TheInfoList

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, the Lorentz transformations are a six-parameter family of
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...
transformations Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Transf ...
from a coordinate frame in
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch
physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at leas ...

Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical met ...

. The most common form of the transformation, parametrized by the real constant $v,$ representing a velocity confined to the -direction, is expressed as $\begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end$ where and are the coordinates of an event in two frames, where the primed frame is seen from the unprimed frame as moving with speed along the -axis, is the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
, and $\gamma = \left ( \sqrt\right )^$ is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assi ...

. When speed is much smaller than , the Lorentz factor is negligibly different from 1, but as approaches , $\gamma$ grows without bound. The value of must be smaller than for the transformation to make sense. Expressing the speed as $\beta = \frac,$ an equivalent form of the transformation is $\begin ct' &= \gamma \left( c t - \beta x \right) \\ x' &= \gamma \left( x - \beta ct \right) \\ y' &= y \\ z' &= z. \end$ Frames of reference can be divided into two groups:
inertial In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a constan ...
(relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant
angular velocity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

, etc.). The term "Lorentz transformations" only refers to transformations between ''inertial'' frames, usually in the context of special relativity. In each
reference frame In physics, a frame of reference (or reference frame) consists of an abstract coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, wi ...

, an observer can use a local coordinate system (usually
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

in this context) to measure lengths, and a clock to measure time intervals. An
event Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritual A ritual is a sequence of activities involving gestures, words, actions, or objects, performed according to a set sequence. Rituals may be prescribed ...
is something that happens at a point in space at an instant of time, or more formally a point in
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
. The transformations connect the space and time coordinates of an
event Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritual A ritual is a sequence of activities involving gestures, words, actions, or objects, performed according to a set sequence. Rituals may be prescribed ...
as measured by an observer in each frame.One can imagine that in each inertial frame there are observers positioned throughout space, each with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where all reports are collected. When one speaks of a ''particular'' observer, one refers to someone having, at least in principle, a copy of this report. See, e.g., . They supersede the
Galilean transformation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
of Newtonian physics, which assumes an absolute space and time (see
Galilean relativity Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his ''Dialogue Concerning the Two Chief World Systems'' using Galileo's ship, ...
). The Galilean transformation is a good approximation only at relative speeds much less than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different
velocities The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion ...

may measure different distances, , and even different , but always such that the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
is the same in all inertial reference frames. The invariance of light speed is one of the
postulates of special relativity In physics, Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory ...
. Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nan ...

was observed to be independent of the
reference frame In physics, a frame of reference (or reference frame) consists of an abstract coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, wi ...

, and to understand the symmetries of the laws of
electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ...

. The Lorentz transformation is in accordance with
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

's
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
, but was derived first. The Lorentz transformation is a
linear transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In
Minkowski space In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the
spacetime interval In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ...
between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a
hyperbolic rotation In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
of Minkowski space. The more general set of transformations that also includes translations is known as the
Poincaré group The Poincaré group, named after Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to ...
.

# History

Many physicists—including
Woldemar Voigt Woldemar Voigt (; 2 September 1850 – 13 December 1919) was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in 19 ...

, ,
Joseph Larmor Sir Joseph Larmor (11 July 1857 – 19 May 1942) was an Irish and British physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical m ...
, and
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical met ...

himself—had been discussing the physics implied by these equations since 1887. Early in 1889,
Oliver Heaviside Oliver Heaviside Fellow of the Royal Society, FRS (; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who brought complex numbers to Network analysis (electrical circuits), circuit analysis, invented a new technique ...
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
that the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

surrounding a spherical distribution of charge should cease to have
spherical symmetry In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
once the charge is in motion relative to the
luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium Medium may refer to: Science and technology Aviation *Medium bomber, a class of war plane *Tecma Medium, a French hang glider design Communic ...
. 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 being contracted, in order to explain the baffling outcome of the 1887 aether-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 aether hypothesis, also looked for the transformation under which
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
are invariant when transformed from the aether 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 timeLocal time may refer to: * Local mean time ** Apparent solar time * Local time (mathematics) * Local time in the Lorentz ether theory {{disambiguation ...

").
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the s ...
gave a physical interpretation to local time (to first order in ''v''/''c'', the relative velocity of the two reference frames normalized to the speed of light) 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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

published what is now called
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
, by deriving the Lorentz transformation under the assumptions of the
principle of relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ...
and the constancy of the speed of light in any
inertial reference frame In classical physics Classical physics is a group of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies mat ...
, and by abandoning the mechanistic aether as unnecessary.

# Derivation of the group of Lorentz transformations

An ''
event Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritual A ritual is a sequence of activities involving gestures, words, actions, or objects, performed according to a set sequence. Rituals may be prescribed ...
'' is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinate ''ct'' and a set of
Cartesian coordinate A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

s to specify position in space in that frame. Subscripts label individual events. From Einstein's second postulate of relativity (invariance of ''c'') it follows that: in all inertial frames for events connected by ''light signals''. The quantity on the left is called the ''spacetime interval'' between events and . The interval between ''any two'' events, not necessarily separated by light signals, is in fact invariant, i.e., independent of the state of relative motion of observers in different inertial frames, as is shown using homogeneity and isotropy of space. The transformation sought after thus must possess the property that: where are the spacetime coordinates used to define events in one frame, and are the coordinates in another frame. First one observes that () is satisfied if an arbitrary -tuple of numbers are added to events and . Such transformations are called ''spacetime translations'' and are not dealt with further here. Then one observes that a ''linear'' solution preserving the origin of the simpler problem solves the general problem too: (a solution satisfying the left formula automatically satisfies the right one also; see
polarization identity In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
). Finding the solution to the simpler problem is just a matter of look-up in the theory of
classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric and alternating forms, s ...
s that preserve
bilinear form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s of various signature.The separate requirements of the three equations lead to three different groups. The second equation is satisfied for spacetime translations in addition to Lorentz transformations leading to the
Poincaré group The Poincaré group, named after Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to ...
or the ''inhomogeneous Lorentz group''. The first equation (or the second restricted to lightlike separation) leads to a yet larger group, the
conformal group In mathematics, the conformal group of a space is the group (mathematics), group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. ...
of spacetime.
First equation in () can be written more compactly as: where refers to the bilinear form of
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...
on exposed by the right hand side formula in . The alternative notation defined on the right is referred to as the ''relativistic dot product''. Spacetime mathematically viewed as endowed with this bilinear form is known as
Minkowski space In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
. The Lorentz transformation is thus an element of the group , the
Lorentz group In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
or, for those that prefer the other
metric signature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, (also called the Lorentz group).The groups and are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related to and respectively, e.g., the
Clifford algebra In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
s corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.
One has: which is precisely preservation of the bilinear form () which implies (by linearity of and bilinearity of the form) that () is satisfied. The elements of the Lorentz group are
rotations Rotation is the circular movement of an object around an ''axis of rotation''. A three-dimensional object may have an infinite number of rotation axes. If the rotation axis passes internally through the body's own center of mass, then the bo ...
and ''boosts'' and mixes thereof. If the spacetime translations are included, then one obtains the ''inhomogeneous Lorentz group'' or the
Poincaré group The Poincaré group, named after Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to ...
.

# Generalities

The relations between the primed and unprimed spacetime coordinates are the Lorentz transformations, each coordinate in one frame is a
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of all the coordinates in the other frame, and the
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
s are the inverse transformation. Depending on how the frames move relative to each other, and how they are oriented in space relative to each other, other parameters that describe direction, speed, and orientation enter the transformation equations. Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are called ''boosts'', and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformation is rotation in the spatial coordinates only, these like boosts are inertial transformations since there is no relative motion, the frames are simply tilted (and not continuously rotating), and in this case quantities defining the rotation are the parameters of the transformation (e.g.,
axis–angle representation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, or
Euler angle The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189 ...
s, etc.). A combination of a rotation and boost is a ''homogeneous transformation'', which transforms the origin back to the origin. The full Lorentz group also contains special transformations that are neither rotations nor boosts, but rather reflections in a plane through the origin. Two of these can be singled out; spatial inversion in which the spatial coordinates of all events are reversed in sign and temporal inversion in which the time coordinate for each event gets its sign reversed. Boosts should not be conflated with mere displacements in spacetime; in this case, the coordinate systems are simply shifted and there is no relative motion. However, these also count as symmetries forced by special relativity since they leave the spacetime interval invariant. A combination of a rotation with a boost, followed by a shift in spacetime, is an ''inhomogeneous Lorentz transformation'', an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.

# Physical formulation of Lorentz boosts

## Coordinate transformation

A "stationary" observer in frame defines events with coordinates . Another frame moves with velocity relative to , and an observer in this "moving" frame defines events using the coordinates . The coordinate axes in each frame are parallel (the and axes are parallel, the and axes are parallel, and the and axes are parallel), remain mutually perpendicular, and relative motion is along the coincident axes. At , the origins of both coordinate systems are the same, . In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in standard configuration, or synchronized. If an observer in records an event , then an observer in records the ''same'' event with coordinates where is the relative velocity between frames in the -direction, is the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
, and $\gamma = \frac$ (lowercase
gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...

) is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assi ...

. Here, is the ''
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...

'' of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity is motion along the positive directions of the axes, zero relative velocity is no relative motion, while negative relative velocity is relative motion along the negative directions of the axes. The magnitude of relative velocity cannot equal or exceed , so only subluminal speeds are allowed. The corresponding range of is . The transformations are not defined if is outside these limits. At the speed of light () is infinite, and
faster than light Faster-than-light (also FTL, superluminal or supercausal) Faster-than-light communication, communications and travel are the conjectural propagation of information or matter faster than the speed of light (). The special theory of relativity impl ...
() is a
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

, each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers. As an
active transformation In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτ ...
, an observer in F′ notices the coordinates of the event to be "boosted" in the negative directions of the axes, because of the in the transformations. This has the equivalent effect of the ''coordinate system'' F′ boosted in the positive directions of the axes, while the event does not change and is simply represented in another coordinate system, a
passive transformation In analytic geometry, spatial transformations in the 3-dimensional Euclidean space \R^3 are distinguished into active or alibi transformations, and passive or alias transformations. An active transformation is a Transformation (mathematics), tr ...
. The inverse relations ( in terms of ) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here is the "stationary" frame while is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from to must take exactly the same form as the transformations from to . The only difference is moves with velocity relative to (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in notes an event , then an observer in notes the ''same'' event with coordinates and the value of remains unchanged. This "trick" of simply reversing the direction of relative velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost in any direction. Sometimes it is more convenient to use (lowercase
beta Beta (, ; uppercase , lowercase , or cursive Cursive (also known as script, among other names) is any style of penmanship Penmanship is the technique of writing Writing is a medium of human communication that involves the represen ...

) instead of , so that $\begin ct' &= \gamma \left( ct - \beta x \right) \,, \\ x' &= \gamma \left( x - \beta ct \right) \,, \\ \end$ which shows much more clearly the symmetry in the transformation. From the allowed ranges of and the definition of , it follows . The use of and is standard throughout the literature. The Lorentz transformations can also be derived in a way that resembles circular rotations in 3d space using the
hyperbolic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s. For the boost in the direction, the results are where (lowercase
zeta Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek Demotic Greek or Dimotiki ( el, Δημοτική Γλώσσα, , , lit. "language of the people") was a colloquial vernacular form of Modern Greek, in c ...

) is a parameter called ''
rapidity In Theory of relativity, relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being ...
'' (many other symbols are used, including ). Given the strong resemblance to rotations of spatial coordinates in 3d space in the Cartesian xy, yz, and zx planes, a Lorentz boost can be thought of as a
hyperbolic rotation In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4d
Minkowski space In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
. The parameter is the
hyperbolic angle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a
Minkowski diagram A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation In physics and Th ...

. The hyperbolic functions arise from the ''difference'' between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking or in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying , which parametrizes the curves according to the identity $\cosh^2\zeta - \sinh^2\zeta = 1 \,.$ Conversely the and axes can be constructed for varying coordinates but constant . The definition $\tanh\zeta = \frac \,,$ provides the link between a constant value of rapidity, and the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

of the axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor $\cosh\zeta = \frac \,.$ Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between , , and are $\begin \beta &= \tanh\zeta \,, \\ \gamma &= \cosh\zeta \,, \\ \beta \gamma &= \sinh\zeta \,. \end$ Taking the inverse hyperbolic tangent gives the rapidity $\zeta = \tanh^\beta \,.$ Since , it follows . From the relation between and , positive rapidity is motion along the positive directions of the axes, zero rapidity is no relative motion, while negative rapidity is relative motion along the negative directions of the axes. The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity since this is equivalent to negating the relative velocity. Therefore, The inverse transformations can be similarly visualized by considering the cases when and . So far the Lorentz transformations have been applied to ''one event''. If there are two events, there is a spatial separation and time interval between them. It follows from the
linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...
of the Lorentz transformations that two values of space and time coordinates can be chosen, the Lorentz transformations can be applied to each, then subtracted to get the Lorentz transformations of the differences; $\begin \Delta t' &= \gamma \left( \Delta t - \frac \right) \,, \\ \Delta x' &= \gamma \left( \Delta x - v \, \Delta t \right) \,, \end$ with inverse relations $\begin \Delta t &= \gamma \left( \Delta t' + \frac \right) \,, \\ \Delta x &= \gamma \left( \Delta x' + v \, \Delta t' \right) \,. \end$ where (uppercase
delta Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, a landform at the mouth of a river * D (NATO phonetic alphabet: "Delta"), the fourth letter of the modern English alphabet * Delta Air Lines, an Ame ...
) indicates a difference of quantities; e.g., for two values of coordinates, and so on. These transformations on ''differences'' rather than spatial points or instants of time are useful for a number of reasons: * in calculations and experiments, it is lengths between two points or time intervals that are measured or of interest (e.g., the length of a moving vehicle, or time duration it takes to travel from one place to another), * the transformations of velocity can be readily derived by making the difference infinitesimally small and dividing the equations, and the process repeated for the transformation of acceleration, * if the coordinate systems are never coincident (i.e., not in standard configuration), and if both observers can agree on an event in and in , then they can use that event as the origin, and the spacetime coordinate differences are the differences between their coordinates and this origin, e.g., , , etc.

## Physical implications

A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in the equation for a pulse of light along the direction is , then in the Lorentz transformations give , and vice versa, for any . For relative speeds much less than the speed of light, the Lorentz transformations reduce to the
Galilean transformation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
$\begin t' &\approx t \\ x' &\approx x - vt \end$ in accordance with the
correspondence principle In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical mechanics, classical physics in the limit of large quantum numbers. In o ...
. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance". Three counterintuitive, but correct, predictions of the transformations are: ;
Relativity of simultaneity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

: Suppose two events occur simultaneously () along the x axis, but separated by a nonzero displacement . Then in , we find that $\Delta t\text{'} = \gamma \frac$, so the events are no longer simultaneous according to a moving observer. ;
Time dilation In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

: Suppose there is a clock at rest in . If a time interval is measured at the same point in that frame, so that , then the transformations give this interval in by . Conversely, suppose there is a clock at rest in . If an interval is measured at the same point in that frame, so that , then the transformations give this interval in F by . Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor than the time interval between ticks of his own clock. ;
Length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more compl ...
: Suppose there is a rod at rest in aligned along the x axis, with length . In , the rod moves with velocity , so its length must be measured by taking two simultaneous () measurements at opposite ends. Under these conditions, the inverse Lorentz transform shows that . In the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in . So each observer measures the distance between the end points of a moving rod to be shorter by a factor than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.

## Vector transformations

The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative
velocity vector The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...
with a magnitude that cannot equal or exceed , so that . Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial
position vector In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

as measured in , and as measured in , each into components perpendicular (⊥) and parallel ( ‖ ) to , $\mathbf=\mathbf_\perp+\mathbf_\, \,,\quad \mathbf' = \mathbf_\perp' + \mathbf_\, ' \,,$ then the transformations are $\begin t' &= \gamma \left(t - \frac \right) \\ \mathbf_\, ' &= \gamma (\mathbf_\, - \mathbf t) \\ \mathbf_\perp' &= \mathbf_\perp \end$ where is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
. The Lorentz factor retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition with magnitude is also used by some authors. Introducing a
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
in the direction of relative motion, the relative velocity is with magnitude and direction , and
vector projection The vector projection of a vector a on (or onto) a nonzero vector b, sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of a in the direction of b), is the orthogonal projection In linear ...
and rejection give respectively $\mathbf_\parallel = (\mathbf\cdot\mathbf)\mathbf\,,\quad \mathbf_\perp = \mathbf - (\mathbf\cdot\mathbf)\mathbf$ Accumulating the results gives the full transformations, The projection and rejection also applies to . For the inverse transformations, exchange and to switch observed coordinates, and negate the relative velocity (or simply the unit vector since the magnitude is always positive) to obtain The unit vector has the advantage of simplifying equations for a single boost, allows either or to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing and . It is not convenient for multiple boosts. The vectorial relation between relative velocity and rapidity is $\boldsymbol = \beta \mathbf = \mathbf \tanh\zeta \,,$ and the "rapidity vector" can be defined as $\boldsymbol = \zeta\mathbf = \mathbf\tanh^\beta \,,$ each of which serves as a useful abbreviation in some contexts. The magnitude of is the absolute value of the rapidity scalar confined to , which agrees with the range .

## Transformation of velocities

Defining the coordinate velocities and Lorentz factor by :$\mathbf = \frac \,,\quad \mathbf\text{'} = \frac \,,\quad \gamma_\mathbf = \frac$ taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to : The velocities and are the velocity of some massive object. They can also be for a third inertial frame (say ''F''′′), in which case they must be ''constant''. Denote either entity by X. Then X moves with velocity relative to F, or equivalently with velocity relative to F′, in turn F′ moves with velocity relative to F. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange and , and change to . The transformation of velocity is useful in
stellar aberration In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses m ...

, the
Fizeau experiment The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Fizeau used a special interferometer arrangement to measure the effect of movement of a medium upon the speed of light. A ...
, and the
relativistic Doppler effect The relativistic Doppler effect is the change in frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of exi ...
. The Lorentz transformations of acceleration can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.

## Transformation of other quantities

In general, given four quantities and and their Lorentz-boosted counterparts and , a relation of the form $A^2 - \mathbf\cdot\mathbf = ^2 - \mathbf'\cdot\mathbf'$ implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates; $\begin A' &= \gamma \left(A - \frac \right) \,, \\ \mathbf' &= \mathbf + (\gamma-1)(\mathbf\cdot\mathbf)\mathbf - \frac \,. \end$ The decomposition of (and ) into components perpendicular and parallel to is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange and to switch observed quantities, and reverse the direction of relative motion by the substitution ). The quantities collectively make up a ''
four-vector In special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in oth ...
'', where is the "timelike component", and the "spacelike component". Examples of and are the following: For a given object (e.g., particle, fluid, field, material), if or correspond to properties specific to the object like its
charge density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

,
mass density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass per unit volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter Rho (letter), rho), although the L ...
,
spin Spin or spinning may refer to: Businesses * SPIN (cable system) or South Pacific Island Network * Spin (company), an American scooter-sharing system * SPiN, a chain of table tennis lounges Computing * SPIN model checker, Gerard Holzmann's tool f ...
, etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a
rest energy The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in
relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré group, Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocity, velocities up to those comparable to t ...
spin depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity , however a boosted observer will perceive a nonzero timelike component and an altered spin. Not all quantities are invariant in the form as shown above, for example orbital
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

does not have a timelike quantity, and neither does the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

nor the
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

. The definition of angular momentum is , and in a boosted frame the altered angular momentum is . Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out transforms with another vector quantity related to boosts, see
relativistic angular momentum In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thr ...
for details. For the case of the and fields, the transformations cannot be obtained as directly using vector algebra. The
Lorentz force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

is the definition of these fields, and in it is while in it is . A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra, given below.

# Mathematical formulation

Throughout, italic non-bold capital letters are 4×4 matrices, while non-italic bold letters are 3×3 matrices.

## Homogeneous Lorentz group

Writing the coordinates in column vectors and the
Minkowski metric In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...

as a square matrix $X' = \begin c\,t' \\ x' \\ y' \\ z' \end \,, \quad \eta = \begin -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end \,, \quad X = \begin c\,t \\ x \\ y \\ z \end$ the spacetime interval takes the form (superscript denotes
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

) $X \cdot X = X^\mathrm \eta X = ^\mathrm \eta$ and is invariant under a Lorentz transformation $X' = \Lambda X$ where is a square matrix which can depend on parameters. The set of all Lorentz transformations Λ in this article is denoted $\mathcal$. This set together with matrix multiplication forms a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, in this context known as the ''
Lorentz group In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
''. Also, the above expression is a
quadratic form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the
indefinite orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimension of a vector space, dimensional real vector space that leave invariant a nondegenerate form, nondegenerate, symmetric bilinear for ...
O(3,1), a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are
matrix Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operat ...
s. In this context the operation of composition amounts to
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. From the invariance of the spacetime interval it follows $\eta = \Lambda^\mathrm \eta \Lambda$ and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of the equation using the product ruleFor two square matrices and , gives immediately Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form, $\eta = \begin-1 & 0 \\ 0 & \mathbf\end \,, \quad \Lambda=\begin\Gamma & -\mathbf^\mathrm\\-\mathbf & \mathbf\end \,,$ carrying out the block matrix multiplications obtains general conditions on to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results $\Gamma^2 = 1 + \mathbf^\mathrm\mathbf$ is useful; always so it follows that $\Gamma^2 \geq 1 \quad \Rightarrow \quad \Gamma \leq - 1 \,,\quad \Gamma \geq 1$ The negative inequality may be unexpected, because multiplies the time coordinate and this has an effect on time symmetry. If the positive equality holds, then is the Lorentz factor. The determinant and inequality provide four ways to classify Lorentz Transformations (''herein LTs for brevity''). Any particular LT has only one determinant sign ''and'' only one inequality. There are four sets which include every possible pair given by the
intersection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
s ("n"-shaped symbol meaning "and") of these classifying sets. where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities. The full Lorentz group splits into the union ("u"-shaped symbol meaning "or") of four
disjoint set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s $\mathcal = \mathcal_^\uparrow \cup \mathcal_^\uparrow \cup \mathcal_^\downarrow \cup \mathcal_^\downarrow$ A
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
of a group must be closed under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations and from a particular set, the composite Lorentz transformations and must be in the same set as and . This is not always the case: the composition of two antichronous Lorentz transformations is orthochronous, and the composition of two improper Lorentz transformations is proper. In other words, while the sets $\mathcal_+^\uparrow$, $\mathcal_+$, $\mathcal^\uparrow$, and $\mathcal_0 = \mathcal_+^\uparrow \cup \mathcal_^\downarrow$ all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g. $\mathcal_+^\downarrow$, $\mathcal_^\downarrow$, $\mathcal_^\uparrow$) do not form subgroups.

## Proper transformations

If a Lorentz covariant 4-vector is measured in one inertial frame with result $X$, and the same measurement made in another inertial frame (with the same orientation and origin) gives result $X\text{'}$, the two results will be related by $X' = B(\mathbf)X$ where the boost matrix $B\left(\mathbf\right)$ represents the Lorentz transformation between the unprimed and primed frames and $\mathbf$ is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by $B(\mathbf) = \begin \gamma &-\gamma v_x/c &-\gamma v_y/c &-\gamma v_z/c \\ -\gamma v_x/c&1+(\gamma-1)\dfrac & (\gamma-1)\dfrac& (\gamma-1)\dfrac \\ -\gamma v_y/c& (\gamma-1)\dfrac&1+(\gamma-1)\dfrac & (\gamma-1)\dfrac \\ -\gamma v_z/c& (\gamma-1)\dfrac& (\gamma-1)\dfrac&1+(\gamma-1)\dfrac \end,$ where $v=\sqrt$ is the magnitude of the velocity and $\gamma=\frac$ is the Lorentz factor. This formula represents a passive transformation, as it describes how the coordinates of the measured quantity changes from the unprimed frame to the primed frame. The active transformation is given by $B\left(-\mathbf\right)$. If a frame is boosted with velocity relative to frame , and another frame is boosted with velocity relative to , the separate boosts are $X'' = B(\mathbf)X' \,, \quad X' = B(\mathbf)X$ and the composition of the two boosts connects the coordinates in and , $X'' = B(\mathbf)B(\mathbf)X \,.$ Successive transformations act on the left. If and are
collinear In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
(parallel or antiparallel along the same line of relative motion), the boost matrices commute: . This composite transformation happens to be another boost, , where is collinear with and . If and are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute: and are not equal. Also, each of these compositions is ''not'' a single boost, but they are still Lorentz transformations they each preserve the spacetime interval. It turns out the composition of any two Lorentz boosts is equivalent to a boost followed or preceded by a rotation on the spatial coordinates, in the form of or . The and are composite velocities, while and are rotation parameters (e.g. axis-angle variables,
Euler angles The Euler angles are three angles introduced by Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics ( ...
, etc.). The rotation in
block matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
form is simply $\quad R(\boldsymbol) = \begin 1 & 0 \\ 0 & \mathbf(\boldsymbol) \end \,,$ where is a 3d
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \th ...
, which rotates any 3d vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is ''not'' simple to connect and (or and ) to the original boost parameters and . In a composition of boosts, the matrix is named the Wigner rotation, and gives rise to the
Thomas precession In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
. These articles give the explicit formulae for the composite transformation matrices, including expressions for . In this article the axis-angle representation is used for . The rotation is about an axis in the direction of a
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, through angle (positive anticlockwise, negative clockwise, according to the
right-hand rule In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

). The "axis-angle vector" $\boldsymbol = \theta \mathbf$ will serve as a useful abbreviation. Spatial rotations alone are also Lorentz transformations they leave the spacetime interval invariant. Like boosts, successive rotations about different axes do not commute. Unlike boosts, the composition of any two rotations is equivalent to a single rotation. Some other similarities and differences between the boost and rotation matrices include: *
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
s: (relative motion in the opposite direction), and (rotation in the opposite sense about the same axis) *
identity transformation image:Function-x.svg, Graph of a function, Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function t ...
for no relative motion/rotation: * unit
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

: . This property makes them proper transformations. * matrix symmetry: is symmetric (equals
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

), while is nonsymmetric but
orthogonal In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...
(transpose equals
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
, ). The most general proper Lorentz transformation includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, and . An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here. Nevertheless, closed form expressions for the transformation matrices will be given below using group theoretical arguments. It will be easier to use the rapidity parametrization for boosts, in which case one writes and .

### The Lie group SO+(3,1)

The set of transformations $\$ with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the special indefinite orthogonal group SO+(3,1). (The plus sign indicates that it preserves the orientation of the temporal dimension). For simplicity, look at the infinitesimal Lorentz boost in the x direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by the Taylor expansion of the boost matrix to first order about , $B_x = I + \zeta \left. \frac \_ + \cdots$ where the higher order terms not shown are negligible because is small, and is simply the boost matrix in the ''x'' direction. The matrix calculus, derivative of the matrix is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated at , $\left. \frac \_ = - K_x \,.$ For now, is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a matrix exponential is obtained $B_x =\lim_\left(I-\fracK_x\right)^ = e^$ where the Exponential function#Formal definition, limit definition of the exponential has been used (see also characterizations of the exponential function). More generallyExplicitly, $\boldsymbol \cdot\mathbf = \zeta_x K_x + \zeta_y K_y + \zeta_z K_z$ $\boldsymbol \cdot\mathbf = \theta_x J_x + \theta_y J_y + \theta_z J_z$ $B(\boldsymbol) = e^ \, , \quad R(\boldsymbol) = e^ \,.$ The axis-angle vector and rapidity vector are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are and , each vectors of matrices with the explicit formsIn quantum mechanics,
relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré group, Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocity, velocities up to those comparable to t ...
, and quantum field theory, a different convention is used for these matrices; the right hand sides are all multiplied by a factor of the imaginary unit .
$K_x = \begin 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end\,,\quad K_y = \begin 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end\,,\quad K_z = \begin 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 \end$ $J_x = \begin 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ \end\,,\quad J_y = \begin 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end\,,\quad J_z = \begin 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end$ These are all defined in an analogous way to above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime: are the ''rotation generators'' which correspond to
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

, and are the ''boost generators'' which correspond to the motion of the system in spacetime. The derivative of any smooth curve with in the group depending on some group parameter with respect to that group parameter, evaluated at , serves as a definition of a corresponding group generator , and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always map smoothly back into the group via for all ; this curve will yield again when differentiated at . Expanding the exponentials in their Taylor series obtains $B()=I-\sinh \zeta (\mathbf \cdot \mathbf )+(\cosh \zeta -1)(\mathbf \cdot \mathbf )^2$ $R(\boldsymbol )=I+\sin \theta (\mathbf \cdot \mathbf )+(1-\cos \theta )(\mathbf \cdot \mathbf )^2\,.$ which compactly reproduce the boost and rotation matrices as given in the previous section. It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At the ''infinitesimal'' level the product $\begin \Lambda &= (I - \boldsymbol \cdot \mathbf + \cdots )(I + \boldsymbol \cdot \mathbf + \cdots ) \\ &= (I + \boldsymbol \cdot \mathbf + \cdots )(I - \boldsymbol \cdot \mathbf + \cdots ) \\ &= I - \boldsymbol \cdot \mathbf + \boldsymbol \cdot \mathbf + \cdots \end$ is commutative because only linear terms are required (products like and count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential $\Lambda (\boldsymbol, \boldsymbol) = e^.$ The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular, $e^ \ne e^ e^,$ because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotation ''in principle'' (this usually does not yield an intelligible expression in terms of generators and ), see Wigner rotation. If, on the other hand, ''the decomposition is given'' in terms of the generators, and one wants to find the product in terms of the generators, then the Baker–Campbell–Hausdorff formula applies.

### The Lie algebra so(3,1)

Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the set of all Lorentz generators $V = \$ together with the operations of ordinary matrix addition and matrix multiplication#Scalar multiplication, multiplication of a matrix by a number, forms a vector space over the real numbers.Until now the term "vector" has exclusively referred to "Euclidean vector", examples are position , velocity , etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., see linear algebra and vector space for details. The generators of a Lie group also form a vector space over a field (mathematics), field of numbers (e.g. real numbers,
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s), since a linear combination of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3d space.
The generators form a basis (linear algebra), basis set of ''V'', and the components of the axis-angle and rapidity vectors, , are the coordinate vector, coordinates of a Lorentz generator with respect to this basis.In ordinary 3d position space, the position vector is expressed as a linear combination of the Cartesian unit vectors which form a basis, and the Cartesian coordinates are coordinates with respect to this basis. Three of the commutation relations of the Lorentz generators are $[ J_x, J_y ] = J_z \,,\quad [ K_x, K_y ] = -J_z \,,\quad [ J_x, K_y ] = K_z \,,$ where the bracket is known as the ''commutator'', and the other relations can be found by taking cyclic permutations of x, y, z components (i.e. change x to y, y to z, and z to x, repeat). These commutation relations, and the vector space of generators, fulfill the definition of the Lie algebra $\mathfrak\left(3, 1\right)$. In summary, a Lie algebra is defined as a vector space ''V'' over a field (mathematics), field of numbers, and with a binary operation [ , ] (called a Lie bracket in this context) on the elements of the vector space, satisfying the axioms of Bilinear map, bilinearity, alternatization, and the Jacobi identity. Here the operation [ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators ''V'' as given previously, and the field is the set of real numbers. Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense. The exponential map (Lie theory), exponential map from the Lie algebra to the Lie group, $\exp \, : \, \mathfrak(3,1) \to \mathrm(3,1),$ provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. It the case of the Lorentz group, the exponential map is just the matrix exponential. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is surjective function, surjective (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.

## Improper transformations

Lorentz transformations also include parity inversion $P = \begin 1 & 0 \\ 0 & - \mathbf \end$ which negates all the spatial coordinates only, and T-symmetry, time reversal $T = \begin - 1 & 0 \\ 0 & \mathbf \end$ which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here is the 3d identity matrix. These are both symmetric, they are their own inverses (see involution (mathematics)), and each have determinant −1. This latter property makes them improper transformations. If is a proper orthochronous Lorentz transformation, then is improper antichronous, is improper orthochronous, and is proper antichronous.

## Inhomogeneous Lorentz group

Two other spacetime symmetries have not been accounted for. For the spacetime interval to be invariant, it can be shown that it is necessary and sufficient for the coordinate transformation to be of the form $X' = \Lambda X + C$ where ''C'' is a constant column containing translations in time and space. If ''C'' ≠ 0, this is an inhomogeneous Lorentz transformation or Poincaré transformation. If ''C'' = 0, this is a homogeneous Lorentz transformation. Poincaré transformations are not dealt further in this article.

# Tensor formulation

## Contravariant vectors

Writing the general matrix transformation of coordinates as the matrix equation $\begin ^0 \\ ^1 \\ ^2 \\ ^3 \end = \begin _0 & _1 & _2 & _3 \\ _0 & _1 & _2 & _3 \\ _0 & _1 & _2 & _3 \\ _0 & _1 & _2 & _3 \\ \end \begin x^0 \\ x^1 \\ x^2 \\ x^3 \end$ allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., tensors or spinors of any order in 4d spacetime, to be defined. In the corresponding tensor index notation, the above matrix expression is $^\nu = _\mu x^\mu,$ where lower and upper indices label covariance and contravariance of vectors, covariant and contravariant components respectively, and the summation convention is applied. It is a standard convention to use Greek alphabet, Greek indices that take the value 0 for time components, and 1, 2, 3 for space components, while Latin alphabet, Latin indices simply take the values 1, 2, 3, for spatial components. Note that the first index (reading left to right) corresponds in the matrix notation to a ''row index''. The second index corresponds to the column index. The transformation matrix is universal for all
four-vector In special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in oth ...
s, not just 4-dimensional spacetime coordinates. If is any four-vector, then in tensor index notation $^\nu = _\mu A^\mu \,.$ Alternatively, one writes $A^ = _\mu A^\mu \,.$ in which the primed indices denote the indices of A in the primed frame. This notation cuts risk of exhausting the Greek alphabet roughly in half. For a general -component object one may write $^\alpha = _\beta X^\beta \,,$ where is the appropriate Representation theory of the Lorentz group, representation of the Lorentz group, an matrix for every . In this case, the indices should ''not'' be thought of as spacetime indices (sometimes called Lorentz indices), and they run from to . E.g., if is a bispinor, then the indices are called ''Dirac indices''.

## Covariant vectors

There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of ''lowering an index''; e.g., $x_\nu = \eta_x^\mu,$ where is the metric tensor. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given by $x^\mu = \eta^x_\nu,$ where, when viewed as matrices, is the inverse of . As it happens, . This is referred to as ''raising an index''. To transform a covariant vector , first raise its index, then transform it according to the same rule as for contravariant -vectors, then finally lower the index; $_\nu = \eta_ _\sigma \eta^A_\mu.$ But $\eta_ _\sigma \eta^ = _\nu,$ That is, it is the -component of the ''inverse'' Lorentz transformation. One defines (as a matter of notation), $^\mu \equiv _\nu,$ and may in this notation write $_\nu = ^\mu A_\mu.$ Now for a subtlety. The implied summation on the right hand side of $_\nu = ^\mu A_\mu = _\nu A_\mu$ is running over ''a row index'' of the matrix representing . Thus, in terms of matrices, this transformation should be thought of as the ''inverse transpose'' of acting on the column vector . That is, in pure matrix notation, $A' = \left(\Lambda^\right)^\mathrm A.$ This means exactly that covariant vectors (thought of as column matrices) transform according to the dual representation of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replace with .

## Tensors

If and are linear operators on vector spaces and , then a linear operator may be defined on the tensor product of and , denoted according to From this it is immediately clear that if and are a four-vectors in , then transforms as The second step uses the bilinearity of the tensor product and the last step defines a 2-tensor on component form, or rather, it just renames the tensor . These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space can be written as a sum of a coefficient (component!) times tensor products of basis vectors and basis covectors, one arrives at the transformation law for any tensor quantity . It is given by where is defined above. This form can generally be reduced to the form for general -component objects given above with a single matrix () operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.

### Transformation of the electromagnetic field

Lorentz transformations can also be used to illustrate that the
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

and
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

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. * An observer measures a charge at rest in 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 does not observe any magnetic field. * The other observer in frame F′ moves at velocity relative to F and the charge. ''This'' observer sees a different electric field because the charge moves at velocity in their rest frame. The motion of the charge corresponds to an electric current, and thus the observer in frame F′ also sees a magnetic field. The electric and magnetic fields transform differently from space and time, but exactly the same way as relativistic angular momentum and the boost vector. The electromagnetic field strength tensor is given by $F^ = \begin 0 & -\fracE_x & -\fracE_y & -\fracE_z \\ \fracE_x & 0 & -B_z & B_y \\ \fracE_y & B_z & 0 & -B_x \\ \fracE_z & -B_y & B_x & 0 \end \text(+,-,-,-)\text.$ in SI units. In relativity, the Gaussian units, Gaussian system of units is often preferred over SI units, even in texts whose main choice of units is SI units, because in it the electric field and the magnetic induction have the same units making the appearance of the Electromagnetic tensor, electromagnetic field tensor more natural. Consider a Lorentz boost in the -direction. It is given by $_\nu = \begin \gamma & -\gamma\beta & 0 & 0\\ -\gamma\beta & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end, \qquad F^ = \begin 0 & E_x & E_y & E_z \\ -E_x & 0 & B_z & -B_y \\ -E_y & -B_z & 0 & B_x \\ -E_z & B_y & -B_x & 0 \end \text(-,+,+,+)\text,$ where the field tensor is displayed side by side for easiest possible reference in the manipulations below. The general transformation law becomes $F^ = _\mu _\nu F^.$ For the magnetic field one obtains $\begin B_ &= F^ = _\mu _\nu F^ = _2 _3 F^ = 1 \times 1 \times B_x \\ &= B_x, \\ B_ &= F^ = _\mu _\nu F^ = _3 _\nu F^ = _3 _0 F^ + _3 _1 F^ \\ &= 1 \times (-\beta\gamma) (-E_z) + 1 \times \gamma B_y = \gamma B_y + \beta\gamma E_z \\ &= \gamma\left(\mathbf - \boldsymbol \times \mathbf\right)_y \\ B_ &= F^ = _\mu _\nu F^ = _\mu _2 F^ = _0 _2 F^ + _1 _2 F^ \\ &= (-\gamma\beta) \times 1\times E_y + \gamma \times 1 \times B_z = \gamma B_z - \beta\gamma E_y \\ &= \gamma\left(\mathbf - \boldsymbol \times \mathbf\right)_z \end$ For the electric field results $\begin E_ &= F^ = _\mu _\nu F^ = _1 _0 F^ + _0 _1 F^ \\ &= (-\gamma\beta)(-\gamma\beta)(-E_x) + \gamma\gamma E_x = -\gamma^2\beta^2(E_x) + \gamma^2 E_x = E_x(1 - \beta^2)\gamma^2 \\ &= E_x, \\ E_ &= F^ = _\mu _\nu F^ = _\mu _2 F^ = _0 _2 F^ + _1 _2 F^ \\ &= \gamma \times 1 \times E_y + (-\beta\gamma) \times 1 \times B_z = \gamma E_y - \beta\gamma B_z \\ &= \gamma\left(\mathbf + \boldsymbol \times \mathbf\right)_y \\ E_ &= F^ = _\mu _\nu F^ = _\mu _3 F^ = _0 _3 F^ + _1 _3 F^ \\ &= \gamma \times 1 \times E_z - \beta\gamma \times 1 \times (-B_y) = \gamma E_z + \beta\gamma B_y \\ &= \gamma\left(\mathbf + \boldsymbol \times \mathbf\right)_z. \end$ Here, is used. These results can be summarized by $\begin \mathbf_ &= \mathbf_\parallel \\ \mathbf_ &= \mathbf_\parallel \\ \mathbf_ &= \gamma \left( \mathbf_\bot + \boldsymbol \times \mathbf_\bot \right) = \gamma \left( \mathbf + \boldsymbol \times \mathbf \right)_\bot,\\ \mathbf_ &= \gamma \left( \mathbf_\bot - \boldsymbol \times \mathbf_\bot \right) = \gamma \left( \mathbf - \boldsymbol \times \mathbf \right)_\bot, \end$ and are independent of the metric signature. For SI units, substitute . refer to this last form as the view as opposed to the ''geometric view'' represented by the tensor expression $F^ = _\mu _\nu F^,$ and make a strong point of the ease with which results that are difficult to achieve using the view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact under ''any'' smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object in ''spacetime'' as opposed to two interdependent, but separate, 3-vector fields in ''space'' and ''time''. The fields (alone) and (alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations and that immediately yield . One should note that the primed and unprimed tensors refer to the ''same event in spacetime''. Thus the complete equation with spacetime dependence is $F^\left(x'\right) = _\mu _\nu F^\left(\Lambda^ x'\right) = _\mu _\nu F^(x).$ Length contraction has an effect on
charge density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

and current density , and time dilation has an effect on the rate of flow of charge (current), so charge and current distributions must transform in a related way under a boost. It turns out they transform exactly like the space-time and energy-momentum four-vectors, $\begin \mathbf' &= \mathbf - \gamma\rho v\mathbf + \left( \gamma - 1 \right)(\mathbf \cdot \mathbf)\mathbf \\ \rho' &= \gamma \left(\rho - \mathbf \cdot \frac\right), \end$ or, in the simpler geometric view, $j^ = _\mu j^\mu.$ One says that charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector. The Maxwell equations are invariant under Lorentz transformations.

## Spinors

Equation hold unmodified for any representation of the Lorentz group, including the bispinor representation. In one simply replaces all occurrences of by the bispinor representation , The above equation could, for instance, be the transformation of a state in Fock space describing two free electrons.

### Transformation of general fields

A general ''noninteracting'' multi-particle state (Fock space state) in quantum field theory transforms according to the rule where is the Wigner rotation and is the representation of .

# References

* *

## Papers

English translation
* * * eqn (55). * * * * *

## Books

* * * * * * * * * * * * * * * * * * * * * * * * * * *

* * *

Derivation of the Lorentz transformations
This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.

This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.

– a chapter from an online textbook
Warp Special Relativity Simulator
A computer program demonstrating the Lorentz transformations on everyday objects. * visualizing the Lorentz transformation.
MinutePhysics video
on YouTube explaining and visualizing the Lorentz transformation with a mechanical Minkowski diagram
Interactive graph
on Desmos (graphing) showing Lorentz transformations with a virtual Minkowski diagram
Interactive graph
on Desmos showing Lorentz transformations with points and hyperbolas

''from John de Pillis.'' Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, ''etc''. {{Authority control Special relativity Theoretical physics Mathematical physics Spacetime Coordinate systems Hendrik Lorentz