快度

Rapidity

First,we know: \[ arctanh(x)=\frac{1}{2}\ln\frac{1+x}{1-x} \]

Then define:

rapidity \[ w=arctanh{\frac{v}{c}}=arctanh{\frac{|p|c}{E}}=\frac{1}{2}\ln\frac{E+|p|c}{E-|p|c} \]

pseudorapidity \[ \eta=arctanh{\frac{v_z}{v}}=arctanh{\frac{|p_z|}{|p|}}=\frac{1}{2}\ln\frac{|p|+|p_z|}{|p|-|p_z|}=-\ln(\tan(\frac{\theta}{2})) \]

rapidity in high energy experiment \[ y=arctanh{\frac{v_z}{c}}=arctanh{\frac{|p_z|c}{E}}=\frac{1}{2}\ln\frac{E+|p_z|c}{E-|p_z|c} \]