{"id":445,"date":"2022-01-07T15:23:52","date_gmt":"2022-01-07T06:23:52","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=445"},"modified":"2022-06-07T15:51:03","modified_gmt":"2022-06-07T06:51:03","slug":"%e5%85%b1%e5%a4%89%e5%be%ae%e5%88%86%e3%81%ae%e5%ae%9a%e7%be%a9%e3%81%a8%e3%83%aa%e3%83%83%e3%83%81%e3%81%ae%e6%81%92%e7%ad%89%e5%bc%8f","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\/%e5%85%b1%e5%a4%89%e5%be%ae%e5%88%86%e3%81%ae%e5%ae%9a%e7%be%a9%e3%81%a8%e3%83%aa%e3%83%83%e3%83%81%e3%81%ae%e6%81%92%e7%ad%89%e5%bc%8f\/","title":{"rendered":"\u5171\u5909\u5fae\u5206\u306e\u5b9a\u7fa9\u3068\u30ea\u30c3\u30c1\u306e\u6052\u7b49\u5f0f"},"content":{"rendered":"

\u672c\u7a3f\u3067\u306f\uff0c\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3092\u4f7f\u308f\u305a\u306b\uff0c\u30d9\u30af\u30c8\u30eb\u3092\u5fae\u5206\u3059\u308b\u969b\u306b\u306f\u6210\u5206\u3060\u3051\u3067\u306a\u304f\u57fa\u672c\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3082\u5fae\u5206\u3059\u308b\u306e\u3060\u3068\u3044\u3046\u65b9\u91dd\u3067\u901a\u5e38\u306e\u504f\u5fae\u5206\u306e\u307f\u3067\u901a\u3057\u305f\u304c\uff0c\u5171\u5909\u5fae\u5206<\/strong><\/span>\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/h3>\n

\u3042\u308b\u4e16\u754c\u7dda<\/strong><\/span> \\(x^{\\mu}(v)\\) \u306e\u63a5\u30d9\u30af\u30c8\u30eb\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002
\n$$\\boldsymbol{u} = u^{\\mu}(x) \\,\\boldsymbol{e}_{\\mu} \\equiv \\frac{dx^{\\mu}}{dv} \\,\\boldsymbol{e}_{\\mu}$$<\/p>\n

\u6e2c\u5730\u7dda<\/strong><\/span>\u3068\u306f\uff0c\u3053\u306e\u4e16\u754c\u7dda\u306b\u305d\u3063\u3066\u63a5\u30d9\u30af\u30c8\u30eb\u304c\u4e00\u5b9a\uff0c\u3064\u307e\u308a
\n$$\\frac{d\\boldsymbol{u}}{dv} = \\boldsymbol{0}$$\u3067\u3042\u308b\u3088\u3046\u306a\u7dda\u3067\u3042\u308a\uff0c\u3053\u306e\u5f0f\u3092\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u547c\u3076\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306e\u5de6\u8fba\u306e\u8a08\u7b97\u3092\u3082\u3046\u5c11\u3057\u7d9a\u3051\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{d\\boldsymbol{u}}{dv} &=&
\n\\frac{d x^{\\nu}}{dv} \\frac{\\partial}{\\partial x^{\\nu}} \\left( u^{\\mu}\\,\\boldsymbol{e}_{\\mu}\\right) \\\\
\n&=& \\left\\{ \\frac{\\partial u^{\\mu}}{\\partial x^{\\nu}}\\,\\boldsymbol{e}_{\\mu} +
\nu^{\\mu} \\,\\frac{\\partial\\boldsymbol{e}_{\\mu}}{\\partial x^{\\nu}}\\right\\}\u00a0 \\frac{d x^{\\nu}}{dv} \\\\
\n&=& \\left( u^{\\mu}_{\\ \\, ,\\nu} \\,\\boldsymbol{e}_{\\mu} +
\nu^{\\mu} \\,\\boldsymbol{e}_{\\mu,\\nu} \\right)\u00a0 u^{\\nu} \\\\
\n&=& \\left( u^{\\rho}_{\\ \\, ,\\nu} \\,\\boldsymbol{e}_{\\rho} +
\nu^{\\mu} \\,\\varGamma^{\\rho}_{\\ \\,\u00a0 \\mu\\nu}\\,\\boldsymbol{e}_{\\rho} \\right)\u00a0 u^{\\nu} \\\\
\n&=& \\left( u^{\\rho}_{\\ \\, ,\\nu}\u00a0 +
\n\\varGamma^{\\rho}_{\\ \\,\u00a0 \\mu\\nu} u^{\\mu} \\right)\u00a0 u^{\\nu}\\,\\boldsymbol{e}_{\\rho}
\n\\end{eqnarray}<\/p>\n

(1,0)-\u578b\u30c6\u30f3\u30bd\u30eb\u3059\u306a\u308f\u3061\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/h3>\n

\u4e0a\u8a18\u306e\u3088\u3046\u306b\uff0c\u30d9\u30af\u30c8\u30eb\u3092\u5fae\u5206\u3059\u308b\u3068\u304d\u306b\u306f\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3060\u3051\u3067\u306a\u304f\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3082\u5fae\u5206\u3059\u308b\u3053\u3068\u3092\u5fd8\u308c\u306a\u3051\u308c\u3070\uff0c\u4e16\u306e\u4e2d\u306b\u306f\u666e\u901a\u306e\u504f\u5fae\u5206\u3060\u3051\u3067\u3053\u3068\u304c\u8db3\u308a\u308b\u3002<\/p>\n

\u3057\u304b\u3057\uff0c\u4e16\u306e\u4e2d\u306b\u306f\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u3092\u7701\u7565\u3057\u3066\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u300c\u3060\u3051\u300d\u3067\u30d9\u30af\u30c8\u30eb\u3092\u8868\u3059\u4eba\u3082\u3044\u308b\u3002\u305d\u3093\u306a\uff08\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u7701\u7565\u6d3e\u306e\uff09\u4eba\u3005\u306e\u305f\u3081\u306b\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n

$$\\nabla_{\\nu} u^{\\rho} \\equiv u^{\\rho}_{\\ \\, ;\\nu} \\equiv u^{\\rho}_{\\ \\, ,\\nu}\u00a0 +
\n\\varGamma^{\\rho}_{\\ \\,\u00a0 \\mu\\nu} u^{\\mu} $$<\/p>\n

\u3053\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u305f\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3092\u4f7f\u3046\u3068\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f<\/p>\n

$$\\frac{d\\boldsymbol{u}}{dv} = u^{\\rho}_{\\ \\, ;\\nu} u^{\\nu} \\boldsymbol{e}_{\\rho} = \\boldsymbol{0}$$
\n\u3057\u305f\u304c\u3063\u3066\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306e\u6210\u5206\u306f\uff0c
\n$$ u^{\\rho}_{\\ \\, ;\\nu} u^{\\nu} = 0$$ \u3068\u306a\u308b\u3002<\/p>\n

\u4e16\u306e\u4e2d\u306b\u306f\uff0c\u666e\u901a\u306e\u504f\u5fae\u5206\u300c\u4ee5\u5916\u300d\u306b\uff0c\u4f55\u304b\u5225\u306e\u3082\u306e\u3067\u3042\u308b\u5171\u5909\u5fae\u5206<\/strong><\/span>\u304c\u3042\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3002\u4e16\u306e\u4e2d\u306b\u3042\u308b\u306e\u306f\u666e\u901a\u306e\u504f\u5fae\u5206\u306e\u307f\u3067\u3042\u308b\u3002\u5171\u5909\u5fae\u5206\u3068\u306f\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u7701\u7565\u6d3e\u306e\u305f\u3081\u306e\u4fbf\u6cd5\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u305f\u3082\u306e<\/strong><\/span>\uff0c\u3068\u3044\u3046\u7406\u89e3\u3082\u3042\u308b\u304b\u3068\u601d\u3044\u307e\u3059\u304c\uff0c\u3044\u304b\u304c\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n

\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/h3>\n

\u30d9\u30af\u30c8\u30eb\u306f\uff0c(1, 0)-\u578b\u306e\u30c6\u30f3\u30bd\u30eb\uff0c\u3064\u307e\u308a\u6210\u5206\u304c1\u3064\u306e\u4e0a\u6dfb\u5b57\u3092\u6301\u3064\u30c6\u30f3\u30bd\u30eb\u3067\u3042\u308b\u3002\u3053\u308c\u304b\u3089\uff0c\u4e00\u822c\u306e (m,n)-\u578b\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u5b9a\u7fa9\u3057\u3066\u3044\u304f\u3002<\/p>\n

(0, 0)-\u578b\u30c6\u30f3\u30bd\u30eb\u3059\u306a\u308f\u3061\u30b9\u30ab\u30e9\u30fc\u306e\u5171\u5909\u5fae\u5206<\/h4>\n

\u30b9\u30ab\u30e9\u30fc \\(\\phi \\) \u306b\u5bfe\u3059\u308b\u5171\u5909\u5fae\u5206<\/strong><\/span>\u306f\uff0c\u901a\u5e38\u306e\u504f\u5fae\u5206\u3068\u540c\u3058\u3068\u3059\u308b\u3002<\/p>\n

$$ \\nabla_{\\nu} \\phi = \\phi_{;\\nu} = \\phi_{,\\nu}$$<\/p>\n

(0, 1)-\u578b\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/h4>\n

\\( a_{\\mu} \\) \u306e\u3088\u3046\u306a (0, 1)-\u578b\u306e\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306b\u3064\u3044\u3066\u306f\uff0c\\( a_{\\mu} \\,u^{\\mu} \\) \u304c\u30b9\u30ab\u30e9\u30fc\u3068\u306a\u308b\u3053\u3068\u304b\u3089\uff0c
\n$$\\left( a_{\\mu} \\,u^{\\mu} \\right)_{;\\mu} = \\left( a_{\\mu} \\,u^{\\mu} \\right)_{,\\mu}
\n= a_{\\mu, \\nu}\\,u^{\\mu} + a_{\\mu} \\,u^{\\mu}_{\\ \\, ,\\nu}$$
\n\u4e00\u65b9\u3067\uff0c\u5171\u5909\u5fae\u5206<\/strong><\/span>\u306b\u5bfe\u3057\u3066\u3082\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\u304c\u6210\u308a\u7acb\u3064\u3068\u3057\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\n\\left( a_{\\mu} \\,u^{\\mu} \\right)_{;\\mu} &=& a_{\\mu; \\nu}\\,u^{\\mu} + a_{\\mu} \\,u^{\\mu}_{\\ \\, ; \\nu} \\\\
\n&=& a_{\\mu; \\nu}\\,u^{\\mu} + a_{\\mu} \\left( u^{\\mu}_{\\ \\, ,\\nu} + \\varGamma^{\\mu}_{\\ \\, \\rho\\nu} u^{\\rho}\\right) \\\\
\n&=& a_{\\mu; \\nu}\\,u^{\\mu} + a_{\\mu}\\,u^{\\mu}_{\\ \\, ,\\nu} + a_{\\rho}\\, \\varGamma^{\\rho}_{\\ \\, \\mu\\nu}\\, u^{\\mu}
\n\\end{eqnarray}<\/p>\n

2\u3064\u306e\u8868\u793a\u3092\u6bd4\u8f03\u3057\u3066\uff0c\u4ee5\u4e0b\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n

$$a_{\\mu; \\nu} = a_{\\mu, \\nu} – \\varGamma^{\\rho}_{\\ \\, \\mu\\nu}\\, a_{\\rho}$$<\/p>\n

(0, 2)-\u578b\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/h4>\n

\\(b_{\\mu\\nu}\\) \u306e\u3088\u3046\u306a (0, 2)-\u578b\u30c6\u30f3\u30bd\u30eb\u306b\u3064\u3044\u3066\u306f\uff0c\\(b_{\\mu\\nu}\\, u^{\\nu}= a_{\\mu} \\) \u3068\u3057\u3066 (0, 1)-\u578b\u306e\u5171\u5909\u5fae\u5206<\/strong><\/span>\u306e\u898f\u5247\u3092\u9069\u7528\u3057\u3066<\/p>\n

\\begin{eqnarray}
\n\\left( b_{\\mu\\nu}\\, u^{\\nu}\\right)_{; \\sigma} &=& \\left( b_{\\mu\\nu}\\, u^{\\nu}\\right)_{, \\sigma} – \\varGamma^{\\rho}_{\\ \\, \\mu\\sigma} \\, b_{\\rho\\nu}\\, u^{\\nu} \\\\
\n&=& b_{\\mu\\nu, \\sigma}\\, u^{\\nu} + b_{\\mu\\nu}\\, u^{\\nu}_{\\ \\, ,\\sigma} – \\varGamma^{\\rho}_{\\ \\, \\mu\\sigma} \\, b_{\\rho\\nu}\\, u^{\\nu}
\n\\end{eqnarray}
\n\u4e00\u65b9\u3067\uff0c\u5171\u5909\u5fae\u5206<\/strong><\/span>\u306e\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\n\\left( b_{\\mu\\nu}\\, u^{\\nu}\\right)_{; \\sigma} &=&
\nb_{\\mu\\nu; \\sigma}\\, u^{\\nu} + b_{\\mu\\nu}\\, u^{\\nu}_{\\ \\,; \\sigma} \\\\
\n&=& b_{\\mu\\nu; \\sigma}\\, u^{\\nu} +
\nb_{\\mu\\nu}\\left(u^{\\nu}_{\\ \\, ,\\sigma} + \\varGamma^{\\nu}_{\\ \\, \\rho\\sigma} \\,u^{\\rho} \\right) \\\\
\n&=& b_{\\mu\\nu; \\sigma}\\, u^{\\nu} +
\nb_{\\mu\\nu}\\,u^{\\nu}_{\\ \\, ,\\sigma} + b_{\\mu\\rho} \\varGamma^{\\rho}_{\\ \\, \\nu\\sigma} \\,u^{\\nu}
\n\\end{eqnarray} 2\u3064\u306e\u8868\u793a\u3092\u6bd4\u8f03\u3057\u3066\uff0c\u4ee5\u4e0b\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n

$$ b_{\\mu\\nu; \\sigma} = b_{\\mu\\nu, \\sigma} – \\varGamma^{\\rho}_{\\ \\, \\mu\\sigma} \\, b_{\\rho\\nu} – \\varGamma^{\\rho}_{\\ \\, \\nu\\sigma} \\, b_{\\mu\\rho}$$<\/p>\n

<\/h4>\n

(1, 1)-\u578b\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/h4>\n

\\(c^{\\mu}_{\\ \\, \\nu}\\) \u306e\u3088\u3046\u306a (1, 1)-\u578b\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3082\uff0c\u305f\u3068\u3048\u3070 \\(c^{\\mu}_{\\ \\, \\nu} \\,u^{\\nu} \\) \u304c\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3068\u540c\u7b49\u3067\u3042\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

$$c^{\\mu}_{\\ \\, \\nu; \\sigma} = c^{\\mu}_{\\ \\, \\nu, \\sigma} + \\varGamma^{\\mu}_{\\ \\,\\rho\\sigma}\\,c^{\\rho}_{\\ \\, \\nu} – \\varGamma^{\\rho}_{\\ \\, \\nu\\sigma}\\,c^{\\mu}_{\\ \\,\\rho}$$<\/p>\n

 <\/p>\n

\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u5171\u5909\u5fae\u5206\u306f\u30bc\u30ed<\/h3>\n

\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span> \\(g_{\\mu\\nu} \\) \u3082 (0, 2)-\u578b\u30c6\u30f3\u30bd\u30eb\u3067\u3042\u308b\u306e\u3067\uff0c\u4e0a\u5f0f\u306e\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3092\u9069\u7528\u3059\u308b\u3068\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

$$g_{\\mu\\nu; \\sigma} = g_{\\mu\\nu, \\sigma} – \\varGamma^{\\rho}_{\\ \\, \\mu\\sigma} \\, g_{\\rho\\nu} – \\varGamma^{\\rho}_{\\ \\, \\nu\\sigma} \\, g_{\\mu\\rho} = 0$$<\/p>\n

\u4e0a\u8a18\u306e\u7d50\u679c\u306f\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u305d\u3082\u305d\u3082\u306e\u5b9a\u7fa9\u304b\u3089\u304d\u3066\u3044\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb \\(g_{\\mu\\nu} \\equiv
\n\\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu}\\) \u3092 \\(x^{\\sigma}\\) \u3067\u504f\u5fae\u5206\u3059\u308b\u3068\uff0c
\n\\begin{eqnarray}
\ng_{\\mu\\nu, \\sigma} &=&
\n\\boldsymbol{e}_{\\mu, \\sigma}\\cdot\\boldsymbol{e}_{\\nu}
\n+ \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu,\\sigma} \\nonumber \\\\
\n&=& \\varGamma^{\\rho}_{\\ \\ \\mu\\sigma}\\boldsymbol{e}_{\\rho}
\n\\cdot\\boldsymbol{e}_{\\nu}
\n+ \\varGamma^{\\rho}_{\\
\n\\nu\\sigma}\\boldsymbol{e}_{\\rho}\\cdot\\boldsymbol{e}_{\\mu}\\nonumber\\\\
\n&=& g_{\\rho\\nu} \\varGamma^{\\rho}_{\\ \\ \\mu\\sigma} + g_{\\rho\\mu} \\varGamma^{\\rho}_{\\ \\ \\nu\\sigma}
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u9006\u884c\u5217<\/strong><\/span> \\( g^{\\mu\\nu} \\) \u306e\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3082\u30bc\u30ed\u3068\u306a\u308b\u3002
\n$$g^{\\mu\\nu}_{\\ \\ \\\u00a0\u00a0 ;\\sigma} = 0$$
\n\u3053\u308c\u306f\u9006\u884c\u5217\u306e\u5b9a\u7fa9
\n$$g^{\\mu\\nu}\\,g_{\\nu\\lambda} = \\delta^{\\mu}_{\\ \\,\\lambda}$$ \u306e\u4e21\u8fba\u306e\u5171\u5909\u5fae\u5206<\/strong><\/span>\u304b\u3089\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u3067 \\(\\displaystyle \\delta^{\\mu}_{\\ \\,\\lambda}\\) \u306f\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u306e\u30c7\u30eb\u30bf<\/strong><\/span>\u3067\u3042\u308a\uff0c\\(\\mu=\\lambda\\) \u306e\u3068\u304d \\(1\\)\uff0c\u305d\u308c\u4ee5\u5916\u306f \\(0\\) \u3092\u4e0e\u3048\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u30c6\u30f3\u30bd\u30eb\u306e\u6dfb\u5b57\u306e\u4e0a\u3052\u4e0b\u3052\u3068\u5171\u5909\u5fae\u5206\u306f\u53ef\u63db<\/h4>\n

\u4e00\u822c\u306b (m, n)-\u578b\u30c6\u30f3\u30bd\u30eb\u306e\u6dfb\u5b57\u306e\u4e0a\u3052\u4e0b\u3052\u306f \\(g^{\\mu\\nu}\\) \u3068 \\(g_{\\mu\\nu}\\) \u3092\u4f7f\u3063\u3066\u884c\u3046\u306e\u3067\uff0c\\(g^{\\mu\\nu}_{\\ \\ \\\u00a0 ;\\sigma} = 0, \\ \\ g_{\\mu\\nu;\\sigma} = 0\\) \u304b\u3089\uff0c\u6dfb\u5b57\u306e\u4e0a\u3052\u4e0b\u3052\u3068\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3068\u306f\u53ef\u63db<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u3064\u307e\u308a\uff0c\u5148\u306b\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u6dfb\u5b57\u3092\u4e0a\u3052\u4e0b\u3052\u3057\u3066\u3042\u3068\u304b\u3089\u305d\u306e\u6210\u5206\u3092\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3057\u3088\u3046\u304c\uff0c\u5148\u306b\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u3092\u5171\u5909\u5fae\u5206<\/strong><\/span>\u3057\u3066\u304b\u3089\u3042\u3068\u3067\u305d\u306e\u6210\u5206\u306e\u6dfb\u5b57\u3092\u4e0a\u3052\u4e0b\u3052\u3057\u3088\u3046\u304c\uff0c\u7b54\u3048\u306f\u540c\u3058\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n

\u5171\u5909\u5fae\u5206\u306e\u975e\u53ef\u63db\u6027\u3068\u30ea\u30c3\u30c1\u306e\u6052\u7b49\u5f0f<\/h3>\n

\u504f\u5fae\u5206\u306f\u53ef\u63db<\/strong><\/span>\uff08\u4ea4\u63db\u53ef\u80fd\uff0c\u5fae\u5206\u306e\u9806\u5e8f\u3092\u5165\u308c\u66ff\u3048\u3066\u3082\u540c\u3058\u7d50\u679c\u306b\u306a\u308b\u3068\u3044\u3046\u3053\u3068\uff09\u3067\u3042\u308b\u304c\uff0c\u4e0a\u8a18\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u305f\u5171\u5909\u5fae\u5206\u306f\u53ef\u63db\u3067\u306f\u306a\u3044<\/strong><\/span>\u3002<\/p>\n

\u30d9\u30af\u30c8\u30eb\uff08\u5f8c\u3067\u4f7f\u3046\u306e\u3067\uff0c\u3053\u3053\u3067\u306f (0,1)-\u578b\u30c6\u30f3\u30bd\u30eb\u3068\u3059\u308b\uff09\u306e2\u968e\u5171\u5909\u5fae\u5206\u306e\u975e\u53ef\u63db\u6027<\/strong><\/span>\u3092\u8868\u3059\u53cd\u5bfe\u79f0\u90e8\u5206\u306f\uff0c\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n

$$k_{\\alpha; \\mu\\nu} – k_{\\alpha; \\nu\\mu} = R_{\\beta\\alpha\\mu\\nu} k^{\\beta}$$<\/p>\n

\u3053\u308c\u3092\uff08\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u304c\u73fe\u308c\u308b\u306e\u306b\u3082\u304b\u304b\u308f\u3089\u305a\uff09\u30ea\u30c3\u30c1\u306e\u6052\u7b49\u5f0f<\/strong><\/span>\u3068\u3044\u3063\u305f\u308a\u3059\u308b\u304c\uff0c\u3053\u308c\u3092\u793a\u3057\u3066\u304a\u304f\u3002<\/p>\n

\u307e\u305a\uff0c<\/p>\n

\\begin{eqnarray}
\nk_{\\alpha; \\mu} &=& k_{\\alpha, \\mu} – \\varGamma^{\\beta}_{\\ \\ \\ \\alpha\\mu} k_{\\beta}\\\\
\nk_{\\alpha; \\mu\\nu} &=& \\left(k_{\\alpha, \\mu} – \\varGamma^{\\beta}_{\\ \\ \\ \\alpha\\mu} k_{\\beta} \\right)_{, \\nu} \\\\
\n&&\\qquad – \\varGamma^{\\lambda}_{\\ \\ \\ \\alpha\\nu} \\left(k_{\\lambda, \\mu} – \\varGamma^{\\beta}_{\\ \\ \\ \\lambda\\mu} k_{\\beta} \\right)
\n– \\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu} \\left(k_{\\alpha, \\lambda} – \\varGamma^{\\beta}_{\\ \\ \\ \\alpha\\lambda} k_{\\beta} \\right) \\\\
\n&=& k_{\\alpha, {\\color{red}{\\mu\\nu}}} – \\varGamma^{\\beta}_{\\ \\ \\ \\alpha\\mu, \\nu} k_{\\beta}
\n– \\varGamma^{\\beta}_{\\ \\ \\ \\alpha{\\color{red}{\\mu}}} k_{\\beta, {\\color{red}{\\nu}}}
\n– \\varGamma^{\\lambda}_{\\ \\ \\ \\alpha{\\color{red}{\\nu}}} k_{\\lambda, {\\color{red}{\\mu}}} \\\\
\n&&\\qquad + \\varGamma^{\\lambda}_{\\ \\ \\ \\alpha\\nu} \\varGamma^{\\beta}_{\\ \\ \\ \\lambda\\mu} k_{\\beta}
\n– \\varGamma^{\\lambda}_{\\ \\ \\ {\\color{red}{\\mu\\nu}}} \\left(k_{\\alpha, \\lambda} – \\varGamma^{\\beta}_{\\ \\ \\ \\alpha\\lambda} k_{\\beta} \\right) \\\\
\n\\end{eqnarray}<\/p>\n

\u8d64\u8272<\/span>\u306e\u90e8\u5206\u306f ${\\color{red}{\\mu \\nu}}$ \u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u90e8\u5206\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n

\\begin{eqnarray}
\n\\therefore\\ \\ k_{\\alpha; \\mu\\nu}\u00a0 – k_{\\alpha; \\nu\\mu}&=&
\n\\left(\\varGamma^{\\beta}_{\\ \\ \\ \\alpha\\nu, \\mu}
\n– \\varGamma^{\\beta}_{\\ \\ \\ \\alpha\\mu, \\nu}
\n+ \\varGamma^{\\beta}_{\\ \\ \\ \\lambda\\mu}\\varGamma^{\\lambda}_{\\ \\ \\ \\alpha\\nu}
\n– \\varGamma^{\\beta}_{\\ \\ \\ \\lambda\\nu} \\varGamma^{\\lambda}_{\\ \\ \\ \\alpha\\mu} \\right) k_{\\beta}\\\\
\n&=& R^{\\beta}_{\\ \\ \\ \\alpha\\mu\\nu} k_{\\beta} \\\\
\n&=& R_{\\beta\\alpha\\mu\\nu} k^{\\beta}
\n\\end{eqnarray}<\/p>\n

\u30ea\u30c3\u30c1\u306e\u6052\u7b49\u5f0f<\/strong><\/span>\u306f\u307e\u305f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u65b9\u304c\u30dd\u30d4\u30e5\u30e9\u30fc\u304b\u3082\u3057\u308c\u306a\u3044\u3002\uff08\u672c\u7a3f\u3067\u306f\uff0c\u5171\u5909\u5fae\u5206\u3092\u5b9a\u7fa9\u305b\u305a\u306b\u6e2c\u5730\u7dda\u504f\u5dee\u306e\u5f0f\u304b\u3089\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u3092\u5b9a\u7fa9\u3057\u305f<\/a><\/strong><\/span>\u304c\uff0c\u5171\u5909\u5fae\u5206\u306e\u5b9a\u7fa9\u304b\u3089\u59cb\u307e\u308b\u30c6\u30ad\u30b9\u30c8\u3067\u306f\uff0c\u3053\u306e\u5f0f\u3092\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u5b9a\u7fa9<\/strong><\/span>\u3068\u3059\u308b\u5834\u5408\u3082\u591a\u3044\u304b\u3068\u3002\uff09<\/p>\n

$$k^{\\alpha}_{\\ \\ ;\\mu\\nu} – k^{\\alpha}_{\\ \\ ;\\nu\\mu} = R^{\\alpha}_{\\ \\ \\ \\beta \\nu\\mu} k^{\\beta}$$<\/p>\n

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