Editing Lorentz transformation (section)

===Transformation of velocities===

{{Further|differential of a function|velocity addition formula}}

[[File:Lorentz transformation of velocity including velocity addition.svg|upright=1.75|thumb|The transformation of velocities provides the definition [[velocity addition formula|relativistic velocity addition]] {{math|⊕}}, the ordering of vectors is chosen to reflect the ordering of the addition of velocities; first {{math|'''v'''}} (the velocity of {{mvar|F′}} relative to {{mvar|F}}) then {{math|'''u′'''}} (the velocity of {{mvar|X}} relative to {{mvar|F′}}) to obtain {{math|'''u''' {{=}} '''v''' ⊕ '''u′'''}} (the velocity of {{mvar|X}} relative to {{mvar|F}}).]]

Defining the coordinate velocities and Lorentz factor by

:<math>\mathbf{u} = \frac{d\mathbf{r}}{dt} \,,\quad \mathbf{u}' = \frac{d\mathbf{r}'}{dt'} \,,\quad \gamma_\mathbf{v} = \frac{1}{\sqrt{1-\dfrac{\mathbf{v}\cdot\mathbf{v}}{c^2}}}</math>

taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to

:<math>\mathbf{u}' = \frac{1}{ 1 - \frac{\mathbf{v}\cdot\mathbf{u}}{c^2} }\left[\frac{\mathbf{u}}{\gamma_\mathbf{v}} - \mathbf{v} + \frac{1}{c^2}\frac{\gamma_\mathbf{v}}{\gamma_\mathbf{v} + 1}\left(\mathbf{u}\cdot\mathbf{v}\right)\mathbf{v}\right] </math>

The velocities {{math|'''u'''}} and {{math|'''u′'''}} are the velocity of some massive object. They can also be for a third inertial frame (say {{mvar|F′′}}), in which case they must be ''constant''. Denote either entity by {{mvar|X}}. Then {{mvar|X}} moves with velocity {{math|'''u'''}} relative to {{mvar|F}}, or equivalently with velocity {{math|'''u′'''}} relative to {{mvar|F′}}, in turn {{mvar|F′}} moves with velocity {{math|'''v'''}} relative to {{mvar|F}}. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange {{math|'''u'''}} and {{math|'''u′'''}}, and change {{math|'''v'''}} to {{math|−'''v'''}}.

The transformation of velocity is useful in [[stellar aberration]], the [[Fizeau experiment]], and the [[relativistic Doppler effect]].

The [[Acceleration (special relativity)#Three-acceleration|Lorentz transformations of acceleration]] can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.