<\/p>\n
4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206 $u^{\\mu}$ \u306e\u5171\u5909\u5fae\u5206\u306f<\/p>\n
$$\\nabla_{\\nu} u^{\\mu} = \\partial_{\\nu} u^{\\mu} + \\varGamma^{\\mu}_{\\ \\ \\ \\rho \\nu} u^{\\rho}$$<\/p>\n
\u3068\u66f8\u3051\u308b\u306e\u3067\u3042\u3063\u305f\u3002\u5ff5\u306e\u305f\u3081\uff0c\u504f\u5fae\u5206\u306e\u8868\u8a18\u306f<\/p>\n
$$\\partial_{\\nu} \\equiv \\frac{\\partial}{\\partial x^{\\nu}}$$<\/p>\n
\u3053\u308c\u304b\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u300c\u767a\u6563\u300d\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\nabla_{\\mu} u^{\\mu} &=& \\partial_{\\mu} u^{\\mu} + \\varGamma^{\\mu}_{\\ \\ \\ \\rho \\mu} u^{\\rho} \\\\
\n&=& \\partial_{\\mu} u^{\\mu} + \\frac{1}{2} g^{\\mu\\nu}\\left( g_{\\nu\\rho, \\mu}+g_{\\nu\\mu, \\rho}-g_{\\rho\\mu, \\nu}\\right) u^{\\rho} \\\\
\n&=& \\partial_{\\mu} u^{\\mu} + \\frac{1}{2} g^{\\mu\\nu}g_{\\mu\\nu, \\rho}\\,u^{\\rho} \\\\
\n&=& \\partial_{\\mu} u^{\\mu} + \\frac{1}{2}\\frac{1}{g}\\, \\left(\\partial_{\\rho} g\\right)\\, u^{\\rho}\\\\
\n&=& \\partial_{\\mu} u^{\\mu} + \\frac{1}{2}\\frac{1}{g}\\, \\left(\\partial_{\\mu} g\\right)\\, u^{\\mu}
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067 $g$ \u306f\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb $g_{\\mu\\nu}$ \u3092$4\\times 4$ \u6b63\u65b9\u884c\u5217\u3068\u3057\u3066\u3068\u304d\u306e\u884c\u5217\u5f0f<\/p>\n
$$g = \\det(g_{\\mu\\nu})$$<\/p>\n
\u3067\u3042\u308a\uff0c\u76f8\u5bfe\u8ad6\u3067\u4f7f\u30464\u6b21\u5143\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306b\u3064\u3044\u3066\u306f signature $(-, +, +, +)$ \uff08\u307e\u305f\u306f $(+, -, -, -)$ \u306e\u6d41\u5100\u306e\u4eba\u3082\u3042\u308b\uff09\u3067\u3042\u308b\u304b\u3089 $g < 0$ \u3002\u5f93\u3063\u3066<\/p>\n
$$\\frac{1}{2}\\frac{1}{g}\\, \\left(\\partial_{\\mu} g\\right) = \\frac{1}{\\sqrt{-g}} \\partial_{\\mu} \\left( \\sqrt{-g}\\right)$$<\/p>\n
\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\therefore\\ \\ \\nabla_{\\mu} u^{\\mu} &=& \\partial_{\\mu} u^{\\mu} + \\frac{1}{2}\\frac{1}{g}\\, \\left(\\partial_{\\mu} g\\right)\\, u^{\\mu} \\\\
\n&=& \\partial_{\\mu} u^{\\mu} +\\frac{1}{\\sqrt{-g}} \\partial_{\\mu} \\left( \\sqrt{-g}\\right)\\, u^{\\mu} \\\\
\n&=& \\frac{1}{\\sqrt{-g}} \\partial_{\\mu} \\left( \\sqrt{-g}\\, u^{\\mu}\\right)
\n\\end{eqnarray}<\/p>\n
3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u767a\u6563\u3082\u5168\u304f\u540c\u69d8\u306b\u66f8\u3051\u308b\u30023\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3092 $v^i$ \u3068\u3059\u308b\u3068\uff0c\u300c\u767a\u6563\u300d\u306f<\/p>\n
$$\\nabla_i v^i = \\partial_i v^i + \\frac{1}{2} \\frac{1}{g} \\left( \\partial_i\u00a0 g\\right) v^i$$<\/p>\n
\u5ff5\u306e\u305f\u3081\uff0c\u504f\u5fae\u5206\u306f<\/p>\n
$$\\partial_i \\equiv \\frac{\\partial}{\\partial x^i}$$<\/p>\n
\u3067\u3042\u308a\uff0c$x^i = (x^1, x^2, x^3)$ \u306f $x$ \u306e2\u4e57\u3068\u304b3\u4e57\u3068\u304b\u306e\u610f\u5473\u3067\u306f\u306a\u304f\uff0c1\u756a\u76ee\u306e\u5ea7\u6a19\uff0c2\u756a\u76ee\u306e\u5ea7\u6a19\uff0c3\u756a\u76ee\u306e\u5ea7\u6a19\u3068\u3044\u3046\u610f\u5473\u3067\u3057\u305f\u3002<\/p>\n
\u3053\u3053\u3067 $g$ \u306f\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb $g_{ij}$ \u3092$3\\times 3$ \u6b63\u65b9\u884c\u5217\u3068\u3057\u3066\u3068\u304d\u306e\u884c\u5217\u5f0f<\/p>\n
$$g = \\det(g_{ij})$$<\/p>\n
\u3067\u3042\u308a\uff0c3\u6b21\u5143\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306b\u3064\u3044\u3066\u306f signature $(+, +, +)$ \u3067\u3042\u308b\u304b\u3089 $g > 0$ \u3002\u5f93\u3063\u3066<\/p>\n
$$\\frac{1}{2}\\frac{1}{g}\\, \\left(\\partial_{i} g\\right) = \\frac{1}{\\sqrt{g}} \\partial_{i} \\left( \\sqrt{g}\\right)$$<\/p>\n
\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\therefore\\ \\ \\nabla_{i} v^{i} &=& \\partial_i v^i + \\frac{1}{\\sqrt{g}} \\partial_{i} \\left( \\sqrt{g}\\right)\\, v^i \\\\
\n&=& \\frac{1}{\\sqrt{g}} \\partial_{i} \\left( \\sqrt{g}\\, v^{i}\\right)
\n\\end{eqnarray}<\/p>\n
\u7279\u306b $v^i = \\nabla^i \\varphi = g^{ij} \\nabla_j \\varphi = g^{ij} \\partial_j \\varphi$ \uff08\u30b9\u30ab\u30e9\u30fc\u95a2\u6570 $\\varphi$ \u306b\u5bfe\u3057\u3066\u306f\u5171\u5909\u5fae\u5206\u306f\u504f\u5fae\u5206\u3068\u540c\u3058\uff09\u3068\u3059\u308b\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\nabla_{i} v^{i} = \\nabla_{i} \\nabla^i \\varphi \\equiv \\nabla^2 \\varphi &=&
\n\\frac{1}{\\sqrt{g}} \\partial_{i} \\left( \\sqrt{g}\\, g^{ij} \\partial_j \\varphi\\right)
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u304c\u4efb\u610f\u306e $\\varphi$ \u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3064\u306e\u3067\u3042\u308b\u304b\u3089\uff0c3\u6b21\u5143\uff082\u6b21\u5143\u3067\u3082OK\uff09\u4e00\u822c\u5ea7\u6a19\u306b\u304a\u3051\u308b\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f<\/p>\n
$$\\nabla^2 = \\frac{1}{\\sqrt{g}} \\partial_{i} \\left( \\sqrt{g}\\, g^{ij} \\partial_j \\right)$$<\/p>\n
\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u306b\uff0c2\u6b21\u5143\u304a\u3088\u30733\u6b21\u5143\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059\u3068\u3069\u3046\u306a\u308b\u304b\u304c\u66f8\u3044\u3066\u3042\u308b\u3002\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206\u6cd5\u306b\u5f93\u3063\u3066\u611a\u76f4\u306b\u3072\u305f\u3059\u3089\u5b9f\u76f4\u306b\u8a08\u7b97\u3059\u308c\u3070\u6c42\u3081\u3089\u308c\u308b\u304c\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u304a\u3088\u3073\u5171\u5909\u5fae\u5206\u3092\u5b66\u3093\u3060\u3089\uff0c\u3082\u3046\u5c11\u3057\u7c21\u5358\u306b\u6c42\u3081\u3089\u308c\u308b\u3093\u3060\u3088\uff0c\u3068\u3044\u3046\u3053\u3068\u3092\u3053\u3053\u3067\u306f\u8a00\u3044\u305f\u3044\u308f\u3051\u3067\u3059\u3002<\/p>\n
\u4e0a\u306e\u8a08\u7b97\u3067\u884c\u5217\u5f0f $g$ \u306e\u5fae\u5206\u304c<\/p>\n
$$\\frac{1}{g} \\partial_{\\rho} g = g^{\\mu\\nu} \\partial_{\\rho} g_{\\mu\\nu}$$<\/p>\n
\u3068\u306a\u308b\u3053\u3068\u3092\u4f7f\u3063\u305f\u3002\u3053\u3053\u3067 $g^{\\mu\\nu}$ \u306f\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb $g_{\\mu\\nu}$ \u306e\u9006\u884c\u5217\u3067\u3042\u308b\u3002\u3053\u306e\u5f0f\u306f\uff0c\u9006\u884c\u5217\u3084\u884c\u5217\u5f0f\u306e\u5b9a\u7fa9\u3092\u4f7f\u3063\u3066\u8a3c\u660e\u3067\u304d\u308b\u3002\u3053\u3053\u3067\u306f\u8a3c\u660e\u306f\u7701\u7565\u3057\u3066\uff0c\u4f8b\u3068\u3057\u30663\u6b21\u5143\u6975\u5ea7\u6a19\u306e\u5834\u5408\u306b\uff0c\u78ba\u304b\u306b\u6210\u308a\u7acb\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u3066\u307f\u308b\u3002<\/p>\n
\u307e\u305a\uff0c3\u6b21\u5143\u6975\u5ea7\u6a19 $(x^1, x^2, x^3) = (r, \\theta, \\phi)$ \u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306f<\/p>\n
$$
\ng_{ij} = \\left(\\begin{array}{ccc}
\ng_{rr} & g_{r\\theta} & g_{r\\phi} \\\\
\ng_{\\theta r} & g_{\\theta\\theta} & g_{\\theta\\phi} \\\\
\ng_{\\phi r} & g_{\\phi\\theta} & g_{\\phi\\phi}
\n\\end{array}\\right)
\n=\u00a0 \\left(\\begin{array}{ccc}
\n1 & 0& 0 \\\\
\n0 & r^2& 0 \\\\
\n0 & 0 & r^2 \\sin^2\\theta
\n\\end{array}\\right)
\n$$<\/p>\n
\u3067\u3042\u308a\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3092\u884c\u5217\u3068\u307f\u306a\u3057\u305f\u3068\u304d\u306e\u884c\u5217\u5f0f $g$ \u306f<\/p>\n
$$g \\equiv \\det (g_{ij}) = r^4 \\sin^2\\theta$$<\/p>\n
\u307e\u305f\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u9006\u884c\u5217 $g^{ij}$ \u306f<\/p>\n
$$
\ng^{ij} = \\left(\\begin{array}{ccc}
\ng^{rr} & g^{r\\theta} & g^{r\\phi} \\\\
\ng^{\\theta r} & g^{\\theta\\theta} & g^{\\theta\\phi} \\\\
\ng^{\\phi r} & g^{\\phi\\theta} & g^{\\phi\\phi}
\n\\end{array}\\right)
\n=\u00a0 \\left(\\begin{array}{ccc}
\n1 & 0& 0 \\\\
\n0 & \\dfrac{1}{r}& 0 \\\\
\n0 & 0 & \\dfrac{1}{r^{2} \\sin^{2}\\theta}
\n\\end{array}\\right)$$<\/p>\n
\u3053\u308c\u3089\u3092\u4f7f\u3046\u3068<\/p>\n
\\begin{eqnarray}
\n\\frac{1}{g} \\partial_{r} g &=& \\frac{1}{r^4 \\sin^2\\theta} \\partial_r \\left( r^4 \\sin^2\\theta\\right) \\\\
\n&=& \\frac{4}{r} \\\\
\n\\frac{1}{g} \\partial_{\\theta} g &=& \\frac{1}{r^4 \\sin^2\\theta} \\partial_{\\theta} \\left( r^4 \\sin^2\\theta\\right) \\\\
\n&=& 2\\frac{\\cos\\theta}{\\sin\\theta}
\n\\end{eqnarray}<\/p>\n
\u4e00\u65b9\uff0c<\/p>\n
\\begin{eqnarray}
\ng^{ij} \\partial_r g_{ij} &=& g^{rr} \\partial_r g_{rr} + g^{\\theta\\theta} \\partial_r g_{\\theta\\theta} + g^{\\phi\\phi} \\partial_r g_{\\phi\\phi}\\\\
\n&=& 0 + \\frac{1}{r} \\cdot 2 r + \\frac{1}{r^2 \\sin^2\\theta} \\cdot 2 r \\sin^2\\theta \\\\
\n&=& \\frac{4}{r} \\\\
\ng^{ij} \\partial_{\\theta} g_{ij} &=& g^{rr} \\partial_{\\theta} g_{rr} + g^{\\theta\\theta} \\partial_{\\theta} g_{\\theta\\theta}+ g^{\\phi\\phi} \\partial_{\\theta} g_{\\phi\\phi}\\\\
\n&=&0 + 0 + \u00a0 \\frac{1}{r^2 \\sin^2\\theta} \\cdot 2 r^2 \\sin\\theta\\cos\\theta \\\\
\n&=& 2\\frac{\\cos\\theta}{\\sin\\theta}
\n\\end{eqnarray}<\/p>\n
\u3068\u306a\u3063\u3066\u78ba\u304b\u306b<\/p>\n
\\begin{eqnarray}
\n\\frac{1}{g} \\partial_{r} g &=&g^{ij} \\partial_r g_{ij}\\\\
\n\\frac{1}{g} \\partial_{\\theta} g &=& g^{ij} \\partial_{\\theta} g_{ij} \\\\
\n\\frac{1}{g} \\partial_{\\phi} g &=& g^{ij} \\partial_{\\phi} g_{ij} \\ \\\u00a0 (= 0)
\n\\end{eqnarray}<\/p>\n
\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":445,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-10364","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