{"id":3042,"date":"2022-06-20T15:21:18","date_gmt":"2022-06-20T06:21:18","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3042"},"modified":"2024-06-10T11:27:03","modified_gmt":"2024-06-10T02:27:03","slug":"%e5%8f%82%e8%80%83%ef%bc%9a4%e5%85%83%e6%b3%a2%e6%95%b0%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%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","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%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad\/%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e6%b8%ac%e5%9c%b0%e7%b7%9a%e6%96%b9%e7%a8%8b%e5%bc%8f\/%e5%8f%82%e8%80%83%ef%bc%9a4%e5%85%83%e6%b3%a2%e6%95%b0%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%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\/","title":{"rendered":"\u53c2\u8003\uff1a4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u5171\u5909\u6210\u5206\u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa\u65b9\u6cd5"},"content":{"rendered":"

4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206\u3067\u3042\u308b\u300c\u5171\u5909\u6210\u5206<\/strong>\u300d<\/span> \\(k_{\\mu} \\equiv g_{\\mu\\nu} k^{\\nu}\\) \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d k_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}$$<\/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

\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\u5149\u306e\u4e16\u754c\u7dda \\(x(v)\\) \u306e\u63a5\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{k} = k^{\\mu} \\boldsymbol{e}_{\\mu}\\) \u306e\u4e0b\u4ed8\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 \\(k_{\\mu} \\equiv g_{\\mu\\nu} k^{\\nu}\\) \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d k_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}$$<\/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\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

\u30e9\u30f3\u30c0\u30a6\u30fb\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u5834\u306e\u53e4\u5178\u8ad6\u300d<\/a>\u306e p.274 \u306e\u811a\u6ce8\u306b\uff0c\u30cc\u30eb\u6e2c\u5730\u7dda\u3067\u306f\u306a\u304f\uff0c\u81ea\u7531\u7c92\u5b50\u306e4\u5143\u901f\u5ea6\u306e\u5171\u5909\u6210\u5206\u3067\u8868\u3057\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u3068\u3057\u3066<\/p>\n

$$\\frac{d u_i}{ds} -\\frac{1}{2} \\frac{\\partial g_{kl}}{\\partial x^i} u^k u^l = 0$$<\/p>\n

\u304c\u7d39\u4ecb\u3055\u308c\u3066\u3044\u308b\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

\u5149\u306e\u5834\u5408\u306f\u30cc\u30eb \\(ds^2 \\equiv -d\\tau^2 = 0\\) \u3067\u3042\u308b\u305f\u3081\uff0c\u56fa\u6709\u6642\u9593 \\(\\tau\\) \u3092\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u3068\u3057\u3066\u4f7f\u3046\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u3002\u305d\u3053\u3067\uff0c\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf\u3092 \\(v\\) \u3068\u3057\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30e9\u30b0\u30e9\u30f3\u30b8\u30a2\u30f3\u3092\u4f7f\u3046\u5834\u5408\u304c\u591a\u3044\u3002\uff08\u4f8b\u3048\u3070 Schneider, Ehlers, and Falco, “Gravitational Lenses”<\/a> \u306a\u3069\uff09\uff08\u95a2\u4fc2\u306a\u3044\u3051\u3069\uff0c20\u4e16\u7d00\u306e\u5927\u6614\uff0c\u5927\u5b66\u9662\u306e\u540c\u671f\u306e Y \u304c\u300c\u30e9\u30b0\u30e9\u30f3<\/strong><\/span>\u30b8\u30a2\u30f3\u300d\u3068\u8a00\u3048\u306a\u304f\u3066\uff0c\u3042\u308b\u3044\u306f\u8a00\u308f\u306a\u304f\u3066\u300c\u30e9\u30b0\u30e9\u30b8\u30a2\u30f3\u300d\u3068\u8a00\u3063\u3066\u3044\u305f\u306e\u304c\u61d0\u304b\u3057\u3044\u3002\u672c\u4eba\u306f\u5426\u5b9a\u3059\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\u5c11\u306a\u304f\u3068\u3082\u79c1\u306b\u306f\u3044\u3064\u3082\u300c\u30e9\u30b0\u30e9\u30b8\u30a2\u30f3\u300d\u3068\u805e\u3053\u3048\u305f\u3002\uff09<\/p>\n

$${\\cal{L}}\\left(x^{\\alpha}, \\frac{d x^{\\alpha}}{dv} \\right) \\equiv \\frac{1}{2} g_{\\alpha\\beta}\\frac{dx^{\\alpha}}{dv} \\frac{dx^{\\beta}}{dv} $$<\/p>\n

\u5909\u5206\u539f\u7406 \\(\\displaystyle \\delta \\int {\\cal{L}} dv = 0\\) \u3088\u308a\u5c0e\u304b\u308c\u308b\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u65b9\u7a0b\u5f0f\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{d}{dv}\\left( \\frac{\\partial {\\cal{L}}}{\\partial\\left( \\frac{dx^{\\mu}}{dv}\\right)} \\right) -\\frac{\\partial {\\cal{L}}}{\\partial x^{\\mu}}
\n&=& \\frac{d}{dv}\\left( g_{\\mu\\nu} \\frac{dx^{\\nu}}{dv}\\right) -\\frac{1}{2} g_{\\alpha\\beta, \\mu}\\frac{dx^{\\alpha}}{dv} \\frac{dx^{\\beta}}{dv} \\\\
\n&=& 0 \\\\ \\ \\\\
\n\\therefore\\ \\ \\frac{d}{dv}\\left( g_{\\mu\\nu} \\frac{dx^{\\nu}}{dv}\\right) &=& \\frac{1}{2} g_{\\alpha\\beta, \\mu}\\frac{dx^{\\alpha}}{dv} \\frac{dx^{\\beta}}{dv}
\n\\end{eqnarray}<\/p>\n

\\(\\displaystyle k^{\\mu} \\equiv \\frac{dx^{\\mu}}{dv}, \\ k_{\\mu} \\equiv g_{\\mu\\nu} k^{\\nu}\\) \u3068\u3059\u308c\u3070<\/p>\n

$$\\frac{d k_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}$$<\/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 \\({\\cal{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 \\(k_{\\mu} \\) \u304c\u4fdd\u5b58\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u8868\u3057\u3066\u3044\u308b\u3002<\/p>\n

\u5171\u5909\u6210\u5206\u306e\u5171\u5909\u5fae\u5206\u304b\u3089\u306e\u5c0e\u51fa<\/h3>\n

\u5149\u306e\u7d4c\u8def\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b \\(v\\) \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

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

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

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

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

$$ k^{\\mu}_{\\ \\ ;\\nu} k^{\\nu} = 0$$<\/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} k^{\\lambda}_{\\ \\ ;\\nu} k^{\\nu} &=&
\nk_{\\mu ;\\nu} k^{\\nu} \\\\
\n&=& \\left\\{k_{\\mu, \\nu} -\\varGamma^{\\alpha}_{\\ \\ \\ \\mu\\nu} k_{\\alpha} \\right\\} k^{\\nu} \\\\
\n&=& \\frac{d k_{\\mu}}{dv} -\\frac{1}{2} g^{\\alpha\\beta} \\left( g_{\\beta\\mu,\\nu} + g_{\\beta\\nu, \\mu} -g_{\\mu\\nu, \\beta} \\right) k_{\\alpha} k^{\\nu} \\\\
\n&=& \\frac{d k_{\\mu}}{dv} -\\frac{1}{2} \\left( g_{\\beta\\mu,\\nu} + g_{\\beta\\nu, \\mu} -g_{\\mu\\nu, \\beta} \\right) k^{\\beta} k^{\\nu} \\\\
\n&=& \\frac{d k_{\\mu}}{dv} -\\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}\u00a0 \\\\
\n&=& 0\\\\ \\ \\\\
\n\\therefore\\ \\ \\frac{d k_{\\mu}}{dv} &=& \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}
\n\\end{eqnarray}<\/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\u3057\u3066…<\/h3>\n

\u901a\u5e38\u306e\u30c6\u30ad\u30b9\u30c8\u7684\u306b\u306f\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

\\begin{eqnarray}
\n\\frac{dk^{\\lambda}}{dv} + \\varGamma^{\\lambda}_{\\ \\ \\mu\\nu} k^{\\mu}k^{\\nu} &=&0
\n\\end{eqnarray}<\/p>\n

\u306e\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\u3092\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u504f\u5fae\u5206\u3067\u8868\u3057\u305f\u5f0f\uff08\u300c\u88dc\u8db3\uff1a\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7<\/a>\u300d\u3092\u53c2\u7167\uff09<\/p>\n

$$\\varGamma^{\\lambda}_{\\ \\ \\mu\\nu} = \\frac{1}{2} g^{\\lambda\\sigma}(g_{\\sigma\\mu, \\nu} + g_{\\sigma\\nu, \\mu} -g_{\\mu\\nu, \\sigma})$$<\/p>\n

\u3092\u4f7f\u3063\u3066\u6574\u7406\u3057\u3066\uff0c<\/p>\n

$$\\frac{d k_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}$$<\/p>\n

\u3092\u6c42\u3081\u308b\u3067\u3042\u308d\u3046\u3002\uff08$k_{\\mu}$ \u306f\u5149\u306e4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u3092\u60f3\u5b9a\u3057\u3066\u3044\u308b\u304c\uff0c\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 $u_{\\mu}$ \u306e\u5834\u5408\u3082\uff0c\u30cc\u30eb\u304b\u6642\u9593\u7684\u304b\u3068\u3044\u3046\u9055\u3044\u306e\u307f\u3067\u5c0e\u51fa\u306f\u540c\u69d8\u3002\uff09\u3053\u308c\u306f\u7c21\u5358\u306a\u306e\u3067\uff0c\u7df4\u7fd2\u554f\u984c\u3068\u3057\u3066\u5404\u81ea\u3084\u3063\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206\u3067\u3042\u308b\u300c\u5171\u5909\u6210\u5206\u300d \\(k_{\\mu} \\equiv g_{\\mu\\nu} k^{\\nu}\\) \u306b\u5bfe\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{d k_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}$$<\/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":513,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-3042","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