event Lorentz transformation Lorentz factor speed of light one-parameter group linear mappings rapidity Lorentz transformations spacetime
Define an event to have spacetime coordinates (t, x, y, z) in system S and (t′, x′, y′, z′) in a reference frame moving at a velocity v with respect to that frame, S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:









t
′




=
γ
 
(
t
−
v
x

/


c

2


)





x
′




=
γ
 
(
x
−
v
t
)





y
′




=
y





z
′




=
z
,






{\displaystyle {\begin{aligned}t'&=\gamma \ (t-vx/c^{2})\\x'&=\gamma \ (x-vt)\\y'&=y\\z'&=z,\end{aligned}}}


where 



γ
=


1

1
−



v

2



c

2









{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}

 is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′, relative to S, is parallel to the x-axis. For simplicity, the y and z coordinates are unaffected; only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.
