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Lorentz transformation equations

The Lorentz transformation is a set of equations that take Special relativity into account when converting the location of an event in one system of coordinates to another system of coordinates that is moving at a constant velocity with respect to the first. They are as follows:

  • <math>t = \gamma (t^\prime+v\frac{x^\prime}{c})</math>
  • <math>x = \gamma (x^\prime+vt^\prime)</math>
  • <math>y = y^\prime</math>
  • <math>z = z^\prime</math>

where <math>\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}</math>

x, y, z, and t are the coordinates of the event in the frame we consider stationary. Their primed counterparts are the coordinates of the event in a reference frame that is moving at velocity v in the positive x direction. We assume that t = t' = 0 at the moment when the origins of the two systems coincide.

An equation which remains identical under a Lorentz transformation is known as Lorentz invariant. It it believed that all equations which describe physical objects are Lorentz invariant.

Under these transformations, the speed of light is the same in all reference frames, as postulated by special relativity. Although the equations are associated with special relativity, they were developed before special relativity and were proposed by Lorentz in 1904 as a means of explaining the Michelson-Morley experiment through contraction of lengths.

This is in contrast to the more intuitive Galilean transformation, which is sufficient at non-relativistic speeds:

  • t=t
  • x= x'+v t
  • y=y
  • z=z

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