{"id":1463,"date":"2022-01-25T13:10:34","date_gmt":"2022-01-25T04:10:34","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1463"},"modified":"2024-07-10T17:40:21","modified_gmt":"2024-07-10T08:40:21","slug":"%e5%ae%9a%e6%9b%b2%e7%8e%87%e7%a9%ba%e9%96%93%e3%81%ae%e8%a8%88%e9%87%8f","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%e5%ae%87%e5%ae%99%e8%ab%96\/%e5%ae%9a%e6%9b%b2%e7%8e%87%e7%a9%ba%e9%96%93%e3%81%ae%e8%a8%88%e9%87%8f\/","title":{"rendered":"\u5b9a\u66f2\u7387\u7a7a\u9593\u306e\u8a08\u91cf"},"content":{"rendered":"<p>FLRW \u306e3\u6b21\u5143\u7a7a\u9593\u306f\u5b9a\u66f2\u7387\u7a7a\u9593\u3067\u3042\u3063\u305f\u3002\u5b9a\u66f2\u7387\u7a7a\u9593\u3092\u8868\u3059\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u308b\u3002<!--more--><\/p>\n<h3>3\u6b21\u5143\u5b9a\u66f2\u7387\u7a7a\u9593<\/h3>\n<p>\u5225\u30da\u30fc\u30b8\u3067\uff0c3\u6b21\u5143\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb \\(\\gamma_{ij}\\) \u304b\u3089\u8a08\u7b97\u3055\u308c\u308b\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb \\({}^{(3)}\\! R^{i}_{\\ \\\u00a0 j} \\)\u00a0 \u304c\u66f2\u7387\u5b9a\u6570 \\(k\\) \u3092\u4f7f\u3063\u3066<br \/>\n$${}^{(3)}\\! R^{i}_{\\ \\\u00a0 j} = 2 k \\delta^i_{\\ \\ j}$$\u3068\u8868\u3055\u308c\u308b\u3068\u304d\uff0c\u3053\u306e\u7a7a\u9593\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5b9a\u66f2\u7387\u7a7a\u9593<\/strong><\/span>\u3067\u3042\u308b\uff0c\u3068\u3044\u3063\u305f\u304c\uff0c\u672c\u6765\u306e\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n<hr \/>\n<p style=\"padding-left: 40px;\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u304c\uff08\u66f2\u7387\u5b9a\u6570 \\(k\\) \u3068\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb \\(\\gamma_{ij}\\) \u3060\u3051\u3092\u4f7f\u3063\u3066\uff09<br \/>\n$${}^{(3)}\\! R_{ijkl} = k(\\gamma_{ik} \\gamma_{jl} -\\gamma_{il} \\gamma_{jk})$$<br \/>\n\u3068\u8868\u3055\u308c\u308b\u7a7a\u9593\u3092<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5b9a\u66f2\u7387\u7a7a\u9593<\/strong><\/span>\u3068\u3044\u3046\u3002<\/p>\n<hr \/>\n<p>\u3053\u308c\u304c\u540c\u3058\u3053\u3068\u3092\u8a00\u3063\u3066\u3044\u308b\u306e\u3060\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u306e\u3053\u3068\u304b\u3089\u308f\u304b\u308b\u3002<\/p>\n<p>3\u6b21\u5143\u7a7a\u9593\u3067\u306f\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u81ea\u7531\u5ea6\u3068\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u306e\u81ea\u7531\u5ea6\u304c\u7b49\u3057\u3044\u305f\u3081\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306f\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\uff08\u3068\u30ea\u30c3\u30c1\u30b9\u30ab\u30e9\u30fc \\( {}^{(3)}\\! R\\)\uff09\u3092\u4f7f\u3063\u3066\u66f8\u3051\u3066\u3057\u307e\u3046\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n{}^{(3)}\\! R_{ijkl} &amp;=&amp; \\gamma_{ik} {}^{(3)}\\! R_{jl} -\\gamma_{il} {}^{(3)}\\! R_{jk} -\\gamma_{jk} {}^{(3)}\\! R_{il} + \\gamma_{jl} {}^{(3)}\\! R_{ik} \\\\<br \/>\n&amp;&amp; \\quad + \\frac{1}{2} \\left( \\gamma_{il} \\gamma_{jk}\u00a0 -\\gamma_{ik} \\gamma_{jl} \\right) {}^{(3)}\\! R<br \/>\n\\end{eqnarray}<\/p>\n<p>${}^{(3)}\\! R^{i}_{\\ \\\u00a0 j} = 2 k \\delta^i_{\\ \\ j}$ \u3092\u4ee3\u5165\u3059\u308b\u3068<br \/>\n\\begin{eqnarray}<br \/>\n{}^{(3)}\\! R_{ijkl} &amp;=&amp; \\gamma_{ik} \\cdot 2k\\gamma_{jl} -\\gamma_{il} \\cdot 2k\\gamma_{jk} -\\gamma_{jk} \\cdot 2k\\gamma_{il} + \\gamma_{jl}\\cdot 2 k \\gamma_{ik} \\\\<br \/>\n&amp;&amp; \\quad + \\frac{1}{2} \\left( \\gamma_{il} \\gamma_{jk}\u00a0 -\\gamma_{ik} \\gamma_{jl} \\right) \\cdot 6 k\\\\<br \/>\n&amp;=&amp; k(\\gamma_{ik} \\gamma_{jl} -\\gamma_{il} \\gamma_{jk})<br \/>\n\\end{eqnarray}<br \/>\n\u3068\u306a\u308a\uff0c\u540c\u3058\u3053\u3068\u3092\u8a00\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<h3>\u5171\u5f62\u5e73\u5766\u6027<\/h3>\n<p>\u3055\u3066\uff0c3\u6b21\u5143\u5b9a\u66f2\u7387\u7a7a\u9593\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5171\u5f62\u5e73\u5766<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u3082\u308f\u304b\u308b\u3002\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u304c\u5b9a\u6570 \\(k\\) \u3068\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3060\u3051\u3067\u66f8\u304b\u308c\u308b\u306e\u3067\u3042\u308b\u304b\u3089\uff0c\u30b3\u30c3\u30c8\u30f3\u30c6\u30f3\u30bd\u30eb\u306f\u30bc\u30ed\u306b\u306a\u308b\u3002\u3088\u3063\u3066\u5171\u5f62\u5e73\u5766\u3067\u3042\u308b\u3002\uff08\u5b9a\u66f2\u7387\u7a7a\u9593\u4ee5\u5916\u3067\u3082\u5171\u5f62\u5e73\u5766\u306a\u7a7a\u9593\u306f\u5b58\u5728\u3059\u308b\u3067\u3057\u3087\u3046\u306d\u3047\u3002\u4f8b\u306e\u300c\u5341\u5206\u6761\u4ef6\u3067\u306f\u3042\u308b\u304c\u5fc5\u8981\u6761\u4ef6\u3067\u306f\u306a\u3044\u300d\u3068\u3044\u3046\u3084\u3064\u3067\u3059\u304b&#8230; \uff09<\/p>\n<ul>\n<li><a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%82%B3%E3%83%83%E3%83%88%E3%83%B3%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB\">\u30b3\u30c3\u30c8\u30f3\u30c6\u30f3\u30bd\u30eb &#8211; Wikipedia<\/a><\/li>\n<\/ul>\n<p>3\u6b21\u5143\u7a7a\u9593\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5171\u5f62\u5e73\u5766<\/strong><\/span>\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u3064\u307e\u308a\uff0c\u7dda\u7d20\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306e\u7dda\u7d20\u306b\u6bd4\u4f8b\u3059\u308b\u5f62\u306b\u66f8\u3051\u308b<\/strong><\/span>\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u305f\u3068\u3048\u3070<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\gamma_{ij} dx^i dx^j &amp;=&amp; F(r_1) \\left( dx^2 + dy^2 + dz^2\\right) \\\\<br \/>\n&amp;=&amp; F(r_1) \\left( dr_1^2 + r_1^2 d\\theta^2 + r_1^2 \\sin^2\\theta d\\phi^2\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u3053\u3067 \\(r_1 = \\sqrt{x^2 + y^2 + z^2}\\) \u3002\u3055\u3089\u306b \\(F(r_1) r_1^2 \\equiv r^2 \\) \u3068\u3044\u3046\u5909\u6570\u5909\u63db\u3092\u884c\u3048\u3070\u7dda\u7d20\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3060\u308d\u3046\u3002<\/p>\n<p>\uff08\u8ffd\u8a18\uff1a\u3053\u306e\u8ad6\u7406\u5c55\u958b\u3060\u3068\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5b9a\u66f2\u7387\u7a7a\u9593<\/strong><\/span>\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u5171\u5f62\u5e73\u5766<\/strong><\/span>\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306e\u4ed6\u306b\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7403\u5bfe\u79f0<\/strong><\/span>\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3082\u4f7f\u3063\u3066\u3044\u308b\u306a\u3041\u3002\u7403\u5bfe\u79f0\u6027\u306f\u3069\u306e\u3088\u3046\u306b\u3057\u3066\u4fdd\u8a3c\u3055\u308c\u308b\u306e\u3067\u3042\u308d\u3046\u304b&#8230; \uff09<\/p>\n<p>$$\\gamma_{ij} dx^i dx^j = \\frac{dr^2}{f(r)} + r^2 \\left( d\\theta^2 + \\sin^2\\theta d\\phi^2\\right)$$<\/p>\n<p>\u3053\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u304b\u3089\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u3092\u8a08\u7b97\u3057\uff0c\\(R_{ij} = 2 k \\gamma_{ij}\\) \u306b\u306a\u308b\u3068\u3044\u3046\u6761\u4ef6\u3067 \\(f(r)\\) \u3092\u6c42\u3081\u308b\u3068\uff0c<br \/>\n$$ f(r) = 1 -kr^2$$<br \/>\n\u3068\u306a\u308b\u3002\uff08\u7a4d\u5206\u5b9a\u6570\u306f $r = 0$ \u3067 $f(0) = 1$ \u3068\u3057\u305f\u3002\uff09<\/p>\n<h4>Maxima \u306e ctensor \u3092\u4f7f\u3063\u305f\u8a08\u7b97\u4f8b<\/h4>\n<hr \/>\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\">$$\\gamma_{ij} dx^i dx^j = \\frac{dr^2}{f(r)} + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)$$\u3068\u4eee\u5b9a\u3057\u3066\uff0c\u5b9a\u66f2\u7387\u7a7a\u9593<br \/>\n$$R_{ij} = 2 k \\gamma_{ij}$$<br \/>\n\u3068\u306a\u308b\u3088\u3046\u306b $f(r)$ \u3092\u6c42\u3081\u308b\u3002<\/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-maxima\">\n<pre><span class=\"nf\">load<\/span><span class=\"p\">(<\/span><span class=\"nv\">ctensor<\/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 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-maxima\">\n<pre><span class=\"nf\">csetup<\/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 \">Enter the dimension of the coordinate system:<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>3;\r\n<\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">Do you wish to change the coordinate names?<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>y;\r\n<\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">Enter a list containing the names of the coordinates in order<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>[r, theta, phi];\r\n<\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">Do you want to<br \/>\n1. Enter a new metric?<br \/>\n2. Enter a metric from a file?<br \/>\n3. Approximate a metric with a Taylor series?<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>1;\r\n\r\nIs the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General\r\n\r\nIs the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General\r\nXAnswer 1, 2, 3 or 4 : \\Answer 1, 2, 3 or 4 : 1;\r\n\r\nXRow 1 Column 1: \\Row 1 Column 1: 1\/f(r);\r\n\r\nXRow 2 Column 2: \\Row 2 Column 2: r**2;\r\n\r\nXRow 3 Column 3: \\Row 3 Column 3: r**2 * sin(theta)**2;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">Enter functional dependencies with DEPENDS or &#8216;N&#8217; if none<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Matrix entered.\r\nN;\r\n<\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">Do you wish to see the metric?<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>y;\r\n<\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\begin{pmatrix}\\frac{1}{f\\left(r\\right)} &amp; 0 &amp; 0 \\\\ 0 &amp; r^2 &amp; 0 \\\\ 0 &amp; 0 &amp; r^2\\,\\sin ^2\\vartheta \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[2]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{2}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[3]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">ricci<\/span><span class=\"p\">(<\/span><span class=\"no\">true<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{3}$}{\\it ric}_{1,1}=-\\frac{\\left(f\\left(r\\right)\\right)_{r}}{r\\,f\\left(r\\right)}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{4}$}{\\it ric}_{2,2}=-\\frac{r\\,\\left(f\\left(r\\right)\\right)_{r}}{2}-f\\left(r\\right)+1\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{5}$}{\\it ric}_{3,3}=-\\frac{r\\,\\left(f\\left(r\\right)\\right)_{r}\\,\\sin ^2\\vartheta}{2}-f\\left(r\\right)\\,\\sin ^2\\vartheta+\\sin ^2\\vartheta\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[3]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{5}$}\\mathbf{done}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$\\displaystyle \\frac{R_{11}}{g_{11}} = 2 k$ \u3092\u89e3\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[4]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">eq<\/span><span class=\"o\">:<\/span> <span class=\"nv\">ric<\/span><span class=\"p\">[<\/span><span class=\"mi\">1<\/span>,1<span class=\"p\">]<\/span><span class=\"o\">\/<\/span><span class=\"nv\">lg<\/span><span class=\"p\">[<\/span><span class=\"mi\">1<\/span>,1<span class=\"p\">]<\/span> <span class=\"o\">=<\/span> 2<span class=\"o\">*<\/span><span class=\"nv\">k<\/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\">\\[\\tag{${\\it \\%o}_{6}$}-\\frac{\\left(f\\left(r\\right)\\right)_{r}}{r}=2\\,k\\]<\/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[5]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">sol<\/span><span class=\"o\">:<\/span> <span class=\"nf\">ode2<\/span><span class=\"p\">(<\/span><span class=\"nv\">eq<\/span>, <span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">r<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">r<\/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\">\\[\\tag{${\\it \\%o}_{7}$}f\\left(r\\right)={\\it \\%c}-k\\,r^2\\]<\/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[6]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">f<\/span> <span class=\"o\">=<\/span> <span class=\"nf\">rhs<\/span><span class=\"p\">(<\/span><span class=\"nv\">sol<\/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\">\\[\\tag{${\\it \\%o}_{8}$}f={\\it \\%c}-k\\,r^2\\]<\/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[7]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">ic1<\/span><span class=\"p\">(<\/span><span class=\"nv\">%<\/span>, <span class=\"nv\">r<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">0<\/span>, <span class=\"nv\">f<\/span> <span class=\"o\">=<\/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[7]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{9}$}f=1-k\\,r^2\\]<\/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>\u5099\u8003\uff1a<\/p>\n<p>$\\displaystyle \\frac{R_{22}}{g_{22}} = 2k$ \u3084 $\\displaystyle \\frac{R_{33}}{g_{33}} = 2k$ \u3092\u89e3\u304b\u305b\u308b\u3068\u9593\u9055\u3063\u305f\u89e3\u3092\u51fa\u3059\uff08\u304b\u3082\u3057\u308c\u306a\u3044\uff09\u306e\u3067\u8981\u6ce8\u610f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<hr \/>\n<h4>EinsteinPy \u3068 SymPy \u3092\u4f7f\u3063\u305f\u8a08\u7b97\u4f8b<\/h4>\n<hr \/>\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\">$$\\gamma_{ij} dx^i dx^j = \\frac{dr^2}{f(r)} + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)$$\u3068\u4eee\u5b9a\u3057\u3066\uff0c\u5b9a\u66f2\u7387\u7a7a\u9593<br \/>\n$$R_{ij} = 2 k \\gamma_{ij}$$<br \/>\n\u3068\u306a\u308b\u3088\u3046\u306b $f(r)$ \u3092\u6c42\u3081\u308b\u3002<\/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>EinsteinPy \u3068 SymPy \u306e\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u306f\uff08\u5f18\u5927 JupyterHub \u3067\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u6e08\u307f\u3067\u3059\u3002\uff09<\/p>\n<div class=\"highlight\">\n<pre><span class=\"o\">(<\/span>sudo<span class=\"o\">)<\/span> pip install einsteinpy sympy\r\n<\/pre>\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[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<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\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 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=\"n\">r<\/span><span class=\"p\">,<\/span> <span class=\"n\">theta<\/span><span class=\"p\">,<\/span> <span class=\"n\">phi<\/span> <span class=\"o\">=<\/span> <span class=\"n\">symbols<\/span><span class=\"p\">(<\/span><span class=\"s1\">'r, theta, phi'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">f<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'f'<\/span><span class=\"p\">)(<\/span><span class=\"n\">r<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">Metric<\/span> <span class=\"o\">=<\/span> <span class=\"n\">diag<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"n\">f<\/span><span class=\"p\">,<\/span> <span class=\"n\">r<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">,<\/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<span class=\"n\">gamma<\/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\">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<\/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[3]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">gamma<\/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[3]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\left[\\begin{matrix}\\frac{1}{f{\\left(r \\right)}} &amp; 0 &amp; 0\\\\0 &amp; r^{2} &amp; 0\\\\0 &amp; 0 &amp; r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/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[4]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ric<\/span><span class=\"o\">=<\/span><span class=\"n\">RicciTensor<\/span><span class=\"o\">.<\/span><span class=\"n\">from_metric<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ric<\/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\">$\\displaystyle \\left[\\begin{matrix} -\\frac{\\frac{d}{d r} f{\\left(r \\right)}}{r f{\\left(r \\right)}} &amp; 0 &amp; 0\\\\0 &amp; -\\frac{r \\frac{d}{d r} f{\\left(r \\right)}}{2} -f{\\left(r \\right)} + 1 &amp; 0\\\\0 &amp; 0 &amp; \\left(-\\frac{r \\frac{d}{d r} f{\\left(r \\right)}}{2} -f{\\left(r \\right)} + 1\\right) \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/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<p>$ R_{ij} = 2 k \\gamma_{ij}$ \u3092\u89e3\u304f\u3002<\/p>\n<p>Python \u306f zero \u306f\u3058\u307e\u308a\u306a\u306e\u3067\uff0c$R_{11}$ \u306f <code>ric[0,0]<\/code>\uff0c$g_{11}$ \u306f <code>gamma[0, 0]<\/code> \u3068\u6307\u5b9a\u3057\u307e\u3059\u3002<\/p>\n<p>\u307e\u305a\u306f<br \/>\n$\\displaystyle\\frac{R_{11}}{g_{11}} -2 k = 0$ \u3064\u307e\u308a $\\frac{\\texttt{ric[0,0]}}{\\texttt{gamma[0,0]}} -2 k = 0$ \u3092 $f(r)$ \u306b\u95a2\u3059\u308b\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u3057\u3066\u89e3\u304f\u3002<br \/>\n\u7a4d\u5206\u5b9a\u6570\u306f $r = 0$ \u3067 $f(0) = 1$ \u3068\u3044\u3046\u5883\u754c\u6761\u4ef6\u304b\u3089\u6c7a\u3081\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[5]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">k<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Symbol<\/span><span class=\"p\">(<\/span><span class=\"s1\">'k'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">ric<\/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\">gamma<\/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=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">k<\/span><span class=\"p\">,<\/span> <span class=\"n\">f<\/span><span class=\"p\">,<\/span> <span class=\"n\">ics<\/span><span class=\"o\">=<\/span><span class=\"p\">{<\/span><span class=\"n\">f<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">r<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/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[5]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle f{\\left(r \\right)} = -k r^{2} + 1$<\/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$$\\frac{R_{22}}{g_{22}}\u00a0 -2 k = 0, \\quad \\frac{R_{33}}{g_{33}} -2 k = 0$$\u3082\u89e3\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[6]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">ric<\/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=\"o\">\/<\/span><span class=\"n\">gamma<\/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=\"o\">-<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">k<\/span><span class=\"p\">,<\/span> <span class=\"n\">f<\/span><span class=\"p\">,<\/span> <span class=\"n\">ics<\/span><span class=\"o\">=<\/span><span class=\"p\">{<\/span><span class=\"n\">f<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">r<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/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\">$\\displaystyle f{\\left(r \\right)} = -k r^{2} + 1$<\/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[7]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">ric<\/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\">gamma<\/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=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">k<\/span><span class=\"p\">,<\/span> <span class=\"n\">f<\/span><span class=\"p\">,<\/span> <span class=\"n\">ics<\/span><span class=\"o\">=<\/span><span class=\"p\">{<\/span><span class=\"n\">f<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">r<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/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[7]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle f{\\left(r \\right)} = -k r^{2} + 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<hr \/>\n<h3>\u5b9a\u66f2\u7387\u7a7a\u9593\u8a08\u91cf\u306e\u3044\u304f\u3064\u304b\u306e\u8868\u793a\u4f8b<\/h3>\n<p>\u3044\u3063\u305f\u3093\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u66f2\u7387\u7a7a\u9593\u306e\u8a08\u91cf\u304c\u66f8\u304b\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3068<br \/>\n$$\\gamma_{ij} dx^i dx^j = \\frac{dr^2}{1-kr^2} + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)$$<\/p>\n<p>\u7c21\u5358\u306a\u5909\u6570\u5909\u63db\u3067\uff0c\u30c6\u30ad\u30b9\u30c8\u306b\u3088\u304f\u73fe\u308c\u308b\u5225\u306e\u8868\u793a\u4f8b\u306b\u5909\u63db\u3067\u304d\u308b\u3002\u305f\u3068\u3048\u3070\uff0c<\/p>\n<p>$$r =\\frac{r_1}{1 + \\frac{k}{4} r_1^2}$$\u3068\u304a\u304f\u3068\uff0c<br \/>\n$$r^2 = \\frac{r_1^2}{\\left(1 + \\frac{k}{4} r_1^2\\right)^2},<br \/>\n\\quad \\frac{dr^2}{1 -k r^2} = \\frac{dr_1^2}{\\left(1 + \\frac{k}{4} r_1^2\\right)^2}$$\u3068\u306a\u308a\uff0c<br \/>\n\\begin{eqnarray}<br \/>\n\\gamma_{ij} dx^i dx^j &amp;=&amp; \\frac{1}{\\left(1 + \\frac{k}{4} r_1^2\\right)^2}<br \/>\n\\left(dr_1^2 + r_1^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)\u00a0 \\right)\\\\<br \/>\n&amp;=&amp; \\frac{1}{\\left(1 + \\frac{k}{4} r_1^2\\right)^2}<br \/>\n\\left(dx^2 + dy^2 + dz^2\u00a0 \\right)<br \/>\n\\end{eqnarray}<br \/>\n\uff08\u3053\u3053\u3067\uff0c\\(r_1 = \\sqrt{x^2 + y^2 + z^2}\\)\uff09\u307e\u3055\u306b\u5171\u5f62\u5e73\u5766\u306a\u5f62\u306b\u8868\u3059\u3053\u3068\u3082\u3067\u304d\u308b\u3002<\/p>\n<p>\u307e\u305f\uff0c<br \/>\n$$\\frac{dr}{\\sqrt{1 -k r^2}}\u00a0 = d\\chi,<br \/>\n\\quad \\therefore \\ \\\u00a0 r = \\sigma(\\chi) \\equiv \\frac{\\sin(\\sqrt{k} \\chi)}{\\sqrt{k}}$$<br \/>\n\u3068\u304a\u304f\u3068\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\gamma_{ij} dx^i dx^j &amp;=&amp; d\\chi^2 + \\sigma(\\chi)^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u8868\u3059\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u3061\u306a\u307f\u306b\uff0c\u66f2\u7387\u5b9a\u6570 \\(k\\) \u304c\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308b\u3068\u3057\u3066<\/p>\n<p>$$\\sigma(\\chi) = \\left \\{ \\begin{array}{cl} \\sin \\chi\u00a0 &amp; \\mbox{for}\\\u00a0 k = +1 \\\\\u00a0 \\chi &amp; \\mbox{for}\\\u00a0 k = 0 \\\\ \\sinh \\chi &amp; \\mbox{for}\\\u00a0 k = -1\\end{array} \\right.$$\u3068\u66f8\u304f\u5834\u5408\u3082\u591a\u3044\u304c\uff0c\u672c\u7a3f\u3067\u306f $k$ \u3092\u305d\u306e\u307e\u307e\u6b8b\u3057\u305f $\\sigma(\\chi)$ \u306e\u5b9a\u7fa9\u3092\u63a1\u7528\u3059\u308b\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n<p>$k &gt; 0$ \u306e\u3068\u304d\u306f\u305d\u306e\u307e\u307e\uff0c<\/p>\n<p>$$\\sigma(\\chi) = \\frac{\\sin(\\sqrt{k} \\chi)}{\\sqrt{k}}\\quad \\mbox{for}\\\u00a0 k &gt; 0$$<\/p>\n<p>$k &lt; 0$ \u306e\u3068\u304d\u3082\u305d\u306e\u307e\u307e\u4f7f\u3063\u3066<br \/>\n$$\\sigma(\\chi) = \\frac{\\sin(\\sqrt{k} \\chi)}{\\sqrt{k}}=<br \/>\n\\frac{\\sin(i \\sqrt{|k|} \\chi)}{i \\sqrt{|k| }} = \\frac{\\sinh( \\sqrt{|k|} \\chi)}{\\sqrt{|k| }}\\quad\\mbox{for}\\ k &lt; 0$$<\/p>\n<p>$k = 0$ \u306e\u5834\u5408\u306f\uff0c\u305d\u306e\u307e\u307e\u306e\u5f0f\u306e\u6975\u9650\u3092\u3068\u3063\u3066<\/p>\n<p>$$\\sigma(\\chi) = \\lim_{k\\rightarrow 0}\\frac{\\sin(\\sqrt{k} \\chi)}{\\sqrt{k}} = \\chi\\quad\\mbox{for}\\ k = 0$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>FLRW \u306e3\u6b21\u5143\u7a7a\u9593\u306f\u5b9a\u66f2\u7387\u7a7a\u9593\u3067\u3042\u3063\u305f\u3002\u5b9a\u66f2\u7387\u7a7a\u9593\u3092\u8868\u3059\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306b\u3064\u3044\u3066\u307e\u3068\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%e5%ae%87%e5%ae%99%e8%ab%96\/%e5%ae%9a%e6%9b%b2%e7%8e%87%e7%a9%ba%e9%96%93%e3%81%ae%e8%a8%88%e9%87%8f\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":1430,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1463","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=\"FLRW \u306e3\u6b21\u5143\u7a7a\u9593\u306f\u5b9a\u66f2\u7387\u7a7a\u9593\u3067\u3042\u3063\u305f\u3002\u5b9a\u66f2\u7387\u7a7a\u9593\u3092\u8868\u3059\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u308b\u3002 3\u6b21\u5143\u5b9a\u66f2\u7387\u7a7a\u9593\" \/>\n\t<meta name=\"robots\" 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