Lorentz transformation

A visualisation of the Lorentz transformation. 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 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 (1905) named the Lorentz transformations after the Dutch physicist and mathematician Hendrik Lorentz (1853-1928). They form the mathematical basis for Albert Einstein's theory of special relativity. The Lorentz transformations remove contradictions between the theories of electromagnetism and classical mechanics. They were derived by Joseph Larmor (1897) and Lorentz (1899, 1904). In 1905 Einstein derived them under the assumptions of Lorentz covariance and the constancy of the speed of light in any inertial reference frame.

Lorentz transformation for frames in standard configuration

Views of spacetime along the world line of a rapidly accelerating observer moving in a 1-dimensional (straight line) "universe".

Vertical direction indicates time. 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.

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 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:

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

where $\gamma ={1 \over {\sqrt {1-v^{2}/c^{2}}}},$ 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

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

or more generally for the x, y, and z-directions:

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

where $\beta ={\frac {v}{c}}={\frac {\|{\vec {v}}\|}{c}}$ and $\gamma ={\frac {1}{\left(1-\beta ^{2}\right)^{\frac {1}{2}}}}$.

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:

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

Equivalently:

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

Then the Lorentz transformation in standard configuration is:

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

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

$y=y'\ ,$

$z=z'\ .$

Hyperbolic trigonometric expressions

It can also be shown that:

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

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

and therefore,

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

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.

General boosts

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

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

${\vec {r'}}={\vec {r}}_{\perp }+\gamma ({\vec {r}}_{\|}-{\vec {v}}t)$

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

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

These equations can be expressed in matrix form as

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

where $I$ is the identity matrix.

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 space-time translation. When $C^{a}\,=0$, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.

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

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

Lorentz transformations with $\det({\Lambda ^{\mu }}_{\nu })=+1$ are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with $\det({\Lambda ^{\mu }}_{\nu })=-1$ 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.

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 on a distance" $c\rightarrow \infty$.

History

See also History of Lorentz transformations.

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.

Actually 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 must not necessarily be 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 the vacuum. It is interesting to know that the need for locality in physical theories was already seen by Newton (see Koestler's "The Sleepwalkers"), who considered "philosophically absurd" the notion of an action at a distance and believed that gravity must be transmitted by an agent (interstellar aether) which obeys certain physical laws.

In an 1964 paper, Erik Christopher Zeeman showed that a, in a mathematical sense, weaker condition, the causality preserving property, is enough to assure that the coordinate transformations be the Lorentz-transformations.

From group postulates

Group Postulate Derivation

Following is a classical derivation based on group postulates and isotropy of the space.

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 inverse transformation. On one hand the inverse transformation is done simply by the inverse matrix,

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

On the other hand the inverse transformation is the one where $v$ is substituted by $-v$,

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

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 comapring the two matrices, we get

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

At last 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, in particular the diagonal elements should be equal. Calculating the product of two transformation matrices, one with $v$ the other with $v'$ and comparing the diagonal elements gives

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

Since this holds for any arbitrary $v$ and $v'$ this combination of function must be a universal constant, one and the same for all inertial frames. Let's define this constant as ${\frac {v\gamma (v)}{\delta (v)}}=-c^{2}$ where $c$ has a dimension of velocity (we have not yet assumed, that $c^{2}>0$). Using the equation from the inverse transformation we finally get $\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$ and the transformation matrix is given by

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

Apparently $c^{2}$ cannot be negative because otherwise there would be a transformation which transforms time into spatial coordinate and vice versa. This is no good (at least in special relativity) since time can only run in the positive direction while coordinates in both. If then $c^{2}>0$ it is apparently the highest achievable velocity. Theoretically it can be either infinitely large, which gives Galilean transformation and Euclidean world with absolute time, or it can be finite, which gives Lorentz transformation and Minkowski world of special relativity. The experiment tells us that it is finite, $c=$299792458m/s.

See also

• Electromagnetic field
• Galilean transformation
• Principle of relativity
• Hyperbolic rotation
• Invariance mechanics
• Velocity-addition formula

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.

References

• Giulini, Domenico. "Algebraic and geometric structures of Special Relativity". arXiv eprint server. Retrieved February 19. {{cite web}}: Check date values in: |accessdate= (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help)
• Ernst, A. and Hsu, J.-P. (2001) “First proposal of the universal speed of light by Voigt 1887”, Chinese Journal of Physics, 39(3), 211-230.
• Langevin, P. (1911) "L'évolution de l'espace et du temps", Scientia, X, 31-54
• Larmor, J. (1897) "Dynamical Theory of the Electric and Luminiferous Medium" Philosophical Transactions of the Royal Society, 190, 205-300.
• Larmor, J. (1900) Aether and Matter, Cambridge University Press
• Lorentz, H. A. (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
• Lorentz, H. A. (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
• Lorentz, H. A. (1913) The theory of electrons (book)
• Poincaré, H. (1905) "Sur la dynamique de l'électron", Comptes Rendues, 140, 1504-08.
• Voigt, W. (1887) "Über das Doppler'sche princip" Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen, 2, 41-51.
• Thornton, S., Marion, J., (2004) Classical Dynamics of Particles and Systems Fifth Edition, Thomson Learning, 546-579.