{"id":420,"date":"2022-01-07T14:23:34","date_gmt":"2022-01-07T05:23:34","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=420"},"modified":"2024-11-11T15:56:50","modified_gmt":"2024-11-11T06:56:50","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%83%aa%e3%83%bc%e3%83%9e%e3%83%b3%e3%83%86%e3%83%b3%e3%82%bd%e3%83%ab%e3%81%ae%e7%8b%ac%e7%ab%8b%e3%81%aa%e6%88%90%e5%88%86%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6","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%b9%b3%e8%a1%8c%e7%b7%9a%e3%81%ae%e5%85%ac%e7%90%86%e3%81%ae%e7%a0%b4%e3%82%8c%e3%81%a8%e3%83%aa%e3%83%bc%e3%83%9e%e3%83%b3%e3%83%86%e3%83%b3%e3%82%bd%e3%83%ab\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%83%aa%e3%83%bc%e3%83%9e%e3%83%b3%e3%83%86%e3%83%b3%e3%82%bd%e3%83%ab%e3%81%ae%e7%8b%ac%e7%ab%8b%e3%81%aa%e6%88%90%e5%88%86%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6\/","title":{"rendered":"\u88dc\u8db3\uff1a\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u72ec\u7acb\u306a\u6210\u5206\u306b\u3064\u3044\u3066"},"content":{"rendered":"

\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u305f\u3002<\/p>\n

$$R^{\\lambda}_{\\ \\, \\nu\\alpha\\beta} \\boldsymbol{e}_{\\lambda} \\equiv \\boldsymbol{e}_{\\nu, \\beta\\alpha} – \\boldsymbol{e}_{\\nu, \\alpha\\beta}$$<\/p>\n

\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u306e\u6dfb\u5b57\u306e\u5bfe\u79f0\u6027<\/h3>\n

\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306f4\u3064\u306e\u6dfb\u5b57\u3092\u3082\u3064\u305f\u3081\uff0c4\u6b21\u5143\u6642\u7a7a\u3067\u306f \\(4^4 = 256 \\) \u500b\u306e\u6210\u5206\u304c\u3042\u308b\u3053\u3068\u306b\u306a\u308b\u304c\uff0c\u4ee5\u4e0b\u3067\u793a\u3059\u3088\u3046\u306a\u6dfb\u5b57\u306e\u5bfe\u79f0\u6027\u306e\u305f\u3081\u306b\uff0c\u771f\u306b\u72ec\u7acb\u306a\u6210\u5206\u306e\u500b\u6570\u306f\u6700\u7d42\u7684\u306b 20 \u500b\u306b\u306a\u308b\u3002<\/p>\n

\u307e\u305a\uff0c\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306e\u5b9a\u7fa9\u5f0f\u306b \\(\\boldsymbol{e}_{\\mu} \\) \u3092\u304b\u3051\u3066\u5185\u7a4d\u3092\u3068\u308b\u3068\uff0c
\n\\begin{eqnarray}
\n\\boldsymbol{e}_{\\mu}\\cdot \\left( R^{\\lambda}_{\\ \\, \\nu\\alpha\\beta} \\boldsymbol{e}_{\\lambda}\\right) &=& \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\lambda} R^{\\lambda}_{\\ \\, \\nu\\alpha\\beta}\u00a0 \\\\
\n&=& g_{\\mu\\lambda} R^{\\lambda}_{\\ \\, \\nu\\alpha\\beta} \\\\
\n&\\equiv& R_{\\mu\\nu\\alpha\\beta} = \\boldsymbol{e}_{\\mu}\\cdot\\left( \\boldsymbol{e}_{\\nu, \\beta\\alpha} – \\boldsymbol{e}_{\\nu, \\alpha\\beta}\\right)
\n\\end{eqnarray} \u3059\u306a\u308f\u3061
\n$$R_{\\mu\\nu\\alpha\\beta} \\equiv \\boldsymbol{e}_{\\mu}\\cdot\\left( \\boldsymbol{e}_{\\nu, \\beta\\alpha} – \\boldsymbol{e}_{\\nu, \\alpha\\beta}\\right)$$
\n4\u3064\u304c\u5168\u3066\u4e0b\u6dfb\u5b57\u3067\u3042\u308b\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206 \\( R_{\\mu\\nu\\alpha\\beta} \\) \u306b\u3064\u3044\u3066\uff0c\u6dfb\u5b57\u306e\u5bfe\u79f0\u6027\u3092\u4ee5\u4e0b\u3067\u8abf\u3079\u308b\u3002<\/p>\n

(1) \\(\\displaystyle\u00a0 R_{\\mu\\nu\\alpha\\beta} = – R_{\\mu\\nu\\beta\\alpha}\\)<\/h4>\n

\u3053\u308c\u306f\u5b9a\u7fa9\u3088\u308a\u660e\u3089\u304b\u3060\u304c\uff0c\u5ff5\u306e\u305f\u3081<\/p>\n

\\begin{eqnarray}
\nR_{\\mu\\nu\\alpha\\beta} + R_{\\mu\\nu\\beta\\alpha} &=& \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu, \\beta\\alpha} – \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu, \\alpha\\beta} \\\\
\n&& + \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu, \\alpha\\beta} – \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu, \\beta\\alpha} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

 <\/p>\n

(2) \\(\\displaystyle R_{\\mu\\nu\\alpha\\beta} = – R_{\\nu\\mu\\alpha\\beta}\\)<\/h4>\n

 <\/p>\n

\\begin{eqnarray}
\nR_{\\mu\\nu\\alpha\\beta} + R_{\\nu\\mu\\alpha\\beta} &=& \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu, \\beta\\alpha} – \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu, \\alpha\\beta} \\\\
\n&& + \\boldsymbol{e}_{\\nu}\\cdot\\boldsymbol{e}_{\\mu, \\beta\\alpha} – \\boldsymbol{e}_{\\nu}\\cdot\\boldsymbol{e}_{\\mu, \\alpha\\beta} \\\\
\n&=& \\left( \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu}\\right)_{,\\beta\\alpha} –\u00a0 \\left( \\boldsymbol{e}_{\\mu}\\cdot\\boldsymbol{e}_{\\nu}\\right)_{,\\alpha\\beta} \\\\
\n&=& g_{\\mu\\nu, \\beta\\alpha} – g_{\\mu\\nu, \\alpha\\beta} = 0
\n\\end{eqnarray}<\/p>\n

\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206 \\(g_{\\mu\\nu}\\) \u306f\u666e\u901a\u306e\u95a2\u6570\u3067\u3042\u308b\u304b\u3089\u504f\u5fae\u5206\u306f\u4ea4\u63db\u53ef\u80fd\u3067\u3042\u308b\u3002<\/p>\n

 <\/p>\n

(3) \\(\\displaystyle\u00a0 R_{\\mu\\nu\\alpha\\beta} + R_{\\mu\\alpha\\beta\\nu}+ R_{\\mu\\beta\\nu\\alpha} = 0\\)<\/h4>\n

\\begin{eqnarray}
\nR_{\\mu\\nu\\alpha\\beta} &=& \\boldsymbol{e}_{\\mu}\\cdot\\left(\\color{red}{\\boldsymbol{e}_{\\nu, \\beta\\alpha}} \\color{blue}{- \\boldsymbol{e}_{\\nu, \\alpha\\beta}}\\right) \\\\
\nR_{\\mu\\alpha\\beta\\nu} &=& \\boldsymbol{e}_{\\mu}\\cdot\\left(\\boldsymbol{e}_{\\alpha, \\nu\\beta} – \\boldsymbol{e}_{\\alpha, \\beta\\nu}\\right)
\n=\\boldsymbol{e}_{\\mu}\\cdot\\left(\\color{blue}{\\boldsymbol{e}_{\\nu, \\alpha\\beta}} \\color{green}{- \\boldsymbol{e}_{\\alpha, \\beta\\nu}}\\right) \\\\
\nR_{\\mu\\beta\\nu\\alpha} &=& \\boldsymbol{e}_{\\mu}\\cdot\\left(\\boldsymbol{e}_{\\beta, \\alpha\\nu} – \\boldsymbol{e}_{\\beta, \\nu\\alpha}\\right)
\n= \\boldsymbol{e}_{\\mu}\\cdot\\left(\\color{green}{\\boldsymbol{e}_{\\alpha, \\beta\\nu} }\\color{red}{- \\boldsymbol{e}_{\\nu, \\beta\\alpha}}\\right)
\n\\end{eqnarray}<\/p>\n

3\u672c\u306e\u5f0f\u3092\u8db3\u3057\u5408\u308f\u305b\u308b\u3068\u540c\u3058\u8272\u540c\u58eb\u304c\u30ad\u30e3\u30f3\u30bb\u30eb\u3057\u3066\u30bc\u30ed\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u306e\u72ec\u7acb\u306a\u6210\u5206\u306e\u500b\u6570\uff08\u81ea\u7531\u5ea6\uff09<\/h3>\n

\u3044\u3088\u3044\u3088\uff0c4\u6b21\u5143\u6642\u7a7a\u306e\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u72ec\u7acb\u306a\u6210\u5206\u306e\u500b\u6570\u306f20\u500b\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n

(1) \u306e\u7d50\u679c\u304b\u3089\uff0c\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306e\u5f8c2\u3064\u306e\u6dfb\u5b57 \\((\\alpha\u00a0 \\beta)\\) \u306e\u7d44\u307f\u5408\u308f\u305b\u306b\u3064\u3044\u3066\u306f \\(4\\times 4\\) \u306e\u53cd\u5bfe\u79f0\u884c\u5217\u306e\u81ea\u7531\u5ea6\u3068\u540c\u30586\u500b\u306e\u81ea\u7531\u5ea6\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u305f\u3068\u3048\u3070\uff0c\\((\\alpha \\beta) = (01), (02), (03), (12), (13), (23)\\)<\/p>\n

(2) \u306e\u7d50\u679c\u304b\u3089\uff0c\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306e\u524d2\u3064\u306e\u6dfb\u5b57 \\((\\mu\u00a0 \\nu)\\) \u306e\u7d44\u307f\u5408\u308f\u305b\u306b\u3064\u3044\u3066\u3082\uff0c \\(4\\times 4\\) \u306e\u53cd\u5bfe\u79f0\u884c\u5217\u306e\u81ea\u7531\u5ea6\u3068\u540c\u30586\u500b\u306e\u81ea\u7531\u5ea6\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u305f\u3068\u3048\u3070\uff0c\\((\\mu\u00a0 \\nu) = (01), (02), (03), (12), (13), (23)\\)<\/p>\n

\u3064\u307e\u308a\uff0c(1) \u3068 (2) \u306e\u6027\u8cea\u304b\u3089\uff0c\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u72ec\u7acb\u306a\u6210\u5206\u306e\u500b\u6570\u306f \\( 6\\times 6 = 36\\) \u307e\u3067\u843d\u3068\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u3053\u3053\u304b\u3089\u3055\u3089\u306b (3) \u306e\u6761\u4ef6\u5f0f\u3067\uff0c\u81ea\u7531\u5ea6\u304c\u6e1b\u3063\u3066\u3044\u304f\u3002\u3055\u3066 (3) \u306e\u6761\u4ef6\u5f0f\u306e\u72ec\u7acb\u306a\u81ea\u7531\u5ea6\u306f\u3044\u304f\u3064\u3067\u3042\u308d\u3046\u304b\u3002<\/p>\n

\\(\\mu\\) \u306f4\u901a\u308a\u3068\u308c\u308b\u3002\u6b8b\u308a\u306e \\([\\nu\u00a0 \\alpha\u00a0 \\beta]\\) \u306b\u3064\u3044\u3066\u306f\uff0c\u3069\u308c\u304b2\u3064\u304c\u540c\u3058\u306a\u3089 (1) \u3068 (2) \u306e\u6027\u8cea\u304b\u3089\u81ea\u660e\u306b\u6e80\u305f\u3055\u308c\u308b\u5f0f\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u306e\u3067\uff0c\\([\\nu\u00a0 \\alpha \\beta]\\) \u306f\u3059\u3079\u3066\u7570\u306a\u3063\u3066\u3044\u308b\u5fc5\u8981\u304c\u3042\u308a\uff0c\u305d\u306e\u72ec\u7acb\u306a\u81ea\u7531\u5ea6\u306f 4 \u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\uff080 \u304b\u3089 3 \u307e\u3067\u306e4\u500b\u306e\u3046\u3061\u306e\u4f7f\u308f\u306a\u30441\u500b\u3092\u9078\u3076\u306e\u3060\u304b\u30894\u901a\u308a\uff09\u3002<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c(3) \u306e\u6761\u4ef6\u5f0f\u306f \\(4\\times 4 = 16\\) \u306e\u72ec\u7acb\u306a\u81ea\u7531\u5ea6\u304c\u3042\u308b\u308f\u3051\u3060\u3002<\/p>\n

\u6700\u7d42\u7684\u306b\u306f\uff0c\\(6\\times 6 – 4 \\times 4 = 36 – 16 = 20 \\) \u3068\u306a\u308a\uff0c\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u72ec\u7acb\u306a\u6210\u5206\u306e\u500b\u6570\u306f \\(20\\) \u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u306e\u5bfe\u79f0\u6027\u3078<\/h3>\n

(4)\\( \\ R_{\\mu\\nu\\alpha\\beta} = R_{\\alpha\\beta\\mu\\nu} \\)<\/h4>\n

\u6700\u5f8c\u306b\uff0c(1) \u304b\u3089 (3) \u307e\u3067\u306e\u6761\u4ef6\u3092\u4f7f\u3063\u3066\uff0c\\(\\displaystyle R_{\\mu\\nu\\alpha\\beta} = R_{\\alpha\\beta\\mu\\nu} \\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u306f\u5f8c\u3067\u8ff0\u3079\u308b\u3088\u3046\u306b\uff0c\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u304c\u5bfe\u79f0\u30c6\u30f3\u30bd\u30eb\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u969b\u306b\u7528\u3044\u3089\u308c\u308b\u3002<\/p>\n

\u8a3c\u660e\u306e\u305f\u3081\u306b\uff0c(3) \u306e\u7d50\u679c\u3092\u6dfb\u5b57\u3092\u305a\u3089\u3057\u30664\u672c\u66f8\u304f\u3002<\/p>\n

\\begin{eqnarray}
\nR_{\\mu\\nu\\alpha\\beta} \\quad \\color{red}{+ \\quad R_{\\mu\\alpha\\beta\\nu}} \\quad \\color{blue}{+ \\quad\u00a0 R_{\\mu\\beta\\nu\\alpha}} &=& 0 \\\\
\n\\color{green}{-R_{\\nu\\alpha\\beta\\mu}} \\color{cyan}{\\quad – \\quad R_{\\nu\\beta\\mu\\alpha}} \\quad \\color{black}{- \\quad R_{\\nu\\mu\\alpha\\beta}} &=& 0 \\\\
\n-R_{\\alpha\\beta\\mu\\nu}\\color{red}{ \\quad – \\quad R_{\\alpha\\mu\\nu\\beta}}\\color{green}{\\quad – \\quad\u00a0 R_{\\alpha\\nu\\beta\\mu}} &=& 0 \\\\
\n\\color{blue}{R_{\\beta\\mu\\nu\\alpha} } \\quad\\color{cyan}{ + \\quad R_{\\beta\\nu\\alpha\\mu}} \\quad \\color{black}{+ \\quad R_{\\beta\\alpha\\mu\\nu}} &=& 0
\n\\end{eqnarray}<\/p>\n

(1) \u304a\u3088\u3073 (2) \u306e\u6027\u8cea\u3092\u4f7f\u3046\u3068\u540c\u3058\u8272\u540c\u58eb\u304c\u30ad\u30e3\u30f3\u30bb\u30eb\u3057\u3066\u6d88\u3048\uff0c\u9ed2\u5b57\u90e8\u5206\u306e\u307f\u304c\u6b8b\u308a\uff0c<\/p>\n

$$2 \\left(R_{\\mu\\nu\\alpha\\beta}\u00a0\u00a0 -R_{\\alpha\\beta\\mu\\nu}\\right)\u00a0 = 0$$
\n$$\\therefore \\ R_{\\mu\\nu\\alpha\\beta} = R_{\\alpha\\beta\\mu\\nu} $$<\/p>\n

\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206 \\( R_{\\nu\\beta} \\) \u306f\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306e1\u756a\u76ee\u306e\u6dfb\u5b57\u30683\u756a\u76ee\u306e\u6dfb\u5b57\u3092\u7e2e\u7d04\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u306e\u3067\uff0c<\/p>\n

$$ R_{\\nu\\beta} \\equiv g^{\\mu\\alpha} R_{\\mu\\nu\\alpha\\beta} = g^{\\mu\\alpha}R_{\\alpha\\beta\\mu\\nu} = g^{\\alpha\\mu}R_{\\alpha\\beta\\mu\\nu} = R_{\\beta\\nu} $$<\/p>\n

$$\\therefore\\ \\ R_{\\nu\\beta} =R_{\\beta\\nu} $$<\/p>\n

 <\/p>\n

\u3068\u306a\u308a\uff0c\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u306e\u6dfb\u5b57\u306b\u5bfe\u3059\u308b\u6027\u8cea\u304b\u3089\uff0c\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u304c2\u3064\u306e\u4e0b\u6dfb\u5b57\u306b\u3064\u3044\u3066\u5bfe\u79f0\u3067\u3042\u308b\u3053\u3068\u304c\u5c0e\u304b\u308c\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":337,"menu_order":30,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-420","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\/420","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=420"}],"version-history":[{"count":17,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/420\/revisions"}],"predecessor-version":[{"id":9542,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/420\/revisions\/9542"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/337"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}