In physics, the Lorentz transformation (or transformations) is named after the Dutch physicist Hendrik Lorentz. It was 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. The Lorentz transformation is in accordance with special relativity, but was derived well before special relativity.
The transformations describe how measurements of space and time by two observers are related. They reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events. They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light.
The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost.
In the Minkowski space, the Lorentz transformations preserve the spacetime interval between any two events. They describe only the transformations in which the spacetime 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.
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 the conjecture that bodies in motion are getting contracted, in order to explain the baffling outcome of the 1887 ether-wind experiment of Michelson and Morley. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called FitzGerald–Lorentz contraction hypothesis. Their explanation was widely known before 1905.
Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous ether hypothesis, were also seeking the transformation under which Maxwell's equations were invariant when transformed from the ether to a moving frame. They extended the FitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("local time"). Henri Poincaré gave a physical interpretation to local time (to first order in v/c) as the consequence of clock synchronization under the assumption that the speed of light is constant in moving frames. Larmor is credited to have been the first to understand the crucial time dilation property inherent in his equations.
In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz. Later in the same year Albert Einstein published what is now called special relativity, by deriving the Lorentz transformation under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame, and by abandoning the mechanical aether.
Lorentz transformation for frames in standard configuration 
Consider two observers O and O′, each using their own Cartesian coordinate system to measure space and time intervals. O uses (t, x, y, z) and O′ uses (t′, x′, y′, z′). Assume further that the coordinate systems are oriented so that, in 3 dimensions, the x-axis and the x′-axis are collinear, the y-axis is parallel to the y′-axis, and the z-axis parallel to the z′-axis. The relative velocity between the two observers is v along the common x-axis; O measures O′ to move at velocity v along the coincident xx′ axes, while O′ measures O to move at velocity −v along the coincident xx′ axes. Also assume that the origins of both coordinate systems are the same, that is, coincident times and positions. If all these hold, then the coordinate systems are said to be in standard configuration.
The inverse of a Lorentz transformation relates the coordinates the other way round; from the coordinates O′ measures (t′, x′, y′, z′) to the coordinates O measures (t, x, y, z), so t, x, y, z are in terms of t′, x′, y′, z′. The mathematical form is nearly identical to the original transformation; the only difference is the negation of the uniform relative velocity (from v to −v), and exchange of primed and unprimed quantities, because O′ moves at velocity v relative to O, and equivalently, O moves at velocity −v relative to O′. This symmetry makes it effortless to find the inverse transformation (carrying out the exchange and negation saves a lot of rote algebra), although more fundamentally; it highlights that all physical laws should remain unchanged under a Lorentz transformation.
Below, the Lorentz transformations are called "boosts" in the stated directions.
Boost in the x-direction 
- 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 standard throughout the literature. For the remainder of the article – they will be also used throughout unless otherwise stated. Since the above is a linear system of equations (more technically a linear transformation), they can be written in matrix form:
According to the principle of relativity, there is no privileged frame of reference, so the inverse transformations frame F′ to frame F must be given by simply negating v:
where the value of γ remains unchanged.
Boost in the y or z directions 
The above collection of equations apply only for a boost in the x-direction. The standard configuration works equally well in the y or z directions instead of x, and so the results are similar.
For the y-direction:
where v and so β are now in the y-direction.
For the z-direction:
where v and so β are now in the z-direction.
The Lorentz transform for a boost in one of the above directions can be compactly written as a single matrix equation:
Boost in any direction 
Vector form 
For a boost in an arbitrary direction with velocity v, that is, O observes O′ to move in direction v in the F coordinate frame, while O′ observes O to move in direction −v in the F′ coordinate frame, it is convenient to decompose the spatial vector r into components perpendicular and parallel to v:
are "warped" by the Lorentz factor:
The parallel and perpendicular components can be eliminated, by substituting into r′:
Since r‖ and v are parallel we have
where geometrically and algebraically:
- v/v is a dimensionless unit vector pointing in the same direction as r‖,
- r‖ = (r • v)/v is the projection of r into the direction of v,
substituting for r‖ and factoring v gives
This method, of eliminating parallel and perpendicular components, can be applied to any Lorentz transformation written in parallel-perpendicular form.
Matrix forms 
These equations can be expressed in block matrix form as
and β is the magnitude of β:
More explicitly stated:
The transformation Λ can be written in the same form as before,
which has the structure:
and the components deduced from above are:
where δij is the Kronecker delta, and by convention: Latin letters for indices take the values 1, 2, 3, for spatial components of a 4-vector (Greek indices take values 0, 1, 2, 3 for time and space components).
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. 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.
Composition of two boosts 
- B(v) is the 4 × 4 matrix that uses the components of v, i.e. v1, v2, v3 in the entries of the matrix, or rather the components of v/c in the representation that is used above,
- is the velocity-addition,
- Gyr[u,v] (capital G) is the rotation arising from the composition. If the 3 × 3 matrix form of the rotation applied to spatial coordinates is given by gyr[u,v], then the 4 × 4 matrix rotation applied to 4-coordinates is given by:
- gyr (lower case g) is the gyrovector space abstraction of the gyroscopic Thomas precession, defined as an operator on a velocity w in terms of velocity addition:
- for all w.
The composition of two Lorentz transformations L(u, U) and L(v, V) which include rotations U and V is given by:
Visualizing the transformations in Minkowski space 
The yellow axes are the rest frame of an observer, the blue axes correspond to the frame of a moving observer
The red lines are world lines, a continuous sequence of events: straight for an object travelling at constant velocity, curved for an object accelerating. Worldlines of light form the boundary of the light cone.
The purple hyperbolae indicate this is a hyperbolic rotation, the hyperbolic angle ϕ is called rapidity (see below). The greater the relative speed between the reference frames, the more "warped" the axes become. The relative velocity cannot exceed c.
The black arrow is a displacement four-vector between two events (not necessarily on the same world line), showing that in a Lorentz boost; time dilation (fewer time intervals in moving frame) and length contraction (shorter lengths in moving frame) occur. The axes in the moving frame are orthogonal (even though they do not look so).
Then the Lorentz transformation in standard configuration is:
Hyperbolic expressions 
From the above expressions for eφ and e−φ
Hyperbolic rotation of coordinates 
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 parameter ϕ represents the hyperbolic angle of rotation, often referred to as rapidity. This transformation is sometimes illustrated with a Minkowski diagram, as displayed above.
Transformation of other physical quantities 
or in tensor index notation:
in which the primed indices denote indices of Z in the primed frame.
where is the inverse matrix of
Special relativity 
The crucial insight of Einstein's clock-setting method is the idea that time is relative. In essence, each observer's frame of reference is associated with a unique set of clocks, the result being that time as measured for a location passes at different rates for different observers. This was a direct result of the Lorentz transformations and is called time dilation. We can also clearly see from the Lorentz "local time" transformation that the concept of the relativity of simultaneity and of the relativity of length contraction are also consequences of that clock-setting hypothesis.
Transformation of the electromagnetic field 
Lorentz transformations can also be used to prove that magnetic and electric fields are simply different aspects of the same force — the electromagnetic force, as a consequence of relative motion between electric charges and observers. The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment:
- 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 any magnetic field.
- Consider another observer in frame F′ moving at relative velocity v (relative to F and the charge). This observer will see a different electric field because the charge is moving at velocity −v in their rest frame. Further, in frame F′ the moving charge constitutes an electric current, and thus the observer in frame F′ will also see a magnetic field.
This shows that the Lorentz transformation also applies to electromagnetic field quantities when changing the frame of reference, given below in vector form.
The correspondence principle 
The correspondence limit is usually stated mathematically as: as v → 0, c → ∞. In words: as velocity approaches 0, the speed of light (seems to) approach infinity. Hence, it is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".
Spacetime interval 
In a given coordinate system xμ, 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 xμ → x′ μ. Then, the interval in this coordinate system is given by
It is a result of special relativity that the interval is an invariant. That is, s2 = s′ 2. For this to hold, it can be shown that it is necessary (but not sufficient) for the coordinate transformation to be of the form
Here, Cμ is a constant vector and Λμν a constant matrix, where we require that
Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation. The Ca represents a spacetime translation. When Ca = 0, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.
Taking the determinant of
The cases are:
- Proper Lorentz transformations have det(Λμν) = +1, and form a subgroup called the special orthogonal group SO(1,3).
- Improper Lorentz transformations are det(Λμν) = −1, which do not form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation.
From the above definition of Λ it can be shown that (Λ00)2 ≥ 1, so either Λ00 ≥ 1 or Λ00 ≤ −1, called orthochronous and non-orthochronous respectively. An important subgroup of the proper Lorentz transformations are the proper orthochronous Lorentz transformations which consist purely of boosts and rotations. Any Lorentz transform can be written as a proper orthochronous, together with one or both of the two discrete transformations; space inversion P and time reversal T, whose non-zero elements are:
The set of Poincaré transformations satisfies the properties of a group and is called the Poincaré group. Under the Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all Lorentz transformations forms a group, called the Lorentz group.
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
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 (such as an interstellar aether) which obeys certain physical laws.
Michelson and Morley in 1887 designed an experiment, employing an interferometer and a half-silvered mirror, that was accurate enough to detect aether flow. The mirror system reflected the light back into the interferometer. If there were an aether drift, it would produce a phase shift and a change in the interference that would be detected. However, no phase shift was ever found. The negative outcome of the Michelson–Morley experiment left the concept of 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, Erik Christopher Zeeman showed that the causality preserving property, a condition that is weaker in a mathematical sense than the invariance of the speed of light, is enough to assure that the coordinate transformations are the Lorentz transformations.
From physical principles 
The problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and z coordinates do not intervene. The following is similar to that of Einstein. As in the Galilean transformation, the Lorentz transformation is linear since the relative velocity of the reference frames is constant as a vector; otherwise, inertial forces would appear. They are called inertial or Galilean reference frames. According to relativity no Galilean reference frame is privileged. Another condition is that the speed of light must be independent of the reference frame, in practice of the velocity of the light source.
Galilean and Einstein's relativity
- 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 of reference.
In Einstein's relativity, the main difference from Galilean relativity is that space and time coordinates are intertwined, and in different inertial frames t ≠ t′.
Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship 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 the transformation must yield x′ = 0 if x = vt. Therefore, b = −γv and the first equation is written as
- Principle of relativity
According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame R′ to frame R should have the same form as the original. To take advantage of this, we arrange by reversing the axes that R′ sees R moving towards positive x′ (i.e. just as R sees R′ moving towards positive x ), so that we can write
which, when multiplied through by −1, becomes
- 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 must guarantee 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
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 system of coordinates.
Einstein's popular derivation
In his popular book Einstein derived the Lorentz transformation by arguing that there must be two non-zero coupling constants λ and μ 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
Substituting x′ = 0 corresponding to x = vt and noting that the relative velocity is v = bc/γ, this gives
The constant γ can be evaluated as was previously shown above.
The Lorentz transformations can also be derived by simple application of the special relativity postulates and using hyperbolic identities. It is sufficient to derive the result in for a boost in the one direction, since for an arbitrary direction the decomposition of the position vector into parallel and perpendicular components can be done after, and generalizations therefrom follow, as outlined above.
- Relativity postulates
Start from the equations of the spherical wave front of a light pulse, centred at the origin:
which take the same form in both frames because of the special relativity postulates. Next, consider relative motion along the x-axes of each frame, in standard configuration above, so that y = y′, z = z′, which simplifies to
Now assume that the transformations take the linear form:
where A, B, C, D are to be found. If they were non-linear, they would not take the same form for all observers, since fictitious forces (hence accelerations) would occur in one frame even if the velocity was constant in another, which is inconsistent with inertial frame transformations.
Substituting into the previous result:
and comparing coefficients of x2, t2, xt:
- Hyperbolic rotation
The formulae resemble the hyperbolic identity
Introducing the rapidity parameter ϕ as a parametric hyperbolic angle allows the self-consistent identifications
where the signs after the square roots are chosen so that x and t increase. The hyperbolic transformations have been solved for:
If the signs were chosen differently the position and time coordinates would need to be replaced by −x and/or −t so that x and t increase not decrease.
To find what ϕ actually is, from the standard configuration the origin of the primed frame x′ = 0 is measured in the unprimed frame to be x = vt (or the equivalent and opposite way round; the origin of the unprimed frame is x = 0 and in the primed frame it is at x′ = −vt):
and manipulation of hyperbolic identities leads to
so the transformations are also:
From group postulates 
Following is a classical derivation (see, e.g.,  and references therein) based on group postulates and isotropy of the space.
- Coordinate transformations as a group
The coordinate transformations between inertial frames form a group (called the proper Lorentz group) with the group operation being the composition of transformations (performing one transformation after another). Indeed the four group axioms are satisfied:
- Closure: the composition of two transformations is a transformation: consider a composition of transformations from the inertial frame K to inertial frame K′, (denoted as K → K′), and then from K′ to inertial frame K′′, [K′ → K′′], there exists a transformation, [K → K′][K′ → K′′], directly from an inertial frame K to inertial frame K′′.
- Associativity: the result of ([K → K′][K′ → K′′])[K′′ → K′′′] and [K → K′]([K′ → K′′][K′′ → K′′′]) is the same, K → K′′′.
- Identity element: there is an identity element, a transformation K → K.
- Inverse element: for any transformation K → K′ there exists an inverse transformation K′ → K.
- Transformation matrices consistent with group axioms
Let us consider two inertial frames, K and K′, the latter moving with velocity 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, z) = (0, 0) and (t′, z′) = (0, 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 the 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,
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, because γ(v) is always nonzero;
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. Define this constant as δ(v)/vγ(v) = κ where κ has the dimension of 1/v2. Solving
we finally get
and thus the transformation matrix, consistent with the group axioms, is given by
If κ > 0, then there would be transformations (with κv2 ≫ 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:
- with the universal constant κ = 0, and
- with κ < 0.
- Galilean transformations
If κ = 0 then we get the Galilean-Newtonian kinematics with the Galilean transformation,
where time is absolute, t′ = t, and the relative velocity v of two inertial frames is not limited.
- Lorentz transformations
where the speed of light is a finite universal constant determining the highest possible relative velocity between inertial frames.
If v ≪ c 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.
See also 
- Ricci calculus
- Electromagnetic field
- Galilean transformation
- Hyperbolic rotation
- Invariance mechanics
- Lorentz group
- Principle of relativity
- Velocity-addition formula
- Algebra of physical space
- Relativistic aberration
- Prandtl–Glauert transformation
- O'Connor, John J.; Robertson, Edmund F., A History of Special Relativity
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- 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
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- Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons Ltd, ISBN 978-0-470-01460-8
- http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html. Hyperphysics, web-based physics matrial hosted by Georgia State University, USA.
- Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0
- Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
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- eq. (55), Thomas rotation and the parameterization of the Lorentz transformation group, AA Ungar – Foundations of Physics Letters, 1988
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- An Introduction to Mechanics, D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, ISBN 978-0-521-19821-9
Further reading 
- Einstein, Albert (1961), Relativity: The Special and the General Theory, New York: Three Rivers Press (published 1995), ISBN 0-517-88441-0
- Ernst, A.; Hsu, J.-P. (2001), "First proposal of the universal speed of light by Voigt 1887", 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
|Wikisource has original works on the topic: Relativity|
|Wikibooks has a book on the topic of: special relativity|
- Derivation of the Lorentz transformations. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
- The Paradox of Special Relativity. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
- Relativity – a chapter from an online textbook
- Special Relativity: The Lorentz Transformation, The Velocity Addition Law on Project PHYSNET
- Warp Special Relativity Simulator. A computer program demonstrating the Lorentz transformations on everyday objects.
- Animation clip visualizing the Lorentz transformation.
- Lorentz Frames Animated from John de Pillis. Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, etc.