Index

Lorentz transformation 1

Lorentz transformation

Lorentz transformations on the Minkowski light cone spacetime diagram, for one spatialdimension. The greater the relative speed between the inertial frames, the more "warped"

the axes become. The red diagonal lines are world lines for light - the relative velocitycannot exceed c. The hyperbolae indicate this is a hyperbolic rotation, the hyperbolic

angle ϕ is called rapidity - see below.

In physics, the Lorentztransformation orLorentz-Fitzgerald transformationdescribes how, according to the theoryof special relativity, differentmeasurements of space and time bytwo observers can be converted intothe measurements observed in eitherframe of reference.

It is named after the Dutch physicistHendrik Lorentz. It reflects the factthat observers moving at differentvelocities may measure differentdistances, elapsed times, and evendifferent orderings of events.

The Lorentz transformation wasoriginally the result of attempts byLorentz and others to explain how thespeed of light was observed to beindependent of the reference frame,and to understand the symmetries ofthe laws of electromagnetism. AlbertEinstein later re-derived thetransformation from his postulates ofspecial relativity. The Lorentz transformation supersedes the Galilean transformation of Newtonian physics, whichassumes an absolute space and time (see Galilean relativity). According to special relativity, the Galileantransformation 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. It may include a rotationof space; a rotation-free Lorentz transformation is called a Lorentz boost. Since relativity postulates that the speedof light is the same for all observers, the Lorentz transformation must preserve the spacetime interval between anytwo events in Minkowski space. The Lorentz transformation describes only the transformations in which thespacetime event at the origin is left fixed, so they can be considered as a hyperbolic rotation of Minkowski space.The more general set of transformations that also includes translations is known as the Poincaré group.

HistorySee also History of Lorentz transformations.

Many physicists, including Woldemar Voigt, George FitzGerald, Joseph Larmor, Hendrik Lorentz had been discussing the physics behind these equations since 1887.[1][2] 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

Lorentz transformation 2

applied to a theory of intermolecular forces. Some months later, FitzGerald published his conjecture in Science toexplain the baffling outcome of the 1887 ether-wind experiment of Michelson and Morley. This idea was extendedby Lorentz[3] and Larmor[4] over several years, and became known as the FitzGerald-Lorentz explanation of theMichelson-Morley null result, known early on through the writings of Lodge, Lorentz, Larmor, and FitzGerald.[5]

Their explanation was widely known before 1905.[6] Larmor is also credited to have been the first to understand thecrucial time dilation property inherent in his equations.[7]

In 1905, Henri Poincaré was the first to recognize that the transformation has the properties of a mathematical group,and named it after Lorentz.[8] Later in the same year Einstein derived the Lorentz transformation under theassumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame,[9]

obtaining results that were algebraically equivalent to Larmor's (1897) and Lorentz's (1899, 1904), but with adifferent interpretation.Paul Langevin (1911) said of the transformation:[10]

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

Lorentz transformation for frames in standard configuration

An observer O, situated at the origin of a local set of coordinates - aframe of reference F. The observer in this frame uses the coordinates

(x, y, z, t) to describe a spacetime event, shown as a star.

Consider two observers O and O' , each using their ownCartesian coordinate system to measure space and timeintervals. O uses (t, x, y, z) and O ' uses (t' , x' , y' , z' ).Assume further that the coordinate systems are orientedso that, in 3 dimensions, the x-axis and the x' -axis arecollinear, the y-axis is parallel to the y' -axis, and thez-axis parallel to the z' -axis. The relative velocitybetween the two observers is v along the commonx-axis. Also assume that the origins of both coordinatesystems are the same, that is, coincident times andpositions.

If all these hold, then the coordinate systems are said tobe in standard configuration. A symmetricpresentation between the forward LorentzTransformation and the inverse Lorentz Transformationcan be achieved if coordinate systems are in symmetricconfiguration. The symmetric form highlights that allphysical laws should remain unchanged under aLorentz transformation.

Lorentz transformation 3

The space-time coordinates of an event, as measured by each observer in their inertialreference frame (in standard configuration) are shown in the speech bubbles.Top: frameF' moves at velocity v along the x-axis of frame F.Bottom: frame F moves at velocity −valong the x'-axis of frame F'.University Physics – With Modern Physics (12th Edition),

H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson International),1st Edition: 1949, 12th Edition: 2008, ISBN (10-) 0-321-50130-6, ISBN (13-)

978-0-321-50130-1

Below the Lorentz transformations arecalled "boosts" in the stated directions.

Boost in the x-direction

These are the simplest forms. TheLorentz transformation for frames instandard configuration can be shown tobe (see for example [12] and [13]):

where:• v is the relative velocity between

frames in the x-direction,• c is the speed of light,

• is the Lorentz

factor (Greek lowercase gamma),• (Greek lowercase beta),

again for the x-direction.The use of β and γ is standardthroughout the literature.[14] For the remainder of the article - they will be also used throughout unless otherwisestated. Since the above is a linear system of equations (more technically a linear transformation), they can be writtenin matrix form:

Boost in the y or z directionsThe above collection of equations apply only for a boost in the x-direction. If the standard configuration used the y orz directions instead of x, the results would be similar.For the y-direction:

summarized by

Lorentz transformation 4

where v and so β are now in the y-direction. For the z-direction:

summarized by

where v and so β are now in the z-direction. These are easily obtained by Cyclic permutations of x, y, z. If wecouldn't do this - it would imply the laws of physics would be different in each direction. This is not the case, byexperimentation and observation.The Lorentz or boost matrix is usually denoted by Λ (Greek capital lambda). Above the transformations have beenapplied to the four-position R,

The Lorentz transform for a boost in one of the above directions can be compactly written as a single matrixequation:

However, the transformation matrix is universal for all four-vectors.[15] If A is any four-vector, then:

Boost in any one direction by index permutationThe previous sets of equations can be summarized using index notation, rather than Cartesian coordinates [15]:

where:• x0 is the time coordinate,• xi, xj, xk are spatial coordinates,• βi and vi are in the direction of relative motion,• The indices i, j, k each correspond a direction mutually perpendicular to the others, so xi is mutually perpendicular

to xj and xk, xj mutually perpendicular to xi and xk etc., for all cyclic permutations of i, j, k.

Lorentz transformation 5

Boost in any directionMore generally for a boost in any arbitrary direction at velocity v = (vx, vy, vz), or equivalently β = (βx, βy, βz),

where:

• (cartesian notation) equivalently written (component notation),• equivalently written • equivalently written • where

• applies for the resultant velocity v, not only one component.

Again the transformation can be written in the same form as before,

Although the matrix Λ is symmetric, it appears daunting and unwieldy. To make it easier to remember and use, wecould simply write the matrix in terms of components.[15]

The above transformation has the structure:

where the components are:

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

Composition of two boostsThe composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by:[16][17]

,where is the velocity-addition, and Gyr[u,v] (capital G) is the rotation arising from the composition, gyr(lower case g) being the gyrovector space abstraction of the gyroscopic Thomas precession, and B(v) is the 4×4matrix 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 therepresentation 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:[18]

Lorentz transformation 6

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

.[16]

Views of spacetime along the world line of arapidly accelerating observer (center) moving in a

1-dimensional (straight line) "universe". Thevertical direction indicates time, while the

horizontal indicates distance, the dashed line isthe spacetime trajectory ("world line") of the

observer. The small dots are specific events inspacetime. If one imagines these events to be theflashing 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 slopeof the world line (deviation from being vertical)gives the relative velocity to the observer. Notehow the view of spacetime changes when the

observer accelerates.

For a boost in an arbitrary direction with velocity , it is convenientto decompose the spatial vector into components perpendicular andparallel 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, vT is its transpose (a row vector) and is itsversor.

Lorentz transformation 7

RapidityThe Lorentz transformation can be cast into another useful form by defining a parameter called the rapidity (aninstance of hyperbolic angle) such that

so that

Equivalently:

Then the Lorentz transformation in standard configuration is:

Hyperbolic trigonometric expressionsFrom the above expressions for eφ and e−φ

and therefore,

Hyperbolic rotation of coordinatesSubstituting 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 theparameter ϕ represents the hyperbolic angle of rotation, often referred to as rapidity. This transformation issometimes illustrated with a Minkowski diagram, as shown at the beginning of the article.

Lorentz transformation 8

Lorentz transformation of the electromagnetic fieldThe fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thoughtexperiment:•• Consider an observer measuring a charge at rest in a reference frame F. The observer will detect a static electric

field. As the charge is stationary in this frame, there is no electric current, so the observer will not observe anymagnetic field.

• Consider another observer in frame F' moving at relative velocity v (relative to F and the charge). This observerwill see a different electric field because the charge is moving at velocity −v in their rest frame. Further, in frameF' the moving charge constitutes an electric current, and thus the observer in frame F' will also see a magneticfield.

This shows that the Lorentz transformation also applies to electromagnetic field quantities when changing the frameof reference.For the electric and magnetic field quantities, the following transformations apply:[19]

In non-relativistic approximation, i. e. for speeds , the relativistic factor , so that there is no need todistinguish between the spatial and temporal coordinates in Maxwell's equations. This yields the followingtransformations:

Spacetime intervalIn a given coordinate system ( ), if two events and 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,

Lorentz transformation 9

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 isgiven by

or

It is a result of special relativity that the interval is an invariant. That is, . For this to hold, it can beshown[20] that it is necessary (but not sufficient) for 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.[21] The represents a spacetime translation. When , the transformation is called an homogeneous Lorentztransformation, or simply a Lorentz transformation.

Taking the determinant of gives us

Lorentz transformations with form a subgroup called proper Lorentz transformations which isthe special orthogonal group . Those with are called improper Lorentz

transformations which is not a subgroup, as the product of any two improper Lorentz transformations will be aproper Lorentz transformation. From the above definition of it can be shown that , so either

or , called orthochronous and non-orthochronous respectively. An important subgroup ofthe proper Lorentz transformations are the proper orthochronous Lorentz transformations which consist purelyof boosts and rotations. Any Lorentz transform can be written as a proper orthochronous, together with one or bothof the two discrete transformations; space inversion ( ) and time reversal ( ), whose non-zero elements are:

The set of Poincaré transformations satisfies the properties of a group and is called the Poincaré group. Under theErlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combinesLorentz 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.

Lorentz transformation 10

Special relativityOne of the most astounding consequences of Einstein's clock-setting method is the idea that time is relative. Inessence, each observer's frame of reference is associated with a unique set of clocks, the result being that time passesat different rates for different observers.[22] This was a direct result of the Lorentz transformations and is called timedilation. We can also clearly see from the Lorentz "local time" transformation that the concept of the relativity ofsimultaneity 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 thesame force — the electromagnetic force. If we have one charge or a collection of charges which are all stationarywith respect to each other, we can observe the system in a frame in which there is no motion of the charges. In thisframe, 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 principleFor relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galileantransformation in accordance with the correspondence principle.The correspondence limit is usually stated mathematically as: as , . In words: as velocityapproaches 0, the speed of light (seems to) approach infinity. Hence, it is sometimes said that nonrelativistic physicsis a physics of "instantaneous action at a distance".[22]

DerivationThe usual treatment (e.g., Einstein's original work) is based on the invariance of the speed of light. However, this isnot necessarily the starting point: indeed (as is exposed, for example, in the second volume of the Course ofTheoretical Physics by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes thatthe influence that one particle, say, exerts on another can not be transmitted instantaneously. Hence, there exists atheoretical maximal speed of information transmission which must be invariant, and it turns out that this speedcoincides 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" andbelieved that gravity must be transmitted by an agent (such as an interstellar aether) which obeys certain physicallaws.Michelson and Morley in 1887 designed an experiment, employing an interferometer and a half-silvered mirror, thatwas accurate enough to detect aether flow. The mirror system reflected the light back into the interferometer. If therewere 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 conceptof aether (or its drift) undermined. There was consequent perplexity as to why light evidently behaves like a wave,without any detectable medium through which wave activity might propagate.In a 1964 paper,[23] Erik Christopher Zeeman showed that the causality preserving property, a condition that isweaker in a mathematical sense than the invariance of the speed of light, is enough to assure that the coordinatetransformations are the Lorentz transformations.

Lorentz transformation 11

From group postulatesFollowing is a classical derivation (see, e.g., [24] and references therein) based on group postulates and isotropy ofthe space.

Coordinate transformations as a group

The coordinate transformations between inertial frames form a group (called the proper Lorentz group) with thegroup operation being the composition of transformations (performing one transformation after another). Indeed thefour group axioms are satisfied:1. Closure: the composition of two transformations is a transformation: consider a composition of transformations

from the inertial frame to inertial frame , (denoted as ), and then from to inertial frame, , there exists a transformation, , directly from an inertial frame to inertial

frame .2. Associativity: the result of and

is the same, .3. Identity element: there is an identity element, a transformation .4. Inverse element: for any transformation there exists an inverse transformation .

Transformation matrices consistent with group axioms

Let us consider two inertial frames, K and K', the latter moving with velocity with respect to the former. Byrotations 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 theperpendicular coordinates and we can just omit them for brevity. Now since the transformation we are looking afterconnects 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 and are some yet unknown functions of the relative velocity .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 theK 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 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 coordinatesystem to the coordinate system. The first is to apply the inverse of the transform matrix to the coordinates:

Lorentz transformation 12

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

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

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

In the original transform matrix, the main diagonal elements are both equal to , hence, for the combined transformmatrix 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 is always nonzero, as . If v=0 we have theidentity matrix which coincides with putting v=0 in the matrix we get at the end of this derivation for the othervalues 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 . 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 >> 1) which transform time into a spatialcoordinate 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 =0and ii) with <0.

Lorentz transformation 13

Galilean transformations

If then we get the Galilean-Newtonian kinematics with the Galilean transformation,

where time is absolute, , and the relative velocity of two inertial frames is not limited.

Lorentz transformations

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 betweeninertial frames.If the Galilean transformation is a good approximation to the Lorentz transformation.Only experiment can answer the question which of the two possibilities, =0 or <0, is realised in our world. Theexperiments measuring the speed of light, first performed by a Danish physicist Ole Rømer, show that it is finite, andthe Michelson–Morley experiment showed that it is an absolute speed, and thus that <0.

From physical principlesThe problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and zcoordinates do not intervene. It is similar to that of Einstein.[22][25] As in the Galilean transformation, the Lorentztransformation is linear since the relative velocity of the reference frames is constant as a vector; otherwise, inertialforces would appear. They are called inertial or Galilean reference frames. According to relativity no Galileanreference frame is privileged. Another condition is that the speed of light must be independent of the referenceframe, 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, the transformation is:x = x′ + vt, or x′ = x − vt. This relationship is linear for a constant v, that is when R and R' are Galilean frames ofreference.In Einstein's relativity, the main difference with Galilean relativity is that space is a function of time and vice-versa:t ≠ t′. Since space is assumed to be homogeneous, the transformation must be linear. The most general linearrelationship is obtained with four constant coefficients, A, B, γ, and b:

The Lorentz transformation becomes the Galilean transformation when γ = B = 1 , b = -v and A = 0.An object at rest in the R' frame at position x′=0 moves with constant velocity v in the R frame. Hence thetransformation must yield x′=0 if x=v t. Therefore, b=-γ v and the first equation is written as:

Lorentz transformation 14

Principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference. Therefore, the inversetransformation for the position from frame R′ to frame R must be

with the same value of γ (which must therefore be an even function of v).

The speed of light is constant

Since the speed of light is the same in all frames of reference, for the case of a light signal, the transformation mustguarantee that t =  x/c and t' = x'/c.Substituting for t and t ′ in the preceding equations gives:

Multiplying these two equations together gives,

At any time after t = t' = 0, xx' is not zero, so dividing both sides of the equation by xx' results in

which is called the "Lorentz factor".

Transformation of time

The transformation equation for time can be easily obtained by considering the special case of a light signal,satisfying

Substituting term by term into the earlier obtained equation for the spatial coordinate

gives

so that

which determines the transformation coefficients A and B as

So A and B are the unique coefficients necessary to preserve the constancy of the speed of light in the primed systemof coordinates.

Lorentz transformation 15

Einstein's popular derivation

In his popular book[22] Einstein derived the Lorentz transformation by arguing that there must be two non-zerocoupling constants and such that

that correspond to light traveling along the positive and negative x-axis, respectively. For light x = ct if and only if x'= ct'. Adding and subtracting the two equations and defining

gives

Substituting x' = 0 corresponding to x = vt and noting that the relative velocity is v = bc/γ, this gives

The constant can be evaluated as was previously shown above.

References[1] O'Connor, John J.; Robertson, Edmund F., A History of Special Relativity (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ HistTopics/

Special_relativity. html),[2] Sinha, Supurna (2000), "Poincaré and the Special Theory of Relativity" (http:/ / www. ias. ac. in/ resonance/ Feb2000/ pdf/ Feb2000p12-15.

pdf), Resonance 5 (2): 12–15, doi:10.1007/BF02838818,[3] See History of Special Relativity. The work is contained within Lorentz, Hendrik Antoon (1895), Attempt of a Theory of Electrical and

Optical Phenomena in Moving Bodies, Leiden, [The Netherlands]: E.J. Brill; Lorentz, Hendrik Antoon (1899), "Simplified Theory ofElectrical and Optical Phenomena in Moving Systems", Proc. Acad. Science Amsterdam I: 427–443; and 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

[4] Larmor, J. (1897), "On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with material media", PhilosophicalTransactions of the Royal Society 190: 205–300, Bibcode 1897RSPTA.190..205L, doi:10.1098/rsta.1897.0020

[5] Brown, Harvey R., Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited (http:/ / philsci-archive. pitt. edu/ id/ eprint/ 987),[6] Rothman, Tony (2006), "Lost in Einstein's Shadow" (http:/ / www. americanscientist. org/ libraries/ documents/ 200622102452_866. pdf),

American Scientist 94 (2): 112f.,[7] Macrossan, Michael N. (1986), "A Note on Relativity Before Einstein" (http:/ / espace. library. uq. edu. au/ view. php?pid=UQ:9560), Brit.

Journal Philos. Science 37: 232–34,[8] The reference is within the following paper: Poincaré, Henri (1905), "On the Dynamics of the Electron", Comptes rendus hebdomadaires des

séances de l'Académie des sciences 140: 1504–1508[9] Einstein, Albert (1905-06-30), "Zur Elektrodynamik bewegter Körper" (http:/ / www. pro-physik. de/ Phy/ pdfs/ ger_890_921. pdf), Annalen

der Physik 17 (10): 891–921, Bibcode 1905AnP...322..891E, doi:10.1002/andp.19053221004, , retrieved 2009-02-02.[10] The citation is within the following paper: Langevin, P. (1911), "L'évolution de l'éspace et du temps", Scientia X: 31–54[11] University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson

International), 1st Edition: 1949, 12th Edition: 2008, ISBN (10-) 0-321-50130-6, ISBN (13-) 978-0-321-50130-1[12] Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons Ltd, ISBN 978-0-470-01460-8[13] http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ hframe. html. Hyperphysics, web-based physics matrial hosted by Georgia State University,

USA.[14][14] Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0[15] Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0[16] Ungar, A. A: The relativistic velocity composition paradox and the Thomas rotation. (http:/ / www. springerlink. com/ content/

g157304vh4434413/ ) Found. Phys. 19, 1385–1396 (1989)[17] The relativistic composite-velocity reciprocity principle (http:/ / citeseerx. ist. psu. edu/ viewdoc/ download?doi=10. 1. 1. 35. 1131&

rep=rep1& type=pdf), AA Ungar - Foundations of Physics, 2000 - Springer

Lorentz transformation 16

[18][18] eq. (55), Thomas rotation and the parameterization of the Lorentz transformation group, AA Ungar - Foundations of Physics Letters, 1988[19] Daniel, Herbert (1997), "4.5.1" (http:/ / books. google. com/ books?id=8vAC8YG41goC), Physik: Elektrodynamik, relativistische Physik,

Walter de Gruyter, pp. 360–361, ISBN 3-11-015777-2, , Extract of pages 360-361 (http:/ / books. google. com/ books?id=8vAC8YG41goC&pg=PA360)

[20] Weinberg, Steven (1972), Gravitation and Cosmology, New York, [NY.]: Wiley, ISBN 0-471-92567-5: (Section 2:1)[21] Weinberg, Steven (1995), The quantum theory of fields (3 vol.), Cambridge, [England] ; New York, [NY.]: Cambridge University Press,

ISBN 0-521-55001-7 : volume 1.[22] Einstein, Albert (1916). "Relativity: The Special and General Theory" (http:/ / www. archive. org/ stream/ cu31924011804774#page/ n35/

mode/ 2up) (PDF). . Retrieved 2012-01-23.[23] Zeeman, Erik Christopher (1964), "Causality implies the Lorentz group", Journal of Mathematical Physics 5 (4): 490–493,

Bibcode 1964JMP.....5..490Z, doi:10.1063/1.1704140[24] http:/ / arxiv. org/ abs/ gr-qc/ 0107091[25] Stauffer, Dietrich; Stanley, Harry Eugene (1995). From Newton to Mandelbrot: A Primer in Theoretical Physics (http:/ / books. google.

com/ books?id=o8rvAAAAMAAJ) (2nd enlarged ed.). Springer-Verlag. p. 80,81. ISBN 978-3-540-59191-7. .

Further reading• Einstein, Albert (1961), Relativity: The Special and the General Theory (http:/ / www. marxists. org/ reference/

archive/ einstein/ works/ 1910s/ relative/ ), 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" (http:/ / psroc. phys.

ntu. edu. tw/ cjp/ v39/ 211. pdf), Chinese Journal of Physics 39 (3): 211–230, Bibcode 2001ChJPh..39..211E• 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 (http:/ / www2. physics. umd. edu/ ~yakovenk/ teaching/ Lorentz. pdf).

This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on groupproperties.

• The Paradox of Special Relativity (http:/ / casa. colorado. edu/ ~ajsh/ sr/ paradox. html). This webpage poses aproblem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.

• Relativity (http:/ / www. lightandmatter. com/ html_books/ 0sn/ ch07/ ch07. html) - a chapter from an onlinetextbook

• Special Relativity: The Lorentz Transformation, The Velocity Addition Law (http:/ / physnet. org/ home/ modules/pdf_modules/ m12. pdf) on Project PHYSNET (http:/ / www. physnet. org)

• Warp Special Relativity Simulator (http:/ / www. adamauton. com/ warp/ ). A computer program demonstratingthe Lorentz transformations on everyday objects.

• Animation clip (http:/ / www. youtube. com/ watch?v=C2VMO7pcWhg) visualizing the Lorentz transformation.• Lorentz Frames Animated (http:/ / math. ucr. edu/ ~jdp/ Relativity/ Lorentz_Frames. html) from John de Pillis.

Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, etc.

Article Sources and Contributors 17

Article Sources and ContributorsLorentz transformation  Source: http://en.wikipedia.org/w/index.php?oldid=494416300  Contributors: 84user, AManWithNoPlan, AN(Ger), Aaagmnr, Adoniscik, Ahoerstemeier, Ali Obeid,Altenmann, Ap, Archelon, Arjayay, Army1987, Artichoker, AugPi, BD2412, Bakken, Bender235, Bobo123456, Boethius65, Booyabazooka, Bopomofo, Boud, Brion VIBBER, Bryan Derksen,Bsadowski1, Bschaeffer, CES1596, Chris Howard, Chunminghan, Complexica, Connor Behan, Conversion script, CountingPine, Crazyjimbo, D.H, DMacks, DVdm, Dario Gnani, David-SarahHopwood, Davidslindsey, Ddcampayo, Delaszk, Dispenser, DrBob, Dratman, DreamGuy, E.Shubee, E23, E4mmacro, Eleider, Ems57fcva, Enormousdude, Epbr123, Erianna, Estudiarme,F=q(E+v^B), Fashionslide, Fel64, First Harmonic, Fiskbil, FlorianMarquardt, Fly by Night, Francvs, Fresheneesz, Fropuff, Geoffrey.landis, Giftlite, GoldenBoar, Goudzovski, Grandfatherclok,Grav-universe, Greg park avenue, H2g2bob, Hankwang, HappyCamper, Harald88, Headbomb, Hurfunkel, Iiar, Imrahil, Isocliff, JCSantos, JDoolin, JRSpriggs, JYOuyang, JabberWok, Jeepday,Jhausauer, Jj1236, Jmhodges, JocK, Johanvdv, Josh Cherry, Joshk, Jowr, Kaliumfredrik, Kasuga, Kilom691, Korte, Ksoileau, Kurykh, Lambiam, Laurentius, Lethe, LiDaobing, Light current,LokiClock, Loodog, Looxix, MOBle, Magencalc, Maksim-e, MarSch, Martlet1215, Mas2265, Maschen, MathKnight, Mattblack82, Maxdlink, Mercy11, Messier35, Mestizo777, Mgiganteus1,Michael Hardy, Michael Lenz, Mosaffa, MovGP0, Mpatel, Muhandes, Multipole, NOrbeck, NakedCelt, Natkuhn, Nbarth, Neparis, Netheril96, Nicobn, Niout, Optokinetics, Owlbuster, PAR,Philip Trueman, Phys, Pmetzger, Postscript07, Pt, Pulu, Qartis, R'n'B, RG2, RJFJR, Randallbsmith, Rathemis, RatnimSnave, Red Act, Rgdboer, Rhkramer, Ricky81682, Rob Hooft, Ros Power,Ryulong, Sam Hocevar, Sangwine, Sargon3, Schlafly, Sctfn, SebastianHelm, Shawnc, Smite-Meister, Stanmorgan, Star trooper man, Starshipenterprise, Stevenj, Stiangk, Stijn Vermeeren,SudoGhost, Svenlafe, TStein, Tassos Kan., Teorth, The Anome, The Cunctator, Thegreenj, Thurth, Tiddly Tom, Tim Shuba, TimothyRias, Tishchen, TobiasS, Totallyweb, TreasuryTag,Turgidson, Udirock, Ugncreative Usergname, Unfree, Uthbrian, VQuakr, Viridiflavus, Whit537, Wikien, Willandbeyond, Wwheaton, XCelam, XJamRastafire, Xaos, Yoctobarryc, Zzyzx11, 桜咲, 261 anonymous edits

Image Sources, Licenses and ContributorsFile:Minkowski lightcone lorentztransform.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Minkowski_lightcone_lorentztransform.svg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: User:MaschenFile:Referance frame and observer.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Referance_frame_and_observer.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:MaschenFile:Lorentz transforms 2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Lorentz_transforms_2.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:MaschenFile:Lorentz transform of world line.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Lorentz_transform_of_world_line.gif  License: GNU Free Documentation License Contributors: Aushulz, Cyp, Darapti, Jarekt, Julia W, Melchoir, Pieter Kuiper, Rovnet, Schekinov Alexey Victorovich, TommyBee, 4 anonymous edits

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