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Shell 1 is travelling at a speed $v_1$ with Lorentz factor $\Gamma_1$, and shell 2 at speed $v_2$ with Lorentz factor $\Gamma_2$. Just before the two shells meet, the relative lorentz factor $\Gamma_r$ is given by: $$\gamma_r = \Gamma_1 \Gamma_2 (1-v_1v_2/c^2) $$

as detailed in equation 141 of page 102 here.

Can anyone explain why this should be so?

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Use the Lorentz Boost parameter ($\lambda$) for each shell, where

$$ v/c=\tanh(\lambda) $$ $$ \gamma = \cosh(\lambda) $$ $$ \beta \gamma = \sinh(\lambda) $$

For boosts in the same direction the boost parameters just add or subtract. So $\lambda_1 -\lambda_2$ is the boost with which one shell is approaching the other. Consider the trigonometric identity

$$ \cosh(\lambda_1 -\lambda_2)= \cosh(\lambda_1) \cosh(\lambda_2)- \sinh(\lambda_1) \sinh(\lambda_2) $$ $$ = \cosh(\lambda_1) \cosh(\lambda_2)(1- \tanh(\lambda_1) \tanh(\lambda_2)) $$

Then putting $\gamma$ and $v/c$ in for cosh and tanh gives

$$ \gamma_{12}=\gamma_1 \gamma_2 (1-(v_1/c)(v_2/c)) $$

Using Lorentz Boost parameters makes most Special Relativity problems much easier!

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Lorentz Transformation $\:O x y z \longrightarrow O_1 x_1 y_1 z_1\:$ and its inverse $\:O_1 x_1 y_1 z_1 \longrightarrow O x y z \:$, $\:\left(y_1 = y, z_1 = z\right)\:$

\begin{equation} \begin{bmatrix} x_1\\ \\ \\ c t_1 \end{bmatrix} = \gamma_1 \begin{bmatrix} 1 & -\dfrac{v_1}{c}\\ & \\ -\dfrac{v_1}{c} & 1 \end{bmatrix} \begin{bmatrix} x\\ \\ \\ c t \end{bmatrix} \Longleftrightarrow \begin{bmatrix} x\\ \\ \\ c t \end{bmatrix} =\gamma_1 \begin{bmatrix} 1 & \dfrac{v_1}{c}\\ & \\ \dfrac{v_1}{c} & 1 \end{bmatrix} \begin{bmatrix} x_1\\ \\ \\ c t_1 \end{bmatrix} \tag{01} \end{equation} Lorentz Transformation $\:O x y z \longrightarrow O_2 x_2 y_2 z_2\:$, $\:\left(y_2 = y, z_2 = z\right)\:$ and its composition with $\:O_1 x_1 y_1 z_1 \longrightarrow O x y z \:$

\begin{equation} \begin{bmatrix} x_2\\ \\ \\ c t_2 \end{bmatrix} = \gamma_2 \begin{bmatrix} 1 & -\dfrac{v_2}{c}\\ & \\ -\dfrac{v_2}{c} & 1 \end{bmatrix} \begin{bmatrix} x\\ \\ \\ c t \end{bmatrix} =\gamma_2 \begin{bmatrix} 1 & -\dfrac{v_2}{c}\\ & \\ -\dfrac{v_2}{c^2} & 1 \end{bmatrix} \gamma_1 \begin{bmatrix} 1 & \dfrac{v_1}{c}\\ & \\ \dfrac{v_1}{c} & 1 \end{bmatrix} \begin{bmatrix} x_1\\ \\ \\ c t_1 \end{bmatrix} \tag{02} \end{equation} So,

Lorentz Transformation $\:O_1 x_1 y_1 z_1 \longrightarrow O_2 x_2 y_2 z_2\:$, $\:\left(y_2 = y_1, z_2 = z_1\right)\:$ \begin{equation} \begin{bmatrix} x_2\\ \\ \\ \\ \\ c t_2 \end{bmatrix} = \gamma_1 \gamma_2\left(1-\dfrac{v_1 v_2}{c^2} \right) \begin{bmatrix} 1 & - \dfrac {\left(v_2 - v_1\right)}{c\left(1-\dfrac{v_1 v_2}{c^2}\right)}\\ & \\ - \dfrac {\left(v_2 - v_1\right)}{c\left(1-\dfrac{v_1 v_2}{c^2}\right)} & 1 \end{bmatrix} \begin{bmatrix} x_1\\ \\ \\ \\ \\ c t_1 \end{bmatrix} \tag{03} \end{equation} that is the velocity of $\:O_2 x_2 y_2 z_2\:$ relative to $\:O_1 x_1 y_1 z_1\:$ is

\begin{equation} v_{2\: relative\; to\: 1}= v_{21}=\dfrac {v_2 - v_1}{1-\dfrac{v_1 v_2}{c^2} } \tag{04} \end{equation} with $\:\gamma-$factor \begin{equation} \gamma_{21}=\left[1-\left(\dfrac { v_{21}}{c}\right)^{2}\right]^{-\frac{1}{2}} =\left[1-\left(\dfrac{\dfrac {v_2 - v_1}{1-\dfrac{v_1 v_2}{c^2}}}{c}\right)^{2}\right]^{-\frac{1}{2}}= \gamma_1 \gamma_2\left(1-\dfrac{v_1 v_2}{c^2} \right) \tag{05} \end{equation}

There exists a quantity called rapidity defined by $\:\cosh \zeta = \gamma \:$, that is

\begin{equation} \zeta =\cosh^{-1}\gamma =\cosh^{-1}\left[\left(1-\dfrac{v^2}{c^2}\right)^{-\frac{1}{2}}\right] \tag{06} \end{equation}

Of course \begin{equation} \tanh \zeta = \dfrac{v}{c} \tag{07} \end{equation}

Having in mind that \begin{equation} \cosh \zeta = \cos \left(i\zeta\right)\;, \quad \sinh \zeta = -i\sin \left(i\zeta\right) \tag{08} \end{equation}

equation (01) is expressed as

\begin{equation} \begin{bmatrix} x_1\\ \\ i c t_1 \end{bmatrix} = \begin{bmatrix} \cos \left(i\zeta_1 \right) & - \sin \left(i\zeta_1 \right)\\ & \\ \sin \left(i\zeta_1 \right) & \cos \left(i\zeta\right) \end{bmatrix} \begin{bmatrix} x\\ \\ i c t \end{bmatrix} \tag{09} \end{equation}

or defining like an angle \begin{equation} \theta_1 = i\zeta_1 \tag{10} \end{equation}

\begin{equation} \begin{bmatrix} x_1\\ \\ i c t_1 \end{bmatrix} = \begin{bmatrix} \cos \theta_1 & - \sin \theta_1 \\ & \\ \sin \theta_1 & \cos \theta_1 \end{bmatrix} \begin{bmatrix} x\\ \\ i c t \end{bmatrix} \tag{11} \end{equation}

which reminds us a rotation through the (imaginary) angle $\:\theta_1 \:$.

Then equation (05) for the $\:\gamma-$factor is the well-known identity for the $\:\cos \left(\theta_2 - \theta_1\right)\:$

\begin{equation} \cos \left(\theta_2 - \theta_1\right)=\cos \theta_2 \cos \theta_1 + \sin \theta_2 \sin \theta_1 \tag{12} \end{equation}


SR 1

SR 1 3d

SR 2 3d

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