{"id":185,"date":"2022-01-05T13:26:43","date_gmt":"2022-01-05T04:26:43","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=185"},"modified":"2022-01-10T12:09:26","modified_gmt":"2022-01-10T03:09:26","slug":"%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e7%89%b9%e6%ae%8a%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%8a%b9%e6%9e%9c%e3%81%ae%e7%90%86-2","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e7%9b%b8%e5%af%be%e8%ab%96%e3%81%ae%e7%90%86%e8%a7%a3\/%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e7%89%b9%e6%ae%8a%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%8a%b9%e6%9e%9c%e3%81%ae%e7%90%86-2\/","title":{"rendered":"3\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247"},"content":{"rendered":"

\u89b3\u6e2c\u8005 \\(A\\) (\\(\\boldsymbol{u}\\)) \u306b\u5bfe\u3057\u3066\uff0c\u89b3\u6e2c\u8005 \\(B\\) (\\(\\bar{\\boldsymbol{u}}\\)) \u306f\u901f\u3055 \\(V\\) \u3067 \\(\\boldsymbol{e}\\) \u65b9\u5411\u306b\u904b\u52d5\u3057\uff0c\u3053\u306e \\(B\\) \u306b\u5bfe\u3057\u3066\uff0c\u89b3\u6e2c\u8005 \\(C\\) (\\( \\tilde{\\boldsymbol{u}}\\)) \u306f\u901f\u3055 \\(W\\) \u3067 \\(\\bar{\\boldsymbol{e}}\\) \u65b9\u5411\u306b\u904b\u52d5\u3059\u308b\u3002<\/p>\n

\u89b3\u6e2c\u8005 \\(A\\) \u3068\u89b3\u6e2c\u8005 \\(B\\) \u306b\u5bfe\u3059\u308b4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h3>\n

$$ \\bar{\\boldsymbol{u}} = \\frac{1}{\\sqrt{1-V^2}} \\left( \\boldsymbol{u}+ V \\boldsymbol{e}\\right)$$
\n$$ \\bar{\\boldsymbol{e}} = \\frac{1}{\\sqrt{1-V^2}} \\left( \\boldsymbol{e} + V \\boldsymbol{u} \\right)$$<\/p>\n

 <\/p>\n

\u89b3\u6e2c\u8005\\(B\\) \u3068\u89b3\u6e2c\u8005 \\(C\\) \u306b\u5bfe\u3059\u308b4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h3>\n

$$ \\tilde{\\boldsymbol{u}} = \\frac{1}{\\sqrt{1-W^2}} \\left( \\bar{\\boldsymbol{u}} + W \\bar{\\boldsymbol{e}} \\right)$$
\n$$ \\tilde{\\boldsymbol{e}} = \\frac{1}{\\sqrt{1-W^2}} \\left( \\bar{\\boldsymbol{e}} + W \\bar{\\boldsymbol{u}} \\right)$$<\/p>\n

 <\/p>\n

\u89b3\u6e2c\u8005\\(A\\) \u3068\u89b3\u6e2c\u8005 \\(C\\) \u306b\u5bfe\u3059\u308b4\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/h3>\n

\u4e0a\u5f0f\u304b\u3089 \\(\\bar{\\boldsymbol{u}}\\) \u3068 \\(\\bar{\\boldsymbol{e}}\\) \u3092\u6d88\u53bb\u3057\u3066 \\(\\tilde{\\boldsymbol{u}} \\) \u3092\u76f4\u63a5 \\(\\boldsymbol{u}\\) \u3068 \\(\\boldsymbol{e}\\) \u3092\u4f7f\u3063\u3066\u8868\u3059\u3068
\n\\begin{eqnarray} \\tilde{\\boldsymbol{u}} &=& \\frac{1}{\\sqrt{1-W^2}}
\n\\left\\{ \\frac{1}{\\sqrt{1-V^2}}\\left(\\boldsymbol{u} + V \\boldsymbol{e}\\right) +
\nW \\frac{1}{\\sqrt{1-V^2}} \\left(\\boldsymbol{e} + V \\boldsymbol{u} \\right)
\n\\right\\}\\\\
\n&=& \\frac{1}{\\sqrt{(1-V^2)(1-W^2)}}
\n\\left\\{(1+VW) \\boldsymbol{u} + (V + W) \\boldsymbol{e}
\n\\right\\}\\\\
\n&=& \\frac{1+VW}{\\sqrt{(1-V^2)(1-W^2)}}\\left\\{
\n\\boldsymbol{u} + \\frac{V + W}{1+VW} \\boldsymbol{e}
\n\\right\\}\\\\
\n&=& \\frac{1}{\\sqrt{1 – \\left(\\frac{V+W}{1+VW}\\right)^2}} \\left\\{
\n\\boldsymbol{u}+ \\frac{V + W}{1+VW} \\boldsymbol{e}
\n\\right\\}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c$$ U \\equiv \\frac{V+W}{1+VW} $$
\n\u3068\u3059\u308b\u3068\uff0c\u7d50\u5c40\uff0c\u89b3\u6e2c\u8005 \\(C\\) (\\(\\tilde{\\boldsymbol{u}}\\)) \u306f\uff0c\u89b3\u6e2c\u8005 \\(A\\)\uff08\\(\\boldsymbol{u}\\)\uff09\u306b\u5bfe\u3057\u3066\u901f\u5ea6 \\(U \\boldsymbol{e}\\) \u3067\u904b\u52d5\u3059\u308b\u3053\u3068\uff0c\u3059\u306a\u308f\u3061
\n$$\\tilde{\\boldsymbol{u}}=\\frac{1}{\\sqrt{1-U^2}} (\\boldsymbol{u} + U \\boldsymbol{e})$$ \u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n

\u3053\u306e\u3088\u3046\u306b\u3057\u3066\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u7684\u306a3\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/strong><\/span>$$ U \\equiv \\frac{V+W}{1+VW} $$\u304c\u5f97\u3089\u308c\u305f\u3002\u3061\u306a\u307f\u306b\uff0c\u5149\u901f \\(c\\) \u3092\u3042\u304b\u3089\u3055\u307e\u306b\u66f8\u304f\u3068
\n$$ U \\equiv \\frac{V+W}{1+\\frac{VW}{c^2}} $$
\n\u3068\u306a\u308a\uff0c\u30ac\u30ea\u30ec\u30a4\u5909\u63db\u306b\u57fa\u3065\u3044\u305f\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u306a\u901f\u5ea6\u5408\u6210\u5247<\/strong><\/span>\u3068\u306f\u5206\u6bcd\u306e \\(\\frac{VW}{c^2}\\) \u306e\u9805\u306e\u307f\u304c\u7570\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\\(\\frac{1}{c^2}\\) \u306b\u6bd4\u4f8b\u3059\u308b\u9805\u3092\u7121\u8996\u3059\u308b\u3068\u3044\u3046\uff0c\u3044\u308f\u3086\u308b\u30cb\u30e5\u30fc\u30c8\u30f3\u8fd1\u4f3c<\/strong><\/span>\u306e\u7bc4\u56f2\u5185\u3067\u306f\u7279\u6b8a\u76f8\u5bfe\u8ad6\u7684\u306a3\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u306a\u901f\u5ea6\u5408\u6210\u5247\u306b\u4e00\u81f4\u3059\u308b<\/strong><\/span>\u3002<\/p>\n

\u53c2\u8003\u307e\u3067\u306b\uff0c
\n\\begin{eqnarray} \\frac{1+VW}{\\sqrt{(1-V^2)(1-W^2)}} &=&
\n\\frac{1}{\\sqrt{\\frac{1-V^2 -W^2 + V^2 W^2}{(1+VW)^2}}}\\\\
\n&=& \\frac{1}{\\sqrt{\\frac{(1+ 2 VW + V^2 W^2) -(V^2 + 2 VW + W^2)}{(1+VW)^2}}}\\\\
\n&=& \\frac{1}{\\sqrt{\\frac{(1+\u00a0 VW)^2 – (V + W)^2}{(1+VW)^2}}} \\\\
\n&=& \\frac{1}{\\sqrt{1 – \\left(\\frac{V + W}{1+VW}\\right)^2}}
\n\\end{eqnarray}<\/p>\n

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