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Lorentz transformation

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The Lorentz transformation, named after its discoverer, a Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, that has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. It can be used (for example) to calculate how a particle trajectory looks like if viewed from a inertial reference frame that is moving with constant velocity (with respect to the initial reference frame). It replaces the earlier Galilean transformation. The velocity of light, c, enters as a parameter in the Lorentz transformation. If c is taken to be infinite, the Galilean transformation is recovered, such that it may be indentified as a limiting case.

The Lorentz transformation is a group transformation that is used to transform the space and time coordinates (or in general any four-vector) of one inertial reference frame, <math>S</math>, into those of another one, <math>S'</math>, with <math>S'</math> traveling at a relative speed of <math>{\mathbf u}</math> to <math>S</math>. If an event has space-time coordinates of <math>(x, y, z, t)</math> in <math>S</math> and <math>(x', y', z', t')</math> in <math>S'</math>, then these are related according to the Lorentz transformation in the following way:

<math>x' = \gamma (x - ut)</math>
<math>y' = y</math>
<math>z' = z</math>
<math>t' = \gamma \left(t - \frac{u x}{c^{2}} \right)</math>

where

<math>\gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}}</math>
and <math>c</math> is the speed of light.

These equations only work if <math>{\mathbf u}</math> is pointed along the x-axis of <math>S</math>. In cases where <math>{\mathbf u}</math> does not point along the x-axis of <math>S</math>, it is generally easier to perform a rotation so that <math>{\mathbf u}</math> does point along the x-axis of <math>S</math> than to bother with the general case of the Lorentz transformation. Another limiting factor of the above transformation is that the "position" of the origins must coincide at 0. What this means is that <math>(0, 0, 0, 0)</math> in frame <math>S</math> must be the same as <math>(0, 0, 0, 0)</math> in <math>S'</math>.

Lorentz invariance

Quantities which remain the same under Lorentz transforms are said to be Lorentz invariant. The space-time interval is a Lorentz-invariant quantity[?]. In addition, if every solution to some equation of motion or some field equation remains a solution of the same equation after Lorentz transformation, then the equation is said to be Lorentz invariant (i.e. the same equation may be used to describe physics in any reference frame). All fundamental equations of physics are Lorentz invariant, including in particular Maxwell's equations of electromagnetism, where the Lorentz transformation and the special theory of relativity have been first discovered.

History

Lorentz discovered in 1900 that the transformation preserved Maxwell's equations. Lorentz believed the luminiferous aether hypothesis; it was Albert Einstein who developed the theory of relativity to provide a proper foundation for its application.



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