Editing Special relativity (section)

=== Lorentz transformation of velocities ===
{{See also|Velocity-addition formula}}

Consider two frames ''S'' and {{prime|''S''}} in standard configuration. A particle in ''S'' moves in the x direction with velocity vector {{tmath|1= \mathbf{u} }}. What is its velocity <math>\mathbf{u'}</math> in frame {{prime|''S''}}?

We can write
{{NumBlk2||<math display="block"> \mathbf{|u|} = u = dx / dt \, . </math>|7}}
{{NumBlk2||<math display="block"> \mathbf{|u'|} = u' = dx' / dt' \, . </math>|8}}

Substituting expressions for <math>dx'</math> and <math>dt'</math> from {{EquationNote|5|Equation&nbsp;5}} into {{EquationNote|8|Equation&nbsp;8}}, followed by straightforward mathematical manipulations and back-substitution from {{EquationNote|7|Equation&nbsp;7}} yields the Lorentz transformation of the speed <math>u</math> to {{tmath|1= u' }}:
{{NumBlk2||<math display="block">u' = \frac{dx'}{dt'}=\frac{\gamma(dx-v\,dt)}{\gamma \left(dt-\dfrac{v\,dx}{c^2} \right)} = \frac{\dfrac{dx}{dt}-v}{1 - \dfrac{v}{c^2} \, \dfrac{dx}{dt} } =\frac{u-v}{1- \dfrac{uv}{c^2}}. </math>|9}}

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing <math> v </math> with {{tmath|1= -v }}.
{{NumBlk2||<math display="block">u=\frac{u'+v}{1+ u'v / c^2}.</math>|10}}

For <math>\mathbf{u}</math> not aligned along the x-axis, we write:<ref name="Rindler0"/>{{rp|47–49}}
{{NumBlk2||<math display="block"> \mathbf{u} = (u_1, \ u_2,\ u_3 ) = ( dx / dt, \ dy/dt, \ dz/dt) \ . </math>|11}}
{{NumBlk2||<math display="block"> \mathbf{u'} = (u_1', \ u_2', \ u_3') = ( dx' / dt', \ dy'/dt', \ dz'/dt') \ . </math>|12}}

The forward and inverse transformations for this case are:
{{NumBlk2||<math display="block">u_1'=\frac{u_1 -v}{1-u_1 v / c^2 } \ , \qquad u_2'=\frac{u_2}{\gamma \left( 1-u_1 v / c^2 \right) } \ , \qquad u_3'=\frac{u_3}{\gamma \left( 1- u_1 v / c^2 \right) } \ . </math>|13}}
{{NumBlk2||<math display="block">u_1=\frac{u_1' +v}{1+ u_1' v / c^2 } \ , \qquad u_2=\frac{u_2'}{ \gamma \left( 1+ u_1' v / c^2 \right) } \ , \qquad u_3=\frac{u_3'}{\gamma \left( 1+ u_1' v / c^2 \right)} \ . </math>|14}}

{{EquationNote|10|Equation&nbsp;10}} and {{EquationNote|14|Equation&nbsp;14}} can be interpreted as giving the ''resultant'' <math> \mathbf{u} </math> of the two velocities <math> \mathbf{v} </math> and {{tmath|1= \mathbf{u'} }}, and they replace the formula {{tmath|1= \mathbf{u = u' + v} }}. which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the ''relativistic velocity addition (or composition) formulas'', valid for the three axes of ''S'' and {{prime|''S''}} being aligned with each other (although not necessarily in standard configuration).<ref name="Rindler0"/>{{rp|47–49}}

We note the following points:
* If an object (e.g., a [[photon]]) were moving at the speed of light in one frame {{nowrap|1=(i.e., ''u'' = ±''c''}} {{nowrap|1=or {{prime|''u''}} = ±''c'')}}, then it would also be moving at the speed of light in any other frame, moving at {{nowrap|{{abs|''v''}} < ''c''}}.
* The resultant speed of two velocities with magnitude less than ''c'' is always a velocity with magnitude less than ''c''.
* If both {{abs|''u''}} and {{abs|''v''}} (and then also {{abs|{{prime|''u''}}}} and {{abs|{{prime|''v''}}}}) are small with respect to the speed of light (that is, e.g., {{nowrap|{{abs|{{sfrac|''u''|''c''}}}} ≪ {{math|1}})}}, then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
* Attaching a frame to a photon (''riding a light beam'' like Einstein considers) requires special treatment of the transformations.

There is nothing special about the ''x'' direction in the standard configuration. The above [[Formalism (mathematics)|formalism]] applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See ''[[Velocity-addition formula]]'' for details.