{"id":5205,"date":"2023-01-29T12:15:41","date_gmt":"2023-01-29T03:15:41","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=5205"},"modified":"2025-02-04T12:35:39","modified_gmt":"2025-02-04T03:35:39","slug":"einsteinpy-%e3%81%a8-sympy-%e3%81%a7%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e8%a7%a3%e3%81%84%e3%81%a6-kottler-%e8%a7%a3%e3%82%92","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%a8-sympy-%e3%81%a7%e3%82%a2%e3%82%a4%e3%83%b3%e3%82%b7%e3%83%a5%e3%82%bf%e3%82%a4%e3%83%b3%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e8%a7%a3%e3%81%84%e3%81%a6-kottler-%e8%a7%a3%e3%82%92\/","title":{"rendered":"EinsteinPy \u3068 SymPy \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066 Kottler \u89e3\u3092\u6c42\u3081\u308b"},"content":{"rendered":"
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EinsteinPy \u3092\u4f7f\u3063\u3066\u7403\u5bfe\u79f0\u306a\u8a08\u91cf\u304b\u3089\u771f\u7a7a\u306e\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

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

\u3092\u89e3\u304d\uff0cKottler \u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy<\/span> import<\/span> *<\/span>\r\nfrom<\/span> einsteinpy.symbolic<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7403\u5bfe\u79f0\u306a\u8a08\u91cf<\/h3>\n
\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u306b\u305d\u3063\u3066\uff0c\u7403\u5bfe\u79f0\u306a\u8a08\u91cf\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u304a\u304f\u3002\uff08$\\lambda$ \u306f\u4e88\u7d04\u8a9e\uff1f\u306a\u306e\u3067 $\\mu$ \u306b\u3057\u305f\u3002\uff09<\/p>\n
$$ds^2 = – e^{\\nu(t,r)} dt^2 + e^{\\mu(t,r)} dr^2 + r^2 \\left(d\\theta^2 + \\sin^2\\theta\\,d\\phi^2 \\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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var<\/span>(<\/span>'t, r, theta, phi'<\/span>)<\/span>\r\nmu<\/span> =<\/span> Function<\/span>(<\/span>'mu'<\/span>)<\/span>\r\nnu<\/span> =<\/span> Function<\/span>(<\/span>'nu'<\/span>)<\/span>\r\n\r\nMetric<\/span> =<\/span> diag<\/span>(<\/span>-<\/span>exp<\/span>(<\/span>nu<\/span>(<\/span>t<\/span>,<\/span>r<\/span>)),<\/span> exp<\/span>(<\/span>mu<\/span>(<\/span>t<\/span>,<\/span>r<\/span>)),<\/span> r<\/span>**<\/span>2<\/span>,<\/span> r<\/span>**<\/span>2<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>)<\/span>.<\/span>tolist<\/span>()<\/span>\r\ng<\/span> =<\/span> MetricTensor<\/span>(<\/span>Metric<\/span>,<\/span> [<\/span>t<\/span>,<\/span> r<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span>])<\/span>\r\ng<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- e^{\\nu{\\left(t,r \\right)}} & 0 & 0 & 0\\\\0 & e^{\\mu{\\left(t,r \\right)}} & 0 & 0\\\\0 & 0 & r^{2} & 0\\\\0 & 0 & 0 & r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb<\/h3>\n
$\\displaystyle G^{\\mu}_{\\ \\ \\nu} = R^{\\mu}_{\\ \\ \\nu} – \\frac{1}{2} R \\delta^{\\mu}_{\\ \\ \\nu} $ = ein<\/code> \u3068\u304a\u304f\u3002\uff08.change_config('ul')<\/code> \u3067\u4e0a\u4ed8\u4e0b\u4ed8\u306b\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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ein<\/span>=<\/span>EinsteinTensor<\/span>.<\/span>from_metric<\/span>(<\/span>g<\/span>)<\/span>.<\/span>change_config<\/span>(<\/span>'ul'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/h3>\n
$$G^{\\mu}_{\\ \\ \\nu} + \\Lambda \\delta^{\\mu}_{\\ \\ \\nu} = 0$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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var<\/span>(<\/span>'Lambda'<\/span>)<\/span>\r\n\r\ndef<\/span> EinEq<\/span>(<\/span>a<\/span>,<\/span> b<\/span>):<\/span>\r\n    return<\/span> Eq<\/span>(<\/span>expand<\/span>(<\/span>ein<\/span>[<\/span>a<\/span>,<\/span>b<\/span>])<\/span> +<\/span> Lambda<\/span> *<\/span>KroneckerDelta<\/span>(<\/span>a<\/span>,<\/span> b<\/span>),<\/span> 0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\mu$ \u306f\u6642\u9593\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3053\u3068<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 0} = 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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EinEq<\/span>(<\/span>1<\/span>,<\/span>0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
$\\displaystyle \\frac{e^{- \\mu{\\left(t,r \\right)}} \\frac{\\partial}{\\partial t} \\mu{\\left(t,r \\right)}}{r} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 0} = 0$ \u3088\u308a<\/p>\n
$$\\frac{\\partial \\mu}{\\partial t} = 0, \\quad \\therefore\\ \\ \\mu(t, r) \\Rightarrow \\mu(r)$$<\/p>\n
$\\mu(r)$ \u3068\u3057\u3066\uff0c\u3042\u3089\u305f\u3081\u3066\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb\u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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var<\/span>(<\/span>'t, r, theta, phi'<\/span>)<\/span>\r\nmu<\/span> =<\/span> Function<\/span>(<\/span>'mu'<\/span>)<\/span>\r\nnu<\/span> =<\/span> Function<\/span>(<\/span>'nu'<\/span>)<\/span>\r\n\r\nMetric<\/span> =<\/span> diag<\/span>(<\/span>-<\/span>exp<\/span>(<\/span>nu<\/span>(<\/span>t<\/span>,<\/span>r<\/span>)),<\/span> exp<\/span>(<\/span>mu<\/span>(<\/span>r<\/span>)),<\/span> r<\/span>**<\/span>2<\/span>,<\/span> r<\/span>**<\/span>2<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>)<\/span>.<\/span>tolist<\/span>()<\/span>\r\ng<\/span> =<\/span> MetricTensor<\/span>(<\/span>Metric<\/span>,<\/span> [<\/span>t<\/span>,<\/span> r<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span>])<\/span>\r\ng<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- e^{\\nu{\\left(t,r \\right)}} & 0 & 0 & 0\\\\0 & e^{\\mu{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & r^{2} & 0\\\\0 & 0 & 0 & r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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ein<\/span>=<\/span>EinsteinTensor<\/span>.<\/span>from_metric<\/span>(<\/span>g<\/span>)<\/span>.<\/span>change_config<\/span>(<\/span>'ul'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{0}_{\\ \\ 0} +\\Lambda =0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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EinEq<\/span>(<\/span>0<\/span>,<\/span> 0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
$\\displaystyle \\Lambda – \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{d}{d r} \\mu{\\left(r \\right)}}{r} – \\frac{1}{r^{2}} + \\frac{e^{- \\mu{\\left(r \\right)}}}{r^{2}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 1} +\\Lambda =0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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EinEq<\/span>(<\/span>1<\/span>,<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle \\Lambda + \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}}{r} – \\frac{1}{r^{2}} + \\frac{e^{- \\mu{\\left(r \\right)}}}{r^{2}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{2}_{\\ \\ 2} +\\Lambda =0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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EinEq<\/span>(<\/span>2<\/span>,<\/span> 2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
$\\displaystyle \\Lambda – \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{d}{d r} \\mu{\\left(r \\right)} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}}{4} + \\frac{e^{- \\mu{\\left(r \\right)}} \\left(\\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right)^{2}}{4} + \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{\\partial^{2}}{\\partial r^{2}} \\nu{\\left(t,r \\right)}}{2} – \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{d}{d r} \\mu{\\left(r \\right)}}{2 r} + \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}}{2 r} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{3}_{\\ \\ 3} +\\Lambda = 0$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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EinEq<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
$\\displaystyle \\Lambda – \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{d}{d r} \\mu{\\left(r \\right)} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}}{4} + \\frac{e^{- \\mu{\\left(r \\right)}} \\left(\\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right)^{2}}{4} + \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{\\partial^{2}}{\\partial r^{2}} \\nu{\\left(t,r \\right)}}{2} – \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{d}{d r} \\mu{\\left(r \\right)}}{2 r} + \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}}{2 r} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{2}_{\\ \\ 2} = \\displaystyle G^{3}_{\\ \\ 3}$ \u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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ein<\/span>[<\/span>2<\/span>,<\/span>2<\/span>]<\/span> -<\/span> ein<\/span>[<\/span>3<\/span>,<\/span>3<\/span>]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
$\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\nu = – \\mu$ \u3068\u304a\u3051\u308b\u3053\u3068<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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EinEq<\/span>(<\/span>1<\/span>,<\/span> 1<\/span>)<\/span>.<\/span>lhs<\/span> -<\/span> EinEq<\/span>(<\/span>0<\/span>,<\/span> 0<\/span>)<\/span>.<\/span>lhs<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
$\\displaystyle \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{d}{d r} \\mu{\\left(r \\right)}}{r} + \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}}{r}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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simplify<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
$\\displaystyle \\frac{\\left(\\frac{d}{d r} \\mu{\\left(r \\right)} + \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right) e^{- \\mu{\\left(r \\right)}}}{r}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$G^1_{\\ \\ 1} – G^0_{\\ \\ 0} = 0$ \u3088\u308a\uff0c<\/p>\n
$$\\frac{\\partial}{\\partial r}\\left(\\mu(r) + \\nu(t, r) \\right) = 0$$<\/p>\n
\u3053\u308c\u304b\u3089\uff0c<\/p>\n
$$\\nu(t, r) = – \\mu(r) + f(t)$$<\/p>\n
\u3068\u306a\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\therefore\\ \\ e^{\\nu(t, r)} dt^2 &=& e^{- \\mu(r) + f(t)} dt^2 \\\\
\n&=& e^{- \\mu(r)} \\left( e^{\\frac{f(t)}{2}} dt\\right)^2
\n\\end{eqnarray}<\/p>\n
\u6642\u9593 $t$ \u306e\u307f\u306e\u4efb\u610f\u95a2\u6570 $f(t)$ \u306e\u81ea\u7531\u5ea6\u306f\uff0c$e^{\\frac{f(t)}{2}} dt \\Rightarrow dt’$ \u306a\u308b\u65b0\u3057\u3044\u6642\u9593\u5ea7\u6a19\u306e\u5b9a\u7fa9\u306b\u3088\u3063\u3066\u5438\u53ce\u3067\u304d\u308b\u306e\u3067\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f $f(t) = 0$ \u3059\u306a\u308f\u3061<\/p>\n
$$\\nu(t, r) = – \\mu(r)$$<\/p>\n
\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u3042\u3089\u305f\u3081\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306b\u5bfe\u3057\u3066\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30fb\u30c6\u30f3\u30bd\u30eb\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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var<\/span>(<\/span>'t, r, theta, phi'<\/span>)<\/span>\r\nmu<\/span> =<\/span> Function<\/span>(<\/span>'mu'<\/span>)<\/span>\r\n\r\nMetric<\/span> =<\/span> diag<\/span>(<\/span>-<\/span>exp<\/span>(<\/span>-<\/span>mu<\/span>(<\/span>r<\/span>)),<\/span> exp<\/span>(<\/span>mu<\/span>(<\/span>r<\/span>)),<\/span> r<\/span>**<\/span>2<\/span>,<\/span> r<\/span>**<\/span>2<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>**<\/span>2<\/span>)<\/span>.<\/span>tolist<\/span>()<\/span>\r\ng<\/span> =<\/span> MetricTensor<\/span>(<\/span>Metric<\/span>,<\/span> [<\/span>t<\/span>,<\/span> r<\/span>,<\/span> theta<\/span>,<\/span> phi<\/span>])<\/span>\r\ng<\/span>.<\/span>tensor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- e^{- \\mu{\\left(r \\right)}} & 0 & 0 & 0\\\\0 & e^{\\mu{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & r^{2} & 0\\\\0 & 0 & 0 & r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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ein<\/span>=<\/span>EinsteinTensor<\/span>.<\/span>from_metric<\/span>(<\/span>g<\/span>)<\/span>.<\/span>change_config<\/span>(<\/span>'ul'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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EinEq<\/span>(<\/span>0<\/span>,<\/span> 0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
$\\displaystyle \\Lambda – \\frac{e^{- \\mu{\\left(r \\right)}} \\frac{d}{d r} \\mu{\\left(r \\right)}}{r} – \\frac{1}{r^{2}} + \\frac{e^{- \\mu{\\left(r \\right)}}}{r^{2}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\uff0c$\\mu(r)$ \u3092\u6c42\u3081\u308b<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[18]:<\/div>\n
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sol<\/span> =<\/span> dsolve<\/span>(<\/span>EinEq<\/span>(<\/span>0<\/span>,<\/span> 0<\/span>),<\/span> mu<\/span>(<\/span>r<\/span>))<\/span>\r\nsol<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
$\\displaystyle \\mu{\\left(r \\right)} = \\log{\\left(\\frac{r}{C_{1} – \\Lambda r^{3} + 3 r} \\right)} + \\log{\\left(3 \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$f(r) \\equiv e^{-\\mu(r)}$ \u3068\u3057\u3066\uff0c$f(r)$ \u306b\u3064\u3044\u3066\u66f8\u3044\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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exp<\/span>(<\/span>-<\/span>sol<\/span>.<\/span>rhs<\/span>)<\/span>.<\/span>expand<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
$\\displaystyle \\frac{C_{1}}{3 r} – \\frac{\\Lambda r^{2}}{3} + 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7a4d\u5206\u5b9a\u6570 $C_1$ \u306f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u306b\u306a\u308b\u3053\u3068\u304b\u3089\uff0c<\/p>\n
$$C_1 = -3 r_g$$<\/p>\n
\u3068\u306a\u308b\u3002<\/p>\n
\u6700\u7d42\u7684\u306b<\/p>\n
$$ds^2 = -\\left(1 \u2013 \\frac{r_g}{r} -\\frac{\\Lambda}{3} r^2\\right) dt^2 + \\frac{dr^2}{1 \u2013 \\frac{r_g}{r}-\\frac{\\Lambda}{3} r^2} + r^2 \\left(d\\theta^2 + \\sin^2\\theta \\,d\\phi^2 \\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
EinsteinPy \u3092\u4f7f\u3063\u3066\u7403\u5bfe\u79f0\u306a\u8a08\u91cf\u304b\u3089\u771f\u7a7a\u306e\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/p>\n
$$G^{\\mu}_{\\ \\ \\nu} + \\Lambda \\delta^{\\mu}_{\\ \\ \\nu} = 0$$<\/p>\n
\u3092\u89e3\u304d\uff0cKottler \u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":5114,"menu_order":25,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-5205","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t