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Lorentz transformation

From Academic Kids

The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which 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 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 v is low enough with respect to c then <math> v/c \to 0<math>, and the Galilean transformation is recovered, so it may be identified 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>{v}<math> to <math>S<math> along the x-axis. If an event has space-time coordinates of <math>(t, x, y, z)<math> in <math>S<math> and <math>(t', x', y', z')<math> in <math>S'<math>, then these are related according to the Lorentz transformation in the following way:

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

where

<math>\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}<math>

is called the Lorentz factor and <math>c<math> is the speed of light in a vacuum.

The four equations above can be expressed together in matrix form as

<math>

\begin{bmatrix} t' \\x' \\y' \\z' \end{bmatrix} = \begin{bmatrix} \gamma&-\frac{v}{c^2} \gamma&0&0\\ -v \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} t\\x\\y\\z \end{bmatrix} <math> or alternatively as

<math>

\begin{bmatrix} c t' \\x' \\y' \\z' \end{bmatrix} = \begin{bmatrix} \gamma&-\frac{v}{c} \gamma&0&0\\ -\frac{v}{c} \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c t\\x\\y\\z \end{bmatrix}. <math> The first matrix formulation has the advantage of being easily seen to collapse to the Galilean transformation in the limit <math> v/c \to 0<math>. The second matrix formulation has the advantage of being easily seen to preserve the spacetime interval <math>ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2<math>, which is a fundamental invariant in special relativity.

These equations only work if <math>{v}<math> is pointed along the x-axis of <math>S<math>. In cases where <math>{v}<math> does not point along the x-axis of <math>S<math>, it is generally easier to perform a rotation so that <math>{v}<math> does point along the x-axis of <math>S<math> than to bother with the general case of the Lorentz transformation.

For a boost in an arbitrary direction it is convenient to decompose the spatial vector <math>\mathbf{x}<math> into components perpendicular and parallel to the velocity <math>\mathbf{v}<math>: <math>\mathbf{x}=\mathbf{x}_\perp+\mathbf{x}_\|<math>. Only the component <math>\mathbf{x}_\|<math> in direction of <math>\mathbf{v}<math> is warped by the factor <math>\gamma<math>:

<math>t' = \gamma \left(t - \frac{v x_\|}{c^{2}} \right)<math>
<math>\mathbf{x}' = \mathbf{x}_\perp + \gamma (\mathbf{x}_\| - \mathbf{v} t)<math>

These equations can be expressed in matrix form as

<math>

\begin{bmatrix} c t' \\ \mathbf{x}' \end{bmatrix} = \begin{bmatrix} \gamma&-\frac{\mathbf{v^T}}{c}\gamma\\ -\frac{\mathbf{v}}{c}\gamma&\mathbf{1}+\frac{\mathbf{v}\cdot\mathbf{v^T}}{v^2}(\gamma-1)\\ \end{bmatrix} \begin{bmatrix} c t\\\mathbf{x} \end{bmatrix} <math>.

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>. A generalization of Lorentz transformations that relaxes this restriction is the Poincar transformations.

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

<math>g=

\begin{bmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{bmatrix}<math> and X is the 4-vector describing spacetime displacements, <math>X\rightarrow \Lambda X<math> 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 Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincar 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.

The Lorentz transformations were first published in 1904, but their formalism was at the time imperfect. Henri Poincar, the French mathematician, revised Lorentz's formalism to make the four equations into the coherent, self-consistent whole we know today.da:Lorentz-transformation de:Lorentz-Transformation fr:Transformation de Lorentz ko:로렌츠변환 he:טרנספורמציות לורנץ it:Trasformazioni di Lorentz nl:Lorentztransformatie pl:Transformacja Lorentza ru:Преобразования Лоренца sv:Lorentz-transformation zh:洛仑兹变换

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