Suppose that two inertial systems $K$ and $K'$ are related by an arbitrary Lorentz transformation $\Lambda$, what is the best way to compute the relative velocity between the two reference frames?

One idea would be to decompose $\Lambda$ into the product $\Lambda=R(\theta)\Lambda_x(\psi)R(\varphi)$ and extract the relative velocity from the boost transformation $\Lambda_x(\psi)$. But for a general Lorentz transformation I think this decomposition would take quite long, and there is probably a far quicker method. In my special relativity class we learned relativistic velocity addition only for reference frames which were in special configurations such that they were related by a Lorentz boost along a single spatial direction, how does this generalize?


A general Lorentz transformaiton $\Lambda$ can be written as

$$\Lambda = R B$$

where $R$ is a rotation and $B$ is a boost. Thus, denoting matrix transposition with a superscript $T$,

$$\Lambda^T\Lambda = B^2$$

since $R^TR=I$ and $B^T=B$. But then $B$ has the general form

$$B=\begin{pmatrix} \cosh\varphi&-u^T\sinh\varphi \\ -u\sinh\varphi& uu^T\cosh\varphi + I-uu^T \end{pmatrix}$$

where $u$ is a unit 3-vector giving the direction of the boost and $\varphi$ is the so-called rapidity. So

$$B^2 = \begin{pmatrix} 1&-u^T\sinh2\varphi \\ -u\sinh2\varphi& \cdots \end{pmatrix}.$$

As you can see, you can read $u$ and $\varphi$ by inspection of $B^2$ and then the velocity you seek is just

$$v = u\tanh\varphi.$$


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