A set of equations proposed by H. A. Lorentz for transforming the position and motion parameters from a frame of reference with origin at O and coordinates (x,y,z) to a frame moving relative to it with origin at O′ and coordinates (x′,y′,z′). They replace the Galilean transformations used in Newtonian mechanics and are used in relativistic mechanics. They are:

where v is the relative velocity of separation of O and O′, c is the speed of light, and β=1/√(1 − v²/c²)‎. The above equations apply for constant v in the xx′ direction with O and O′ coinciding at t=t′=0.