{"id":5116,"date":"2024-05-30T12:30:05","date_gmt":"2024-05-30T03:30:05","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=5116"},"modified":"2025-02-04T11:25:14","modified_gmt":"2025-02-04T02:25:14","slug":"5116-2","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\/5116-2\/","title":{"rendered":"EinsteinPy \u3068 SymPy \u3067\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u3092\u6c42\u3081\u308b"},"content":{"rendered":"
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\nEinsteinPy \u3092\u4f7f\u3063\u3066\u7403\u5bfe\u79f0\u306a\u8a08\u91cf\u304b\u3089\u771f\u7a7a\u306e\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

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

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

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\u30e9\u30a4\u30d6\u30e9\u30ea\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\nfrom<\/span> einsteinpy.symbolic<\/span> import<\/span> *<\/span>\r\ninit_printing<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7403\u5bfe\u79f0\u306a\u8a08\u91cf<\/h3>\n
\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u306e\u8a18\u8ff0\u306b\u305d\u3063\u3066\uff08\u3057\u304b\u3057\uff0csignature \u306f $(-, +, +, +)$ \u306b\u3057\u3066\uff09\uff0c\u7403\u5bfe\u79f0\u306a\u8a08\u91cf\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u304a\u304f\u3002\uff08\u30ae\u30ea\u30b7\u30e3\u6587\u5b57\u306e $\\lambda$ \u306f lambda<\/code> \u3067\u306f\u306a\u304f\uff0csympy.abc<\/code> \u306e\u4e2d\u3067 lamda<\/code> \u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u3002lambda<\/code> \u306f Python \u306e\u4e88\u7d04\u8a9e\uff0c\u5909\u6570\u3068\u3057\u3066\u4f7f\u7528\u4e0d\u53ef\u3002\uff09<\/p>\n
$$ds^2 = -e^{\\nu(t,r)} dt^2 + e^{\\lambda(t,r)} dr^2 + r^2 \\left(d\\theta^2 + \\sin^2\\theta\\,d\\phi^2 \\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# from sympy.abc import * \u3059\u308b\u3068\u4f7f\u3048\u308b<\/span>\r\nlamda<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
$\\displaystyle \\lambda$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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lamda<\/span> =<\/span> Function<\/span>(<\/span>'lamda'<\/span>)(<\/span>t<\/span>,<\/span> r<\/span>)<\/span>\r\nnu<\/span> =<\/span> Function<\/span>(<\/span>'nu'<\/span>)(<\/span>t<\/span>,<\/span> r<\/span>)<\/span>\r\n\r\nMetric<\/span> =<\/span> diag<\/span>(<\/span>-<\/span>exp<\/span>(<\/span>nu<\/span>),<\/span> exp<\/span>(<\/span>lamda<\/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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$\\displaystyle \\left[\\begin{matrix} -e^{\\nu{\\left(t,r \\right)}} & 0 & 0 & 0\\\\0 & e^{\\lambda{\\left(t,r \\right)}} & 0 & 0\\\\0 & 0 & r^{2} & 0\\\\0 & 0 & 0 & r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/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}\u00a0 -\\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[4]:<\/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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$\\lambda$ \u306f\u6642\u9593\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3053\u3068<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 0}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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ein<\/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
$\\displaystyle \\frac{e^{ -\\lambda{\\left(t,r \\right)}} \\frac{\\partial}{\\partial t} \\lambda{\\left(t,r \\right)}}{r}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 0} = 0$ \u3088\u308a<\/p>\n
$$\\frac{\\partial \\lambda}{\\partial t} = 0, \\quad \\therefore\\ \\ \\lambda(t, r) \\Rightarrow \\lambda(r)$$<\/p>\n
$\\lambda(r)$ \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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lamda<\/span> =<\/span> Function<\/span>(<\/span>'lamda'<\/span>)(<\/span>r<\/span>)<\/span>\r\nnu<\/span> =<\/span> Function<\/span>(<\/span>'nu'<\/span>)(<\/span>t<\/span>,<\/span> r<\/span>)<\/span>\r\n\r\nMetric<\/span> =<\/span> diag<\/span>(<\/span>-<\/span>exp<\/span>(<\/span>nu<\/span>),<\/span> exp<\/span>(<\/span>lamda<\/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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$\\displaystyle \\left[\\begin{matrix} -e^{\\nu{\\left(t,r \\right)}} & 0 & 0 & 0\\\\0 & e^{\\lambda{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & r^{2} & 0\\\\0 & 0 & 0 & r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/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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$\\displaystyle G^{0}_{\\ \\ 0}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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ein<\/span>[<\/span>0<\/span>,<\/span>0<\/span>]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\left( -r \\frac{d}{d r} \\lambda{\\left(r \\right)}\u00a0 -e^{\\lambda{\\left(r \\right)}} + 1\\right) e^{ -\\lambda{\\left(r \\right)}}}{r^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{1}_{\\ \\ 1}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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ein<\/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
$\\displaystyle \\frac{\\left(r \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\u00a0 -e^{\\lambda{\\left(r \\right)}} + 1\\right) e^{ -\\lambda{\\left(r \\right)}}}{r^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{2}_{\\ \\ 2}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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ein<\/span>[<\/span>2<\/span>,<\/span>2<\/span>]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\left( -r \\frac{d}{d r} \\lambda{\\left(r \\right)} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)} + r \\left(\\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right)^{2} + 2 r \\frac{\\partial^{2}}{\\partial r^{2}} \\nu{\\left(t,r \\right)}\u00a0 -2 \\frac{d}{d r} \\lambda{\\left(r \\right)} + 2 \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right) e^{ -\\lambda{\\left(r \\right)}}}{4 r}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{3}_{\\ \\ 3}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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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[11]:<\/div>\n
$\\displaystyle \\frac{\\left( -r \\frac{d}{d r} \\lambda{\\left(r \\right)} \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)} + r \\left(\\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right)^{2} + 2 r \\frac{\\partial^{2}}{\\partial r^{2}} \\nu{\\left(t,r \\right)}\u00a0 -2 \\frac{d}{d r} \\lambda{\\left(r \\right)} + 2 \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right) e^{ -\\lambda{\\left(r \\right)}}}{4 r}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle G^{2}_{\\ \\ 2} = \\displaystyle G^{3}_{\\ \\ 3}$ \u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3002$\\displaystyle G^{2}_{\\ \\ 2}\u00a0 -\\displaystyle G^{3}_{\\ \\ 3}$ \u304c\u30bc\u30ed\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[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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$\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\nu =\u00a0 -\\lambda$ \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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simplify<\/span>(<\/span>ein<\/span>[<\/span>1<\/span>,<\/span>1<\/span>]<\/span>-<\/span>ein<\/span>[<\/span>0<\/span>,<\/span>0<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\left(\\frac{d}{d r} \\lambda{\\left(r \\right)} + \\frac{\\partial}{\\partial r} \\nu{\\left(t,r \\right)}\\right) e^{ -\\lambda{\\left(r \\right)}}}{r}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$G^1_{\\ \\ 1}\u00a0 -G^2_{\\ \\ 2} = 0$ \u3088\u308a\uff0c\u4ee5\u4e0b\u306e\u5f0f\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n
$$\\frac{\\partial}{\\partial r}\\left(\\lambda(r) + \\nu(t, r) \\right) = 0$$<\/p>\n
\u3053\u308c\u304b\u3089\uff0c<\/p>\n
$$\\nu(t, r) =\u00a0 -\\lambda(r) + f(t)$$<\/p>\n
\u3068\u306a\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\therefore\\ \\ e^{\\nu(t, r)} dt^2 &=& e^{ -\\lambda(r) + f(t)} dt^2 \\\\
\n&=& e^{ -\\lambda(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
$$\\nu(t, r) \\Rightarrow\u00a0 -\\lambda(r)$$<\/p>\n
\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\u30d0\u30fc\u30b3\u30d5\u306e\u5b9a\u7406<\/h4>\n
\u3053\u3053\u307e\u3067\u306f\uff0c\u7403\u5bfe\u79f0\u771f\u7a7a\u89e3\u306f metric \u304c\u6642\u9593\u306b\u3088\u3089\u306a\u3044\uff0c\u3064\u307e\u308a\u9759\u7684\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u308f\u3051\u3067\uff0c\u30d0\u30fc\u30b3\u30d5\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u30d0\u30fc\u30b3\u30d5\u306e\u5b9a\u7406\u306e\u3082\u3046\u4e00\u3064\u306e\u5e30\u7d50\u3067\u3042\u308b\u6f38\u8fd1\u7684\u5e73\u5766\u6027\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u3067\u793a\u3059\u3088\u3046\u306b\u89e3\u304c $e^{-\\mu(r)} = \\displaystyle 1\u00a0 -\\frac{r_g}{r}$ \u3068\u306a\u308b\u3053\u3068\u3067 $r \\rightarrow \\infty$ \u3067 $e^{-\\mu(r)} \\rightarrow 1$ \u3068\u306a\u308b\u3053\u3068\u304b\u3089\u308f\u304b\u308b\u3002<\/p>\n