Lorentz transformation

A visualisation of the Lorentz transformation (full animation). Only one space coordinate is considered. The thin solid lines crossing at right angles depict the time and distance coordinates of an observer at rest; the skewed solid straight lines depict the coordinate grid of a moving observer.

In physics, the Lorentz transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik Lorentz. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.

The Lorentz transformation was originally the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. Albert Einstein later re-derived the transformation from his postulates of special relativity. The Lorentz transformation supersedes the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). According to special relativity, this is a good approximation only at relative speeds much smaller than the speed of light.

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 transformation describes only the transformations in which the spacetime event at the origin 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.

Lorentz transformation for frames in standard configuration

Standard configuration of coordinate systems for Lorentz transformations.

Assume there are two observers O and Q, each using their own Cartesian coordinate system to measure space and time intervals. O uses $(t,x,y,z)$ and Q uses $(t',x',y',z')$. Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis are collinear, 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:

${\begin{cases}t'&=\gamma \left(t-vx/c^{2}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{cases}}$

where $\ \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ 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

${\begin{bmatrix}ct'\\x'\\y'\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}{\begin{bmatrix}c\,t\\x\\y\\z\end{bmatrix}}\ .$

More generally for a boost in any arbitrary direction $(\beta _{x},\beta _{y},\beta _{z})$,

${\begin{bmatrix}c\,t'\\x'\\y'\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &-\beta _{x}\,\gamma &-\beta _{y}\,\gamma &-\beta _{z}\,\gamma \\-\beta _{x}\,\gamma &1+(\gamma -1){\dfrac {\beta _{x}^{2}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{x}\beta _{y}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{x}\beta _{z}}{\beta ^{2}}}\\-\beta _{y}\,\gamma &(\gamma -1){\dfrac {\beta _{y}\beta _{x}}{\beta ^{2}}}&1+(\gamma -1){\dfrac {\beta _{y}^{2}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{y}\beta _{z}}{\beta ^{2}}}\\-\beta _{z}\,\gamma &(\gamma -1){\dfrac {\beta _{z}\beta _{x}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{z}\beta _{y}}{\beta ^{2}}}&1+(\gamma -1){\dfrac {\beta _{z}^{2}}{\beta ^{2}}}\\\end{bmatrix}}{\begin{bmatrix}c\,t\\x\\y\\z\end{bmatrix}}\ ,$

where $\beta ={\frac {v}{c}}={\frac {|{\vec {v}}|}{c}}$ and $\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}$.

Note that this transformation is only the "boost," i.e., a transformation between two frames whose $x,y$, and $z$ axis are parallel and whose spacetime origins coincide (see The "Standard configuration" Figure). The most general proper Lorentz transformation also contains a rotation of the three axes, because the composition of two boosts is not a pure boost but is a boost followed by a rotation. The rotation gives rise to Thomas precession. The boost is given by a symmetric matrix, but the general Lorentz transformation matrix need not be symmetric.

The composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by:

$B(\mathbf {u} )B(\mathbf {v} )=B(\mathbf {u} \oplus \mathbf {v} )gyr[\mathbf {u} ,\mathbf {v} ]=gyr[\mathbf {u} ,\mathbf {v} ]B(\mathbf {v} \oplus \mathbf {u} )$,

where u$\oplus$v is the velocity-addition, and gyr[u,v] is the rotation arising from the composition, gyr being the gyrovector space abstraction of the gyroscopic Thomas precession into an operator, and B(v) is the 4x4 matrix that uses the components of v, i.e. v₁, v₂, v₃ in the entries of the matrix, or rather the components of v/c in the representation that is used above.

The composition of two Lorentz transformations L(u,U) and L(v,V) which include rotations U and V is given by:

$L(\mathbf {u} ,U)L(\mathbf {u} ,V)=L(\mathbf {u} \oplus U\mathbf {v} ,gyr[\mathbf {u} ,U\mathbf {v} ]UV)$

If the 3x3 matrix form of the rotation applied to spatial coordinates is given by gyr[u,v], then the 4x4 matrix rotation applied to 4-coordinates is given by:

${\begin{pmatrix}1&0\\0&gyr[\mathbf {u} ,\mathbf {v} ]\end{pmatrix}}$.

Views of spacetime along the world line of a rapidly accelerating observer moving in a 1-dimensional (straight line) "universe". The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The small dots are specific events in spacetime. If one imagines these events to be the flashing of a light, then the events that pass the two diagonal lines in the bottom half of the image (the past light cone of the observer in the origin) are the events visible to the observer. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates.

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:

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

where now $\gamma \equiv {\frac {1}{\sqrt {1-{\vec {v}}\cdot {\vec {v}}/c^{2}}}}$. 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

${\begin{bmatrix}ct'\\\mathbf {r'} \end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma {\dfrac {\mathbf {v} ^{\mathrm {T} }}{c}}\\-\displaystyle {\frac {\gamma \mathbf {v} }{c}}&\mathbf {I} +(\gamma -1){\mathbf {\hat {v}} \mathbf {\hat {v}} ^{\mathrm {T} }}\\\end{bmatrix}}{\begin{bmatrix}ct\\\mathbf {r} \end{bmatrix}}{\text{,}}$

where I is the identity matrix, v is velocity written as a column vector, v^T is its transpose (a row vector) and $\mathbf {\hat {v}}$ is its versor.

Rapidity

The Lorentz transformation can be cast into another useful form by defining a parameter $\scriptstyle {\boldsymbol {\phi }}$ called the rapidity (an instance of hyperbolic angle) such that

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

so that

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

Equivalently:

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

Then the Lorentz transformation in standard configuration is:

${\begin{cases}ct-x=e^{-\phi }(ct'-x')\\ct+x=e^{\phi }(ct'+x')\\y=y'\\z=z'.\end{cases}}$

Hyperbolic trigonometric expressions

From the above expressions for e^φ and e^−φ

$\gamma =\cosh \phi ={e^{\phi }+e^{-\phi } \over 2},$

$\beta \gamma =\sinh \phi ={e^{\phi }-e^{-\phi } \over 2},$

and therefore,

$\beta =\tanh \phi ={e^{\phi }-e^{-\phi } \over e^{\phi }+e^{-\phi }}.$

Hyperbolic rotation of coordinates

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

${\begin{bmatrix}ct'\\x'\\y'\\z'\end{bmatrix}}={\begin{bmatrix}\cosh \phi &-\sinh \phi &0&0\\-\sinh \phi &\cosh \phi &0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}{\begin{bmatrix}ct\\x\\y\\z\end{bmatrix}}\ .$

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.

Spacetime interval

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

$(\Delta t,\Delta x,\Delta y,\Delta z)=(t_{B}-t_{A},x_{B}-x_{A},y_{B}-y_{A},z_{B}-z_{A})\ ,$

the spacetime interval between them is given by

$s^{2}=-c^{2}(\Delta t)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\ .$

This can be written in another form using the Minkowski metric. In this coordinate system,

$\eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\ .$

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,

$s^{2}=\eta _{\mu \nu }x^{\mu }x^{\nu }\ .$

Now suppose that we make a coordinate transformation $x^{\mu }\rightarrow x'^{\mu }$. 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

$s'^{2}=\eta _{\mu \nu }x'^{\mu }x'^{\nu }\ .$

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

$x'^{\mu }=x^{\nu }{\Lambda ^{\mu }}_{\nu }+C^{\mu }\ .$

Here, $C^{\mu }\$ is a constant vector and ${\Lambda ^{\mu }}_{\nu }$ a constant matrix, where we require that

$\eta _{\mu \nu }{\Lambda ^{\mu }}_{\alpha }{\Lambda ^{\nu }}_{\beta }=\eta _{\alpha \beta }\ .$

Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation. The $C^{a}$ represents a spacetime translation. When $C^{a}\,=0$, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.

Taking the determinant of $\eta _{\mu \nu }{\Lambda ^{\mu }}_{\alpha }{\Lambda ^{\nu }}_{\beta }=\eta _{\alpha \beta }$ gives us

$\det({\Lambda ^{a}}_{b})=\pm 1\ .$

Lorentz transformations with $\det({\Lambda ^{\mu }}_{\nu })=+1$ form a subgroup called proper Lorentz transformations which is the special orthogonal group $SO(1,3)$. Those with $\det({\Lambda ^{\mu }}_{\nu })=-1$ are called improper Lorentz transformations which is not a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation. From the above definition of $\Lambda$ it can be shown that $({\Lambda ^{0}}_{0})^{2}\geq 1$, so either ${\Lambda ^{0}}_{0}\geq 1$ or ${\Lambda ^{0}}_{0}\leq -1$, called orthochronos and non-orthochronos respectively. An important subgroup of the proper Lorentz transformations are the proper orthochronos Lorentz transformations which consist purely of boosts and rotations. Any Lorentz transform can be written as a proper orthochronos, together with one or both of the two discrete transformations; space inversion ($P$) and time reversal ($T$), whose non-zero elements are:

${P^{0}}_{0}=1,{P^{1}}_{1}={P^{2}}_{2}={P^{3}}_{3}=-1$

${T^{0}}_{0}=-1,{T^{1}}_{1}={T^{2}}_{2}={T^{3}}_{3}=1$

The set of Poincaré transformations satisfies the properties of a group and is 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 consequences of Einstein's clock-setting method is the idea that time is relative. In essence, each observer's frame of reference is associated with a unique set of clocks, the result being that time passes at different rates for different observers. This was a direct result of the Lorentz transformations and is called time dilation. We can also clearly see from the Lorentz "local time" transformation that the concept of the relativity of simultaneity and of the relativity of length contraction are also consequences of that clock-setting hypothesis.

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 predict that a "magnetic field" is present. This field was initially unified in Maxwell's 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: as $v\rightarrow 0$, $c\rightarrow \infty$. In words: as velocity approaches 0, the speed of light (seems to) approach infinity. Hence, it is sometimes said that nonrelativistic physics is a physics of "instant action at a distance".

History

See also History of Lorentz transformations.

Many physicists, including George FitzGerald, Joseph Larmor, Hendrik Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887. Larmor and Lorentz, who believed the luminiferous ether hypothesis, were seeking the transformation under which Maxwell's equations were invariant when transformed from the ether to a moving frame. Early in 1889, Oliver 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 idea was extended by Lorentz and Larmor over several years, and 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 is also credited to have been the first to understanding the crucial time dilation property inherent in his equations.

In 1905, Henri Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz. Later in the same year Einstein derived the Lorentz transformation under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame, obtaining results that were algebraically equivalent to Larmor's (1897) and Lorentz's (1899, 1904), but with a different interpretation.

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 of 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, employing 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, no phase shift was ever found. The negative outcome of the Michelson-Morley experiment left the whole concept of aether without a reason to exist. Worse still, it created the perplexing situation that light evidently behaved like a wave, yet without any detectable medium through which wave activity might propagate.

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.

From group postulates

Following is a classical derivation (see, e.g., [1] and references therein) based on group postulates and isotropy of the space.

Coordinate transformations as a group

The coordinate transformations between inertial frames form a group (called the proper Lorentz group) with the group operation being the composition of transformations (performing one transformation after another). Indeed the four group axioms are satisfied:

1. Closure: the composition of two transformations is a transformation: consider a composition of transformations from the inertial frame $K$ to inertial frame $K'$, (denoted as $[K\to K']$), and then from $K'$ to inertial frame $K''$, $[K'\to K'']$, there exists a transformation, $[K\to K'']$, directly from an inertial frame $K$ to inertial frame $K''$.
2. Associativity: the result of ${\big (}[K\to K'][K'\to K'']{\big )}[K''\to K''']$ and $[K\to K']{\big (}[K'\to K''][K''\to K''']{\big )}$ is the same, $K\to K'''$.
3. Identity element: there is an identity element, a transformation $K\to K$.
4. Inverse element: for any transformation $K\to K'$ there exists an inverse transformation $K'\to K$.

Transformation matrices consistent with group axioms

Let us consider two inertial frames, K and K', the latter moving with velocity ${\vec {v}}$ with respect to the former. By rotations and shifts we can choose the z and z' axes along the relative velocity vector and also that the events (t=0,z=0) and (t'=0,z'=0) coincide. Since the velocity boost is along the z (and z') axes nothing happens to the perpendicular coordinates and we can just omit them for brevity. Now since the transformation we are looking after connects two inertial frames, it has to transform a linear motion in (t,z) into a linear motion in (t',z') coordinates. Therefore it must be a linear transformation. The general form of a linear transformation is

${\begin{bmatrix}t'\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &\delta \\\beta &\alpha \end{bmatrix}}{\begin{bmatrix}t\\z\end{bmatrix}},$

where $\alpha ,\beta ,\gamma ,$ and $\delta$ are some yet unknown functions of the relative velocity $v$.

Let us now consider the motion of the origin of the frame K'. In the K' frame it has coordinates (t',z'=0), while in the K frame it has coordinates (t,z=vt). These two points are connected by our transformation

${\begin{bmatrix}t'\\0\end{bmatrix}}={\begin{bmatrix}\gamma &\delta \\\beta &\alpha \end{bmatrix}}{\begin{bmatrix}t\\vt\end{bmatrix}},$

from which we get

$\beta =-v\alpha \,$.

Analogously, considering the motion of the origin of the frame K, we get

${\begin{bmatrix}t'\\-vt'\end{bmatrix}}={\begin{bmatrix}\gamma &\delta \\\beta &\alpha \end{bmatrix}}{\begin{bmatrix}t\\0\end{bmatrix}},$

from which we get

$\beta =-v\gamma \,$.

Combining these two gives $\alpha =\gamma$ and the transformation matrix has simplified a bit,

${\begin{bmatrix}t'\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &\delta \\-v\gamma &\gamma \end{bmatrix}}{\begin{bmatrix}t\\z\end{bmatrix}},$

Now let us consider the group postulate inverse element. There are two ways we can go from the $K'$ coordinate system to the $K$ coordinate system. The first is to apply the inverse of the transform matrix to the $K'$ coordinates:

${\begin{bmatrix}t\\z\end{bmatrix}}={\frac {1}{\gamma ^{2}+v\delta \gamma }}{\begin{bmatrix}\gamma &-\delta \\v\gamma &\gamma \end{bmatrix}}{\begin{bmatrix}t'\\z'\end{bmatrix}}.$

The second is, considering that the $K'$ coordinate system is moving at a velocity $v$ relative to the $K$ coordinate system, the $K$ coordinate system must be moving at a velocity $-v$ relative to the $K'$ coordinate system. Replacing $v$ with $-v$ in the transformation matrix gives:

${\begin{bmatrix}t\\z\end{bmatrix}}={\begin{bmatrix}\gamma (-v)&\delta (-v)\\v\gamma (-v)&\gamma (-v)\end{bmatrix}}{\begin{bmatrix}t'\\z'\end{bmatrix}},$

Now the function $\gamma$ can not depend upon the direction of $v$ because it is apparently the factor which defines the relativistic contraction and time dilation. These two (in an isotropic world of ours) cannot depend upon the direction of $v$. Thus, $\gamma (-v)=\gamma (v)$ and comparing the two matrices, we get

$\gamma ^{2}+v\delta \gamma =1.\,$

According to the closure group postulate a composition of two coordinate transformations is also a coordinate transformation, thus the product of two of our matrices should also be a matrix of the same form. Transforming $K$ to $K'$ and from $K'$ to $K''$ gives the following transformation matrix to go from $K$ to $K''$:

${\begin{aligned}{\begin{bmatrix}t''\\z''\end{bmatrix}}&={\begin{bmatrix}\gamma (v')&\delta (v')\\-v'\gamma (v')&\gamma (v')\end{bmatrix}}{\begin{bmatrix}\gamma (v)&\delta (v)\\-v\gamma (v)&\gamma (v)\end{bmatrix}}{\begin{bmatrix}t\\z\end{bmatrix}}\\&={\begin{bmatrix}\gamma (v')\gamma (v)-v\delta (v')\gamma (v)&\gamma (v')\delta (v)+\delta (v')\gamma (v)\\-(v'+v)\gamma (v')\gamma (v)&-v'\gamma (v')\delta (v)+\gamma (v')\gamma (v)\end{bmatrix}}{\begin{bmatrix}t\\z\end{bmatrix}}.\end{aligned}}$

In the original transform matrix, the main diagonal elements are both equal to $\gamma$, hence, for the combined transform matrix above to be of the same form as the original transform matrix, the main diagonal elements must also be equal. Equating these elements and rearranging gives:

$\gamma (v')\gamma (v)-v\delta (v')\gamma (v)=-v'\gamma (v')\delta (v)+\gamma (v')\gamma (v)\,$

$v\delta (v')\gamma (v)=v'\gamma (v')\delta (v)\,$

${\frac {\delta (v)}{v\gamma (v)}}={\frac {\delta (v')}{v'\gamma (v')}}.\,$

The denominator will be nonzero for nonzero v as ${\gamma (v)}$ is always nonzero, as $\gamma ^{2}+v\delta \gamma =1$. If v=0 we have the identity matrix which coincides with putting v=0 in the matrix we get at the end of this derivation for the other values of v, making the final matrix valid for all nonnegative v.

For the nonzero v, this combination of function must be a universal constant, one and the same for all inertial frames. Let's define this constant as ${\frac {\delta (v)}{v\gamma (v)}}\,=\,\kappa \,$ where $\kappa \,$ has the dimension of $1/v^{2}$. Solving

$1=\gamma ^{2}+v\delta \gamma =\gamma ^{2}(1+\kappa v^{2})\,$

we finally get $\gamma =1/{\sqrt {1+\kappa v^{2}}}$ and thus the transformation matrix, consistent with the group axioms, is given by

${\begin{bmatrix}t'\\z'\end{bmatrix}}={\frac {1}{\sqrt {1+\kappa v^{2}}}}{\begin{bmatrix}1&\kappa v\\-v&1\end{bmatrix}}{\begin{bmatrix}t\\z\end{bmatrix}}.$

If $\kappa \,$ were positive, then there would be transformations (with $\kappa v^{2}\gg 1$) which transform time into a spatial coordinate and vice versa. We exclude this on physical grounds, because time can only run in the positive direction. Thus two types of transformation matrices are consistent with group postulates: i) with the universal constant $\kappa =0$ and ii) with $\kappa <0$.

Galilean transformations

If $\kappa \,=\,0\,,$ then we get the Galilean-Newtonian kinematics with the Galilean transformation,

${\begin{bmatrix}t'\\z'\end{bmatrix}}={\begin{bmatrix}1&0\\-v&1\end{bmatrix}}{\begin{bmatrix}t\\z\end{bmatrix}}\;,$

where time is absolute, $t'=t$, and the relative velocity $v$ of two inertial frames is not limited.

Lorentz transformations

If $\kappa \,$ is negative, then we set $c\,=\,{\frac {1}{\sqrt {-\kappa }}}\,$ which becomes the invariant speed, the speed of light in vacuum. This yields $\kappa ={-1 \over c^{2}}\,$ and thus we get special relativity with Lorentz transformation

${\begin{bmatrix}t'\\z'\end{bmatrix}}={\frac {1}{\sqrt {1-{v^{2} \over c^{2}}}}}{\begin{bmatrix}1&{-v \over c^{2}}\\-v&1\end{bmatrix}}{\begin{bmatrix}t\\z\end{bmatrix}}\;,$

where the speed of light is a finite universal constant determining the highest possible relative velocity between inertial frames.

If $v\ll c$ the Galilean transformation is a good approximation to the Lorentz transformation.

Only experiment can answer the question which of the two possibilities, $\kappa =0$ or $\kappa <0$, is realised in our world. The experiments measuring the speed of light, first performed by a Danish physicist Ole Rømer, show that it is finite, and the Michelson–Morley experiment showed that it is an absolute speed, and thus that $\kappa <0$.

From physical principles

The problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and z coordinates do not intervene. It is similar to that of Einstein. More details may be found in As in the Galilean transformation, the Lorentz transformation is linear : the relative velocity of the reference frames is constant. They are called inertial or Galilean reference frames. According to relativity no Galilean reference frame is privileged. Another condition is that the speed of light must be independent of the reference frame, in practice of the velocity of the light source.

Galilean reference frames

In classical kinematics, the total displacement x in the R frame is the sum of the relative displacement x′ in frame R' and of the distance between the two origins x-x'. If v is the relative velocity of R' relative to R, we have v: x = x′ + vt or x′ = x − vt. This relationship is linear for a constant v, that is when R and R' are Galilean frames of reference.

In Einstein's relativity, the main difference with Galilean relativity is that space is a function of time and vice-versa: t ≠ t′. The most general linear relationship is obtained with four constant coefficients, α, β, γ and v:

$x'=\gamma \left(x-vt\right)$

$t'=\beta \left(t+\alpha x\right).$

The Lorentz transformation becomes the Galilean transformation when β = γ = 1 and α = 0.

Speed of light independent of the velocity of the source

Light being independent of the reference frame as was shown by Michelson, we need to have x = ct if x′ = ct′. In other words, light moves at velocity c in both frames. Replacing x and x′ in the preceding equations, one has:

$ct'=\gamma \left(c-v\right)t$

$t'=\beta \left(1+\alpha c\right)t.$

Replacing t′ with the help of the second equation, the first one writes:

$c\beta \left(1+\alpha c\right)t=\gamma \left(c-v\right)t.$

After simplification by t and dividing by cβ, one obtains:

$1+\alpha c={\frac {\gamma }{\beta }}\left(1-{\frac {v}{c}}\right).$

Principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference. One has to find the same Lorentz transformation from frame R to R' or from R' to R. As in the Galilean transformation, the sign of the transport velocity v has to be changed when passing from one frame to the other.

The following derivation uses only the principle of relativity which is independent of light velocity constancy.

The inverse transformation of

$x'=\gamma \left(x-vt\right)$

$t'=\beta \left(t+\alpha x\right)$

is given by

$x={\frac {1}{1+\alpha v}}\left({\frac {x'}{\gamma }}+{\frac {vt'}{\beta }}\right)$

$t={\frac {1}{1+\alpha v}}\left({\frac {t'}{\beta }}-{\frac {\alpha x'}{\gamma }}\right)$.

In accordance with the principle of relativity, the expressions of x and t are

$x=\gamma \left(x'+vt'\right)$

$t=\beta \left(t'-\alpha x'\right)$.

As the right hand sides have to be identical to those obtained by inverting the transformation, we have the identities, valid for any x’ and t’ :

$\gamma \left(x'+vt'\right)={\frac {1}{1+\alpha v}}\left({\frac {x'}{\gamma }}+{\frac {vt'}{\beta }}\right)$

$\beta \left(t'-\alpha x'\right)={\frac {1}{1+\alpha v}}\left({\frac {t'}{\beta }}-{\frac {\alpha x'}{\gamma }}\right).$

Substituting x'=1 and t'=0 in the first identity and x'=0 and t'=1 in the second, we immediately get the equalities

$\beta =\gamma ={\frac {1}{\sqrt {1+\alpha v}}}.$

Expression of the Lorentz transformation

Using the earlier obtained relation

$1+\alpha c={\frac {\gamma }{\beta }}(1-{\frac {v}{c}})$,

one has

$\alpha =-{\frac {v}{c^{2}}}$

and, finally

$\beta =\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$.

We now have all the coefficients needed and, therefore, the Lorentz transformation

$x={\frac {x'+vt'}{\sqrt[{}]{1-{\frac {v^{2}}{c^{2}}}}}}$

$t={\frac {t'+{\frac {vx'}{c^{2}}}}{\sqrt[{}]{1-{\frac {v^{2}}{c^{2}}}}}}$,

or, using the Lorentz factor γ,

$x=\gamma \left(x'+vt'\right)$

$t=\gamma \left(t'+{\frac {vx'}{c^{2}}}\right)$,

and its inverse:

$x'=\gamma \left(x-vt\right)$

$t'=\gamma \left(t-{\frac {vx}{c^{2}}}\right)$.

See also

• Electromagnetic field
• Galilean transformation
• Hyperbolic rotation
• Invariance mechanics
• Lorentz group
• Principle of relativity
• Velocity-addition formula
• Algebra of physical space
• Relativistic aberration

Further reading

• Einstein, Albert (1961), Relativity: The Special and the General Theory, New York: Three Rivers Press (published 1995), ISBN 0-517-88441-0
• Ernst, A.; Hsu, J.-P. (2001), "First proposal of the universal speed of light by Voigt 1887" (PDF), Chinese Journal of Physics, 39 (3): 211–230
• Langevin, P. (1911), "L'évolution de l'éspace et du temps", Scientia, X: 31–54
• Larmor, J. (1897), "Upon a dynamical theory of the electric and luminiferous medium", Philosophical Transactions of the Royal Society, 190: 205–300, doi:10.1098/rsta.1897.0020
• Larmor, J. (1900), Aether and matter, Cambridge, [England]: Cambridge University Press
• Lorentz, Hendrik Antoon (1899), "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I: 427–443
• Lorentz, Hendrik Antoon (1904), "Electromagnetic phenomena in a system moving with any velocity smaller than that of light" , Proc. Acad. Science Amsterdam, IV: 669–678
• Lorentz, Hendrik Antoon (1909), The theory of electrons and its applications to the phenomena of light and radiant heat; a course of lectures delivered in Columbia university, New York, in March and April 1906, Leipzig, [Germany] ; New York, [NY.]: B.G. Teubner ; G.E. Stechert
• Poincaré, Henri (1905), "Sur la dynamique de l'électron" (PDF), Comptes Rendues, 140: 1504–1508
• Thornton, Stephen T.; Marion, Jerry B. (2004), Classical dynamics of particles and systems (5th ed.), Belmont, [CA.]: Brooks/Cole, pp. 546–579, ISBN 0-534-40896-6
• Voigt, Woldemar (1887), "Über das Doppler'sche princip", Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen, 2: 41–51

External links

• Derivation of the Lorentz transformations. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
• The Paradox of Special Relativity. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
• Relativity - a chapter from an online textbook
• Special Relativity: The Lorentz Transformation, The Velocity Addition Law on Project PHYSNET
• Warp Special Relativity Simulator. A computer program demonstrating the Lorentz transformations on everyday objects.
• Animation clip visualizing the Lorentz transformation.
• Lorentz Frames Animated from John de Pillis. Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, etc.