User:Mpatel/sandbox/Lorentz transformation

A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space while leaving the origin fixed. The transformation describes how space and time coordinates are related as measured by observers in different inertial reference frames and are named after the Dutch physicist and mathematician Hendrik Lorentz (1853-1928). They form the mathematical basis for Albert Einstein's theory of special relativity, which was introduced to remove contradictions between the theories of electromagnetism and classical mechanics. The 'Lorentz transformations' were derived by Einstein under the assumptions of Lorentz covariance and the constancy of the speed of light in any inertial reference frame.

In a given coordinate system $$(x^a)$$, the spacetime interval between two events $$A$$ and $$B$$ with coordinates $$(t_1, x_1, y_1, z_1)$$ and ($$t_2, x_2, y_2, z_2)$$ respectively is given by:


 * $$s^2 \,= - (t_2 - t_1)^2 + (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 $$

and is an invariant.:


 * $$x^a x_a \, = x'^a x'_a$$

i.e.,

$$\eta_{ab}'x'^ax'^b \, = \eta_{cd} x^c x^d$$

where $$\eta_{ab}$$ is the Minkowski metric


 * $$\eta_{\mu\nu}=

\begin{bmatrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}$$

and the Einstein summation convention is being used. From this relation follows the linearity of the coordinate transformation:


 * $$x'^a \, = \Lambda ^a{}_b x^b + C^a$$

where $$C^a$$ and $$\, \Lambda ^a{}_b$$ satisfy:


 * $$\Lambda^a{}_b \eta _{ac} \Lambda^c {}_d \,= \eta _{bd}$$


 * $$C^a \eta_{ac} \Lambda ^c{}_b \, = 0$$


 * $$ \eta_{ab} C^a C^b \, = 0$$

Such a transformation is called a Poincaré transformation. The $$C^a$$ represents a space-time translation; when $$C^a \, = 0$$, the transformation is a Lorentz transformation.

Taking the determinant of the first equation gives

$$\det (\Lambda ^a{}_b) \, = \pm 1$$

Lorentz transformations with $$\det (\Lambda ^a{}_b) \, = + 1$$ are called proper Lorentz transformations and consist of spatial rotations and boosts. Those with $$\det (\Lambda ^a{}_b) \, = - 1$$ are called improper Lorentz tranformations and consist of (discrete) space and time reflections.

Lorentz transformation for frames in standard configuration
Given two observers S and S', each using a Cartesian coordinate system to measure space and time intervals, $$\, (t, x, y, z)$$ and $$\, (t', x', y', z')$$, assume that the coordinate systems are oriented so that S' moves with constant speed v relative to S along the common x-x' axis with the y and y' axes parallel (and similarly for the z and z' axes). Also, assume that their origins meet at the common time t=t'=0. Then the frames are said to be in standard configuration (SC). The Lorentz transformation for frames in SC are:


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

where $$\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}$$ is called the Lorentz factor (or gamma factor) and $$c$$ is the speed of light in a vacuum. This Lorentz tranformation is called a boost in the x-direction and is often expressed in matrix form as



\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}. $$

where the coordinate $$t$$ is replaced by $$ct$$ (and similarly for $$t'$$).

The Lorentz transformations in SC may be cast into a more useful form by introducing a parameter $$\phi$$ called the rapidity or hyperbolic parameter through the equation:


 * $$e^{\phi} \equiv \gamma \left( 1 + \frac{v}{c} \right)$$

The Lorentz transformations in SC are then:


 * $$ct'-x' \, = e^{\phi}(ct-x)$$


 * $$ct'+x' \, = e^{- \phi}(ct+x)$$


 * $$y' \, = y$$


 * $$z' \, = z$$

General boosts
For a boost in an arbitrary direction with velocity $$\vec{v}$$, it is convenient to decompose the spatial vector $$\vec{r}$$ into components perpendicular and parallel to the velocity $$\vec{v}$$: $$\vec{r}=\vec{r}_\perp+\vec{r}_\|$$. Then only the component $$\vec{r}_\|$$ in the direction of $$\vec{v}$$ is 'warped' by the gamma factor:


 * $$t' = \gamma \left(t - \frac{\vec{r} \cdot \vec{v}}{c^{2}} \right)$$
 * $$\vec{r'} = \vec{r}_\perp + \gamma (\vec{r}_\| - \vec{v} t)$$

where now $$\gamma \equiv \frac{1}{\sqrt{1 - \vec{v} \cdot \vec{v}/c^2}}$$. The second of these can be written as:


 * $$\vec{r'} = \vec{r} + \left(\frac{\gamma -1}{v^2} (\vec{r} \cdot \vec{v}) - \gamma t \right) \vec{v}$$

These equations can be expressed in matrix form as



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

Lorentz and Poincaré groups
The composition of two Lorentz tranformations is a Lorentz transformation and the set of all Lorentz transformations with the operation of composition forms a gorup called the Lorentz group.

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

Special relativity
One of the most astounding predictions of special relativity was the idea that time is relative. More precisely, each observer carries their own personal clock and time flows different for different observers. This was a direct prediction from the Lorentz transformations and is called time dilation. Other effects can also be derived from the transformations, such as length contraction. The transformation of electric and magnetic fields was also found to be necessary in accordance with the relativity principle.

The correspondence principle
For relative speeds much less than the speed of light, the Lorentz tranformations reduce to the Galilean transformation in accordance with the correspondence principle. The correspondence limit is usually stated mathematically as $$\frac{v}{c} \rightarrow 0$$, or more formally (and less precisely) as $$c= \infty$$.

History
The transformations were first discovered and published by Joseph Larmor in 1897, although Woldemar Voigt had published a slightly different version of them in 1887, for which he showed that Maxwell's equations were invariant. In 1905, Henri Poincaré named them after the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928) who had published a first order version of these transformations in the 1890s and the final version in 1899 and 1904. The development of these transformations was encouraged by the null result of the Michelson-Morley experiment.

The Lorentz transformations were published in 1897 and 1900 by Joseph Larmor and by Hendrik Lorentz in 1899 and 1904. Voigt (1887) had published a form of the equations


 * $$ t' = t - vx/c^2, \;\;x' = x - vt \;\; y' = \frac{y} {\gamma}, z' = \frac{z} {\gamma}$$

which incorporated relativity of simultaneity ("local time") and time dilation. For Voigt, clocks ran slower by the factor $$\gamma ^2$$ which is greater than the now accepted value of $$\gamma$$ predicted by Larmor (1897). Note that Voigt equations have a length expansion in the transverse direction. Voigt derived these transformations as those which would make the speed of light the same in all reference frames. In a similar vein, Larmor and Lorentz were seeking the transformations under which Maxwell's equations were invariant.

Henri Poincaré in 1900 attributed the invention of local time to Lorentz and showed how Lorentz's first version of it (which applies to invariant clock rates) arose when clocks were sychronised by exchanging light signals which were assumed to travel at the same speed against and with the motion of the reference frame (see relativity of simultaneity).

Larmor's (1897) and Lorentz's (1899, 1904) final equations were not in the modern notation and form, but were algebraically equivalent to those published (1905) by Henri Poincaré, the French mathematician, who revised the form to make the four equations into the coherent, self-consistent whole we know today. Both Larmor and Lorentz discovered that the transformation preserved Maxwell's equations. Larmor and Lorentz believed the luminiferous aether hypothesis; it was Albert Einstein who developed the theory of relativity as a foundation for the universal application of the Lorentz transformations.