Rapidity

Remarks

Example: for $v = 0.6 c$, $\varphi = \tfrac{1}{2} \ln \frac{1 + 0.6}{1 - 0.6} = \tfrac{1}{2} \ln 4 = \ln 2$. Check: $\tanh(\ln 2) = (2 - \tfrac{1}{2}) / (2 + \tfrac{1}{2}) = 0.6$. Non-example: $v = c$ has no rapidity, since $\tanh \varphi < 1$ for every real $\varphi$. With $\beta = v / c$ and the Lorentz factor $\gamma$, one has $\gamma = \cosh \varphi$ and $\gamma \beta = \sinh \varphi$. So the boost along $x^1$ of the Lorentz transformation reads $$x'^0 = x^0 \cosh \varphi - x^1 \sinh \varphi, \quad x'^1 = -x^0 \sinh \varphi + x^1 \cosh \varphi,$$ using the hyperbolic functions. Boosts along the same axis with rapidities $\varphi_1$ and $\varphi_2$ compose to the boost with rapidity $\varphi_1 + \varphi_2$, by the addition formulas for $\cosh$ and $\sinh$.