{"id":3021,"date":"2022-06-20T10:14:38","date_gmt":"2022-06-20T01:14:38","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3021"},"modified":"2024-06-18T14:44:59","modified_gmt":"2024-06-18T05:44:59","slug":"%e5%8f%82%e8%80%83%ef%bc%9a4%e5%85%83%e9%80%9f%e5%ba%a6%e3%81%ae%e5%85%b1%e5%a4%89%e6%88%90%e5%88%86%e3%81%ab%e5%af%be%e3%81%99%e3%82%8b%e6%b8%ac%e5%9c%b0%e7%b7%9a%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e3%83%86%e3%82%b9%e3%83%88%e7%b2%92%e5%ad%90%e3%81%ae%e9%81%8b%e5%8b%95\/%e5%9b%ba%e6%9c%89%e6%99%82%e9%96%93%e3%82%92%e3%82%a2%e3%83%95%e3%82%a3%e3%83%b3%e3%83%91%e3%83%a9%e3%83%a1%e3%83%bc%e3%82%bf%e3%81%a8%e3%81%99%e3%82%8b%e6%b8%ac%e5%9c%b0%e7%b7%9a%e6%96%b9%e7%a8%8b\/%e5%8f%82%e8%80%83%ef%bc%9a4%e5%85%83%e9%80%9f%e5%ba%a6%e3%81%ae%e5%85%b1%e5%a4%89%e6%88%90%e5%88%86%e3%81%ab%e5%af%be%e3%81%99%e3%82%8b%e6%b8%ac%e5%9c%b0%e7%b7%9a%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae\/","title":{"rendered":"\u53c2\u8003\uff1a4\u5143\u901f\u5ea6\u306e\u5171\u5909\u6210\u5206\u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa\u65b9\u6cd5"},"content":{"rendered":"

4\u5143\u901f\u5ea6\u306e\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206\u3067\u3042\u308b\u300c\u5171\u5909\u6210\u5206<\/strong><\/span>\u300d\\(u_{\\mu} = g_{\\mu\\nu} u^{\\nu}\\)\u00a0 \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d u_{\\mu}}{d\\tau} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta}$$<\/p>\n

\u306e\u5c0e\u51fa\u65b9\u6cd5\u306e\u304a\u3055\u3089\u3044\u3002<\/p>\n

\u672c\u30b5\u30a4\u30c8\u306e\u57fa\u672c\u65b9\u91dd\u306e\u78ba\u8a8d<\/h3>\n

\u300c\u56fa\u6709\u6642\u9593\u3092\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u3068\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/a><\/strong><\/span>\u300d\u30da\u30fc\u30b8\u306a\u3069\u3092\u307f\u308b\u3068\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u672c\u30b5\u30a4\u30c8\u3067\u306f\u5171\u5909\u5fae\u5206\u306e\u5b9a\u7fa9\u3084\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u504f\u5fae\u5206\u3092\u4f7f\u3063\u305f\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u5177\u4f53\u7684\u306a\u8868\u8a18\u306a\u3069\u3092\u4f7f\u308f\u305a\u306b\uff08\u3053\u306e\u3078\u3093\u304c\u79c1\u306e\u5bc6\u304b\u306a\u3053\u3060\u308f\u308a\u3060\u3063\u305f\u308a\u3059\u308b\u306e\u3060\u304c\uff09\uff0c\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u4e16\u754c\u7dda \\(x(\\tau)\\) \u306e\u63a5\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{u} = u^{\\mu} \\boldsymbol{e}_{\\mu}\\) \u306e\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206\uff08\u6b74\u53f2\u7684\u306b\u300c\u5171\u5909\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3042\u308b\u3044\u306f\u5358\u306b\u5171\u5909\u6210\u5206\u300d\u306a\u3069\u3068\u547c\u3070\u308c\u3066\u304d\u305f\uff09 \\(u_{\\mu} \\equiv g_{\\mu\\nu} u^{\\nu}\\) \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d u_{\\mu}}{d\\tau} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta}$$<\/p>\n

\u3092\u5c0e\u304d\uff0c\u3053\u306e\u5f0f\u3092\u89e3\u3044\u3066\u91cd\u529b\u5834\u4e2d\u306e\u7c92\u5b50\u306e\u904b\u52d5\u3092\u6c42\u3081\u3066\u3044\u308b\u3002\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u3053\u306e\u5f62\u306b\u3057\u3066\u304a\u304f\u3068\uff0c\u4fdd\u5b58\u91cf\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u5927\u5909\u4fbf\u5229\u3067\u3042\u308b\u3053\u3068\u306f\u5ea6\u3005\u5f37\u8abf\u3057\u3066\u3044\u308b\u306e\u3060\u304c\uff0c\u4e16\u306e\u4e2d\u306b\u3042\u307e\u305f\u3042\u308b\u6559\u79d1\u66f8\u3092\u307f\u3066\u3082\uff0c\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u306e\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u306e\u300c\u00a787. \u91cd\u529b\u5834\u306e\u306a\u304b\u3067\u306e\u7c92\u5b50\u306e\u904b\u52d5\u300d\u306e\u30d5\u30c3\u30c8\u30ce\u30fc\u30c8\u306b\u5c0f\u3055\u3044\u30d5\u30a9\u30f3\u30c8\u3067\u7d39\u4ecb\u3055\u308c\u3066\u3044\u308b\u7a0b\u5ea6\u3067\uff0c\u3053\u306e\u5f62\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u7a4d\u6975\u7684\u306b\u6d3b\u7528\u3059\u308b\u66f8\u304d\u3076\u308a\u304c\u3042\u307e\u308a\u898b\u3089\u308c\u306a\u3044\u3088\u3046\u3060\u3002\u3082\u3057\u898b\u843d\u3068\u3057\u3066\u3044\u308b\u30c6\u30ad\u30b9\u30c8\u304c\u3042\u308a\u307e\u3057\u305f\u3089\uff0c\u304a\u624b\u6570\u3067\u3082\u304a\u77e5\u3089\u305b\u304f\u3060\u3055\u3044\u3002<\/p>\n

\u3053\u3053\u3067\u306f\uff0c\u3053\u306e\u5f62\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa\u65b9\u6cd5\u306b\u3064\u3044\u3066\uff0c\uff08\u79c1\u306e\u5bc6\u304b\u306a\u3053\u3060\u308f\u308a\u306f\u3059\u3066\u3066\uff09\u3046\u3061\u306e\u5b66\u79d1\u306e\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\u306e\u95a2\u4fc2\u4e0a\u5c01\u5370\u3057\u3066\u3044\u305f\u5909\u5206\u539f\u7406\uff08\u6700\u5c0f\u4f5c\u7528\u306e\u539f\u7406\uff09\u306b\u3088\u308b\u5c0e\u51fa\u65b9\u6cd5\u3084\uff0c\u504f\u5fae\u5206\u306e\u5b9a\u7fa9\u304a\u3088\u3073\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u306e\u5177\u4f53\u7684\u306a\u8868\u8a18\u3092\u5229\u7528\u3057\u305f\u5c0e\u51fa\u65b9\u6cd5\u306b\u3064\u3044\u3066\u304a\u3055\u3089\u3044\u3057\u3066\u304a\u304f\u3002<\/p>\n

\u5909\u5206\u539f\u7406\u306b\u3088\u308b\u5c0e\u51fa<\/h3>\n

\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u306e\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u306e\u300c\u00a787. \u91cd\u529b\u5834\u306e\u306a\u304b\u3067\u306e\u7c92\u5b50\u306e\u904b\u52d5\u300d\u3092\u53c2\u8003\u306b\u82e5\u5e72 notation \u3092\u5909\u3048\u3066\u3002\uff08\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u3067\u306f\uff0c$\\displaystyle S = -mc \\int ds = -mc^2 \\int d\\tau$\uff09<\/p>\n

\u91cd\u529b\u4ee5\u5916\u306e\u529b\u3092\u53d7\u3051\u306a\u3044\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u904b\u52d5\u306f\uff0c\u4f5c\u7528\u7a4d\u5206<\/p>\n

\\begin{eqnarray}
\nS = – \\int\u00a0 d\\tau\u00a0 &=& – \\int \\sqrt{- g_{\\alpha\\beta}(x) \\frac{dx^{\\alpha}}{d\\tau} \\frac{dx^{\\beta}}{d\\tau}} d\\tau \\\\
\n&=& – \\int \\sqrt{- g_{\\alpha\\beta}(x)\\, \\dot{x}^{\\alpha} \\, \\dot{x}^{\\beta}}\\, d\\tau
\n\\end{eqnarray}<\/p>\n

\u306b\u3064\u3044\u3066 \\(\\delta S = 0\\) \u3068\u3044\u3046\u5909\u5206\u539f\u7406\u304b\u3089\u6c42\u3081\u3089\u308c\u308b\u3002\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3060\u3051\u306a\u3089\u7a4d\u5206\u306e\u524d\u306e\u8ca0\u53f7\u306f\u5fc5\u9808\u3067\u306f\u306a\u3044\u304c\uff0c\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d\u306e\u3088\u3046\u306b\u300c\u6700\u5c0f\u4f5c\u7528\u306e\u539f\u7406<\/strong><\/span>\u300d\uff08\\(S\\) \u304c\u6700\u5c0f\u5024\u3092\u3068\u308b\u3079\u3057\uff09\u306e\u7acb\u5834\u306a\u3089\uff0c\uff08\u6e2c\u5730\u7dda\u306b\u305d\u3063\u305f\u56fa\u6709\u6642\u9593\u306e\u7a4d\u5206\u306f\u6700\u5927\u3068\u306a\u308b\u306e\u3067\uff09\u8ca0\u53f7\u3092\u3064\u3051\u308b\u3002<\/p>\n

\u4e00\u822c\u306b<\/p>\n

$$\\delta S = \\delta \\int L\\left(x, \\dot{x} \\right) d\\tau = 0$$<\/p>\n

\u304b\u3089\u5f97\u3089\u308c\u308b\u30aa\u30a4\u30e9\u30fc\u30fb\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d}{d\\tau}\\left( \\frac{\\partial L}{\\partial \\dot{x}^{\\mu}} \\right) – \\frac{\\partial L}{\\partial x^{\\mu}} = 0$$<\/p>\n

\u3067 \\(\\displaystyle L = – \\sqrt{- g_{\\alpha\\beta}(x) \\,\\dot{x}^{\\alpha}\\, \\dot{x}^{\\beta}} \\) \u3068\u3057\u3066\uff0c\uff08\u5b9f\u306f \\(L = -c\\) \u306a\u306e\u3060\u304c\uff0c\u305d\u308c\u306f\u5909\u5206\u3092\u3068\u3063\u305f\u3042\u3068\uff0c\u3064\u307e\u308a\u504f\u5fae\u5206\u3092\u3057\u305f\u5f8c\u306b \\(L = -c\\) \u3068\u3059\u308b\uff09\u8a08\u7b97\u3057\u3066\u3084\u308b\u3068\uff0c\u307e\u305a\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{d}{d\\tau}\\left( \\frac{\\partial L}{\\partial\\dot{x}^{\\mu}} \\right)
\n&=&\\frac{d}{d\\tau}\\left(-\\frac{1}{2}\\left(- g_{\\alpha\\beta} \\,\\dot{x}^{\\alpha}\\, \\dot{x}^{\\beta} \\right)^{-\\frac{1}{2}} \\left(- g_{\\alpha\\beta}\\right)\\left( \\frac{\\partial\\dot{x}^{\\alpha}}{\\partial\\dot{x}^{\\mu}}\\dot{x}^{\\beta}+\\dot{x}^{\\alpha}\\frac{\\partial\\dot{x}^{\\beta}}{\\partial\\dot{x}^{\\mu}}\\right)\\right) \\\\
\n&=&\\frac{d}{d\\tau}\\left(\\frac{-1}{2 L} g_{\\alpha\\beta}\\left( \\delta^{\\alpha}_{\\ \\ \\mu} \\dot{x}^{\\beta}+\\dot{x}^{\\alpha}\\delta^{\\beta}_{\\ \\ \\mu}\\right)\\right) \\\\
\n&=& \\frac{1}{2 c}\\frac{d}{d\\tau}\\left( g_{\\mu\\beta}\\, \\dot{x}^{\\beta} +g_{\\alpha\\mu}\\, \\dot{x}^{\\alpha}\\right) \\\\
\n&=& \\frac{1}{2 c}\\frac{d}{d\\tau}\\left( 2 g_{\\mu\\nu}\\, \\dot{x}^{\\nu} \\right) \\\\&=& \\frac{1}{c}\\frac{d}{d\\tau}\\left( g_{\\mu \\nu} \\,\\dot{x}^{\\nu}\\right) \\\\
\n\\ \\\\
\n\\frac{\\partial L}{\\partial x^{\\mu}}
\n&=&\\frac{-1}{2 L}
\n\\frac{\\partial g_{\\alpha\\beta}}{\\partial\u00a0 x^{\\mu}} \\,\\dot{x}^{\\alpha}\\, \\dot{x}^{\\beta}\u00a0 \\\\
\n&=&\\frac{1}{2 c}
\ng_{\\alpha\\beta, \\mu} \\,\\dot{x}^{\\alpha}\\, \\dot{x}^{\\beta} \\\\ \\ \\\\
\n\\therefore\\ \\ \\frac{1}{c}\\frac{d}{d\\tau}\\left( g_{\\mu \\nu} \\,\\dot{x}^{\\nu}\\right)&=&
\n\\frac{1}{2 c}
\ng_{\\alpha\\beta, \\mu} \\,\\dot{x}^{\\alpha}\\, \\dot{x}^{\\beta}
\n\\end{eqnarray}<\/p>\n

\\(\\displaystyle u^{\\mu} \\equiv \\dot{x}^{\\mu}, \\ u_{\\mu} \\equiv g_{\\mu\\nu} u^{\\mu}\\) \u3068\u623b\u3057\u3066\u3084\u308c\u3070<\/p>\n

$$\\frac{d u_{\\mu}}{d\\tau} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta}$$<\/p>\n

\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n

\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u3053\u306e\u300c\u5171\u5909\u6210\u5206\u300d\u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\uff0c\u89e3\u6790\u529b\u5b66\u306b\u304a\u3051\u308b\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u65b9\u7a0b\u5f0f\u306b\u76f8\u5f53\u3057\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb \\(g_{\\alpha\\beta} \\) \uff0c\u3057\u305f\u304c\u3063\u3066\u30e9\u30b0\u30e9\u30f3\u30b8\u30a2\u30f3 \\(L\\) \u304c\u7279\u5b9a\u306e\u5ea7\u6a19 \\(x^{\\mu} \\) \u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u5834\u5408\u306f\uff0c \\(x^{\\mu} \\) \u306f\u5faa\u74b0\u5ea7\u6a19\u3068\u547c\u3070\u308c\uff0c\u3053\u306e\u5faa\u74b0\u5ea7\u6a19\u306b\u5171\u5f79\u306a\u4e00\u822c\u5316\u904b\u52d5\u91cf\u3067\u3042\u308b \\(\\dot{x}_{\\mu} = u_{\\mu} \\) \u304c\u4fdd\u5b58\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u8868\u3057\u3066\u3044\u308b\u3002<\/p>\n

\u5352\u7814\u30bc\u30df\u306e K \u541b\u304c<\/p>\n

$\\require{cancel}$
\n\\begin{eqnarray}
\n\\frac{\\partial}{\\partial u^i} \\left( g_{ij} u^i u^j \\right)
\n&=& \\frac{\\partial}{\\partial {\\color{red}{\\cancel{\\color{black}{u^i}}}} }\\left( g_{ij} {\\color{red}{\\cancel{\\color{black}{u^i}}}} u^j \\right)\\\\
\n&=& g_{ij} \\ u^j
\n\\end{eqnarray}<\/p>\n

\u307f\u305f\u3044\u306a\u8a08\u7b97\u3092\u3084\u3063\u3066\u300c1\/2 \u5408\u3044\u307e\u305b\u3093\u300d\u306a\u3069\u3068\u8a00\u3046\u3082\u306e\u3060\u304b\u3089\uff0c\u4e01\u5be7\u306b\u8a08\u7b97\u3092\u793a\u3057\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n

\u3060\u3044\u305f\u3044\u306b\u3057\u3066\uff0c<\/p>\n

$$\\frac{\\partial}{\\partial u^{\\color{red}{i}}} \\left( g_{{\\color{red}{i}}j} u^{\\color{red}{i}} u^j \\right)$$<\/p>\n

\u3068\u3044\u3046\u66f8\u304d\u65b9\u306f\u3054\u6cd5\u5ea6\u3067\uff0c\uff08\u76f8\u5bfe\u8ad6\u306e\u6388\u696d\u3067\u30a2\u30a4<\/strong><\/span>\u3092\u8a9e\u308b\u6559\u54e1\u3067\u3042\u308b\uff09\u79c1\u304c\u5e38\u3005\u8a00\u3063\u3066\u3044\u308b\u683c\u8a00\uff1a<\/p>\n

\u4eba\u751f\u306b\u30a2\u30a4 (i) <\/span>\u306f\u5fc5\u8981\u3060\u3002\u3057\u304b\u3057\u591a\u3059\u304e\u308b\u30a2\u30a4 (i) <\/span>\u306f\u30c8\u30e9\u30d6\u30eb\u306e\u5143\uff01<\/strong><\/span><\/p>\n

\u3092\u809d\u306b\u9298\u3058\u308b\u3053\u3068\u3002\uff08\u30a2\u30a4 (i)<\/span> <\/strong><\/span>\u306f\u30ef\u30f3\u30da\u30a2\u307e\u3067\u304c\u7121\u96e3\u3002\uff09\uff08\u306a\u305c\u3053\u3053\u3060\u3051\u5510\u7a81\u306b greek index \u3067\u306f\u306a\u304f\u3066 latin index \u306a\u306e\u304b\u3068\u3044\u3046\u3068\uff0c\u7121\u7406\u77e2\u7406\u3053\u306e\u683c\u8a00\u3092\u8a00\u3044\u305f\u3044\u305f\u3081… \u3067\u306f\u306a\u304f\uff0clatin index \u304c 0~3 \u306b\u306a\u3063\u3066\u3044\u308b\u4f50\u85e4\u52dd\u5f66\u5148\u751f\u306e\u30c6\u30ad\u30b9\u30c8\u3092\u5f7c\u304c\u4f7f\u3063\u3066\u3044\u308b\u305f\u3081\u3067\u3059\u3088\uff0c\u3068\u8a00\u3044\u8a33\u3002\uff09<\/p>\n

\u6b63\u3057\u304f\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{\\partial}{\\partial u^{\\mu}} \\left( g_{\\alpha\\beta} u^{\\alpha} u^{\\beta} \\right)
\n&=& g_{\\alpha\\beta} \\left( \\frac{\\partial u^{\\alpha}}{\\partial u^{\\mu}} u^{\\beta} +
\nu^{\\alpha} \\frac{\\partial u^{\\beta}}{\\partial u^{\\mu}}\\right) \\\\
\n&=& g_{\\alpha\\beta} \\left( \\delta^{\\alpha}_{\\ \\ \\mu} u^{\\beta} +
\nu^{\\alpha} \\delta^{\\ \\ \\beta}_{\\mu}\\right) \\\\
\n&=& g_{\\mu\\beta} u^{\\beta} + g_{\\alpha\\mu} u^{\\alpha} \\\\
\n&=& g_{\\mu\\beta} u^{\\beta} + g_{\\mu\\alpha} u^{\\alpha} \\\\
\n&=& g_{\\mu\\nu} u^{\\nu} + g_{\\mu\\nu} u^{\\nu} \\\\
\n&=& 2 \\, g_{\\mu\\nu} u^{\\nu}
\n\\end{eqnarray}<\/p>\n

\u5171\u5909\u5fae\u5206\u304b\u3089\u306e\u5c0e\u51fa<\/h3>\n

\u91cd\u529b\u4ee5\u5916\u306e\u529b\u3092\u53d7\u3051\u305a\u306b\u904b\u52d5\u3059\u308b\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u7d4c\u8def\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u56fa\u6709\u6642\u9593 \\(\\tau\\) \u3092\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u3068\u3057\u305f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3067\u8868\u3055\u308c\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

$$ \\frac{d\\boldsymbol{u}}{d\\tau} = \\frac{d}{d\\tau} \\left( u^{\\mu}\\,\\boldsymbol{e}_{\\mu}\\right) = \\frac{d}{d\\tau} \\left( \\frac{dx^{\\mu}}{d\\tau}\\,\\boldsymbol{e}_{\\mu}\\right) = \\boldsymbol{0}$$<\/p>\n

\u53cd\u5909\u6210\u5206 \\(u^{\\mu}\\) \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3068\u3057\u3066\u66f8\u304f\u3068<\/p>\n

\\begin{eqnarray}
\n\\frac{du^{\\mu}}{d\\tau} + \\varGamma^{\\mu}_{\\ \\ \\alpha\\beta} u^{\\alpha}u^{\\beta} &=&
\n\\frac{\\partial u^{\\mu}}{\\partial x^{\\nu}} \\frac{dx^{\\nu}}{d\\tau} + \\varGamma^{\\mu}_{\\ \\ \\ \\nu\\beta} u^{\\beta}u^{\\nu} \\\\
\n&=& \\left\\{u^{\\mu}_{\\ \\ ,\\nu} + \\varGamma^{\\mu}_{\\ \\ \\ \\nu\\beta} u^{\\beta} \\right\\} u^{\\nu} \\\\
\n&=&0
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u306f\u5171\u5909\u5fae\u5206\u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068<\/p>\n

$$ u^{\\mu}_{\\ \\ ;\\nu} u^{\\nu} = 0$$<\/p>\n

\u5171\u5909\u5fae\u5206\u3068\u6dfb\u5b57\u306e\u4e0a\u3052\u4e0b\u3052\u306f\u53ef\u63db\u3060\u304b\u3089<\/p>\n

\\begin{eqnarray}
\ng_{\\mu\\lambda} u^{\\lambda}_{\\ \\ ;\\nu} u^{\\nu} &=&
\nu_{\\mu ;\\nu} u^{\\nu} \\\\
\n&=& \\left\\{u_{\\mu, \\nu} -\\varGamma^{\\alpha}_{\\ \\ \\ \\mu\\nu} u_{\\alpha} \\right\\} u^{\\nu} \\\\
\n&=& \\frac{d u_{\\mu}}{d\\tau} – \\frac{1}{2} g^{\\alpha\\beta} \\left( g_{\\beta\\mu,\\nu} + g_{\\beta\\nu, \\mu} – g_{\\mu\\nu, \\beta} \\right) u_{\\alpha} u^{\\nu} \\\\
\n&=& \\frac{d u_{\\mu}}{d\\tau} – \\frac{1}{2} \\left( g_{\\beta\\mu,\\nu} + g_{\\beta\\nu, \\mu} – g_{\\mu\\nu, \\beta} \\right) u^{\\beta} u^{\\nu} \\\\
\n&=& \\frac{d u_{\\mu}}{d\\tau} – \\frac{1}{2} g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta}\u00a0 \\\\
\n&=& 0\\\\
\n\\therefore\\ \\ \\frac{d u_{\\mu}}{d\\tau} &=& \\frac{1}{2} g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta}
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"

4\u5143\u901f\u5ea6\u306e\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206\u3067\u3042\u308b\u300c\u5171\u5909\u6210\u5206\u300d\\(u_{\\mu} = g_{\\mu\\nu} u^{\\nu}\\)\u00a0 \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d u_{\\mu}}{d\\tau} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta}$$<\/p>\n

\u306e\u5c0e\u51fa\u65b9\u6cd5\u306e\u304a\u3055\u3089\u3044\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":992,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-3021","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\/3021","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=3021"}],"version-history":[{"count":31,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3021\/revisions"}],"predecessor-version":[{"id":8940,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3021\/revisions\/8940"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/992"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=3021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}