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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

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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

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

and

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

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