In physics, the Lorentz transformation, named after the Dutch physicist Hendrik Lorentz, 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 reflects the surprising fact that observers moving at different velocities report different distances, passage of time, and in some cases even different orderings of events.
The Lorentz transformation was originally the result of attempts by Lorentz and others to explain observed properties of light propagating in what was presumed to be the luminiferous aether; Albert Einstein later reinterpreted the transformation to be a statement about the nature of both space and time, and he independently rederived 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 only a good approximation at much smaller relative speeds 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.
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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 xaxis and the x' axis are collinear, the yaxis is parallel to the y' axis, as are the zaxis and the z' axis. The relative velocity between the two observers is v along the common xaxis. 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:
where is called the Lorentz factor. Where light waves move linearly through the void of space at light velocity those wavelengths cannot be lengthened by Doppler shifts because those shifts must occur in the direction of photon motion. The imperative of that fact is that the portion of the wavelength that is Doppler shifted (lengthened) exceed light velocity. However, where light waves travel linearly at light velocity their wavelengths are contracted in the direction of motion in accordance with the LorentzFitzgerald contraction. Doppler shifts of such contracted wavelengths can occur without exceeding light velocity.
This Lorentz transformation is called a "boost" in the xdirection and is often expressed in matrix form as
More generally for a boost in any arbitrary direction (β_{x},β_{y},β_{z}),
where and .
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. The boost is given by a symmetric matrix, but the general Lorentz transformation matrix need not be symmetric.
The Lorentz transformation can be cast into another useful form by defining a parameter called the rapidity (an instance of hyperbolic angle) such that
so that
Equivalently:
Then the Lorentz transformation in standard configuration is:
From the above expressions for e^{φ} and e^{−φ}
and therefore,
Substituting these expressions into the matrix form of the transformation, we have:
Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the rapidity φ represents the hyperbolic angle of rotation.
For a boost in an arbitrary direction with velocity , it is convenient to decompose the spatial vector into components perpendicular and parallel to the velocity : . Then only the component in the direction of is 'warped' by the gamma factor:
where now . The second of these can be written as:
These equations can be expressed in matrix form as
where I is the identity matrix, v is velocity written as a column vector and v^{T} is its transpose (a row vector).
In a given coordinate system (x^{μ}), if two events A and B are separated by
the spacetime interval between them is given by
This can be written in another form using the Minkowski metric. In this coordinate system,
Then, we can write
or, using the Einstein summation convention,
Now suppose that we make a coordinate transformation . Then, the interval in this coordinate system is given by
or
It is a result of special relativity that the interval is an invariant. That is, . It can be shown^{[1]} that this requires the coordinate transformation to be of the form
Here, is a constant vector and a constant matrix, where we require that
Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation.^{[2]} The C^{a} represents a spacetime translation. When , the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.
Taking the determinant of gives us
Lorentz transformations with are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with 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.
One of the most astounding consequences of Einstein's clocksetting 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 clocksetting 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".
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 , so it is usually said that non relativistic physics is a physics of "instant action at a distance" .
Many physicists, including George FitzGerald, Joseph Larmor, Hendrik Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887. ^{[3]} ^{[4]} 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 etherwind experiment of Michelson and Morley. This idea was extended by Lorentz ^{[5]} and Larmor ^{[6]} over several years, and became known as the FitzGeraldLorentz explanation of the MichelsonMorley null result, known early on through the writings of Lodge, Lorentz, Larmor, and FitzGerald. ^{[7]} Their explanation was widely accepted as correct before 1905. ^{[8]} Larmor is also credited to have been the first to understanding the crucial time dilation property inherent in his equations. ^{[9]}
In 1905 Henri Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz. ^{[10]} 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, ^{[11]} 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: ^{[12]}
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 halfsilvered 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 MichaelsonMorely 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,^{[13]} 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.
Following is a classical derivation (see, e.g., [1] and references therein) based on group postulates and isotropy of the space.
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 apparently satisfied:
Let us consider two inertial frames, K and K', the latter moving with velocity 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
where α,β,γ, and δ 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
from which we get
Analogously, considering the motion of the origin of the frame K, we get
from which we get
Combining these two gives α = γ and the transformation matrix has simplified a bit,
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:
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:
Now the function γ 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, γ( − v) = γ(v) and comparing the two matrices, we get
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'':
In the original transform matrix, the main diagonal elements are both equal to γ, 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:
The denominator will be nonzero for nonzero v as γ(v) is always nonzero, as γ^{2} + vδγ = 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 where has the dimension of 1 / v^{2}. Solving
we finally get and thus the transformation matrix, consistent with the group axioms, is given by
If were positive, then there would be transformations (with ) 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 κ = 0 and ii) with κ < 0.
If then we get the GalileanNewtonian kinematics with the Galilean transformation,
where time is absolute, t' = t, and the relative velocity v of two inertial frames is not limited.
If is negative, then we set which becomes the invariant speed, the speed of light in vacuum. This yields and thus we get special relativity with Lorentz transformation
where the speed of light is a finite universal constant determining the highest possible relative velocity between inertial frames.
If the Galilean transformation is a good approximation to the Lorentz transformation.
Only experiment can answer the question which of the two possibilities, κ = 0 or κ < 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 κ < 0.
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.^{[14]} More details may be found in^{[15]} 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.
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 displacement x in frame R. 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 viceversa: t ≠ t′. The most general linear relationship is obtained with four constant coefficients, α, β, γ and v:
The Lorentz transformation becomes the Galilean transformation when β = γ = 1 and α = 0.
Light being independent of the reference frame as was shown by Michelson, we need to have x = ct if x′ = ct′. Replacing x and x′ in the preceding equations, one has:
Replacing t′ with the help of the second equation, the first one writes:
After simplification by t and dividing by cβ, one obtains:
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
is given by
In accordance with the principle of relativity, the expressions of x and t are
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’ :
Substituting x'=1 and t'=0 in the first identity and x'=0 and t'=1 in the second, we immediately get the equalities
Using the earlier obtained relation
one has
and, finally
We now have all the coefficients needed and, therefore, the Lorentz transformation
or, using the Lorentz factor γ,
and its inverse:
In physics, the Lorentz transformation, named after the Dutch physicist Hendrik Lorentz, 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 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 rederived 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 only a good approximation 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.
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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 $(t\text{'},\; x\text{'},\; y\text{'},\; z\text{'})$. Assume further that the coordinate systems are oriented so that the xaxis and the x' axis are collinear, the yaxis is parallel to the y' axis, as are the zaxis and the z' axis. The relative velocity between the two observers is v along the common xaxis. 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:
t' &= \gamma \left( t  v x/c^{2} \right) \\ x' &= \gamma \left( x  v t \right)\\ y' &= y \\ z' &= z \end{cases} where $\backslash \; \backslash gamma\; =\; \backslash frac\{1\}\{\; \backslash sqrt\{1\; \; \{\; \backslash frac\{v^2\}\{c^2\}\}\}\}$ is called the Lorentz factor.
This Lorentz transformation is called a "boost" in the xdirection 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 any arbitrary direction $(\backslash beta\_\{x\},\; \backslash beta\_\{y\},\; \backslash beta\_\{z\})$,
\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+(\gamma1)\dfrac{\beta_{x}^{2}}{\beta^{2}}&(\gamma1)\dfrac{\beta_{x}\beta_{y}}{\beta^{2}}&(\gamma1)\dfrac{\beta_{x}\beta_{z}}{\beta^{2}}\\ \beta_y\,\gamma&(\gamma1)\dfrac{\beta_{y}\beta_{x}}{\beta^{2}}&1+(\gamma1)\dfrac{\beta_{y}^{2}}{\beta^{2}}&(\gamma1)\dfrac{\beta_{y}\beta_{z}}{\beta^{2}}\\ \beta_z\,\gamma&(\gamma1)\dfrac{\beta_{z}\beta_{x}}{\beta^{2}}&(\gamma1)\dfrac{\beta_{z}\beta_{y}}{\beta^{2}}&1+(\gamma1)\dfrac{\beta_{z}^{2}}{\beta^{2}}\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\ , where $\backslash beta\; =\; \backslash frac\{v\}\{c\}=\backslash frac\{\backslash vec\{v\}\}\{c\}$ and $\backslash gamma\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\backslash beta^2\}\}$.
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.
of a rapidly accelerating observer moving in a 1dimensional (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 $\backslash vec\{v\}$, it is convenient to decompose the spatial vector $\backslash vec\{r\}$ into components perpendicular and parallel to the velocity $\backslash vec\{v\}$: $\backslash vec\{r\}=\backslash vec\{r\}\_\backslash perp+\backslash vec\{r\}\_\backslash $. Then only the component $\backslash vec\{r\}\_\backslash $ in the direction of $\backslash vec\{v\}$ 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) \end{cases} where now $\backslash gamma\; \backslash equiv\; \backslash frac\{1\}\{\backslash sqrt\{1\; \; \backslash vec\{v\}\; \backslash cdot\; \backslash vec\{v\}/c^2\}\}$. The second of these can be written as:
These equations can be expressed in matrix form as
\begin{bmatrix} c t' \\ \mathbf{r'} \end{bmatrix} = \begin{bmatrix} \gamma & \gamma \dfrac{\mathbf{v}^\mathrm{T}}{c} \\ \displaystyle\frac{\gamma\mathbf{v}}{c} & I+ (\gamma1) \displaystyle\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 v^{T} is its transpose (a row vector).
The Lorentz transformation can be cast into another useful form by defining a parameter $\backslash scriptstyle\backslash boldsymbol\{\backslash phi\}$ called the rapidity (an instance of hyperbolic angle) such that
so that
Equivalently:
Then the Lorentz transformation in standard configuration is:
c tx = e^{ \phi}(c t'  x') \\ c t+x = e^{\phi}(c t' + x') \\ y = y' \\ z = z'. \end{cases}
From the above expressions for e^{φ} and e^{−φ}
and therefore,
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 $\backslash phi$ represents the hyperbolic angle of rotation.
In a given coordinate system ($x^\backslash mu$), if two events $A$ and $B$ are separated by
the spacetime interval between them is given by
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,
Now suppose that we make a coordinate transformation $x^\backslash mu\; \backslash rightarrow\; x\text{'}^\backslash mu$. 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
It is a result of special relativity that the interval is an invariant. That is, $s^2\; =\; s\text{'}^2\backslash $. It can be shown^{[1]} that this requires the coordinate transformation to be of the form
Here, $C^\backslash mu\backslash $ is a constant vector and $\{\backslash Lambda^\backslash mu\}\_\backslash nu$ a constant matrix, where we require that
Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation.^{[2]} The $C^a$ represents a spacetime translation. When $C^a\; \backslash ,\; =\; 0$, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.
Taking the determinant of $\backslash eta\_\{\backslash mu\backslash nu\}\{\backslash Lambda^\backslash mu\}\_\backslash alpha\{\backslash Lambda^\backslash nu\}\_\backslash beta\; =\; \backslash eta\_\{\backslash alpha\backslash beta\}$ gives us
Lorentz transformations with $\backslash det\; (\{\backslash Lambda^\backslash mu\}\_\backslash nu)=+1$ are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with $\backslash det(\{\backslash Lambda^\backslash mu\}\_\backslash nu)=1$ 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.
One of the most astounding consequences of Einstein's clocksetting 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 clocksetting 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".
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 $v\; \backslash rightarrow\; 0$, so it is usually said that nonrelativistic physics is a physics of "instant action at a distance" $c\; \backslash rightarrow\; \backslash infty$.
Many physicists, including George FitzGerald, Joseph Larmor, Hendrik Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887.^{[3]}^{[4]} 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 etherwind experiment of Michelson and Morley. This idea was extended by Lorentz^{[5]} and Larmor^{[6]} over several years, and became known as the FitzGeraldLorentz explanation of the MichelsonMorley null result, known early on through the writings of Lodge, Lorentz, Larmor, and FitzGerald.^{[7]} Their explanation was widely accepted as correct before 1905.^{[8]} Larmor is also credited to have been the first to understanding the crucial time dilation property inherent in his equations.^{[9]}
In 1905, Henri Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz.^{[10]} 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,^{[11]} 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:^{[12]}
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 halfsilvered 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 MichelsonMorley 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,^{[13]} 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.
Following is a classical derivation (see, e.g., [1] and references therein) based on group postulates and isotropy of the space.
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:
Let us consider two inertial frames, K and K', the latter moving with velocity $\backslash vec\{v\}$ 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 $\backslash alpha,\; \backslash beta,\; \backslash gamma,$ and $\backslash delta$ 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
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
Combining these two gives $\backslash alpha=\backslash gamma$ 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 $K\text{'}$ coordinate system to the $K$ coordinate system. The first is to apply the inverse of the transform matrix to the $K\text{'}$ 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 $K\text{'}$ 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 $K\text{'}$ 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 $\backslash gamma$ 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, $\backslash gamma(v)=\backslash gamma(v)$ 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 $K\text{'}$ and from $K\text{'}$ to $K$ gives the following transformation matrix to go from $K$ to $K$:
\begin{align} \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{align}
In the original transform matrix, the main diagonal elements are both equal to $\backslash gamma$, 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 $\{\backslash gamma(v)\}$ is always nonzero, as $\backslash gamma^2\; +\; v\; \backslash delta\; \backslash gamma\; =\; 1$. 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 $\backslash frac\{\backslash delta(v)\}\{v\backslash gamma(v)\}\; \backslash ,\; =\; \backslash ,\; \backslash kappa\backslash ,$ where $\backslash kappa\backslash ,$ has the dimension of $1/v^2$. Solving
1 = \gamma^2 + v\delta\gamma = \gamma^2 (1 + \kappa v^2) \, we finally get $\backslash gamma=1/\backslash sqrt\{1\; +\; \backslash kappa\; v^2\}$ 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 $\backslash kappa\backslash ,$ were positive, then there would be transformations (with $\backslash kappa\; v^2\; \backslash gg\; 1$) 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 $\backslash kappa=0$ and ii) with $\backslash kappa<0$.
If $\backslash kappa\; \backslash ,\; =\; \backslash ,\; 0\; \backslash ,,$ then we get the GalileanNewtonian 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, $t\text{'}=t$, and the relative velocity $v$ of two inertial frames is not limited.
If $\backslash kappa\backslash ,$ is negative, then we set $c\; \backslash ,\; =\; \backslash ,\; \backslash frac\{1\}\{\backslash sqrt\{\; \backslash kappa\}\}\; \backslash ,$ which becomes the invariant speed, the speed of light in vacuum. This yields $\backslash kappa\; =\; \{1\; \backslash over\; c^2\}\; \backslash ,$ 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 $v\backslash ll\; c$ the Galilean transformation is a good approximation to the Lorentz transformation.
Only experiment can answer the question which of the two possibilities, $\backslash kappa=0$ or $\backslash kappa\; <\; 0$, 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 $\backslash kappa\; <\; 0$.
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.^{[14]}^{[not specific enough to verify]} More details may be found in^{[15]}^{[not specific enough to verify]} 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.
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 xx'. 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 viceversa: t ≠ t′. The most general linear relationship is obtained with four constant coefficients, α, β, γ and v:
The Lorentz transformation becomes the Galilean transformation when β = γ = 1 and α = 0.
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:
Replacing t′ with the help of the second equation, the first one writes:
After simplification by t and dividing by cβ, one obtains:
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
is given by
In accordance with the principle of relativity, the expressions of x and t are
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’ :
Substituting x'=1 and t'=0 in the first identity and x'=0 and t'=1 in the second, we immediately get the equalities
Using the earlier obtained relation
one has
and, finally
We now have all the coefficients needed and, therefore, the Lorentz transformation
or, using the Lorentz factor γ,
and its inverse:
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