{"id":10287,"date":"2025-06-04T14:20:01","date_gmt":"2025-06-04T05:20:01","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=10287"},"modified":"2025-05-31T14:13:47","modified_gmt":"2025-05-31T05:13:47","slug":"%e7%b7%b4%e7%bf%92%e5%95%8f%e9%a1%8c%ef%bc%9a3%e6%ac%a1%e5%85%83%e3%81%ae%e7%b7%9a%e7%b4%a0%e3%82%92%e6%a5%b5%e5%ba%a7%e6%a8%99%e3%81%a7%e8%a1%a8%e3%81%99","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%81%8f%e5%be%ae%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/%e5%85%a8%e5%be%ae%e5%88%86\/%e7%b7%b4%e7%bf%92%e5%95%8f%e9%a1%8c%ef%bc%9a3%e6%ac%a1%e5%85%83%e3%81%ae%e7%b7%9a%e7%b4%a0%e3%82%92%e6%a5%b5%e5%ba%a7%e6%a8%99%e3%81%a7%e8%a1%a8%e3%81%99\/","title":{"rendered":"\u7df4\u7fd2\u554f\u984c\uff1a3\u6b21\u5143\u306e\u7dda\u7d20\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059"},"content":{"rendered":"

\u5168\u5fae\u5206\u306e\u7df4\u7fd2\u554f\u984c\u3068\u3057\u3066\uff0c3\u6b21\u5143\u306e\u300c\u7dda\u7d20\u300d\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3057\u3066\u307f\u308b\u3002<\/p>\n

<\/p>\n

3\u6b21\u5143\u6975\u5ea7\u6a19<\/h3>\n

3\u6b21\u5143\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y, z$ \u3092\u6975\u5ea7\u6a19 $r, \\theta, \\phi$ \u3092\u4f7f\u3063\u3066\u8868\u3059\u3068<\/p>\n

\\begin{eqnarray}
\nx &=& r \\sin\\theta\\, \\cos\\phi \\\\
\ny &=& r \\sin\\theta\\, \\sin\\phi \\\\
\nz &=& r \\cos\\theta
\n\\end{eqnarray}<\/p>\n

1\u968e\u504f\u5fae\u5206<\/h3>\n

\\begin{eqnarray}
\n\\frac{\\partial x}{\\partial r} &=& \\sin\\theta\\, \\cos\\phi \\\\
\n\\frac{\\partial x}{\\partial \\theta} &=& r \\cos\\theta\\, \\cos\\phi \\\\
\n\\frac{\\partial x}{\\partial \\phi} &=& -r \\sin\\theta\\, \\sin\\phi \\\\ \\ \\\\
\n\\frac{\\partial y}{\\partial r} &=& \\sin\\theta\\, \\sin\\phi \\\\
\n\\frac{\\partial y}{\\partial \\theta} &=& r \\cos\\theta\\, \\sin\\phi \\\\
\n\\frac{\\partial y}{\\partial \\phi} &=& r \\sin\\theta\\, \\cos\\phi \\\\ \\ \\\\
\n\\frac{\\partial z}{\\partial r} &=& \\cos\\theta \\\\
\n\\frac{\\partial z}{\\partial \\theta} &=& -r \\sin\\theta\\\\
\n\\frac{\\partial z}{\\partial \\phi} &=& 0
\n\\end{eqnarray}<\/p>\n

\u5168\u5fae\u5206<\/h3>\n

\\begin{eqnarray}
\ndx &=& \\frac{\\partial x}{\\partial r} dr + \\frac{\\partial x}{\\partial \\theta}\u00a0 d\\theta + \\frac{\\partial x}{\\partial \\phi}d\\phi \\\\
\n&=& \\sin\\theta\\, \\cos\\phi\\, dr + r \\cos\\theta\\, \\cos\\phi \\, d\\theta -r \\sin\\theta\\, \\sin\\phi\\,d\\phi \\\\
\ndy &=& \\frac{\\partial y}{\\partial r} dr + \\frac{\\partial y}{\\partial \\theta}\u00a0 d\\theta + \\frac{\\partial y}{\\partial \\phi}d\\phi \\\\
\n&=& \\sin\\theta\\, \\sin\\phi\\, dr + r \\cos\\theta\\, \\sin\\phi \\, d\\theta + r \\sin\\theta\\, \\cos\\phi\\,d\\phi \\\\
\ndz &=& \\frac{\\partial z}{\\partial r} dr + \\frac{\\partial z}{\\partial \\theta}\u00a0 d\\theta + \\frac{\\partial z}{\\partial \\phi}d\\phi \\\\
\n&=& \\cos\\theta\\, dr -r \\sin\\theta\\,\u00a0 d\\theta
\n\\end{eqnarray}<\/p>\n

\u6975\u5ea7\u6a19\u3067\u8868\u3057\u305f\u7dda\u7d20<\/h3>\n

\\begin{eqnarray}
\n\\therefore\\ \\ d\\ell^2 &=& dx^2 + dy^2 + dz^2 \\\\
\n&=& \\left(\\sin\\theta\\, \\cos\\phi\\, dr + r \\cos\\theta\\, \\cos\\phi \\, d\\theta -r \\sin\\theta\\, \\sin\\phi\\,d\\phi \\right)^2 \\\\
\n&& + \\left(\\sin\\theta\\, \\sin\\phi\\, dr + r \\cos\\theta\\, \\sin\\phi \\, d\\theta + r \\sin\\theta\\, \\cos\\phi\\,d\\phi \\right)^2 \\\\
\n&& + \\left(\\cos\\theta\\, dr -r \\sin\\theta\\,\u00a0 d\\theta \\right)^2 \\\\
\n&=& dr^2 + r^2\\,d\\theta^2 + r^2 \\,\\sin^2\\theta\\, d\\phi^2
\n\\end{eqnarray}<\/p>\n

\u53c2\u8003\uff1a3\u6b21\u5143\u6975\u5ea7\u6a19\u306b\u304a\u3051\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206<\/h3>\n

\u3053\u306e\u3053\u3068\u306f\uff0c\u306e\u3061\u306b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u3092\u4f7f\u3063\u3066\u8aac\u660e\u3055\u308c\u308b\u3053\u3068\u306b\u306a\u308b\uff08\u3053\u306e\u3078\u3093<\/strong><\/a>\u3068\u304b\u3053\u306e\u3078\u3093<\/strong><\/a>\u3092\u53c2\u7167\uff09\u3002\u3053\u3053\u3067\u306f\u8aac\u660e\u3092\u7701\u304f\u306e\u3067\u306a\u3093\u306e\u3053\u3068\u304b\u308f\u304b\u3089\u306a\u3044\u3068\u306f\u601d\u3046\u304c\uff0c\u5c06\u6765\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u3092\u5b66\u3093\u3060\u3042\u3068\u3067\u3053\u306e\u30da\u30fc\u30b8\u3092\u898b\u76f4\u3057\u3066\u307f\u308b\u3068\uff0c3\u6b21\u5143\u6975\u5ea7\u6a19\u306b\u304a\u3051\u308b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206\u304c<\/p>\n

$$ \\left(\\begin{array}{ccc}
\ng_{rr} & g_{r\\theta}&g_{r\\phi} \\\\
\ng_{\\theta r} & g_{\\theta\\theta}&g_{\\theta\\phi} \\\\
\ng_{\\phi r} & g_{\\phi\\theta}&g_{\\phi\\phi}
\n\\end{array}\\right)
\n=\u00a0 \\left(\\begin{array}{ccc}
\n1 & 0 &0 \\\\
\n0 & r^2 & 0 \\\\
\n0 & 0 & r^2 \\sin^2\\theta
\n\\end{array}\\right)
\n$$<\/p>\n

\u3067\u3042\u308a\uff0c\u7dda\u7d20\u304c<\/p>\n

$$d\\ell^2 = g_{rr} \\,dr^2 + r_{\\theta\\theta} \\,d\\theta^2 + g_{\\phi\\phi} \\,d\\phi^2$$<\/p>\n

\u306e\u3088\u3046\u306b\u66f8\u3051\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3088\u3046\u306b\u306a\u308b\u3068\u601d\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5168\u5fae\u5206\u306e\u7df4\u7fd2\u554f\u984c\u3068\u3057\u3066\uff0c3\u6b21\u5143\u306e\u300c\u7dda\u7d20\u300d\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3057\u3066\u307f\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2316,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-10287","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