Classical Mechanics (Goldstein) · Problem 3

The Classical Mechanics of the Special Theory of Relativity — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · einstein-addition-law-four-velocity

Problem

The Einstein addition law can also be obtained by remembering that the second velocity is related directly to the space components of a four-velocity, which may then be transformed back to the initial system by a Lorentz transformation. If the second system is moving with a speed v′v' relative to the first in the direction of their zz axes, while a third system is moving relative to the second with an arbitrarily oriented velocity v′′\mathbf{v}'', show by this procedure that the magnitude of the velocity v\mathbf{v} between the first and third system is given by 1−β2=1−β′21−β′′21+β′βz′′,\sqrt{1-\beta^2} = \frac{\sqrt{1-\beta'^2}\sqrt{1-\beta''^2}}{1+\beta'\beta''_z}, and that the components of v\mathbf{v} are βx=βx′′1−β′21+β′βz′′,βy=βy′′1−β′21+β′βz′′,βz=β′+βz′′1+β′βz′′.\beta_x = \frac{\beta''_x\sqrt{1-\beta'^2}}{1+\beta'\beta''_z}, \qquad \beta_y = \frac{\beta''_y\sqrt{1-\beta'^2}}{1+\beta'\beta''_z}, \qquad \beta_z = \frac{\beta' + \beta''_z}{1+\beta'\beta''_z}. Here βx′′=vx′′/c\beta''_x = v''_x/c, and so forth.

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