# Lorentz transformation

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 (i.e. speeds much, much lower than the speed of light).

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 $\displaystyle v/c \to 0$ , 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, $\displaystyle S$ , into those of another one, $\displaystyle S'$ , with $\displaystyle S'$ traveling at a relative speed of $\displaystyle {v}$ to $\displaystyle S$ along the x-axis. If an event has space-time coordinates of $\displaystyle (t, x, y, z)$ in $\displaystyle S$ and $\displaystyle (t', x', y', z')$ in $\displaystyle S'$ , and the origins coincide (in other words (0,0,0,0) in $\displaystyle S$ coincides with (0,0,0,0) in $\displaystyle S'$ ), then these coordinates are related according to the Lorentz transformation in the following way:

$\displaystyle t' = \gamma \left(t - \frac{v x}{c^{2}} \right)$
$\displaystyle x' = \gamma (x - v t)\,$
$\displaystyle y' = y\,$
$\displaystyle z' = z\,$

where

$\displaystyle \gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}$ is called the Lorentz factor and $\displaystyle c$ is the speed of light in a vacuum.

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

$\displaystyle \begin{bmatrix} t' \\x' \\y' \\z' \end{bmatrix} = \begin{bmatrix} \gamma&-\frac{v}{c^2} \gamma&0&0\\ -v \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} t\\x\\y\\z \end{bmatrix}$

or alternatively as

$\displaystyle \begin{bmatrix} c t' \\x' \\y' \\z' \end{bmatrix} = \begin{bmatrix} \gamma&-\frac{v}{c} \gamma&0&0\\ -\frac{v}{c} \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c t\\x\\y\\z \end{bmatrix}.$

The first matrix formulation has the advantage of being easily seen to collapse to the Galilean transformation in the limit $\displaystyle v/c \to 0$ . The second matrix formulation has the advantage of being easily seen to preserve the spacetime interval $\displaystyle ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2$ , which is a fundamental invariant in special relativity.

These equations only work if $\displaystyle {v}$ is pointed along the x-axis of $\displaystyle S$ . In cases where $\displaystyle {v}$ does not point along the x-axis of $\displaystyle S$ , it is generally easier to perform a rotation so that $\displaystyle {v}$ does point along the x-axis of $\displaystyle S$ 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 $\displaystyle \mathbf{x}$ into components perpendicular and parallel to the velocity $\displaystyle \mathbf{v}$ : $\displaystyle \mathbf{x}=\mathbf{x}_\perp+\mathbf{x}_\|$ . Only the component $\displaystyle \mathbf{x}_\|$ in direction of $\displaystyle \mathbf{v}$ is warped by the factor $\displaystyle \gamma$ :

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

These equations can be expressed in matrix form as

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

As noted above, this transformation requires that the origins of the two systems coincide. A generalization of Lorentz transformations that relaxes this restriction is the Poincaré transformations.

Furthermore, we can write Lorentz transformations in the form of a unitary rotation matrix in Minkowskian spacetime, which can be easily obtained from the usual Euclidean one which, in a two dimensional space, has the form:

$\displaystyle \begin{bmatrix} x_1' \\x_2' \end{bmatrix} = \begin{bmatrix} \cos\phi&-\sin\phi\\ \sin\phi &\cos\phi\\ \end{bmatrix} \begin{bmatrix} x_1 \\x_2 \end{bmatrix}$

in the limit $\displaystyle \phi \rightarrow i\psi ; x_2 \rightarrow ix_2$ (writing $\displaystyle x_1 = ct, x_2 = x$ ) so that sines and cosines now become hyperbolic and we get :

$\displaystyle \begin{bmatrix} ct' \\x' \end{bmatrix} = \begin{bmatrix} \cosh\psi& \sinh\psi\\ \sinh\psi &\cosh\psi\\ \end{bmatrix} \begin{bmatrix} ct \\x \end{bmatrix}$

As in Euclidean space, where vectors' absolute values do not vary under rotations, this matrix preserves the invariance of spacetime intervals: in this frame we can easily describe Lorentz transformations as rotations in a Minkowskian space.

More generally, If Λ is any 4x4 matrix such that ΛT$\displaystyle g_{\mu\nu}$ Λ=$\displaystyle g_{\mu\nu}$ , where T stands for transpose and

$\displaystyle g_{\mu\nu}= \begin{bmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{bmatrix}$

and X is the 4-vector describing spacetime displacements, $\displaystyle X\rightarrow \Lambda X$ 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. We often refer to $\displaystyle g_{\mu\nu}$ as the metric tensor of Minkowski space; it generalizes scalar product of 4-vectors and has the remarkable property that its contravariant and covariant components are identical.

Under the Erlangen 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.

The Lorentz transformations were first published in 1904, but their formalism was at the time imperfect. Henri Poincaré, the French mathematician, revised Lorentz's formalism to make the four equations into the coherent, self-consistent whole we know today.