The differential equation of the Lorentz rotor that is consistent with the Lorentz force is d Λ Λ d τ τ = e 2 m c F Λ Λ , such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest u = Λ Λ Λ Λ † † , which can be integrated to find the space-time trajectory x ( τ τ ) with the additional use of d x d τ τ = u .