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I keep getting the wrong results by subbing eq1 into eq2 It’s from Kleppner’s an introduction to mechanics 2nd edition


The quoted text effectively combined two Lorentz transformations into one single transformation to identify the factor $\beta$, which is related to the velocity by $\beta=\frac{v}{c}$, thus the relativistic velocity addition formula. There are a few mistakes, so I'll give you the full proof (this is equivalent to the text).

The first Lorentz transformation can be expressed by a matrix

$$ \Lambda_1 = \begin{bmatrix} \gamma_1 & -\gamma_1 \beta_1 \\ -\gamma_1 \beta_1 & \gamma_1 \\ \end{bmatrix}, \begin{bmatrix}u_1'\\u_4'\end{bmatrix} = \Lambda_1 \begin{bmatrix}u_1\\u_4\end{bmatrix}. $$

Similarly for the second transformation we have

$$ \Lambda_2 = \begin{bmatrix} \gamma_2 & -\gamma_2 \beta_2 \\ -\gamma_2 \beta_2 & \gamma_2 \\ \end{bmatrix}, \begin{bmatrix}u_1''\\u_4''\end{bmatrix} = \Lambda_2 \begin{bmatrix}u_1'\\u_4'\end{bmatrix}. $$

The matrix representation for the combined boost is therefore $$ \Lambda_3 = \Lambda_2 \Lambda_1 = \begin{bmatrix} \gamma_1\gamma_2 (1+\beta_1\beta_2) & -\gamma_1\gamma_2(\beta_1+\beta_2) \\ -\gamma_1\gamma_2(\beta_1+\beta_2) & \gamma_1\gamma_2(1+\beta_1\beta_2) \\ \end{bmatrix}. $$

From this it is clear that

$$ \gamma_3 = \gamma_1\gamma_2(1+\beta_1\beta_2) $$


$$ \beta_3 = \frac{\beta_1+\beta_2}{1+\beta_1\beta_2}. $$


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