{"id":395,"date":"2022-01-07T12:35:49","date_gmt":"2022-01-07T03:35:49","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=395"},"modified":"2024-11-11T15:57:30","modified_gmt":"2024-11-11T06:57:30","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e6%b8%ac%e5%9c%b0%e7%b7%9a%e5%81%8f%e5%b7%ae%e6%96%b9%e7%a8%8b%e5%bc%8f","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%e6%b8%ac%e5%9c%b0%e7%b7%9a%e5%81%8f%e5%b7%ae%e6%96%b9%e7%a8%8b%e5%bc%8f\/","title":{"rendered":"\u88dc\u8db3\uff1a\u6e2c\u5730\u7dda\u504f\u5dee\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa"},"content":{"rendered":"

\u504f\u5dee\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(\\boldsymbol{\\xi} = \\xi^{\\mu} \\boldsymbol{e}_{\\mu}\\) \u306b\u5bfe\u3059\u308b\u5f0f
\n$$ \\frac{d\\boldsymbol{\\xi}}{dv} = \\boldsymbol{u}_{, \\nu}\\, \\xi^{\\nu}, \\quad \\mbox{or}\\quad
\n\\frac{d\\xi^{\\mu}}{dv} = u^{\\mu}_{\\ \\ , \\nu}\\, \\xi^{\\nu}$$\u304b\u3089\uff08\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206\u3092\u4f7f\u3063\u3066\u66f8\u304b\u308c\u308b\u524d\u306e\uff09\u6e2c\u5730\u7dda\u504f\u5dee\u65b9\u7a0b\u5f0f<\/strong><\/span>
\n\\begin{equation}
\n\\frac{d^2\\boldsymbol{\\xi}}{dv^2} =
\n\\left(\\boldsymbol{e}_{\\mu,\\rho\\nu} – \\boldsymbol{e}_{\\mu,\\nu\\rho}\\right)
\nu^{\\mu}u^{\\nu}\\xi^{\\rho}
\n\\end{equation}
\n\u3092\u5c0e\u304f\u3002\u3053\u3053\u3067\uff0c\\(\\boldsymbol{u} = u^{\\mu} \\boldsymbol{e}_{\\mu}\\) \u306f\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>
\n$$\\frac{d\\boldsymbol{u}}{dv} = \\boldsymbol{0}$$\u3092\u6e80\u305f\u3059\u3002<\/p>\n

\\begin{eqnarray}
\n\\frac{d^2\\boldsymbol{\\xi}}{dv^2}
\n&=&\\frac{d}{dv} \\left(\\frac{d\\boldsymbol{\\xi}}{dv}\\right) \\\\
\n&=& \\frac{d}{dv} \\left( \\boldsymbol{u}_{, \\rho}\\, \\xi^{\\rho}\\right)\\\\
\n&=& \\boldsymbol{u}_{, \\rho\\nu} u^{\\nu} \\xi^{\\rho} + \\boldsymbol{u}_{, \\nu} \\frac{d\\xi^{\\nu}}{dv}\\\\
\n&=& \\boldsymbol{u}_{, \\rho\\nu} u^{\\nu} \\xi^{\\rho} + \\boldsymbol{u}_{, \\nu} \\left(u^{\\nu}_{\\ \\ ,\\rho} \\xi^{\\rho}\\right) \\\\
\n&=& \\boldsymbol{u}_{, \\rho\\nu} u^{\\nu} \\xi^{\\rho} + \\left(\\left(\\boldsymbol{u}_{, \\nu} u^{\\nu}\\right)_{ ,\\rho} – \\boldsymbol{u}_{, \\nu\\rho} u^{\\nu}\\right)\\xi^{\\rho} \\\\
\n&=& \\left(\\boldsymbol{u}_{, \\rho\\nu} –\u00a0 \\boldsymbol{u}_{, \\nu\\rho}\\right) u^{\\nu}\\xi^{\\rho}
\n+ \\left( \\frac{d\\boldsymbol{u}}{dv}\\right)_{ ,\\rho}\\xi^{\\rho}\\\\
\n&=& \\left(\\boldsymbol{u}_{, \\rho\\nu} –\u00a0 \\boldsymbol{u}_{, \\nu\\rho}\\right) u^{\\nu}\\xi^{\\rho}
\n\\end{eqnarray}<\/p>\n

\u3068\u3044\u3046\u308f\u3051\u3067\uff0c\u3053\u3053\u307e\u3067\u304d\u305f\u3002
\n$$ \\frac{d^2\\boldsymbol{\\xi}}{dv^2} = \\left(\\boldsymbol{u}_{, \\rho\\nu} –\u00a0 \\boldsymbol{u}_{, \\nu\\rho}\\right) u^{\\nu}\\xi^{\\rho} $$<\/p>\n

\u3053\u3053\u3067\uff0c
\n\\begin{eqnarray}
\n\\boldsymbol{u}_{, \\rho\\nu}
\n&=& \\left( u^{\\mu} \\boldsymbol{e}_{\\mu} \\right)_{, \\rho\\nu}\\\\
\n&=& \\color{red}{u^{\\mu}_{\\ \\ , \\rho\\nu}\\boldsymbol{e}_{\\mu}
\n+ u^{\\mu}_{\\ \\ ,\\rho}\\boldsymbol{e}_{\\mu, \\nu}+ u^{\\mu}_{\\ \\ ,\\nu}\\boldsymbol{e}_{\\mu, \\rho}}
\n\\color{black}{+ u^{\\mu}\\boldsymbol{e}_{\\mu,\\rho\\nu}}
\n\\end{eqnarray}
\n\u3067\u3042\u308a\uff0c\u8d64\u8272\u90e8\u5206<\/span>\u306f\u6dfb\u5b57 \\( \\rho\\) \u3068 \\(\\nu\\) \u306b\u5bfe\u3057\u3066\u5bfe\u79f0\u3067\u3042\u308b\u304b\u3089\u76f4\u3061\u306b
\n$$ \\boldsymbol{u}_{, \\rho\\nu} –\u00a0 \\boldsymbol{u}_{, \\nu\\rho} = \\left( \\boldsymbol{e}_{\\mu, \\rho\\nu} –\u00a0 \\boldsymbol{e}_{\\mu, \\nu\\rho}\\right) u^{\\mu}$$
\n\u3086\u3048\u306b\uff0c
\n$$ \\frac{d^2\\boldsymbol{\\xi}}{dv^2} = \\left( \\boldsymbol{e}_{\\mu, \\rho\\nu} –\u00a0 \\boldsymbol{e}_{\\mu, \\nu\\rho}\\right) u^{\\mu}u^{\\nu}\\xi^{\\rho} $$<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":337,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-395","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