\u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066<\/p>\n
\u3068\u3044\u3046\u72b6\u6cc1\u304c\u306a\u308a\u305f\u3064\u5834\u5408\uff0c\u30e1\u30c8\u30ea\u30c3\u30af\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $\\phi$ \u3092\u4f7f\u3063\u3066<\/p>\n
$$ds^2\u00a0 \\simeq -\\left(1+\\frac{2}{c^2} \\phi \\right) c^2 dt^2 + \\left(\\delta_{ij} + O(c^{-2})\\right)\\, dx^i dx^j$$<\/p>\n
\u3068\u66f8\u3051\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u8fd1\u4f3c\u3092\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u3068\u547c\u3093\u3067\u3044\u308b\u3002<\/p>\n
<\/p>\n
\u9759\u7684\u91cd\u529b\u5834\u3092\u3042\u3089\u308f\u3059\u30e1\u30c8\u30ea\u30c3\u30af\u306f\uff0c\u307e\u305a\u6642\u9593\u5ea7\u6a19\u306b\u3088\u3089\u306a\u3044\uff0c\u3064\u307e\u308a $g_{\\mu\\nu, 0} = 0$ \u304b\u3064\uff0c\u6642\u9593\u53cd\u8ee2\u306b\u5bfe\u3057\u3066\u7dda\u7d20\u304c\u4e0d\u5909\uff0c\u3064\u307e\u308a $g_{0 i} = 0$ \u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n
\u91cd\u529b\u304c\u306a\u3051\u308c\u3070\u6642\u7a7a\u306f\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u3067\u3042\u308b\u3002\u91cd\u529b\u5834\u304c\u5f31\u3044\u3068\u304d\u306e\u6642\u7a7a\u306f\uff0c\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u304b\u3089\u308f\u305a\u304b\u306b\u305a\u308c\u3066\u3044\u308b\u3060\u3051\u3060\u308d\u3046\u304b\u3089\uff0c\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u3068\u540c\u3058\u5ea7\u6a19 $x^{\\mu} = (x^0, x^1, x^2, x^3) = (ct, x, y, z)$ \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3060\u308d\u3046\u3002<\/p>\n
\\begin{eqnarray}
\nds^2 &=& g_{\\mu\\nu} dx^{\\mu} dx^{\\nu} \\\\
\n&=& \\left( \\eta_{\\mu\\nu} + h_{\\mu\\nu} \\right) dx^{\\mu} dx^{\\nu}, \\quad |h_{\\mu\\nu}| \\ll 1 \\\\
\n&=& \\left(-1 + h_{00} \\right) dx^{0} dx^{0} + \\left( \\delta_{ij} + h_{ij} \\right) dx^{i} dx^{j}
\n\\end{eqnarray}<\/p>\n
\u7c92\u5b50\u306e4\u5143\u901f\u5ea6\u3092 $\\displaystyle u^{\\mu} = \\frac{dx^{\\mu}}{d\\tau}$ \u3068\u3059\u308b\u3068\uff0c3\u6b21\u5143\u7684\u306a\u901f\u5ea6 $v^i$ \u306f<\/p>\n
\\begin{eqnarray}
\n\\frac{v^i}{c} = \\frac{1}{c} \\frac{dx^i}{dt} = \\frac{dx^i}{dx^0} = \\frac{u^i}{u^0}
\n\\end{eqnarray}<\/p>\n
\u7c92\u5b50\u306e\u901f\u3055\u304c\u5149\u901f\u306b\u6bd4\u3079\u3066\u5c0f\u3055\u3044\u3068\u306f $\\displaystyle \\left| \\frac{v^i}{c}\\right| \\ll 1$ \u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u304b\u3089\uff0c$\\displaystyle \\left( \\frac{u^i}{u^0}\\right)^2$ \u306f $1$ \u306b\u6bd4\u3079\u3066\u5341\u5206\u306b\u5c0f\u3055\u3044\u306e\u3067\u7121\u8996\u3059\u308b\u3002\uff08\u3042\u308b\u3044\u306f\u540c\u3058\u3053\u3068\u3060\u304c $\\left(u^{0}\\right)^2$ \u306b\u6bd4\u3079\u3066 $\\left(u^{i}\\right)^2$ \u307e\u305f\u306f $u^i u^j$ \u306e\u9805\u306f\u5341\u5206\u5c0f\u3055\u3044\u306e\u3067\u7121\u8996\u3059\u308b\u3068\u8a00\u3044\u63db\u3048\u3066\u3082\u826f\u3044\u3060\u308d\u3046\u3002\uff09\u3053\u306e\u8fd1\u4f3c\u3092\u30b9\u30ed\u30fc\u30e2\u30fc\u30b7\u30e7\u30f3\u8fd1\u4f3c\u3068\u547c\u3076\u3053\u3068\u306b\u3057\u3088\u3046\u3002<\/p>\n
\u3053\u306e\u30b9\u30ed\u30fc\u30e2\u30fc\u30b7\u30e7\u30f3\u8fd1\u4f3c\u3092\u4f7f\u3046\u3068\uff0c4\u5143\u901f\u5ea6\u306e\u898f\u683c\u5316\u6761\u4ef6\u304b\u3089<\/p>\n
\\begin{eqnarray}
\n-c^2 &=& g_{\\mu\\nu} u^{\\mu} u^{\\nu} \\\\
\n&=& g_{00} \\left(u^0\\right)^2 + g_{ij} u^i u^j \\\\
\n&\\simeq&\u00a0 g_{00} \\left(u^0\\right)^2 \\\\
\n&=& g_{00} \\left(\\frac{c dt}{d\\tau}\\right)^2 \\\\
\n\\therefore\\ \\ d\\tau &\\simeq& \\sqrt{-g_{00}} \\ dt
\n\\end{eqnarray}<\/p>\n
\u3068\u3044\u3046\u308f\u3051\u3067\uff0c\u30b9\u30ed\u30fc\u30e2\u30fc\u30b7\u30e7\u30f3\u8fd1\u4f3c\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u305f\u95a2\u4fc2\u5f0f\u306f<\/p>\n
\\begin{eqnarray}
\n\\left(u^0\\right)^2 &\\simeq& \\frac{c^2}{-g_{00}} \\\\
\n\\ d\\tau &\\simeq& \\sqrt{-g_{00}} \\ dt
\n\\end{eqnarray}<\/p>\n
\u3067\u3042\u308a\uff0c\u3053\u308c\u3089\u306f\u6b21\u306e\u9805\u3067\u3059\u3050\u4f7f\u3046\u3002<\/p>\n
\u91cd\u529b\u4ee5\u5916\u306e\u529b\u3092\u53d7\u3051\u305a\u306b\u904b\u52d5\u3059\u308b\u7c92\u5b50\u306e4\u5143\u901f\u5ea6\u306f\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\u3002<\/p>\n
$$\\frac{du^{\\lambda}}{d\\tau} + \\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu} u^{\\mu} u^{\\nu} = 0$$<\/p>\n
\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\u307e\u305f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3082\u66f8\u3051\u308b\u306e\u3067\u3042\u3063\u305f<\/a><\/strong><\/span>\u3002<\/p>\n $$\\frac{d}{d\\tau} \\left(g_{\\mu\\nu} u^{\\nu} \\right) = \\frac{1}{2} g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta} \\tag{1}$$<\/p>\n $(1)$ \u5f0f\u306e\u7a7a\u9593\u6210\u5206 $\\mu = i$ \u306b\u3064\u3044\u3066\u306f\uff0c\u30b9\u30ed\u30fc\u30e2\u30fc\u30b7\u30e7\u30f3\u8fd1\u4f3c\u304b\u3089<\/p>\n \\begin{eqnarray} \u3055\u3089\u306b\u5f31\u5834\u8fd1\u4f3c\u304b\u3089 $-g_{00} \\simeq 1 – h_{00}$ \u3068\u3057\u3066<\/p>\n $$ \\therefore\\ \\ \\left(1-h_{00}\\right) \\frac{1}{\\sqrt{1-h_{00}}}\\frac{d}{dt} \\left(g_{ij}\u00a0 \\frac{1}{\\sqrt{1-h_{00}}}\\frac{dx^j}{dt}\\right) \\simeq \\frac{c^2}{2} h_{00, i} \\tag{2}$$<\/p>\n \u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092 $\\phi$ \u3068\u3057\u305f\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n $$m \\frac{d^2 \\boldsymbol{r}}{dt^2} = \\boldsymbol{F} = – m \\nabla \\phi$$<\/p>\n \u6210\u5206\u3067\u66f8\u304f\u3068<\/p>\n $$\\delta_{ij} \\frac{d^2 x^j}{dt^2} = – \\phi_{, i} \\tag{3}$$<\/p>\n \u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306b\u304a\u3044\u3066\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f $(2)$ \u304c\u904b\u52d5\u65b9\u7a0b\u5f0f $(3)$ \u306b\u5e30\u7740\u3059\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u307e\u305a\u53f3\u8fba\u540c\u58eb\u3092\u898b\u6bd4\u3079\u3066<\/p>\n \\begin{eqnarray} \u3053\u308c\u3092\u4f7f\u3046\u3068 $(2)$ \u5f0f\u306e\u5de6\u8fba\u306b\u3042\u3089\u308f\u308c\u3066\u3044\u308b $h_{00}$ \u306f $c^{-2}$ \u306e\u30aa\u30fc\u30c0\u30fc\u3067\u3042\u308b\u306e\u3067\u7121\u8996\u3067\u304d\u308b\u304b\u3089\uff0c\u5de6\u8fba\u540c\u58eb\u3092\u898b\u6bd4\u3079\u3066<\/p>\n \\begin{eqnarray} \u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306e\u30e1\u30c8\u30ea\u30c3\u30af<\/p>\n \\begin{eqnarray} \u304c\u6c42\u3081\u3089\u308c\u305f\u3002\u30e1\u30c8\u30ea\u30c3\u30af\u306e\u7a7a\u9593\u6210\u5206\u306e $O(c^{-2})$ \u306e\u9805\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3068\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6bd4\u8f03\u3059\u308b\u3060\u3051\u3067\u306f\u6c7a\u307e\u3089\u306a\u3044<\/strong><\/span>\u305f\u3081\u901a\u5e38\u306f\u66f8\u304b\u306a\u3044\u3002<\/p>\n <\/p>\n \u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f $(1)$ \u5f0f\u306e\u6642\u9593\u6210\u5206 $\\mu = 0$ \u306f\u6642\u7a7a\u304c\u9759\u7684\u3067\u30e1\u30c8\u30ea\u30c3\u30af\u304c\u6642\u9593\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3053\u3068\u304b\u3089\uff0c\u4fdd\u5b58\u5247\u3092\u3042\u3089\u308f\u3059\u3002<\/p>\n \\begin{eqnarray} 4\u5143\u901f\u5ea6\u306e\u898f\u683c\u5316\u6761\u4ef6\u304b\u3089<\/p>\n \\begin{eqnarray} \u3064\u307e\u308a\uff0c\u4fdd\u5b58\u91cf $\\varepsilon$ \u306f\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u300c\u9759\u6b62\u8cea\u91cf\u30a8\u30cd\u30eb\u30ae\u30fc\u300d\uff0b\u300c\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u300d\uff0b\u300c\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u300d\u3092\u8868\u3057\u3066\u304a\u308a\uff0c\u300c\u9759\u6b62\u8cea\u91cf\u30a8\u30cd\u30eb\u30ae\u30fc\u300d\u306f\u5b9a\u6570\u3067\u3042\u308b\u306e\u3067\u5f53\u7136\u4e00\u5b9a\u3067\uff0c\u300c\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u300d\u3068\u300c\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u300d\u306e\u548c\uff0c\u3064\u307e\u308a\u529b\u5b66\u7684\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u4fdd\u5b58\u3055\u308c\u308b\u3053\u3068\u3092\u8868\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n \u30e1\u30c8\u30ea\u30c3\u30af\u306e\u7a7a\u9593\u6210\u5206\u306e $h_{ij}$ \u306e\u9805\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3068\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6bd4\u8f03\u3059\u308b\u3060\u3051\u3067\u306f\u6c7a\u307e\u3089\u305a\uff0c\u7d50\u5c40\u306f\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u6c7a\u3081\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u304c\u3093\u3070\u3063\u3066\u8a08\u7b97\u3057\u3066\u307f\u3088\u3046\u3002<\/p>\n \u9759\u7684\u6642\u7a7a\u306e\u4eee\u5b9a\uff08$g_{\\mu\\nu, 0} = 0, \\ g_{0i} = 0$\uff09\u3068\u5f31\u91cd\u529b\u5834\u8fd1\u4f3c\uff08$g_{\\mu\\nu} = \\eta_{\\mu\\nu} + h_{\\mu\\nu}$\uff09\u304b\u3089 $h_{\\mu\\nu}$ \u306e1\u6b21\u307e\u3067\u8a08\u7b97\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \\begin{eqnarray} $\\nabla^2 \\equiv \\delta^{ij} \\partial_i \\partial_j$ \u3068\u3057\uff0c$h^i_{\\ \\ k} = \\delta^{ij} h_{jk}$ \u306a\u3069\u3068\u3057\u3066<\/p>\n \\begin{eqnarray} \u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/p>\n $$R_{\\mu\\nu} – \\frac{1}{2} g_{\\mu\\nu} R = \\frac{8 \\pi G}{c^4} T_{\\mu\\nu}$$<\/p>\n \u306e\u4e21\u8fba\u3092 $g^{\\mu\\nu}$ \u3067\u7e2e\u7d04\u3059\u308b\uff08\u3053\u308c\u3092\u30c8\u30ec\u30fc\u30b9\u3092\u3068\u308b\u3068\u3044\u3046\uff09\u3068\uff0c4\u6b21\u5143\u6642\u7a7a\u3067\u306f $g^{\\mu\\nu} g_{\\mu\\nu} = 4$ \u3067\u3042\u308b\u304b\u3089<\/p>\n $$- R = \\frac{8 \\pi G}{c^4} T\\quad \\mbox{where}\\quad\u00a0 T \\equiv g^{\\mu\\nu}T_{\\mu\\nu} $$<\/p>\n \u3057\u305f\u304c\u3063\u3066\u30ea\u30c3\u30c1\u30b9\u30ab\u30e9\u30fc\u306e\u9805\u3092\u53f3\u8fba\u3078\u79fb\u9805\u3059\u308b\u3068<\/p>\n $$ R_{\\mu\\nu} = \\frac{8 \\pi G}{c^4} \\left( T_{\\mu\\nu} – \\frac{1}{2} g_{\\mu\\nu} T\\right) $$<\/p>\n \u3068\u66f8\u3051\u308b\u3002\u3053\u308c\u3092trace-reversed \u306a\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3068\u547c\u3076\u3002<\/p>\n $T_{\\mu\\nu} = \\rho u_{\\mu}u_{\\nu}$ \u3092\u4f7f\u3046\u3068\uff0c<\/p>\n $$ R_{\\mu\\nu} = \\frac{8 \\pi G}{c^4} \\rho \\left( u_{\\mu}u_{\\nu} + \\frac{c^2}{2} g_{\\mu\\nu}\\right) $$<\/p>\n \u3053\u306e $00$ \u6210\u5206\u306f\u30b9\u30ed\u30fc\u30e2\u30fc\u30b7\u30e7\u30f3\u8fd1\u4f3c\u304b\u3089\u5f97\u3089\u308c\u305f\u95a2\u4fc2 $g_{00} (u^0)^2 \\simeq -c^2$ \u3092\u4f7f\u3063\u3066\uff0c<\/p>\n \\begin{eqnarray} \u30cb\u30e5\u30fc\u30c8\u30f3\u91cd\u529b\u306b\u304a\u3051\u308b\u9023\u7d9a\u7684\u306a\u8cea\u91cf\u5bc6\u5ea6\u5206\u5e03\u306b\u5bfe\u3059\u308b\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247<\/a><\/p>\n $$\\nabla^2 \\phi = 4 \\pi G \\rho$$<\/p>\n \u3068\u898b\u6bd4\u3079\u3066\uff0c\u4eca\u56de\u3082 $\\displaystyle h_{00} = – \\frac{2}{c^2} \\phi$ \u304c\u5c0e\u304b\u308c\u308b\u3002<\/p>\n $jk$ \u6210\u5206\u306f<\/p>\n \\begin{eqnarray} \\begin{eqnarray} <\/p>\n \u4e21\u8fba\u3092\u898b\u6bd4\u3079\u308b\u3068\uff08\u7279\u306b $\\nabla^2$ \u306e\u9805\u3092\u898b\u6bd4\u3079\u3066\uff09\uff0c$h_{ij} = h_{00}\\,\\delta_{ij}$\u00a0 \u3068\u3059\u308c\u3070\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u5b9f\u969b\uff0c\u3053\u308c\u3092\u5de6\u8fba\u306b\u5165\u308c\u3066\u307f\u308b\u3068<\/p>\n \\begin{eqnarray} \u3068\u306a\u308a\uff0c\u53f3\u8fba\u3068\u4e00\u81f4\u3059\u308b\u3002<\/p>\n \u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306e\u30e1\u30c8\u30ea\u30c3\u30af<\/p>\n \\begin{eqnarray} \u304c\u5f97\u3089\u308c\u305f\u3002<\/p>\n \u539f\u70b9\u306b\u304a\u3044\u305f\u8cea\u91cf $M$ \u306e\u8cea\u70b9\u306b\u3088\u3063\u3066\u3064\u304f\u3089\u308c\u308b\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\uff08\u3042\u3068\u3067\u306e\u90fd\u5408\u4e0a\uff0c\u52d5\u5f84\u5ea7\u6a19\u3092 $r$ \u3067\u306f\u306a\u304f $\\rho$ \u3068\u3057\u3066\u304a\u304d\u307e\u3059\uff09<\/p>\n $$\\phi = – \\frac{GM}{\\rho}, \\quad \\rho \\equiv \\sqrt{x^2 + y^2 + z^2}$$<\/p>\n \u3068\u66f8\u3051\u308b\u306e\u3067\u3057\u305f\u306d\u3002\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $\\phi$ \u304c\u52d5\u5f84\u5ea7\u6a19\u3060\u3051\u3067\u66f8\u3051\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u3044\u3063\u305d\u7a7a\u9593\u5ea7\u6a19\u3092\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u304b\u3089\u6975\u5ea7\u6a19\u306b\u5909\u3048\u3066<\/p>\n \\begin{eqnarray} \u3068\u3059\u308c\u3070\uff0c\u6975\u5ea7\u6a19\u3067\u66f8\u3044\u305f\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306e\u30e1\u30c8\u30ea\u30c3\u30af\u306f<\/p>\n \\begin{eqnarray} \u3053\u3053\u3067\uff0c$$r_g \\equiv \\frac{2GM}{c^2}$$<\/p>\n \u3044\u3064\u306e\u65e5\u304b\uff0c\u3053\u308c\u3092\u4f7f\u3046\u65e5\u304c\u6765\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3088\u3002\u305f\u3068\u3048\u3070\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u3068\u3053\u308d\u3067…<\/p>\n <\/p>\n \u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u3092 $r_g$ \u306e1\u6b21\u307e\u3067\u5c55\u958b\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u3053\u306e (B) \u5f0f\u3068\uff0c\u6975\u5ea7\u6a19\u3067\u66f8\u3044\u305f\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306e\u30e1\u30c8\u30ea\u30c3\u30af (A) \u5f0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\uff0c$g_{22}, g_{33}$ \u306e\u90e8\u5206\u304c\u7570\u306a\u308b\u3002(A) \u5f0f\u306e $\\rho, \\theta, \\phi$ \u306f\u7b49\u65b9\u7403\u9762\u5ea7\u6a19\u3068\u547c\u3070\u308c\uff0c(B) \u5f0f\u306e\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u5ea7\u6a19\u3068\u306e\u9593\u306e\uff08\u30d5\u30eb\u306e\uff09\u5909\u63db\u306f\uff0c\u305f\u3068\u3048\u3070\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u7b2c100\u7bc0\u306e\u554f\u984c 4 \u306b\u7b54\u3048\u304c\u66f8\u3044\u3066\u3042\u308b\u3002\u3053\u3053\u3067\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u3068\u306e\u6bd4\u8f03\u3067\u3042\u308b\u306e\u3067\uff0c$r_g$ \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u5ea7\u6a19 (B) \u5f0f\u304b\u3089\u7b49\u65b9\u7a7a\u9593\u5ea7\u6a19 (A) \u5f0f\u3078\u306e\u5909\u63db\u3092\u66f8\u3044\u3066\u304a\u304f\u3002<\/p>\n \\begin{eqnarray} \u3068\u304a\u304f\u3068\uff0c$r_g$ \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067\uff0c<\/p>\n \\begin{eqnarray} \u3057\u305f\u304c\u3063\u3066<\/p>\n \\begin{eqnarray} \u3068\u306a\u308a\uff0c(B) \u5f0f\u304b\u3089 (A) \u5f0f\u306b\u5909\u63db\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u3002\u7df4\u7fd2\u554f\u984c\u3067\u306f\uff0c\u9006\u306b (A) \u5f0f\u304b\u3089 (B) \u5f0f\u3078\u306e\u5ea7\u6a19\u5909\u63db\u3092\u793a\u3057\u3066\u3082\u3089\u3044\u307e\u3059\u3088\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066<\/p>\n \u3068\u3044\u3046\u72b6\u6cc1\u304c\u306a\u308a\u305f\u3064\u5834\u5408\uff0c\u30e1\u30c8\u30ea\u30c3\u30af\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $\\phi$ \u3092\u4f7f\u3063\u3066<\/p>\n $$ds^2\u00a0 \\simeq -\\left(1+\\frac{2}{c^2} \\phi \\right) c^2 dt^2 + \\left(\\delta_{ij} + O(c^{-2})\\right)\\, dx^i dx^j$$<\/p>\n \u3068\u66f8\u3051\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u8fd1\u4f3c\u3092\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u3068\u547c\u3093\u3067\u3044\u308b\u3002<\/p>
\n\\frac{d}{d\\tau} \\left(g_{ij} u^{j} \\right) &=& \\frac{1}{2} g_{00, i} \\left(u^0\\right)^2 + \\frac{1}{2} g_{jk, i} u^i u^j \\\\
\n&\\simeq& \\frac{1}{2} g_{00, i} \\left(u^0\\right)^2 \\\\
\n&\\simeq&\\frac{1}{2} g_{00, i} \\frac{c^2}{-g_{00}} \\\\
\n\\therefore\\ \\ -g_{00} \\frac{d}{d\\tau} \\left(g_{ij} \\frac{dx^j}{d\\tau}\\right) &\\simeq&\\frac{c^2}{2} g_{00, i}\\\\
\n\\therefore\\ \\ -g_{00} \\frac{1}{\\sqrt{-g_{00}}}\\frac{d}{dt} \\left(g_{ij} \\frac{1}{\\sqrt{-g_{00}}}\\frac{dx^j}{dt}\\right) &\\simeq&\\frac{c^2}{2} g_{00, i}
\n\\end{eqnarray}<\/p>\n\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3068\u306e\u6bd4\u8f03<\/h4>\n
\n\\frac{c^2}{2} h_{00, i} &=& – \\phi_{, i} \\\\
\n\\therefore\\ \\\u00a0 h_{00} &=& – \\frac{2}{c^2} \\phi \\\\
\n\\therefore\\ \\ g_{00} &=& -1 + h_{00} = -\\left(1+\\frac{2}{c^2} \\phi \\right)
\n\\end{eqnarray}<\/p>\n
\ng_{ij} &=& \\delta_{ij} + O(c^{-2})
\n\\end{eqnarray}<\/p>\n
\nds^2\u00a0 \\simeq -\\left(1+\\frac{2}{c^2} \\phi \\right) c^2 dt^2 + \\left(\\delta_{ij} + O(c^{-2})\\right)\\, dx^i dx^j
\n\\end{eqnarray}<\/p>\n\u88dc\u8db3\uff1a\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\u5247<\/h4>\n
\n\\frac{d}{d\\tau} \\left(g_{00} u^{0} \\right) &=& \\frac{1}{2} g_{\\alpha\\beta, 0} u^{\\alpha} u^{\\beta} = 0\\\\
\n\\therefore g_{00} u^{0} &=& \\mbox{const.} \\equiv – \\frac{\\varepsilon}{c}
\n\\end{eqnarray}<\/p>\n
\n-c^2 &=& g_{00} \\left(u^0\\right)^2 + g_{ij} u^i u^j \\\\
\n\\therefore\\ \\ \\left(g_{00} u^0\\right)^2 = \\left(- \\frac{\\varepsilon}{c}\\right)^2
\n&=& -c^2 g_{00} – g_{00} g_{ij} u^i u^j \\\\
\n&\\simeq& -c^2 \\left(-1 – \\frac{2}{c^2} \\phi \\right) + \\frac{c^2}{(u^0)^2} \\delta_{ij} u^i u^j \\\\
\n&=& c^2 + 2 \\phi + \\delta_{ij} v^i v^j \\\\
\n\\therefore\\ \\ \\varepsilon &=& c^2 \\sqrt{1 + \\frac{v^2}{c^2} + \\frac{2 \\phi}{c^2}} \\\\
\n&\\simeq& c^2 + \\frac{1}{2} v^2 + \\phi
\n\\end{eqnarray}<\/p>\n\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/h3>\n
\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/h4>\n
\n\\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu} &\\simeq& \\frac{1}{2} \\eta^{\\lambda\\sigma}\\left(
\nh_{\\sigma\\mu, \\nu} + h_{\\sigma\\nu, \\mu} – h_{\\mu\\nu , \\sigma}\\right) \\\\
\n\\varGamma^0_{\\ \\ \\ 00} &\\simeq& \\frac{1}{2} \\eta^{00} \\left(
\nh_{00, 0} + h_{00, 0} – h_{00, 0}\\right) \\\\ &=& 0 \\\\
\n\\varGamma^0_{\\ \\ \\ 0i} &\\simeq& \\frac{1}{2} \\eta^{00} \\left(
\nh_{00, i} + h_{0i, 0} – h_{0i, 0}\\right) \\\\ &=& -\\frac{1}{2} h_{00, i} \\\\
\n\\varGamma^0_{\\ \\ \\ ij} &\\simeq& \\frac{1}{2} \\eta^{00} \\left(
\nh_{0i, j} + h_{0j, i} – h_{ij, 0}\\right) \\\\ &=& 0\\\\
\n\\varGamma^i_{\\ \\ \\ 00} &\\simeq& \\frac{1}{2} \\delta^{ik} \\left(
\nh_{k0,0} + h_{k0, 0} – h_{00, k}\\right) \\\\ &=& -\\frac{1}{2} \\delta^{ik} h_{00, k}\\\\
\n\\varGamma^i_{\\ \\ \\ 0j} &\\simeq& \\frac{1}{2} \\delta^{ik} \\left(
\nh_{k0,j} + h_{kj, 0} – h_{0j, k}\\right) \\\\ &=& 0\\\\
\n\\varGamma^i_{\\ \\ \\ jk} &\\simeq& \\frac{1}{2} \\delta^{i\\ell} \\left(
\nh_{\\ell j,k} + h_{\\ell k, j} – h_{jk, \\ell}\\right)
\n\\end{eqnarray}<\/p>\n\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb<\/h4>\n
\nR_{\\mu\\nu} = R^{\\lambda}_{\\ \\ \\ \\mu\\lambda\\nu} &\\simeq&
\n\\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu, \\lambda} – \\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\lambda, \\nu} \\\\
\n&=& \\varGamma^{i}_{\\ \\ \\ \\mu\\nu, i} – \\varGamma^{0}_{\\ \\ \\ \\mu 0, \\nu}- \\varGamma^{i}_{\\ \\ \\ \\mu i, \\nu}
\n\\end{eqnarray}<\/p>\n
\nR_{00} &\\simeq& \\varGamma^{i}_{\\ \\ \\ 00, i} – \\varGamma^{0}_{\\ \\ \\ 0 0, 0}- \\varGamma^{i}_{\\ \\ \\ 0 i, 0} \\\\
\n&=& -\\frac{1}{2} \\nabla^2 h_{00} \\\\
\nR_{0j} &\\simeq& \\varGamma^{i}_{\\ \\ \\ 0j, i} – \\varGamma^{0}_{\\ \\ \\ 0 0, j}- \\varGamma^{i}_{\\ \\ \\ 0 i, j} \\\\
\n&=& 0 \\\\
\nR_{jk} &\\simeq& \\varGamma^{i}_{\\ \\ \\ jk, i} – \\varGamma^{0}_{\\ \\ \\ j 0, k}- \\varGamma^{i}_{\\ \\ \\ j i, k} \\\\
\n&=& \\frac{1}{2} \\left(
\nh^i_{\\ \\ j, ki} + h^i_{\\ \\ k, ji} – \\nabla^2 h_{jk} – h^i_{\\ \\ i, jk} + h_{00, jk}\\right)
\n\\end{eqnarray}<\/p>\ntrace-reversed \u306a\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/h4>\n
\u30c0\u30b9\u30c8\u7269\u8cea\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb<\/h4>\n
\nR_{00} &=& \\frac{8\\pi G}{c^4} \\rho \\left\\{
\n(g_{00} u^0)^2 + \\frac{c^2}{2} g_{00}\\right\\} \\\\
\n&=& \\frac{8\\pi G}{c^4} \\rho g_{00}\\left\\{
\ng_{00} (u^0)^2 + \\frac{c^2}{2} \\right\\} \\\\
\n&\\simeq& \\frac{8\\pi G}{c^4} \\rho g_{00}\\left\\{
\n-c^2 + \\frac{c^2}{2} \\right\\} \\\\
\n&=&- \\frac{4\\pi G}{c^2} \\rho g_{00} \\\\
\n\\therefore\\ \\ – \\frac{c^2}{2} \\nabla^2 h_{00} &\\simeq& 4\\pi G \\rho
\n\\end{eqnarray}<\/p>\n\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u3068\u306e\u6bd4\u8f03<\/h4>\n
\nR_{jk} &=& \\frac{8 \\pi G}{c^4} \\rho \\left( u_{j}u_{k} + \\frac{c^2}{2} g_{jk}\\right) \\\\
\n&\\simeq& \\frac{4\\pi G}{c^2} \\rho\\, \\delta_{jk} \\\\
\n&\\simeq& – \\frac{1}{2} \\nabla^2 h_{00}\\, \\delta_{jk}
\n\\end{eqnarray}<\/p>\n
\n\\therefore\\ \\ \\frac{1}{2} \\left(
\nh^i_{\\ \\ j, ki} + h^i_{\\ \\ k, ji} – \\nabla^2 h_{jk} – h^i_{\\ \\ i, jk} + h_{00, jk}\\right)
\n&\\simeq& – \\frac{1}{2} \\nabla^2 h_{00}\\, \\delta_{jk}
\n\\end{eqnarray}<\/p>\n
\n\\frac{1}{2}& & \\left(
\nh^i_{\\ \\ j, ki} + h^i_{\\ \\ k, ji} – \\nabla^2 h_{jk} – h^i_{\\ \\ i, jk} + h_{00, jk}\\right) \\\\
\n=
\n\\frac{1}{2}& & \\left(
\nh_{00, jk} + h_{00, jk} – \\nabla^2 h_{00} \\,\\delta_{jk} – 3 h_{00, jk} + h_{00, jk}\\right) \\\\
\n=- \\frac{1}{2} & &\\nabla^2 h_{00} \\,\\delta_{jk}
\n\\end{eqnarray}<\/p>\n\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c\u306e\u30e1\u30c8\u30ea\u30c3\u30af<\/h3>\n
\nds^2 &=& -(1-h_{00}) c^2 dt^2 + (1 + h_{00})\\,\\delta_{ij} dx^i dx^j \\\\
\n&=& -\\left(1 + \\frac{2}{c^2} \\phi\\right) c^2 dt^2 + \\left(1 -\\frac{2}{c^2} \\phi\\right)\\,\\delta_{ij} dx^i dx^j \\\\
\n\\end{eqnarray}<\/p>\n\u539f\u70b9\u306b\u304a\u3044\u305f\u8cea\u91cf $M$ \u304c\u3064\u304f\u308b\u30cb\u30e5\u30fc\u30c8\u30f3\u7684\u91cd\u529b\u5834<\/h4>\n
\n\\delta_{ij} dx^i dx^j &=& dx^2 + dy^2 + dz^2 \\\\
\n&=& d\\rho^2 +\\rho^2 \\left(d\\theta^2 + \\sin^2\\theta d\\phi^2\\right)
\n\\end{eqnarray}<\/p>\n
\nds^2\u00a0 &\\simeq& -\\left(1-\\frac{2GM}{c^2 \\rho}\u00a0 \\right) c^2 dt^2 +
\n\\left(1+\\frac{2GM}{c^2 \\rho}\u00a0 \\right) \\left\\{d\\rho^2+ \\rho^2 \\left(d\\theta^2 + \\sin^2\\theta d\\phi^2\\right) \\right\\} \\\\
\n&=& -\\left(1-\\frac{r_g}{\\rho}\u00a0 \\right) c^2 dt^2 +
\n\\left(1+\\frac{r_g}{\\rho}\u00a0 \\right) \\left\\{d\\rho^2+ \\rho^2 \\left(d\\theta^2 + \\sin^2\\theta d\\phi^2\\right) \\right\\} \\tag{A}
\n\\end{eqnarray}<\/p>\n\n
\u7b49\u65b9\u7403\u9762\u5ea7\u6a19\u3068\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u5ea7\u6a19\u3068\u306e\u9593\u306e\u5909\u63db<\/h4>\n
\nds^2 &=& -\\left(1 \u2013\\frac{r_g}{r} \\right) dt^2 + \\frac{dr^2}{1 \u2013\\frac{r_g}{r}} + r^2 \\left(d\\theta^2 + \\sin^2\\theta \\,d\\phi^2 \\right) \\\\
\n&\\simeq& -\\left(1 \u2013\\frac{r_g}{r} \\right) dt^2 + \\left(1 +\\frac{r_g}{r} \\right) dr^2 + r^2 \\left(d\\theta^2 + \\sin^2\\theta \\,d\\phi^2 \\right) \\tag{B}
\n\\end{eqnarray}<\/p>\n
\nr &\\equiv& \\rho \\,\\left( 1 + \\frac{r_g}{2 \\rho}\\right) \\\\
\n&=& \\rho + \\frac{r_g}{2}
\n\\end{eqnarray}<\/p>\n
\ndr &=& d\\rho \\\\
\n1 \\pm \\frac{r_g}{r} &=& 1 \\pm \\frac{r_g}{\\rho} \\left( 1 + \\frac{r_g}{2 \\rho}\\right)^{-1} \\\\
\n&\\simeq& 1 \\pm \\frac{r_g}{\\rho} \\left( 1 – \\frac{r_g}{2 \\rho}\\right) \\\\
\n&\\simeq& 1 \\pm \\frac{r_g}{\\rho} \\\\ \\ \\\\
\nr^2 &=& \\rho^2 \\left( 1 + \\frac{r_g}{2 \\rho}\\right)^2 \\\\
\n&\\simeq& \\rho^2 \\left( 1 + \\frac{r_g}{\\rho}\\right)
\n\\end{eqnarray}<\/p>\n
\nds^2 &\\simeq& -\\left(1 -\\frac{r_g}{r} \\right) dt^2 + \\left(1 + \\frac{r_g}{r} \\right) dr^2 + r^2 \\left(d\\theta^2 + \\sin^2\\theta \\,d\\phi^2 \\right) \\tag{B}\\\\
\n&\\simeq& -\\left(1 -\\frac{r_g}{\\rho} \\right) dt^2 + \\left(1 + \\frac{r_g}{\\rho} \\right) d\\rho^2 + \\rho^2 \\left( 1 + \\frac{r_g}{\\rho}\\right) \\left(d\\theta^2 + \\sin^2\\theta \\,d\\phi^2 \\right) \\\\
\n&=&-\\left(1-\\frac{r_g}{\\rho}\u00a0 \\right) c^2 dt^2 +
\n\\left(1+\\frac{r_g}{\\rho}\u00a0 \\right) \\left\\{d\\rho^2+ \\rho^2 \\left(d\\theta^2 + \\sin^2\\theta d\\phi^2\\right) \\right\\} \\tag{A}
\n\\end{eqnarray}<\/p>\n