Velocity Addition

We have derived the velocity addition formula using two Lorentz transformations (in the same direction) and rapidity.

$$\beta=\tanh(\phi_1+\phi_2)=\frac{\tanh\phi_1... ...}{1 + \tanh\phi_1 \tanh\phi_2} ={\beta_1+\beta_2\over 1+\beta_1\beta_2}$$

We can also compute this in terms of the usual ${dx_i\over dt}$ going from one frame in which an object is moving to a frame boosted along the $x$ direction. (This should give the same result as above if the particle is moving in the $-x$ direction.)

Let the velocity in the original system be $\vec{u}={d\vec{x}\over dt}$.

$$u'_i={dx'_i\over dt'}$$

$$u'_1={\gamma\left(dx_1-vdt\right)\over \gamma\left(dt-{v\over c^2... ...\over dt}-v\over 1-{v\over c^2}{dx_1\over dt}} = {u_1-v\over 1-{vu_1\over c^2}}$$

$$u'_2={dx_2\over \gamma\left(dt-{v\over c^2}dx_1\right)}={{dx_2\ov... ...er c^2}{dx_1\over dt}\right)} = {u_2\over \gamma\left(1-{vu_1\over c^2}\right)}$$

$$u'_3= {u_3\over \gamma\left(1-{vu_1\over c^2}\right)}$$

This formula gives the same result as above but is more general since it allows us to add velocities that are in different directions.

$$u'_i={dx'_i\over dt'}$$

$$u'_1= {u_1-v\over 1-{vu_1\over c^2}}$$

$$u'_2= {u_2\over \gamma\left(1-{vu_1\over c^2}\right)}$$

$$u'_3= {u_3\over \gamma\left(1-{vu_1\over c^2}\right)}$$