Definitions

# Lorentz transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other. In classical physics (Galilean relativity), the only conversion believed necessary was x′ = x − vt, describing how the origin of one observer's coordinate system slides through space with respect to the other's, at speed v and along the x-axis of each frame. According to special relativity, this is only a good approximation at much smaller speeds than the speed of light, and in general the result is not just an offsetting of the x coordinates; lengths and times are distorted as well.

If space is homogeneous, then the Lorentz transformation must be a linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the spacetime interval between any two events in Minkowski space. The Lorentz transformations describe only the transformations in which the event at x = 0, t = 0 is left fixed, so they can be considered as a rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

Henri Poincaré named the Lorentz transformations after the Dutch physicist and mathematician Hendrik Lorentz (1853–1928) in 1905. They form the mathematical basis for Albert Einstein's theory of special relativity. They were derived by Joseph Larmor in 1897, and Lorentz (1899, 1904). In 1905 Einstein derived them under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame.

## Lorentz transformation for frames in standard configuration

Assume there are two observers O and $Q$, each using their own Cartesian coordinate system to measure space and time intervals. O uses $\left(t, x, y, z\right)$ and Q uses $\left(t\text{'}, x\text{'}, y\text{'}, z\text{'}\right)$. Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis overlap, the y-axis is parallel to the y' -axis, as are the z-axis and the z' -axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same. If all these hold, then the coordinate systems are said to be in standard configuration. A symmetric presentation between the forward Lorentz Transformation and the inverse Lorentz Transformation can be achieved if coordinate systems are in symmetric configuration. The symmetric form highlights that all physical laws should be of such a kind that they remain unchanged under a Lorentz transformation.

The Lorentz transformation for frames in standard configuration can be shown to be:

$begin\left\{cases\right\}$
t' &= gamma left(t - v x/c^{2} right) x' &= gamma left(x - v t right) y' &= y z' &= z end{cases} where $gamma = 1 / sqrt\left\{1 - v^2/c^2\right\}$ is called the Lorentz factor.

### Matrix form

This Lorentz transformation 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&-beta gamma&0&0 -beta gamma&gamma&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c,t x y z end{bmatrix} . More generally for a boost in an arbitrary direction $\left(beta_\left\{x\right\}, beta_\left\{y\right\}, beta_\left\{z\right\}\right)$,

begin{bmatrix} c,t' x' y' z' end{bmatrix} = begin{bmatrix} gamma&-beta_x,gamma&-beta_y,gamma&-beta_z,gamma -beta_x,gamma&1+(gamma-1)frac{beta_{x}^{2}}{beta^{2}}&(gamma-1)frac{beta_{x}beta_{y}}{beta^{2}}&(gamma-1)frac{beta_{x}beta_{z}}{beta^{2}} -beta_y,gamma&(gamma-1)frac{beta_{y}beta_{x}}{beta^{2}}&1+(gamma-1)frac{beta_{y}^{2}}{beta^{2}}&(gamma-1)frac{beta_{y}beta_{z}}{beta^{2}} -beta_z,gamma&(gamma-1)frac{beta_{z}beta_{x}}{beta^{2}}&(gamma-1)frac{beta_{z}beta_{y}}{beta^{2}}&1+(gamma-1)frac{beta_{z}^{2}}{beta^{2}} end{bmatrix} begin{bmatrix} c,t x y z end{bmatrix} , where $beta = frac\left\{v\right\}\left\{c\right\}=frac\left\{|vec\left\{v\right\}|\right\}\left\{c\right\}$ and $gamma = frac\left\{1\right\}\left\{sqrt\left\{1-beta^2\right\}\right\}$.

Note that this is only the "boost", i.e. a transformation between two frames in relative motion. But the most general proper Lorentz transformation also contains a rotation of the three axes. This boost alone is given by a symmetric matrix. But the general Lorentz transformation matrix is not symmetric.

### Rapidity

The Lorentz transformation can be cast into another useful form by introducing a parameter $phi$ called the rapidity (an instance of hyperbolic angle) through the equation:

$e^\left\{phi\right\} = gamma\left(1+beta\right) = gamma left\left(1 + frac\left\{v\right\}\left\{c\right\} right\right) = sqrt frac\left\{1 + v/c\right\}\left\{1 - v/c\right\}$

Equivalently:

$phi = ln left\left[gamma\left(1+beta\right)right\right] , -phi = ln left\left[gamma\left(1-beta\right)right\right] ,$

Then the Lorentz transformation in standard configuration is:

$begin\left\{cases\right\}$
c t-x = e^{- phi}(c t' - x') c t+x = e^{phi}(c t' + x') y = y' z = z' end{cases}

### Hyperbolic trigonometric expressions

It can also be shown that:

$gamma = cosh\left(phi\right) = \left\{ e^\left\{phi\right\} + e^\left\{-phi\right\} over 2 \right\}$
$beta = tanh\left(phi\right) = \left\{ e^\left\{phi\right\} - e^\left\{-phi\right\} over e^\left\{phi\right\} + e^\left\{-phi\right\} \right\}$

and therefore,

$beta gamma = sinh\left(phi\right) = \left\{ e^\left\{phi\right\} - e^\left\{-phi\right\} over 2 \right\}$

### Hyperbolic rotation of coordinates

Substituting these expressions into the matrix form of the transformation, we have:


begin{bmatrix} c t' x' y' z' end{bmatrix} = begin{bmatrix} cosh(phi) &-sinh(phi)&0&0 -sinh(phi) & cosh(phi) &0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c t x y z end{bmatrix} .

Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the rapidity $phi$ represents the hyperbolic angle of rotation.

## General boosts

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

$begin\left\{cases\right\}$
t' = gamma left(t - frac{vec{r} cdot vec{v}}{c^{2}} right) vec{r'} = vec{r}_perp + gamma (vec{r}_| - vec{v} t) end{cases} where now $gamma equiv frac\left\{1\right\}\left\{sqrt\left\{1 - vec\left\{v\right\} cdot vec\left\{v\right\}/c^2\right\}\right\}$. The second of these can be written as:
$vec\left\{r\text{'}\right\} = vec\left\{r\right\} + left\left(frac\left\{gamma -1\right\}\left\{v^2\right\} \left(vec\left\{r\right\} cdot vec\left\{v\right\}\right) - gamma t right\right) vec\left\{v\right\}$

These equations can be expressed in matrix form as


begin{bmatrix} c t' mathbf{r'} end{bmatrix} = begin{bmatrix} gamma & -gamma mathbf{v}^mathrm{T}/c -frac{gammamathbf{v}}{c} & I+ (gamma-1) frac {mathbf{v} mathbf{v}^mathrm{T}}{v^2} end{bmatrix} begin{bmatrix} c t mathbf{r} end{bmatrix}text{,} where I is the identity matrix, v is velocity written as a column vector and vT is its transpose (a row vector).

## Spacetime interval

In a given coordinate system ($x^mu$), if two events $A$ and $B$ are separated by

$\left(Delta t, Delta x, Delta y, Delta z\right) = \left(t_B-t_A, x_B-x_A, y_B-y_A, z_B-z_A\right) ,$
the spacetime interval between them is given by
$s^2 = - c^2\left(Delta t\right)^2 + \left(Delta x\right)^2 + \left(Delta y\right)^2 + \left(Delta z\right)^2 .$
This can be written in another form using the Minkowski metric. In this coordinate system,

eta_{munu} = begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} . Then, we can write

s^2 = begin{bmatrix}c Delta t & Delta x & Delta y & Delta z end{bmatrix} begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c Delta t Delta x Delta y Delta z end{bmatrix} or, using the Einstein summation convention,
$s^2= eta_\left\{munu\right\} x^mu x^nu .$

Now suppose that we make a coordinate transformation $x^mu rightarrow x\text{'}^mu$. Then, the interval in this coordinate system is given by


s'^2 = begin{bmatrix}c Delta t' & Delta x' & Delta y' & Delta z' end{bmatrix} begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c Delta t' Delta x' Delta y' Delta z' end{bmatrix} or
$s\text{'}^2= eta_\left\{munu\right\} x\text{'}^mu x\text{'}^nu .$

It is a result of special relativity that the interval is an invariant. That is, $s^2 = s\text{'}^2$. It can be shown that this requires the coordinate transformation to be of the form

$x\text{'}^mu = x^nu \left\{Lambda^mu\right\}_nu + C^mu .$
Here, $C^mu$ is a constant vector and $\left\{Lambda^mu\right\}_nu$ a constant matrix, where we require that
$eta_\left\{munu\right\}\left\{Lambda^mu\right\}_alpha\left\{Lambda^nu\right\}_beta = eta_\left\{alphabeta\right\} .$
Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation. The $C^a$ represents a space-time translation. When $C^a , = 0$, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.

Taking the determinant of $eta_\left\{munu\right\}\left\{Lambda^mu\right\}_alpha\left\{Lambda^nu\right\}_beta = eta_\left\{alphabeta\right\}$ gives us

$det \left(\left\{Lambda^a\right\}_b\right) = pm 1 .$
Lorentz transformations with $det \left(\left\{Lambda^mu\right\}_nu\right)=+1$ are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with $det\left(\left\{Lambda^mu\right\}_nu\right)=-1$ are called improper Lorentz transformations and consist of (discrete) space and time reflections combined with spatial rotations and boosts. They don't form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation.

The composition of two Poincaré transformations is a Poincaré transformation and the set of all Poincaré transformations with the operation of composition forms a group called the Poincaré group. Under the Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all Lorentz transformations forms a group, called the Lorentz group.

A quantity invariant under Lorentz transformations is known as a Lorentz scalar.

## Special relativity

One of the most astounding predictions of special relativity was the idea that time is relative. In essence, each observer's frame of reference is associated with a unique clock, the result being that time passes at different rates for different observers. This was a direct prediction from the Lorentz transformations and is called time dilation. We can also clearly see from the Lorentz transformations that the concept of simultaneity varies between reference frames. Another startling result is length contraction.

Lorentz transformations can also be used to prove that magnetic and electric fields are simply different aspects of the same force — the electromagnetic force. If we have one charge or a collection of charges which are all stationary with respect to each other, we can observe the system in a frame in which there is no motion of the charges. In this frame, there is only an electric field. If we switch to a moving frame, the Lorentz transformation will give rise to a magnetic field. These two fields are unified in the concept of the electromagnetic field.

## The correspondence principle

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation in accordance with the correspondence principle. The correspondence limit is usually stated mathematically as $v rightarrow 0$, so it is usually said that non relativistic physics is a physics of "instant action at a distance" $c rightarrow infty$.

## History

The transformations were first discovered and published by Joseph Larmor in 1897. 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 1895 and the final version in 1899 and 1904.

Many physicists, including FitzGerald, Larmor, Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887. Larmor and Lorentz, who believed the luminiferous aether hypothesis, were seeking the transformations under which Maxwell's equations were invariant when transformed from the ether to a moving frame. In early 1889, Heaviside had shown from Maxwell's equations that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion relative to the ether. FitzGerald then conjectured that Heaviside’s distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published his conjecture in Science to explain the baffling outcome of the 1887 ether-wind experiment of Michelson and Morley. This became known as the FitzGerald-Lorentz explanation of the Michelson-Morley null result, known early on through the writings of Lodge, Lorentz, Larmor, and FitzGerald. Their explanation was widely accepted as correct before 1905. Larmor gets credit for discovering the basic equations in 1897 and for being first in understanding the crucial time dilation property inherent in his equations.

Larmor's (1897) and Lorentz's (1899, 1904) final equations are algebraically equivalent to those published and interpreted as a theory of relativity by Albert Einstein (1905) but it was the French mathematician Henri Poincaré who first recognized that the Lorentz transformations have the properties of a mathematical group. Both Larmor and Lorentz discovered that the transformation preserved Maxwell's equations. Paul Langevin (1911) said of the transformation:

"It is the great merit of H. A. Lorentz to have seen that the fundamental equations of electromagnetism admit a group of transformations which enables them to have the same form when one passes from one frame of reference to another; this new transformation has the most profound implications for the transformations of space and time".

## Derivation

The usual treatment (e.g., Einstein's original work) is based on the invariance of the speed of light. However, this is not necessarily the starting point: indeed (as is exposed, for example, in the second volume of the Course in Theoretical Physics by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes that the influence that one particle, say, exerts on another can not be transmitted instantaneously. Hence, there exists a theoretical maximal speed of information transmission which must be invariant, and it turns out that this speed coincides with the speed of light in vacuum. The need for locality in physical theories was already noted by Newton (see Koestler's "The Sleepwalkers"), who considered the notion of an action at a distance "philosophically absurd" and believed that gravity must be transmitted by an agent (interstellar aether) which obeys certain physical laws.

Michelson and Morley in 1887 designed an experiment, which employed an interferometer and a half-silvered mirror, that was accurate enough to detect aether flow. The mirror system reflected the light back into the interferometer. If there were an aether drift, it would produce a phase shift and a change in the interference that would be detected. However, given the results were negative, rather than validating the aether, based upon the findings aether was not confirmed. This was a major step in science that eventually resulted in Einstein's Special Theory of Relativity.

In a 1964 paper, Erik Christopher Zeeman showed that the causality preserving property, a condition that is weaker in a mathematical sense than the invariance of the speed of light, is enough to assure that the coordinate transformations are the Lorentz transformations.