{"id":385,"date":"2022-01-07T11:32:22","date_gmt":"2022-01-07T02:32:22","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=385"},"modified":"2023-05-08T12:33:18","modified_gmt":"2023-05-08T03:33:18","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%82%af%e3%83%aa%e3%82%b9%e3%83%88%e3%83%83%e3%83%95%e3%82%a7%e3%83%ab%e8%a8%98%e5%8f%b7","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\/%e6%b8%ac%e5%9c%b0%e7%b7%9a%e3%81%a8%e6%8e%a5%e7%b6%9a%e4%bf%82%e6%95%b0%e3%83%bb%e3%82%af%e3%83%aa%e3%82%b9%e3%83%88%e3%83%83%e3%83%95%e3%82%a7%e3%83%ab%e8%a8%98%e5%8f%b7\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%82%af%e3%83%aa%e3%82%b9%e3%83%88%e3%83%83%e3%83%95%e3%82%a7%e3%83%ab%e8%a8%98%e5%8f%b7\/","title":{"rendered":"\u88dc\u8db3\uff1a\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7"},"content":{"rendered":"

<\/p>\n

\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u504f\u5fae\u5206\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/h3>\n

$$ \\boldsymbol{e}_{\\mu, \\nu} \\equiv \\varGamma^{\\lambda}_{\\ \\ \\mu\\nu} \\boldsymbol{e}_{\\lambda}$$ \u3068\u66f8\u304d\uff0c\\( \\varGamma^{\\lambda}_{\\ \\ \\mu\\nu} \\) \u3092\u4e00\u822c\u306b\u63a5\u7d9a\u4fc2\u6570<\/strong><\/span>\u3068\u547c\u3076\u306e\u3067\u3042\u3063\u305f\u3002\u4e00\u822c\u76f8\u5bfe\u8ad6\u3067\u306f\u4f7f\u308f\u308c\u308b\u63a5\u7d9a\u4fc2\u6570\u306f\uff0c
\n$$ \\varGamma^{\\lambda}_{\\ \\ \\mu\\nu} = \\varGamma^{\\lambda}_{\\ \\ \\nu\\mu}$$
\n\u3059\u306a\u308f\u3061
\n$$\\boldsymbol{e}_{\\mu, \\nu} = \\boldsymbol{e}_{\\nu, \\mu} $$
\n\u306e\u3088\u3046\u306b\u4e0b\u6dfb\u5b57\u306b\u3064\u3044\u3066\u5bfe\u79f0\u3067\u3042\u308b\u3068\u3044\u3046\u6027\u8cea\u3092\u3082\u3061\uff0c\u7279\u306b\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u504f\u5fae\u5206\u3067\u8868\u3059<\/h3>\n

\u3053\u306e\u5b9a\u7fa9\u3068\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306e\u5b9a\u7fa9
\n$$ g_{\\mu\\nu} = \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu} $$ \u304b\u3089
\n$$\\varGamma^{\\lambda}_{\\ \\ \\mu\\nu} = \\frac{1}{2} g^{\\lambda\\sigma}(g_{\\sigma\\mu, \\nu} + g_{\\sigma\\nu, \\mu} – g_{\\mu\\nu, \\sigma})$$ \u3092\u5c0e\u304f\uff0c\u3068\u3044\u3046\u8a71\u3002<\/p>\n

\u307e\u305a\uff0c
\n\\begin{eqnarray}
\ng_{\\mu\\nu, \\sigma} &=& \\boldsymbol{e}_{\\mu, \\sigma}\\cdot \\boldsymbol{e}_{\\nu} + \\boldsymbol{e}_{\\mu}\\cdot \\boldsymbol{e}_{\\nu, \\sigma}\\\\
\n&=&\u00a0 \\varGamma^{\\rho}_{\\ \\ \\mu\\sigma} \\boldsymbol{e}_{\\rho}\\cdot \\boldsymbol{e}_{\\nu} + \\boldsymbol{e}_{\\mu}\\cdot \\varGamma^{\\rho}_{\\ \\ \\nu\\sigma} \\boldsymbol{e}_{\\rho} \\\\
\n&=& g_{\\rho\\nu} \\varGamma^{\\rho}_{\\ \\ \\mu\\sigma} + g_{\\mu\\rho} \\varGamma^{\\rho}_{\\ \\ \\nu\\sigma}
\n\\end{eqnarray}
\n\u3067\u3042\u3063\u305f\u3002\\(\\mu, \\nu, \\rho\\) \u306e\u6dfb\u5b57\u306e\u9806\u756a\u3092\u9069\u5b9c\u5909\u3048\u30663\u672c\u4e26\u3079\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n-g_{\\mu\\nu, \\sigma}
\n&=&\\\u00a0 \\color{red}{-g_{\\rho\\nu} \\varGamma^{\\rho}_{\\ \\ \\mu\\sigma}} \\\u00a0 \\color{blue}{\\\u00a0 – \\\u00a0 \\, g_{\\mu\\rho} \\varGamma^{\\rho}_{\\ \\ \\nu\\sigma}}\\\\
\ng_{\\sigma\\mu, \\nu}
\n&=& \\quad\u00a0 \\color{blue}{g_{\\rho\\mu} \\varGamma^{\\rho}_{\\ \\ \\sigma\\nu}} \\color{black}{\\ + \\ g_{\\sigma\\rho} \\varGamma^{\\rho}_{\\ \\ \\mu\\nu}}\\\\
\ng_{\\sigma\\nu, \\mu}
\n&=& \\quad \u00a0 \\color{red}{g_{\\rho\\nu} \\varGamma^{\\rho}_{\\ \\ \\sigma\\mu}} \\color{black}{\\ +\\\u00a0 g_{\\sigma\\rho} \\varGamma^{\\rho}_{\\ \\ \\nu\\mu}}
\n\\end{eqnarray}<\/p>\n

\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u4e0b\u6dfb\u5b57\u306e\u5bfe\u79f0\u6027\u3092\u4f7f\u3044\uff0c\u4e0a\u5f0f3\u672c\u3092\u8db3\u3057\u5408\u308f\u305b\u308b\u3068\uff0c\u8d64\u8272\u540c\u58eb<\/span>\uff0c\u9752\u8272\u540c\u58eb<\/span>\u306f\u30ad\u30e3\u30f3\u30bb\u30eb\u3057\u3066<\/p>\n

$$g_{\\sigma\\mu, \\nu} + g_{\\sigma\\nu, \\mu} – g_{\\mu\\nu, \\sigma}\u00a0 = 2 \\ g_{\\sigma\\rho} \\varGamma^{\\rho}_{\\ \\ \\mu\\nu}$$<\/p>\n

\u3042\u3068\u306f\uff0c
\n$$ g^{\\lambda\\sigma} g_{\\sigma\\rho} = \\delta^{\\lambda}_{\\ \\ \\rho} $$
\n\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206 \\(g_{\\sigma\\rho} \\) \u306e\u9006\u884c\u5217 \\( g^{\\lambda\\sigma} \\) \u3092\u4f7f\u3046\u3068\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u4e0b\u6dfb\u5b57\u5bfe\u79f0\u6027\u306b\u3064\u3044\u3066\u3082\u3046\u5c11\u3057<\/h3>\n

\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/strong><\/span>\u306e\u4e0b\u6dfb\u5b57\u306e\u5bfe\u79f0\u6027
\n$$\\varGamma^{\\lambda}_{\\ \\\u00a0 \\mu\\nu} = \\varGamma^{\\lambda}_{\\ \\\u00a0 \\nu\\mu}$$
\n\u3059\u306a\u308f\u3061$$\\boldsymbol{e}_{\\mu , \\nu} = \\boldsymbol{e}_{\\nu , \\mu}$$
\n\u306b\u3064\u3044\u3066\u88dc\u8db3\u3057\u3066\u304a\u304f\u3002<\/p>\n

\u4eca\uff0c\u5ea7\u6a19\u5909\u63db\u306b\u3088\u308b\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u5909\u63db\u6027\u3092\u304a\u3055\u3089\u3044\u3059\u308b\u305f\u3081\uff0c\u4ee5\u4e0b\u306e\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(d\\boldsymbol{x}\\) \u3092\u8003\u3048\u308b\u3002
\n$$ d\\boldsymbol{x} \\equiv dx^{\\mu} \\boldsymbol{e}_{\\mu}$$
\n\\( x^{\\mu} \\rightarrow\u00a0 x^{\\mu’} \\) \u306e\u5ea7\u6a19\u5909\u63db\u3092\u8003\u3048\u308b\u3068\uff0c\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u63db\u3055\u308c\u308b\u3002
\n$$ dx^{\\mu} = \\frac{\\partial x^{\\mu}}{\\partial x^{\\nu’}} dx^{\\nu’} $$
\n\u4e00\u65b9\u3067\uff0c\u30d9\u30af\u30c8\u30eb \\(d\\boldsymbol{x}\\) \u305d\u306e\u3082\u306e\u306f\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u306a\u3044\u5e7e\u4f55\u5b66\u7684\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3067\u3042\u308b\u304b\u3089
\n\\begin{eqnarray}
\nd\\boldsymbol{x} &=& dx^{\\lambda} \\boldsymbol{e}_{\\lambda} \\\\
\n&=& \\frac{\\partial x^{\\lambda}}{\\partial x^{\\mu’}} dx^{\\mu’} \\boldsymbol{e}_{\\lambda} \\\\
\n&\\equiv& dx^{\\mu’} \\boldsymbol{e}_{\\mu’}
\n\\end{eqnarray} \u3068\u306a\u308a\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u5909\u63db\u5247\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5c0e\u304b\u308c\u308b\u3002
\n$$\\therefore \\ \\boldsymbol{e}_{\\mu’} = \\frac{\\partial x^{\\lambda}}{\\partial x^{\\mu’}} \\boldsymbol{e}_{\\lambda}$$ \u3053\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u3092 \\(x^{\\nu’}\\) \u3067\u504f\u5fae\u5206\u3059\u308b\u3068
\n\\begin{eqnarray}
\n\\frac{\\partial }{\\partial x^{\\nu’}} \\boldsymbol{e}_{\\mu’} &=& \\boldsymbol{e}_{\\mu’ , \\nu’} \\\\
\n&=& \\frac{\\partial^2 x^{\\lambda}}{\\partial x^{\\nu’} \\partial x^{\\mu’}} \\boldsymbol{e}_{\\lambda} + \\frac{\\partial x^{\\lambda}}{\\partial x^{\\mu’}}\\boldsymbol{e}_{\\lambda , \\sigma}\\frac{\\partial x^{\\sigma}}{\\partial x^{\\nu’}}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c\u6642\u7a7a\u306e\u4efb\u610f\u306e1\u70b9\u3068\u305d\u306e\u8fd1\u508d\u3067\uff0c\u5c40\u6240\u7684\u306b\u7279\u6b8a\u76f8\u5bfe\u8ad6\u304c\u6210\u308a\u7acb\u3064\uff0c\u3064\u307e\u308a\u5c40\u6240\u6163\u6027\u7cfb\u3092\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u308b\uff0c\u3068\u4eee\u5b9a\u3059\u308b\u3002\u70b9 \\(P\\) \u3067\u306e\u5c40\u6240\u6163\u6027\u7cfb\u3092 \\(x^{\\mu}\\) \u3068\u3059\u308b\u3068\uff0c
\n$$\\boldsymbol{e}_{\\lambda , \\sigma}(P) =0 $$ \u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\uff0c\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u3059\u308b\u3068\uff0c
\n\\begin{eqnarray}
\n\\boldsymbol{e}_{\\mu’ , \\nu’}(P) &=& \\frac{\\partial^2 x^{\\lambda}}{\\partial x^{\\nu’} \\partial x^{\\mu’}} \\boldsymbol{e}_{\\lambda}(P)\\\\
\n&=& \\frac{\\partial^2 x^{\\lambda}}{\\partial x^{\\mu’} \\partial x^{\\nu’}} \\boldsymbol{e}_{\\lambda}(P)\\\\
\n&=& \\boldsymbol{e}_{\\nu’ , \\mu’}(P) \\end{eqnarray}<\/p>\n

\u4eca\u5f8c\uff0c\u6642\u7a7a\u306e\u4efb\u610f\u306e1\u70b9\u3068\u305d\u306e\u8fd1\u508d\u3067\u5c40\u6240\u6163\u6027\u7cfb\u3092\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u4eee\u5b9a\u3057\u3066\u3044\u308b\u304b\u3089\uff0c\u4efb\u610f\u306e\u5ea7\u6a19\u7cfb\u30fb\u4efb\u610f\u306e\u70b9\u3067
\n$$\\boldsymbol{e}_{\\mu , \\nu} = \\boldsymbol{e}_{\\nu , \\mu}$$ \u304c\u6210\u308a\u7acb\u3064\u3068\u3057\uff0c\u3053\u306e\u6dfb\u5b57\u5bfe\u79f0\u6027\u3092\u3082\u3064\u63a5\u7d9a\u4fc2\u6570\u3092\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/strong><\/span>\u3068\u547c\u3076\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u81ea\u7531\u5ea6\u306e\u52d8\u5b9a\u3092\u3059\u308b\u3068\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206 \\(g_{\\mu\\nu}\\) \u306f\u5bfe\u79f0\u884c\u5217\u3067\u3042\u308b\u304b\u3089\uff0c4\u6b21\u5143\u6642\u7a7a\u306e\u5834\u5408\u306e\u72ec\u7acb\u306a\u81ea\u7531\u5ea6\u306f10\u500b\u3002<\/p>\n

\u305d\u306e1\u968e\u504f\u5fae\u5206 \\(g_{\\mu\\nu, \\lambda}\\) \u306f \\(\\lambda\\) \u304c4\u500b\u306e\u81ea\u7531\u5ea6\u3092\u3082\u3064\u306e\u3067\uff0c\\( 10 \\times 4 = 40 \\) \u500b\u3002<\/p>\n

\u4e00\u65b9 \\(\\varGamma^{\\lambda}_{\\ \\ \\mu \\nu} = \\varGamma^{\\lambda}_{\\ \\ \\nu \\mu}\\) \u306e\u81ea\u7531\u5ea6\u3082\uff0c\\(\\lambda\\) \u304c4\u500b\uff0c\u4e0b\u6dfb\u5b57\u306e\u5bfe\u79f0\u6027\u304b\u308910\u500b\u3002\u5408\u308f\u305b\u3066 \\(4\\times 10 = 40\\)\u500b\u3067\uff0c\u30e1\u30c8\u30ea\u30c3\u30af\u30fb\u30c6\u30f3\u30bd\u30eb\u306e1\u968e\u504f\u5fae\u5206\u306e\u81ea\u7531\u5ea6\u3068\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u81ea\u7531\u5ea6\u306f\u540c\u3058\u3067\u3042\u308b\u3053\u3068\u3082\uff0c\u597d\u307e\u3057\u3044\u5146\u5019\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":327,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-385","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/385","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=385"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/385\/revisions"}],"predecessor-version":[{"id":6269,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/385\/revisions\/6269"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/327"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=385"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}