\u5b9a\u66f2\u73873\u6b21\u5143\u7a7a\u9593\u306f\u5171\u5f62\u5e73\u5766\u3067\u3042\u308b\u3053\u3068\u306f\u7c21\u5358\u306b\u8a3c\u660e\u3067\u304d\u305d\u3046\u3060\u304c\uff0c\u7403\u5bfe\u79f0\u306a3\u6b21\u5143\u7a7a\u9593\u306f\u5fc5\u305a\u5171\u5f62\u5e73\u5766\u304b\uff0c\u8a3c\u660e\u3067\u304d\u305d\u3046\u306b\u306a\u3044\u3002\u3069\u3063\u304b\u306b\u305d\u3046\u3044\u3046\u5b9a\u7406\u3068\u304b\uff0c\u3042\u308a\u307e\u3059\u304b\u306d\u3047\u3002<\/p>\n
\u4f8b\u3048\u3070\uff0c\u7403\u5bfe\u79f0\u306a\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf\u306e\u7a7a\u9593\u90e8\u5206<\/p>\n
$$d\\ell^2 =\u00a0 \\frac{dr^2}{1 – \\frac{r_g}{r}} + r^2 \\left( d\\theta^2 + \\sin^2 \\theta d\\phi^2\\right)$$<\/p>\n
\u306f\uff0c\uff08\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u306a\u3069\u591a\u304f\u306e\u30c6\u30ad\u30b9\u30c8\u306e\u7df4\u7fd2\u554f\u984c\u306b\u3042\u308b\u3088\u3046\u306b\uff09<\/p>\n
$$ r = \\rho \\left(1 + \\frac{r_g}{4 \\rho} \\right)^2$$<\/p>\n
\u3068\u304a\u3051\u3070<\/p>\n
$$d\\ell^2 = \\left(1 + \\frac{r_g}{4 \\rho} \\right)^4 \\left(d\\rho^2 + \\rho^2 d\\theta^2 + \\rho^2 \\sin^2\\theta d\\phi^2 \\right)$$<\/p>\n
\u3068\u306a\u308b\u3002\u3053\u306e\u6642\u306e $\\rho, \\theta, \\phi$ \u306f\u7b49\u65b9\u7403\u9762\u5ea7\u6a19\u3068\u547c\u3070\u308c\u308b\u3089\u3057\u3044\u3002\u307e\u305f\uff0c<\/p>\n
\\begin{eqnarray}
\nx &=& \\rho \\sin\\theta \\cos\\phi\\\\
\ny &=& \\rho \\sin\\theta \\sin\\phi\\\\
\nz &=& \\rho \\cos\\theta
\n\\end{eqnarray}<\/p>\n
\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u7b49\u65b9\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\uff08\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u3067\u306f\u300c\u7b49\u65b9\u7684\u30ac\u30ea\u30ec\u30a4\u5ea7\u6a19\u300d\u3068\u66f8\u3044\u3066\u3044\u308b\uff09$x, y, z$ \u3092\u4f7f\u3048\u3070<\/p>\n
$$d\\ell^2 = \\left(1 + \\frac{r_g}{4 \\rho} \\right)^4 \\left(dx^2 + dy^2 + dz^2 \\right), \\quad \\rho = \\sqrt{x^2 + y^2 + z^2}$$<\/p>\n
\u3068\u306a\u308b\u3060\u308d\u3046\u3002<\/p>\n
\u307e\u305f\uff0cFLRW \u8a08\u91cf\u306e\u7a7a\u9593\u90e8\u5206\u3082\u7403\u5bfe\u79f0\u3067\u3042\u308a\uff0c<\/p>\n
$$d\\ell^2 = a^2(t) \\left( \\frac{dr^2}{1 – k r^2} + r^2 \\left( d\\theta^2 + \\sin^2 \\theta d\\phi^2\\right)\\right)$$<\/p>\n
\u306f\uff0c<\/p>\n
$$r = \\frac{R}{1 – \\frac{k}{4} R^2}$$<\/p>\n
\u3068\u304a\u3051\u3070<\/p>\n
\\begin{eqnarray}
\nd\\ell^2 &=& \\frac{a^2(t)}{\\left(1 – \\frac{k}{4} R^2\\right)^2}\u00a0 \\left( dR^2 + R^2 \\left( d\\theta^2 + \\sin^2 \\theta d\\phi^2\\right)\\right)\\\\
\n&=&\u00a0\\frac{a^2(t)}{\\left(1 – \\frac{k}{4} R^2\\right)^2}\u00a0 \\left( dX^2 + dY^2 + dZ^2 \\right), \\quad R^2 = X^2 + Y^2 + Z^2
\n\\end{eqnarray}<\/p>\n
\u3068\u306a\u308b\u3002<\/p>\n
\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf\u3082 FLRW \u8a08\u91cf\u3082\uff0c\u7a7a\u9593\u90e8\u5206\u306f\u5171\u5f62\u5e73\u5766\u306b\u3067\u304d\u308b\u306e\u3067\uff0c\u4f55\u304b\uff0c\u7403\u5bfe\u79f0\uff1d\u5171\u5f62\u5e73\u5766\u307f\u306a\u3044\u306a\u5b9a\u7406\u304c\u3042\u3063\u3066\u3082\u3088\u3055\u305d\u3046\u306a\u306e\u3060\u304c\uff0c\u3069\u3046\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5b9a\u66f2\u73873\u6b21\u5143\u7a7a\u9593\u306f\u5171\u5f62\u5e73\u5766\u3067\u3042\u308b\u3053\u3068\u306f\u7c21\u5358\u306b\u8a3c\u660e\u3067\u304d\u305d\u3046\u3060\u304c\uff0c\u7403\u5bfe\u79f0\u306a3\u6b21\u5143\u7a7a\u9593\u306f\u5fc5\u305a\u5171\u5f62\u5e73\u5766\u304b\uff0c\u8a3c\u660e\u3067\u304d\u305d\u3046\u306b\u306a\u3044\u3002\u3069\u3063\u304b\u306b\u305d\u3046\u3044\u3046\u5b9a\u7406\u3068\u304b\uff0c\u3042\u308a\u307e\u3059\u304b\u306d\u3047\u3002<\/p>