# T_NAKAの阿房ブログ

## 一般 Lorentz 変換の積（３）

<<   作成日時 ： 2014/04/22 00:01   >>

 では各要素の計算を続けることにします。 　$l_{10}=-\frac{\gamma_{1}+\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\beta_{1}\gamma_{1}\gamma_{2}+\frac{\beta_{1}\beta_{2}&space;\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\beta_{2}\gamma_{2}=\frac{\beta_{1}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left&space;(\beta_{2}^{2}\gamma_{2}-\gamma_{2}-\gamma_{1}\right)$ 　　　　$=-\frac{\beta_{1}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left&space;\{\gamma_{2}&space;\left(1-\beta_{2}^{2}\right)+\gamma_{1}\right\}=-\frac{\beta_{1}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left(\frac{1}{\gamma_{2}}+\gamma_{1}&space;\right)$ 　　　　$=-\frac{\beta_{1}\gamma_{1}}{\gamma_{1}\gamma_{2}+1}\left(1+\gamma_{1}\gamma_{2}\right)=-\beta_{1}\gamma_{1}$ 　$l_{20}=-\frac{\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\beta_{1}\gamma_{1}\gamma_{2}-\frac{\gamma_{1}+\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\beta_{2}\gamma_{2}=-\frac{\beta_{2}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left(\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}+\gamma_{1}+\gamma_{2}\right)$ 　　　　$=-\frac{\beta_{2}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left(\frac{\gamma_{1}^{2}-1}{\gamma_{1}^{2}&space;}\gamma_{1}^{2}\gamma_{2}+\gamma_{1}+\gamma_{2}\right)=-\frac{\beta_{2}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left(\gamma_{1}^{2}\gamma_{2}-\gamma_{2}+\gamma_{1}+\gamma_{2}\right)$ 　　　　$=-\frac{\beta_{2}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left(\gamma_{1}^{2}\gamma_{2}+\gamma_{1}\right)=-\frac{\gamma_{1}&space;\beta_{2}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}\left(\gamma_{1}\gamma_{2}+1\right)=-\gamma_{1}&space;\beta_{2}\gamma_{2}$ 　$l_{11}=&space;\frac{\gamma_{1}+\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+\frac{\left(\gamma_{1}+\gamma_{2}\right)\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}^{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}-\frac{\beta_{1}^{2}\beta_{2}^{2}\gamma_{1}^{2}\gamma_{2}^{3}}{\left(\gamma_{1}\gamma_{2}+1\right&space;)^{2}}$ 　　　　　$=&space;\frac{\gamma_{1}+\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}^{2}\frac{\gamma_{1}+\gamma_{2}-\beta_{2}^{2}\gamma_{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}=&space;\frac{\gamma_{1}+\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}^{2}\frac{\gamma_{1}+\frac{1}{\gamma_{2}}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}$　　　　　　　 　　　　　$=&space;\frac{\gamma_{1}+\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}\frac{\gamma_{1}\gamma_{2}&space;+1}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}&space;=&space;\frac{\gamma_{1}+\gamma_{2}+\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}&space;=&space;\frac{\gamma_{1}+\gamma_{2}+\left(\gamma_{1}^{2}-1\right)\gamma_{2}}{\gamma_{1}\gamma_{2}+1}$ 　　　　　$=&space;\frac{\gamma&space;_{1}^{2}\gamma_{2}+\gamma_{1}}{\gamma_{1}\gamma_{2}+1}=&space;\gamma_{1}$ 　$l_{21}=&space;\frac{\beta_{1}\beta_{2}\gamma&space;_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+\frac{\beta_{1}^{3}\beta_{2}\gamma_{1}^{3}\gamma_{2}^{3}}{\left(\gamma_{1}\gamma_{2}+1\right)^{2}}+\frac{\left&space;(\gamma&space;_{1}+\gamma_{2}\right)\beta_{1}\beta_{2}\gamma_{1}\gamma&space;_{2}^{2}}{\left(\gamma_{1}\gamma_{2}+1\right)^{2}}$ 　　　　　$=&space;\frac{\beta_{1}\beta_{2}\gamma&space;_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}^{2}&space;\frac{\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}+\gamma&space;_{1}+\gamma_{2}}{\left(\gamma_{1}\gamma_{2}+1\right)^{2}}&space;=&space;\frac{\beta_{1}\beta_{2}\gamma&space;_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}^{2}&space;\frac{\left(\gamma_{1}^{2}-1\right&space;)\gamma_{2}+\gamma&space;_{1}+\gamma_{2}}{\left(\gamma_{1}\gamma_{2}+1\right)^{2}}$ 　　　　　$=&space;\frac{\beta_{1}\beta_{2}\gamma&space;_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}+&space;\frac{\beta_{1}\beta_{2}\gamma_{1}^{2}\gamma_{2}^{2}}{\gamma_{1}\gamma_{2}+1}=&space;\frac{\beta_{1}\beta_{2}\gamma&space;_{1}\gamma_{2}\left(\gamma_{1}\gamma_{2}+1\right)}{\gamma_{1}\gamma_{2}+1}=\beta_{1}\beta_{2}\gamma&space;_{1}\gamma_{2}$ 　$l_{12}=&space;\frac{\left(\gamma_{1}+\gamma_{2}\right)\beta_{1}\beta_{2}\gamma&space;_{1}\gamma_{2}^{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}-\frac{\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}-\frac{\beta_{1}\beta_{2}^{3}\gamma_{1}\gamma_{2}^{3}}{\left(\gamma_{1}\gamma_{2}+1\right)^{2}}$ 　　　　　$=&space;\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}\frac{\gamma_{1}\gamma_{2}+\gamma_{2}^{2}-\beta_{2}^{2}\gamma_{2}^{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}-\frac{\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}$ 　　　　　$=&space;\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}\frac{\gamma_{1}\gamma_{2}+\gamma_{2}^{2}-\left(\gamma_{2}^{2}-1\right&space;)}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}-\frac{\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}&space;=\frac{\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}-\frac{\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}}{\gamma_{1}\gamma_{2}+1}=&space;0$ 　$l_{22}=\frac{\beta_{1}^{2}\beta_{2}^{2}\gamma_{1}^{2}\gamma_{2}^{3}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}+\frac{\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}+\frac{\left&space;(\gamma_{1}+\gamma_{2}&space;\right&space;)\beta_{2}^{2}\gamma_{2}^{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}$ 　　　　　$=\beta_{2}^{2}\gamma_{2}^{2}\frac{\beta_{1}^{2}\gamma_{1}^{2}\gamma_{2}+\gamma_{1}+\gamma_{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}+\frac{\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}&space;=\beta_{2}^{2}\gamma_{2}^{2}\frac{\left(\gamma_{1}^{2}-1\right)\gamma_{2}+\gamma_{1}+\gamma_{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}+\frac{\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}$ 　　　　　$=\beta_{2}^{2}\gamma_{2}^{2}\frac{\gamma_{1}^{2}\gamma_{2}&space;-\gamma_{2}+\gamma_{1}+\gamma_{2}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}+\frac{\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}&space;=\beta_{2}^{2}\gamma_{2}^{2}\frac{\gamma_{1}^{2}\gamma_{2}+\gamma_{1}}{\left&space;(\gamma_{1}\gamma_{2}+1\right)^{2}}+\frac{\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}$ 　　　　　$=\frac{\beta_{2}^{2}\gamma_{1}\gamma_{2}^{2}+\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}=\frac{\left(\gamma_{2}^{2}-1\right)\gamma_{1}+\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}=\frac{\gamma_{1}\gamma_{2}^{2}-\gamma_{1}+\gamma_{1}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}$ 　　　　　$=\frac{\gamma_{1}\gamma_{2}^{2}+\gamma_{2}&space;}{\gamma_{1}\gamma_{2}+1}=\gamma_{2}$ したがって、 　$R\cdot&space;L\left(\boldsymbol{\beta&space;}_{3}\right)=&space;\begin{bmatrix}&space;\gamma_{1}\gamma_{2}&space;&-\beta_{1}\gamma_{1}\gamma_{2}&space;&-\beta_{2}\gamma_{2}&space;&0&space;\\&space;-\beta_{1}\gamma_{1}&space;&\gamma_{1}&space;&0&space;&0&space;\\&space;-\gamma_{1}\beta_{2}\gamma_{2}&space;&\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}&space;&\gamma&space;_{2}&space;&0\\&space;0&space;&0&space;&0&space;&1&space;\end{bmatrix}$ となります。つまり、 　$R\cdot&space;L\left(\boldsymbol{\beta&space;}_{3}\right)=L\left(\boldsymbol{\beta}_{2}\right)L\left(\boldsymbol{\beta}_{1}\right)$ ということになりますが、この意味については後記事で考えましょう。 今日はこの辺で。。

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