Lorentz transformation (original) (raw)

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.

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.

It can be used (for example) to calculate how a particle trajectory looks like if viewed from an 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, , into those of another one, , with traveling at a relative speed of to . If an event has space-time coordinates of in and in , then these are related according to the Lorentz transformation in the following way:

where

and is the speed of light in a vacuum.

These equations only work if is pointed along the x-axis of . In cases where does not point along the x-axis of , it is generally easier to perform a rotation so that does point along the x-axis of 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 in frame must be the same as in . This, of course, simply means we are dealing with Lorentz transformations, not Poincaré ones.

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 (as is the _g_-norm of any 4-vector). 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. It is therefore believed that all equations which describe physical objects are Lorentz invariant.

More generally, If Λ is any 4x4 matrix such that ΛT_g_Λ=g, where T stands for transpose and

and X is the 4-vector describing spacetime displacements, is the most general Lorentz transformation. Such defined matrices Λ form a representation of the group SO(3,1) also known as the Lorentz group.

Under the Erlanger program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations.

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.