{"id":233,"date":"2022-01-05T15:35:52","date_gmt":"2022-01-05T06:35:52","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=233"},"modified":"2022-01-11T10:29:56","modified_gmt":"2022-01-11T01:29:56","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e5%9f%ba%e6%9c%ac%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%8c%e7%a9%ba%e9%96%93%e4%be%9d%e5%ad%98%e6%80%a7%e3%82%92%e3%82%82%e3%81%a4%e4%be%8b","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\/3%e6%ac%a1%e5%85%83%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e5%be%ae%e5%88%86\/%e5%8f%82%e8%80%83%ef%bc%9a%e5%9f%ba%e6%9c%ac%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%8c%e7%a9%ba%e9%96%93%e4%be%9d%e5%ad%98%e6%80%a7%e3%82%92%e3%82%82%e3%81%a4%e4%be%8b\/","title":{"rendered":"\u88dc\u8db3\uff1a\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u304c\u7a7a\u9593\u4f9d\u5b58\u6027\u3092\u3082\u3064\u4f8b"},"content":{"rendered":"

<\/p>\n

2\u6b21\u5143\u6975\u5ea7\u6a19\u7cfb\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u7c21\u5358\u306e\u305f\u3081\uff0c2\u6b21\u5143\u5e73\u9762\u3092\u8003\u3048\uff0c\u8fd1\u63a5\u3057\u305f2\u70b9 \\(P, Q\\) \u3092\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 \\(x, y\\) \u3068\u6975\u5ea7\u6a19 \\(r, \\theta\\) \u3067\u8868\u3059\u5834\u5408\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

\u5ea7\u6a19\u306e\u5909\u63db\u5f0f\u306f
\n$$x = r\\, \\cos\\theta, \\quad \u00a0 y = r\\,\\sin\\theta$$ \u5fae\u5c0f\u5909\u4f4d\u306f
\n$$dx = dr\\, \\cos\\theta – r\\,\\sin\\theta\\, d\\theta, \\quad dy = dr \\,\\sin\\theta + r\\,\\cos\\theta\\,d\\theta$$<\/p>\n

\\(PQ\\) \u9593\u306e\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb \\(d\\vec{x}\\) \u3092\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u8868\u3057\u305f\u306e\u3061\u306b\u5909\u5f62\u3055\u305b\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\nd\\vec{x} &=& dx\\,\\vec{e}_x + dy\\,\\vec{e}_y\u00a0 \\\\
\n&=& (dr\\, \\cos\\theta – r\\,\\sin\\theta\\, d\\theta) \\,\\vec{e}_x + (dr \\,\\sin\\theta + r\\,\\cos\\theta\\,d\\theta)\\,\\vec{e}_y \\\\
\n&=& dr (\\cos\\theta\\,\\vec{e}_x + \\sin\\theta\\,\\vec{e}_y) + d\\theta (-r\\sin\\theta\\,\\vec{e}_x + r\\cos\\theta\\,\\vec{e}_y)
\n\\end{eqnarray}<\/p>\n

\u4e00\u65b9\uff0c\u5fae\u5c0f\u5909\u4f4d\u30d9\u30af\u30c8\u30eb \\(d\\vec{x}\\) \u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059\u3068
\n$$d\\vec{x} = dr\\,\\vec{e}_r + d\\theta\\,\\vec{e}_{\\theta}$$ \u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\n\\vec{e}_r &=& \\cos\\theta\\, \\vec{e}_x + \\sin\\theta\\, \\vec{e}_y \\\\
\n\\vec{e}_{\\theta} &=&-r \\sin\\theta\\, \\vec{e}_x + r \\cos\\theta\\, \\vec{e}_y
\n\\end{eqnarray} \u3068\u306a\u308a\uff0c\u305f\u3068\u3048 \\(\\vec{e}_x, \\vec{e}_y\\) \u306f\u4e00\u5b9a\u306e\u30d9\u30af\u30c8\u30eb\u3067\u3042\u3063\u3066\u3082 \\(\\vec{e}_r,\u00a0 \\vec{e}_{\\theta}\\) \u306f\u4e00\u822c\u306b\u7a7a\u9593\u5ea7\u6a19 \\(r, \\theta\\) \u306e\u4f9d\u5b58\u6027\u3092\u3082\u3064\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb\uff08\u300c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u300d\u306f\u5b66\u90e8\u751f\u5411\u3051\u306e\u7528\u8a9e\u3067\u3042\u308a\uff0c\u4ee5\u5f8c\uff0c\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u3092\u300c\u30c7\u30ab\u30eb\u30c8\u57fa\u5e95\u300d\u306a\u3069\u3068\u547c\u3076\u304b\u3082\u3057\u308c\u306a\u3044\uff09\u306b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6b63\u898f\u76f4\u4ea4\u95a2\u4fc2\u304c\u3042\u308b\u304c\uff0c$$\\vec{e}_x\\cdot \\vec{e}_x = 1, \\quad\\vec{e}_x\\cdot \\vec{e}_y = 0, \\quad\\vec{e}_y\\cdot \\vec{e}_y = 1$$ \u4e00\u822c\u306e\u5ea7\u6a19\u57fa\u5e95\u306e\u5927\u304d\u3055\u306f\u5fc5\u305a\u3057\u3082 \\(1\\) \u306b\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3002\u4f8b\u3048\u3070\uff0c2\u6b21\u5143\u66f2\u5ea7\u6a19\u7cfb\u3067\u306f\uff0c
\n\\begin{eqnarray}
\n\\vec{e}_{\\theta}\\cdot\\vec{e}_{\\theta} &=& \\left( -r \\sin\\theta\\, \\vec{e}_x + r \\cos\\theta\\, \\vec{e}_y\\right)\\cdot\\left( -r \\sin\\theta\\, \\vec{e}_x + r \\cos\\theta\\, \\vec{e}_y\\right) \\\\
\n&=& r^2 \\sin^2\\theta\u00a0 \\,\\vec{e}_x\\cdot\\vec{e}_x – 2 r^2 \\sin\\theta\\cos\\theta \\,\\vec{e}_x\\cdot\\vec{e}_y + r^2 \\cos^2\\theta\u00a0 \\,\\vec{e}_y\\cdot\\vec{e}_y \\\\
\n&=& r^2
\n\\end{eqnarray} \u3068\u306a\u308b\u3002<\/p>\n

\u7a7a\u9593\u5ea7\u6a19\u306b\u3088\u308b\u6975\u5ea7\u6a19\u57fa\u5e95\u306e\u504f\u5fae\u5206\u3092\u3061\u3087\u3063\u3068\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial}{\\partial r} \\vec{e}_r =\\vec{e}_{r,r} &=& \\vec{0}\\\\
\n\\frac{\\partial}{\\partial \\theta} \\vec{e}_r =\\vec{e}_{r,\\theta}&=&\\frac{\\partial}{\\partial \\theta}\\left( \\cos\\theta\\, \\vec{e}_x + \\sin\\theta\\, \\vec{e}_y\\right) \\\\
\n&=& -\\sin\\theta\\, \\, \\vec{e}_x + \\cos\\theta\\, \\vec{e}_y \\\\
\n&=& \\frac{1}{r} \\vec{e}_{\\theta}
\n\\end{eqnarray}<\/p>\n

\\(\\vec{e}_{\\theta,r}, \\vec{e}_{\\theta,\\theta} \\) \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u8a08\u7b97\u3059\u308c\u3070\u3088\u3044\u304c\uff0c\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u5bfe\u79f0\u6027\u304b\u3089
\n$$ \\vec{e}_{\\theta,r} = \\vec{e}_{r, \\theta}$$ \u3067\u3042\u3063\u305f\u304b\u3089\uff0c\u6b8b\u308a\u306f \\( \\vec{e}_{\\theta,\\theta}\\) \u306e\u8a08\u7b97\u3060\u3051\u3002<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial}{\\partial \\theta} \\vec{e}_{\\theta} = \\vec{e}_{\\theta,\\theta}&=& \\frac{\\partial}{\\partial \\theta}\\left( -r \\sin\\theta\\, \\vec{e}_x + r \\cos\\theta\\, \\vec{e}_y\\right) \\\\
\n&=& -r \\cos\\theta\\, \\vec{e}_x – r \\sin\\theta\\, \\vec{e}_y \\\\
\n&=& -r \\,\\vec{e}_r
\n\\end{eqnarray}<\/p>\n

\u4e0a\u8a18\u306e\u8a08\u7b97\u3067\u6ce8\u610f\u3057\u3066\u304a\u304f\u3079\u304d\u3053\u3068\u306f\uff0c\u8a08\u7b97\u306e\u9014\u4e2d\u3053\u305d\u30c7\u30ab\u30eb\u30c8\u57fa\u5e95\u3092\u4f7f\u3063\u3066\u3044\u305f\u304c\uff0c\u6700\u7d42\u7684\u7d50\u679c\u306b\u306f\u30c7\u30ab\u30eb\u30c8\u57fa\u5e95\u306f\u3069\u3053\u306b\u3082\u9854\u3092\u51fa\u3055\u305a\uff0c\u6975\u5ea7\u6a19\u57fa\u5e95\u3060\u3051\u3067\u8868\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u304c\u5927\u4e8b\u3002<\/p>\n

\n

\u3053\u308c\u304c\uff0c\u5225\u9014\u8ff0\u3079\u308b\u300c\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/strong><\/span>\u300d\u306b\u3064\u306a\u304c\u308b\u3002\u4ee5\u4e0b\u306f\uff0c\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u5b9a\u7fa9\u306e\u9805<\/a>\u3092\u8aad\u3093\u3060\u4e0a\u3067\u3042\u3089\u305f\u3081\u3066\u8aad\u307f\u76f4\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

\u6975\u5ea7\u6a19\u57fa\u5e95\u306e\u7a7a\u9593\u504f\u5fae\u5206\u304c\u6975\u5ea7\u6a19\u57fa\u5e95\u3060\u3051\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u306b\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u91cd\u8981\u6027\u304c\u3042\u308b\u3002<\/p>\n

2\u6b21\u5143\u6975\u5ea7\u6a19\u7cfb\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u3092\u76f4\u63a5\u5fae\u5206\u3057\u3066\u53f3\u8fba\u3068\u6bd4\u8f03\u3057\uff0c\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002
\n$$\\vec{e}_{r, r} = \\varGamma^r_{\\ rr} \\vec{e}_{r} + \\varGamma^{\\theta}_{\\ rr} \\vec{e}_{\\theta} , \\quad \\vec{e}_{r, \\theta} = \\varGamma^r_{\\ r\\theta} \\vec{e}_{r} + \\varGamma^{\\theta}_{\\ r\\theta} \\vec{e}_{\\theta}, \\quad \\vec{e}_{\\theta, \\theta} = \\varGamma^r_{\\ \\theta\\theta} \\vec{e}_{r} + \\varGamma^{\\theta}_{\\ \\theta\\theta} \\vec{e}_{\\theta}$$<\/p>\n

$$\\varGamma^r_{\\ rr} = \\varGamma^{\\theta}_{\\ rr} = \\varGamma^r_{\\ r\\theta} = \\varGamma^{\\theta}_{\\ \\theta\\theta} = 0, \\quad \\varGamma^{\\theta}_{\\ r\\theta} = \\frac{1}{r}, \\quad \\varGamma^r_{\\ \\theta\\theta} = -r$$<\/p>\n

\u307e\u305f\uff0c2\u6b21\u5143\u6975\u5ea7\u6a19\u7cfb\u306e\u30e1\u30c8\u30ea\u30c3\u30af\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206 \\(g_{ij}\\) \u306f\uff0c<\/p>\n

$$ g_{rr} = \\vec{e}_r\\cdot\\vec{e}_r = 1, \\quad g_{r\\theta} = \\vec{e}_r\\cdot\\vec{e}_{\\theta}=0, \\quad g_{\\theta\\theta} = \\vec{e}_{\\theta}\\cdot\\vec{e}_{\\theta}=r^2$$<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":217,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-233","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\/233","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=233"}],"version-history":[{"count":7,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/233\/revisions"}],"predecessor-version":[{"id":650,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/233\/revisions\/650"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/217"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}