\u679c\u3092\u5909\u6570 att \u306b\u4ee3\u5165\u3057\uff0c <\/span>\r\n<span class=\"c1\"># a \u306e2\u968e\u5fae\u5206\u306e\u65b9\u7a0b\u5f0f\u3068\u3059\u308b<\/span>\r\n<span class=\"n\">att<\/span> <span class=\"o\">=<\/span> <span class=\"n\">_<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u306e\u5f0f\">\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u306e\u5f0f<\/h5>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>Friedmann<\/code> \u5f0f\u3068\u305d\u306e\u5fae\u5206\uff0c\u305d\u3057\u3066 <code>att<\/code> \u5f0f\u3092\u4f7f\u3063\u3066\uff0c$\\rho$ \u304c\u6e80\u305f\u3059\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3002<\/p>\n<p>$\\displaystyle \\Bigl(\\displaystyle \\frac{d}{dt}$ (<code>Friedmann<\/code>) + $\\displaystyle 2 \\frac{\\dot{a}}{a} \\times$ (<code>Friedmann<\/code>) -$\\displaystyle 2 \\frac{\\dot{a}}{a} \\times$ (<code>att<\/code>) $\\displaystyle \\Bigr) \\times \\frac{3}{8 \\pi G}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[31]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">LHS<\/span> <span class=\"o\">=<\/span> <span class=\"p\">((<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">,<\/span> <span class=\"n\">t<\/span><span class=\"p\">)<\/span>  \r\n        <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span>  \r\n        <span class=\"o\">-<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">att<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span> <span class=\"mi\">3<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">8<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"o\">*<\/span><span class=\"n\">G<\/span><span class=\"p\">))<\/span><span class=\"o\">.<\/span><span class=\"n\">expand<\/span><span class=\"p\">()<\/span>\r\n\r\n<span class=\"n\">RHS<\/span> <span class=\"o\">=<\/span> <span class=\"p\">((<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span><span class=\"p\">,<\/span> <span class=\"n\">t<\/span><span class=\"p\">)<\/span>  \r\n        <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">Friedmann<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span>  \r\n        <span class=\"o\">-<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">a<\/span> <span class=\"o\">*<\/span> <span class=\"n\">att<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span> <span class=\"mi\">3<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">8<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"o\">*<\/span><span class=\"n\">G<\/span><span class=\"p\">))<\/span><span class=\"o\">.<\/span><span class=\"n\">expand<\/span><span class=\"p\">()<\/span>\r\n\r\n<span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">RHS<\/span><span class=\"p\">,<\/span> <span class=\"n\">LHS<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[31]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{3 P{\\left(t \\right)} \\frac{d}{d t} a{\\left(t \\right)}}{a{\\left(t \\right)}} + \\frac{d}{d t} \\rho{\\left(t \\right)} + \\frac{3 \\rho{\\left(t \\right)} \\frac{d}{d t} a{\\left(t \\right)}}{a{\\left(t \\right)}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[32]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">eqrho<\/span> <span class=\"o\">=<\/span> <span class=\"n\">_<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[33]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">display<\/span><span class=\"p\">(<\/span><span class=\"n\">Friedmann<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">display<\/span><span class=\"p\">(<\/span><span class=\"n\">att<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">display<\/span><span class=\"p\">(<\/span><span class=\"n\">eqrho<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">$\\displaystyle \\frac{k}{a^{2}{\\left(t \\right)}} + \\frac{\\left(\\frac{d}{d t} a{\\left(t \\right)}\\right)^{2}}{a^{2}{\\left(t \\right)}} = \\frac{8 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">$\\displaystyle \\frac{\\frac{d^{2}}{d t^{2}} a{\\left(t \\right)}}{a{\\left(t \\right)}} = -4 \\pi G P{\\left(t \\right)} -\\frac{4 \\pi G \\rho{\\left(t \\right)}}{3} + \\frac{\\Lambda}{3}$<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">$\\displaystyle \\frac{3 P{\\left(t \\right)} \\frac{d}{d t} a{\\left(t \\right)}}{a{\\left(t \\right)}} + \\frac{d}{d t} \\rho{\\left(t \\right)} + \\frac{3 \\rho{\\left(t \\right)} \\frac{d}{d t} a{\\left(t \\right)}}{a{\\left(t \\right)}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3068\u3044\u3046\u3053\u3068\u3067\uff0c$P \\neq 0$ \u306e\u5834\u5408\u306b\u4ee5\u4e0b\u306e\u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u305f\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} &amp;=&amp; \\frac{8\\pi G}{3} \\rho + \\frac{\\Lambda}{3}\\\\<br \/>\n\\frac{\\ddot{a}}{a} &amp;=&amp; -\\frac{4\\pi G}{3} (\\rho + 3 P) + \\frac{\\Lambda}{3} \\\\<br \/>\n\\dot{\\rho} + 3 \\frac{\\dot{a}}{a} (\\rho + P) &amp;=&amp; 0<br \/>\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f$$G^{\\mu}_{\\ \\ \\nu} + \\Lambda \\delta^{\\mu}_{\\ \\ \\nu} = 8\\pi G T^{\\mu}_{\\ \\ \\nu}$$\u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3002<\/p><p><a class=\"more-link btn\" href=\"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":5114,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-5189","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<link rel=\"canonical\" href=\"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.0.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"ja_JP\" \/>\n\t\t<meta property=\"og:site_name\" content=\"\u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba -\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b - \u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba\" \/>\n\t\t<meta property=\"og:description\" content=\"EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3\" \/>\n\t\t<meta property=\"og:url\" content=\"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2024-07-16T02:50:28+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2025-02-04T01:39:32+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary\" \/>\n\t\t<meta name=\"twitter:title\" content=\"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b - \u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba\" \/>\n\t\t<meta name=\"twitter:description\" content=\"EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"BreadcrumbList\",\"@id\":\"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\\\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\\\/#breadcrumblist\",\"itemListElement\":[{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity#listItem\",\"position\":1,\"name\":\"\\u30db\\u30fc\\u30e0\",\"item\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"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\\\/#listItem\",\"name\":\"\\u4e00\\u822c\\u76f8\\u5bfe\\u8ad6\\u7684\\u6642\\u7a7a\\u306e\\u8868\\u3057\\u65b9\"}},{\"@type\":\"ListItem\",\"@id\":\"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\\\/#listItem\",\"position\":2,\"name\":\"\\u4e00\\u822c\\u76f8\\u5bfe\\u8ad6\\u7684\\u6642\\u7a7a\\u306e\\u8868\\u3057\\u65b9\",\"item\":\"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\\\/\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"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\\\/#listItem\",\"name\":\"EinsteinPy \\u3084 ctensor \\u3067\\u30a2\\u30a4\\u30f3\\u30b7\\u30e5\\u30bf\\u30a4\\u30f3\\u65b9\\u7a0b\\u5f0f\\u3092\\u8a08\\u7b97\\u3059\\u308b\"},\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity#listItem\",\"name\":\"\\u30db\\u30fc\\u30e0\"}},{\"@type\":\"ListItem\",\"@id\":\"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\\\/#listItem\",\"position\":3,\"name\":\"EinsteinPy \\u3084 ctensor \\u3067\\u30a2\\u30a4\\u30f3\\u30b7\\u30e5\\u30bf\\u30a4\\u30f3\\u65b9\\u7a0b\\u5f0f\\u3092\\u8a08\\u7b97\\u3059\\u308b\",\"item\":\"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\\\/\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"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\\\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\\\/#listItem\",\"name\":\"EinsteinPy \\u3067\\u30d5\\u30ea\\u30fc\\u30c9\\u30de\\u30f3\\u65b9\\u7a0b\\u5f0f\\u3092\\u6c42\\u3081\\u308b\"},\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"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\\\/#listItem\",\"name\":\"\\u4e00\\u822c\\u76f8\\u5bfe\\u8ad6\\u7684\\u6642\\u7a7a\\u306e\\u8868\\u3057\\u65b9\"}},{\"@type\":\"ListItem\",\"@id\":\"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\\\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\\\/#listItem\",\"position\":4,\"name\":\"EinsteinPy \\u3067\\u30d5\\u30ea\\u30fc\\u30c9\\u30de\\u30f3\\u65b9\\u7a0b\\u5f0f\\u3092\\u6c42\\u3081\\u308b\",\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"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\\\/#listItem\",\"name\":\"EinsteinPy \\u3084 ctensor \\u3067\\u30a2\\u30a4\\u30f3\\u30b7\\u30e5\\u30bf\\u30a4\\u30f3\\u65b9\\u7a0b\\u5f0f\\u3092\\u8a08\\u7b97\\u3059\\u308b\"}}]},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity\\\/#organization\",\"name\":\"Understanding Theory of Relativity\",\"url\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity\\\/\"},{\"@type\":\"WebPage\",\"@id\":\"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\\\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\\\/#webpage\",\"url\":\"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\\\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\\\/\",\"name\":\"EinsteinPy \\u3067\\u30d5\\u30ea\\u30fc\\u30c9\\u30de\\u30f3\\u65b9\\u7a0b\\u5f0f\\u3092\\u6c42\\u3081\\u308b - \\u76f8\\u5bfe\\u8ad6\\u306e\\u7406\\u89e3\\u3068\\u305d\\u306e\\u5468\\u8fba\",\"description\":\"EinsteinPy \\u3092\\u4f7f\\u3063\\u3066\\u30d5\\u30ea\\u30fc\\u30c9\\u30de\\u30f3\\u30fb\\u30eb\\u30e1\\u30fc\\u30c8\\u30eb\\u30fb\\u30ed\\u30d0\\u30fc\\u30c8\\u30bd\\u30f3\\u30fb\\u30a6\\u30a9\\u30fc\\u30ab\\u30fc (FLRW) \\u8a08\\u91cf\\u304b\\u3089\\u30a2\\u30a4\\u30f3\",\"inLanguage\":\"ja\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity\\\/#website\"},\"breadcrumb\":{\"@id\":\"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\\\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\\\/#breadcrumblist\"},\"datePublished\":\"2024-07-16T11:50:28+09:00\",\"dateModified\":\"2025-02-04T10:39:32+09:00\"},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity\\\/#website\",\"url\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity\\\/\",\"name\":\"\\u76f8\\u5bfe\\u8ad6\\u306e\\u7406\\u89e3\\u3068\\u305d\\u306e\\u5468\\u8fba\",\"inLanguage\":\"ja\",\"publisher\":{\"@id\":\"https:\\\/\\\/home.hirosaki-u.ac.jp\\\/relativity\\\/#organization\"}}]}\n\t\t<\/script>\n\t\t<!-- All in One SEO -->\n\n","aioseo_head_json":{"title":"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b - \u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba","description":"EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3","canonical_url":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"BreadcrumbList","@id":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/#breadcrumblist","itemListElement":[{"@type":"ListItem","@id":"https:\/\/home.hirosaki-u.ac.jp\/relativity#listItem","position":1,"name":"\u30db\u30fc\u30e0","item":"https:\/\/home.hirosaki-u.ac.jp\/relativity","nextItem":{"@type":"ListItem","@id":"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\/#listItem","name":"\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u6642\u7a7a\u306e\u8868\u3057\u65b9"}},{"@type":"ListItem","@id":"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\/#listItem","position":2,"name":"\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u6642\u7a7a\u306e\u8868\u3057\u65b9","item":"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\/","nextItem":{"@type":"ListItem","@id":"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\/#listItem","name":"EinsteinPy \u3084 ctensor \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u8a08\u7b97\u3059\u308b"},"previousItem":{"@type":"ListItem","@id":"https:\/\/home.hirosaki-u.ac.jp\/relativity#listItem","name":"\u30db\u30fc\u30e0"}},{"@type":"ListItem","@id":"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\/#listItem","position":3,"name":"EinsteinPy \u3084 ctensor \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u8a08\u7b97\u3059\u308b","item":"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\/","nextItem":{"@type":"ListItem","@id":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/#listItem","name":"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b"},"previousItem":{"@type":"ListItem","@id":"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\/#listItem","name":"\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u6642\u7a7a\u306e\u8868\u3057\u65b9"}},{"@type":"ListItem","@id":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/#listItem","position":4,"name":"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b","previousItem":{"@type":"ListItem","@id":"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\/#listItem","name":"EinsteinPy \u3084 ctensor \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u8a08\u7b97\u3059\u308b"}}]},{"@type":"Organization","@id":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/#organization","name":"Understanding Theory of Relativity","url":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/"},{"@type":"WebPage","@id":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/#webpage","url":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/","name":"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b - \u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba","description":"EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3","inLanguage":"ja","isPartOf":{"@id":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/#website"},"breadcrumb":{"@id":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/#breadcrumblist"},"datePublished":"2024-07-16T11:50:28+09:00","dateModified":"2025-02-04T10:39:32+09:00"},{"@type":"WebSite","@id":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/#website","url":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/","name":"\u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba","inLanguage":"ja","publisher":{"@id":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/#organization"}}]},"og:locale":"ja_JP","og:site_name":"\u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba -","og:type":"article","og:title":"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b - \u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba","og:description":"EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3","og:url":"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\/einsteinpy-%e3%81%a7%e3%83%95%e3%83%aa%e3%83%bc%e3%83%89%e3%83%9e%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/","article:published_time":"2024-07-16T02:50:28+00:00","article:modified_time":"2025-02-04T01:39:32+00:00","twitter:card":"summary","twitter:title":"EinsteinPy \u3067\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b - \u76f8\u5bfe\u8ad6\u306e\u7406\u89e3\u3068\u305d\u306e\u5468\u8fba","twitter:description":"EinsteinPy \u3092\u4f7f\u3063\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u30fb\u30ed\u30d0\u30fc\u30c8\u30bd\u30f3\u30fb\u30a6\u30a9\u30fc\u30ab\u30fc (FLRW) \u8a08\u91cf\u304b\u3089\u30a2\u30a4\u30f3"},"aioseo_meta_data":{"post_id":"5189","title":null,"description":null,"keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"WebPage","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":"{\"article\":{\"articleType\":\"BlogPosting\"},\"course\":{\"name\":\"\",\"description\":\"\",\"provider\":\"\"},\"faq\":{\"pages\":[]},\"product\":{\"reviews\":[]},\"recipe\":{\"ingredients\":[],\"instructions\":[],\"keywords\":[]},\"software\":{\"reviews\":[],\"operatingSystems\":[]},\"webPage\":{\"webPageType\":\"WebPage\"}}","pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"ai":null,"created":"2023-01-27 08:11:28","updated":"2025-02-04 01:39:32","seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/5189","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"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=5189"}],"version-history":[{"count":14,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/5189\/revisions"}],"predecessor-version":[{"id":10186,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/5189\/revisions\/10186"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/5114"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=5189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}