In

^{+}(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 ,
$$B\_x\; =\; I\; +\; \backslash zeta\; \backslash left.\; \backslash frac\; \backslash \_\; +\; \backslash cdots$$
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 ,
$$\backslash left.\; \backslash frac\; \backslash \_\; =\; -\; K\_x\; \backslash ,.$$
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
$$B\_x\; =\backslash lim\_\backslash left(I-\backslash fracK\_x\backslash right)^\; =\; e^$$
where the Exponential function#Formal definition, limit definition of the exponential has been used (see also characterizations of the exponential function). More generallyExplicitly,
$$\backslash boldsymbol\; \backslash cdot\backslash mathbf\; =\; \backslash zeta\_x\; K\_x\; +\; \backslash zeta\_y\; K\_y\; +\; \backslash zeta\_z\; K\_z$$
$$\backslash boldsymbol\; \backslash cdot\backslash mathbf\; =\; \backslash theta\_x\; J\_x\; +\; \backslash theta\_y\; J\_y\; +\; \backslash theta\_z\; J\_z$$
$$B(\backslash boldsymbol)\; =\; e^\; \backslash ,\; ,\; \backslash quad\; R(\backslash boldsymbol)\; =\; e^\; \backslash ,.$$
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,

English translation

* * * eqn (55). * * * * *

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

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
$$\backslash begin\; t\text{'}\; \&=\; \backslash gamma\; \backslash left(\; t\; -\; \backslash frac\; \backslash right)\; \backslash \backslash \; x\text{'}\; \&=\; \backslash gamma\; \backslash left(\; x\; -\; v\; t\; \backslash right)\backslash \backslash \; y\text{'}\; \&=\; y\; \backslash \backslash \; z\text{'}\; \&=\; z\; \backslash end$$
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 $\backslash gamma\; =\; \backslash left\; (\; \backslash sqrt\backslash right\; )^$ 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 , $\backslash gamma$ grows without bound. The value of must be smaller than for the transformation to make sense.
Expressing the speed as $\backslash beta\; =\; \backslash frac,$ an equivalent form of the transformation is
$$\backslash begin\; ct\text{'}\; \&=\; \backslash gamma\; \backslash left(\; c\; t\; -\; \backslash beta\; x\; \backslash right)\; \backslash \backslash \; x\text{'}\; \&=\; \backslash gamma\; \backslash left(\; x\; -\; \backslash beta\; ct\; \backslash right)\; \backslash \backslash \; y\text{'}\; \&=\; y\; \backslash \backslash \; z\text{'}\; \&=\; z.\; \backslash end$$
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—includingWoldemar 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 ...

had shown from 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 alinear 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 thespeed 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
$$\backslash gamma\; =\; \backslash frac$$
(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
$$\backslash begin\; ct\text{'}\; \&=\; \backslash gamma\; \backslash left(\; ct\; -\; \backslash beta\; x\; \backslash right)\; \backslash ,,\; \backslash \backslash \; x\text{'}\; \&=\; \backslash gamma\; \backslash left(\; x\; -\; \backslash beta\; ct\; \backslash right)\; \backslash ,,\; \backslash \backslash \; \backslash end$$
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
$$\backslash cosh^2\backslash zeta\; -\; \backslash sinh^2\backslash zeta\; =\; 1\; \backslash ,.$$
Conversely the and axes can be constructed for varying coordinates but constant . The definition
$$\backslash tanh\backslash zeta\; =\; \backslash frac\; \backslash ,,$$
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
$$\backslash cosh\backslash zeta\; =\; \backslash frac\; \backslash ,.$$
Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between , , and are
$$\backslash begin\; \backslash beta\; \&=\; \backslash tanh\backslash zeta\; \backslash ,,\; \backslash \backslash \; \backslash gamma\; \&=\; \backslash cosh\backslash zeta\; \backslash ,,\; \backslash \backslash \; \backslash beta\; \backslash gamma\; \&=\; \backslash sinh\backslash zeta\; \backslash ,.\; \backslash end$$
Taking the inverse hyperbolic tangent gives the rapidity $$\backslash zeta\; =\; \backslash tanh^\backslash beta\; \backslash ,.$$
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;
$$\backslash begin\; \backslash Delta\; t\text{'}\; \&=\; \backslash gamma\; \backslash left(\; \backslash Delta\; t\; -\; \backslash frac\; \backslash right)\; \backslash ,,\; \backslash \backslash \; \backslash Delta\; x\text{'}\; \&=\; \backslash gamma\; \backslash left(\; \backslash Delta\; x\; -\; v\; \backslash ,\; \backslash Delta\; t\; \backslash right)\; \backslash ,,\; \backslash end$$
with inverse relations
$$\backslash begin\; \backslash Delta\; t\; \&=\; \backslash gamma\; \backslash left(\; \backslash Delta\; t\text{'}\; +\; \backslash frac\; \backslash right)\; \backslash ,,\; \backslash \backslash \; \backslash Delta\; x\; \&=\; \backslash gamma\; \backslash left(\; \backslash Delta\; x\text{'}\; +\; v\; \backslash ,\; \backslash Delta\; t\text{'}\; \backslash right)\; \backslash ,.\; \backslash end$$
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 theGalilean 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 ...

$$\backslash begin\; t\text{'}\; \&\backslash approx\; t\; \backslash \backslash \; x\text{'}\; \&\backslash approx\; x\; -\; vt\; \backslash end$$
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 $\backslash Delta\; t\text{'}\; =\; \backslash gamma\; \backslash frac$, 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 relativevelocity 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 ,
$$\backslash mathbf=\backslash mathbf\_\backslash perp+\backslash mathbf\_\backslash ,\; \backslash ,,\backslash quad\; \backslash mathbf\text{'}\; =\; \backslash mathbf\_\backslash perp\text{'}\; +\; \backslash mathbf\_\backslash ,\; \text{'}\; \backslash ,,$$
then the transformations are
$$\backslash begin\; t\text{'}\; \&=\; \backslash gamma\; \backslash left(t\; -\; \backslash frac\; \backslash right)\; \backslash \backslash \; \backslash mathbf\_\backslash ,\; \text{'}\; \&=\; \backslash gamma\; (\backslash mathbf\_\backslash ,\; -\; \backslash mathbf\; t)\; \backslash \backslash \; \backslash mathbf\_\backslash perp\text{'}\; \&=\; \backslash mathbf\_\backslash perp\; \backslash end$$
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
$$\backslash mathbf\_\backslash parallel\; =\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf\backslash ,,\backslash quad\; \backslash mathbf\_\backslash perp\; =\; \backslash mathbf\; -\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf$$
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
$$\backslash boldsymbol\; =\; \backslash beta\; \backslash mathbf\; =\; \backslash mathbf\; \backslash tanh\backslash zeta\; \backslash ,,$$
and the "rapidity vector" can be defined as
$$\backslash boldsymbol\; =\; \backslash zeta\backslash mathbf\; =\; \backslash mathbf\backslash tanh^\backslash beta\; \backslash ,,$$
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 :$\backslash mathbf\; =\; \backslash frac\; \backslash ,,\backslash quad\; \backslash mathbf\text{'}\; =\; \backslash frac\; \backslash ,,\backslash quad\; \backslash gamma\_\backslash mathbf\; =\; \backslash frac$ taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to :$\backslash mathbf\text{\'}\; =\; \backslash frac\backslash left;\; href="/html/ALL/s/frac\_-\_\backslash mathbf\_+\_\backslash frac\backslash frac\backslash left(\backslash mathbf\backslash cdot\backslash mathbf\backslash right)\backslash mathbf\backslash right.html"\; ;"title="frac\; -\; \backslash mathbf\; +\; \backslash frac\backslash frac\backslash left(\backslash mathbf\backslash cdot\backslash mathbf\backslash right)\backslash mathbf\backslash right">frac\; -\; \backslash mathbf\; +\; \backslash frac\backslash frac\backslash left(\backslash mathbf\backslash cdot\backslash mathbf\backslash right)\backslash mathbf\backslash right$ 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 instellar 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 $$A^2\; -\; \backslash mathbf\backslash cdot\backslash mathbf\; =\; ^2\; -\; \backslash mathbf\text{'}\backslash cdot\backslash mathbf\text{'}$$ implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates; $$\backslash begin\; A\text{'}\; \&=\; \backslash gamma\; \backslash left(A\; -\; \backslash frac\; \backslash right)\; \backslash ,,\; \backslash \backslash \; \backslash mathbf\text{'}\; \&=\; \backslash mathbf\; +\; (\backslash gamma-1)(\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf\; -\; \backslash frac\; \backslash ,.\; \backslash end$$ 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 theMinkowski 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
$$X\text{'}\; =\; \backslash begin\; c\backslash ,t\text{'}\; \backslash \backslash \; x\text{'}\; \backslash \backslash \; y\text{'}\; \backslash \backslash \; z\text{'}\; \backslash end\; \backslash ,,\; \backslash quad\; \backslash eta\; =\; \backslash begin\; -1\&0\&0\&0\backslash \backslash \; 0\&1\&0\&0\; \backslash \backslash \; 0\&0\&1\&0\; \backslash \backslash \; 0\&0\&0\&1\; \backslash end\; \backslash ,,\; \backslash quad\; X\; =\; \backslash begin\; c\backslash ,t\; \backslash \backslash \; x\; \backslash \backslash \; y\; \backslash \backslash \; z\; \backslash end$$
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 ...

)
$$X\; \backslash cdot\; X\; =\; X^\backslash mathrm\; \backslash eta\; X\; =\; ^\backslash mathrm\; \backslash eta$$
and is invariant under a Lorentz transformation
$$X\text{'}\; =\; \backslash Lambda\; X$$
where is a square matrix which can depend on parameters.
The set of all Lorentz transformations Λ in this article is denoted $\backslash mathcal$. 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
$$\backslash eta\; =\; \backslash Lambda^\backslash mathrm\; \backslash eta\; \backslash Lambda$$
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
$$\backslash left;\; href="/html/ALL/s/det\_(\backslash Lambda)\backslash right.html"\; ;"title="det\; (\backslash Lambda)\backslash right">det\; (\backslash Lambda)\backslash right$$
Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,
$$\backslash eta\; =\; \backslash begin-1\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash mathbf\backslash end\; \backslash ,,\; \backslash quad\; \backslash Lambda=\backslash begin\backslash Gamma\; \&\; -\backslash mathbf^\backslash mathrm\backslash \backslash -\backslash mathbf\; \&\; \backslash mathbf\backslash end\; \backslash ,,$$
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
$$\backslash Gamma^2\; =\; 1\; +\; \backslash mathbf^\backslash mathrm\backslash mathbf$$
is useful; always so it follows that
$$\backslash Gamma^2\; \backslash geq\; 1\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash Gamma\; \backslash leq\; -\; 1\; \backslash ,,\backslash quad\; \backslash Gamma\; \backslash geq\; 1$$
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
$$\backslash mathcal\; =\; \backslash mathcal\_^\backslash uparrow\; \backslash cup\; \backslash mathcal\_^\backslash uparrow\; \backslash cup\; \backslash mathcal\_^\backslash downarrow\; \backslash cup\; \backslash mathcal\_^\backslash downarrow$$
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 $\backslash mathcal\_+^\backslash uparrow$, $\backslash mathcal\_+$, $\backslash mathcal^\backslash uparrow$, and $\backslash mathcal\_0\; =\; \backslash mathcal\_+^\backslash uparrow\; \backslash cup\; \backslash mathcal\_^\backslash downarrow$ all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g. $\backslash mathcal\_+^\backslash downarrow$, $\backslash mathcal\_^\backslash downarrow$, $\backslash mathcal\_^\backslash uparrow$) 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 $$X\text{'}\; =\; B(\backslash mathbf)X$$ where the boost matrix $B(\backslash mathbf)$ represents the Lorentz transformation between the unprimed and primed frames and $\backslash mathbf$ is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by $$B(\backslash mathbf)\; =\; \backslash begin\; \backslash gamma\; \&-\backslash gamma\; v\_x/c\; \&-\backslash gamma\; v\_y/c\; \&-\backslash gamma\; v\_z/c\; \backslash \backslash \; -\backslash gamma\; v\_x/c\&1+(\backslash gamma-1)\backslash dfrac\; \&\; (\backslash gamma-1)\backslash dfrac\&\; (\backslash gamma-1)\backslash dfrac\; \backslash \backslash \; -\backslash gamma\; v\_y/c\&\; (\backslash gamma-1)\backslash dfrac\&1+(\backslash gamma-1)\backslash dfrac\; \&\; (\backslash gamma-1)\backslash dfrac\; \backslash \backslash \; -\backslash gamma\; v\_z/c\&\; (\backslash gamma-1)\backslash dfrac\&\; (\backslash gamma-1)\backslash dfrac\&1+(\backslash gamma-1)\backslash dfrac\; \backslash end,$$ where $v=\backslash sqrt$ is the magnitude of the velocity and $\backslash gamma=\backslash frac$ 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 $B(-\backslash mathbf)$. If a frame is boosted with velocity relative to frame , and another frame is boosted with velocity relative to , the separate boosts are $$X\text{'}\text{'}\; =\; B(\backslash mathbf)X\text{'}\; \backslash ,,\; \backslash quad\; X\text{'}\; =\; B(\backslash mathbf)X$$ and the composition of the two boosts connects the coordinates in and , $$X\text{'}\text{'}\; =\; B(\backslash mathbf)B(\backslash mathbf)X\; \backslash ,.$$ Successive transformations act on the left. If and arecollinear
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
$$\backslash quad\; R(\backslash boldsymbol)\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash mathbf(\backslash boldsymbol)\; \backslash end\; \backslash ,,$$
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"
$$\backslash boldsymbol\; =\; \backslash theta\; \backslash mathbf$$
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)

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 .
$$K\_x\; =\; \backslash begin\; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; \backslash end\backslash ,,\backslash quad\; K\_y\; =\; \backslash begin\; 0\; \&\; 0\; \&\; 1\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash end\backslash ,,\backslash quad\; K\_z\; =\; \backslash begin\; 0\; \&\; 0\; \&\; 0\; \&\; 1\backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash end$$
$$J\_x\; =\; \backslash begin\; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; -1\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; \backslash end\backslash ,,\backslash quad\; J\_y\; =\; \backslash begin\; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; -1\; \&\; 0\; \&\; 0\; \backslash end\backslash ,,\backslash quad\; J\_z\; =\; \backslash begin\; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; -1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash end$$
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
$$B()=I-\backslash sinh\; \backslash zeta\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf\; )+(\backslash cosh\; \backslash zeta\; -1)(\backslash mathbf\; \backslash cdot\; \backslash mathbf\; )^2$$
$$R(\backslash boldsymbol\; )=I+\backslash sin\; \backslash theta\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf\; )+(1-\backslash cos\; \backslash theta\; )(\backslash mathbf\; \backslash cdot\; \backslash mathbf\; )^2\backslash ,.$$
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
$$\backslash begin\; \backslash Lambda\; \&=\; (I\; -\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; +\; \backslash cdots\; )(I\; +\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; +\; \backslash cdots\; )\; \backslash \backslash \; \&=\; (I\; +\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; +\; \backslash cdots\; )(I\; -\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; +\; \backslash cdots\; )\; \backslash \backslash \; \&=\; I\; -\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; +\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; +\; \backslash cdots\; \backslash end$$
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
$$\backslash Lambda\; (\backslash boldsymbol,\; \backslash boldsymbol)\; =\; e^.$$
The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,
$$e^\; \backslash ne\; e^\; e^,$$
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 $$V\; =\; \backslash $$ 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
$$[\; J\_x,\; J\_y\; ]\; =\; J\_z\; \backslash ,,\backslash quad\; [\; K\_x,\; K\_y\; ]\; =\; -J\_z\; \backslash ,,\backslash quad\; [\; J\_x,\; K\_y\; ]\; =\; K\_z\; \backslash ,,$$
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 $\backslash mathfrak(3,\; 1)$. 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,
$$\backslash exp\; \backslash ,\; :\; \backslash ,\; \backslash mathfrak(3,1)\; \backslash to\; \backslash mathrm(3,1),$$
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 $$P\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; -\; \backslash mathbf\; \backslash end$$ which negates all the spatial coordinates only, and T-symmetry, time reversal $$T\; =\; \backslash begin\; -\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash mathbf\; \backslash end$$ 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 $$X\text{'}\; =\; \backslash Lambda\; X\; +\; C$$ 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 $$\backslash begin\; ^0\; \backslash \backslash \; ^1\; \backslash \backslash \; ^2\; \backslash \backslash \; ^3\; \backslash end\; =\; \backslash begin\; \_0\; \&\; \_1\; \&\; \_2\; \&\; \_3\; \backslash \backslash \; \_0\; \&\; \_1\; \&\; \_2\; \&\; \_3\; \backslash \backslash \; \_0\; \&\; \_1\; \&\; \_2\; \&\; \_3\; \backslash \backslash \; \_0\; \&\; \_1\; \&\; \_2\; \&\; \_3\; \backslash \backslash \; \backslash end\; \backslash begin\; x^0\; \backslash \backslash \; x^1\; \backslash \backslash \; x^2\; \backslash \backslash \; x^3\; \backslash end$$ 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 $$^\backslash nu\; =\; \_\backslash mu\; x^\backslash mu,$$ 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 allfour-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 $$^\backslash nu\; =\; \_\backslash mu\; A^\backslash mu\; \backslash ,.$$
Alternatively, one writes $$A^\; =\; \_\backslash mu\; A^\backslash mu\; \backslash ,.$$ 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 $$^\backslash alpha\; =\; \_\backslash beta\; X^\backslash beta\; \backslash ,,$$ 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., $$x\_\backslash nu\; =\; \backslash eta\_x^\backslash mu,$$ 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 $$x^\backslash mu\; =\; \backslash eta^x\_\backslash nu,$$ 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; $$\_\backslash nu\; =\; \backslash eta\_\; \_\backslash sigma\; \backslash eta^A\_\backslash mu.$$ But $$\backslash eta\_\; \_\backslash sigma\; \backslash eta^\; =\; \_\backslash nu,$$ That is, it is the -component of the ''inverse'' Lorentz transformation. One defines (as a matter of notation), $$^\backslash mu\; \backslash equiv\; \_\backslash nu,$$ and may in this notation write $$\_\backslash nu\; =\; ^\backslash mu\; A\_\backslash mu.$$ Now for a subtlety. The implied summation on the right hand side of $$\_\backslash nu\; =\; ^\backslash mu\; A\_\backslash mu\; =\; \_\backslash nu\; A\_\backslash mu$$ 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, $$A\text{'}\; =\; \backslash left(\backslash Lambda^\backslash right)^\backslash mathrm\; A.$$ 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 themagnetic 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
$$F^\; =\; \backslash begin\; 0\; \&\; -\backslash fracE\_x\; \&\; -\backslash fracE\_y\; \&\; -\backslash fracE\_z\; \backslash \backslash \; \backslash fracE\_x\; \&\; 0\; \&\; -B\_z\; \&\; B\_y\; \backslash \backslash \; \backslash fracE\_y\; \&\; B\_z\; \&\; 0\; \&\; -B\_x\; \backslash \backslash \; \backslash fracE\_z\; \&\; -B\_y\; \&\; B\_x\; \&\; 0\; \backslash end\; \backslash text(+,-,-,-)\backslash text.$$
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
$$\_\backslash nu\; =\; \backslash begin\; \backslash gamma\; \&\; -\backslash gamma\backslash beta\; \&\; 0\; \&\; 0\backslash \backslash \; -\backslash gamma\backslash beta\; \&\; \backslash gamma\; \&\; 0\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\backslash \backslash \; \backslash end,\; \backslash qquad\; F^\; =\; \backslash begin\; 0\; \&\; E\_x\; \&\; E\_y\; \&\; E\_z\; \backslash \backslash \; -E\_x\; \&\; 0\; \&\; B\_z\; \&\; -B\_y\; \backslash \backslash \; -E\_y\; \&\; -B\_z\; \&\; 0\; \&\; B\_x\; \backslash \backslash \; -E\_z\; \&\; B\_y\; \&\; -B\_x\; \&\; 0\; \backslash end\; \backslash text(-,+,+,+)\backslash text,$$
where the field tensor is displayed side by side for easiest possible reference in the manipulations below.
The general transformation law becomes
$$F^\; =\; \_\backslash mu\; \_\backslash nu\; F^.$$
For the magnetic field one obtains
$$\backslash begin\; B\_\; \&=\; F^\; =\; \_\backslash mu\; \_\backslash nu\; F^\; =\; \_2\; \_3\; F^\; =\; 1\; \backslash times\; 1\; \backslash times\; B\_x\; \backslash \backslash \; \&=\; B\_x,\; \backslash \backslash \; B\_\; \&=\; F^\; =\; \_\backslash mu\; \_\backslash nu\; F^\; =\; \_3\; \_\backslash nu\; F^\; =\; \_3\; \_0\; F^\; +\; \_3\; \_1\; F^\; \backslash \backslash \; \&=\; 1\; \backslash times\; (-\backslash beta\backslash gamma)\; (-E\_z)\; +\; 1\; \backslash times\; \backslash gamma\; B\_y\; =\; \backslash gamma\; B\_y\; +\; \backslash beta\backslash gamma\; E\_z\; \backslash \backslash \; \&=\; \backslash gamma\backslash left(\backslash mathbf\; -\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\backslash right)\_y\; \backslash \backslash \; B\_\; \&=\; F^\; =\; \_\backslash mu\; \_\backslash nu\; F^\; =\; \_\backslash mu\; \_2\; F^\; =\; \_0\; \_2\; F^\; +\; \_1\; \_2\; F^\; \backslash \backslash \; \&=\; (-\backslash gamma\backslash beta)\; \backslash times\; 1\backslash times\; E\_y\; +\; \backslash gamma\; \backslash times\; 1\; \backslash times\; B\_z\; =\; \backslash gamma\; B\_z\; -\; \backslash beta\backslash gamma\; E\_y\; \backslash \backslash \; \&=\; \backslash gamma\backslash left(\backslash mathbf\; -\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\backslash right)\_z\; \backslash end$$
For the electric field results
$$\backslash begin\; E\_\; \&=\; F^\; =\; \_\backslash mu\; \_\backslash nu\; F^\; =\; \_1\; \_0\; F^\; +\; \_0\; \_1\; F^\; \backslash \backslash \; \&=\; (-\backslash gamma\backslash beta)(-\backslash gamma\backslash beta)(-E\_x)\; +\; \backslash gamma\backslash gamma\; E\_x\; =\; -\backslash gamma^2\backslash beta^2(E\_x)\; +\; \backslash gamma^2\; E\_x\; =\; E\_x(1\; -\; \backslash beta^2)\backslash gamma^2\; \backslash \backslash \; \&=\; E\_x,\; \backslash \backslash \; E\_\; \&=\; F^\; =\; \_\backslash mu\; \_\backslash nu\; F^\; =\; \_\backslash mu\; \_2\; F^\; =\; \_0\; \_2\; F^\; +\; \_1\; \_2\; F^\; \backslash \backslash \; \&=\; \backslash gamma\; \backslash times\; 1\; \backslash times\; E\_y\; +\; (-\backslash beta\backslash gamma)\; \backslash times\; 1\; \backslash times\; B\_z\; =\; \backslash gamma\; E\_y\; -\; \backslash beta\backslash gamma\; B\_z\; \backslash \backslash \; \&=\; \backslash gamma\backslash left(\backslash mathbf\; +\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\backslash right)\_y\; \backslash \backslash \; E\_\; \&=\; F^\; =\; \_\backslash mu\; \_\backslash nu\; F^\; =\; \_\backslash mu\; \_3\; F^\; =\; \_0\; \_3\; F^\; +\; \_1\; \_3\; F^\; \backslash \backslash \; \&=\; \backslash gamma\; \backslash times\; 1\; \backslash times\; E\_z\; -\; \backslash beta\backslash gamma\; \backslash times\; 1\; \backslash times\; (-B\_y)\; =\; \backslash gamma\; E\_z\; +\; \backslash beta\backslash gamma\; B\_y\; \backslash \backslash \; \&=\; \backslash gamma\backslash left(\backslash mathbf\; +\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\backslash right)\_z.\; \backslash end$$
Here, is used. These results can be summarized by
$$\backslash begin\; \backslash mathbf\_\; \&=\; \backslash mathbf\_\backslash parallel\; \backslash \backslash \; \backslash mathbf\_\; \&=\; \backslash mathbf\_\backslash parallel\; \backslash \backslash \; \backslash mathbf\_\; \&=\; \backslash gamma\; \backslash left(\; \backslash mathbf\_\backslash bot\; +\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\_\backslash bot\; \backslash right)\; =\; \backslash gamma\; \backslash left(\; \backslash mathbf\; +\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\; \backslash right)\_\backslash bot,\backslash \backslash \; \backslash mathbf\_\; \&=\; \backslash gamma\; \backslash left(\; \backslash mathbf\_\backslash bot\; -\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\_\backslash bot\; \backslash right)\; =\; \backslash gamma\; \backslash left(\; \backslash mathbf\; -\; \backslash boldsymbol\; \backslash times\; \backslash mathbf\; \backslash right)\_\backslash bot,\; \backslash end$$
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
$$F^\; =\; \_\backslash mu\; \_\backslash nu\; F^,$$
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
$$F^\backslash left(x\text{'}\backslash right)\; =\; \_\backslash mu\; \_\backslash nu\; F^\backslash left(\backslash Lambda^\; x\text{'}\backslash right)\; =\; \_\backslash mu\; \_\backslash nu\; F^(x).$$
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,
$$\backslash begin\; \backslash mathbf\text{'}\; \&=\; \backslash mathbf\; -\; \backslash gamma\backslash rho\; v\backslash mathbf\; +\; \backslash left(\; \backslash gamma\; -\; 1\; \backslash right)(\backslash mathbf\; \backslash cdot\; \backslash mathbf)\backslash mathbf\; \backslash \backslash \; \backslash rho\text{'}\; \&=\; \backslash gamma\; \backslash left(\backslash rho\; -\; \backslash mathbf\; \backslash cdot\; \backslash frac\backslash right),\; \backslash end$$
or, in the simpler geometric view,
$$j^\; =\; \_\backslash mu\; j^\backslash mu.$$
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 .See also

Footnotes

Notes

References

Websites

* *Papers

* * * * * * * . See alsoEnglish translation

* * * eqn (55). * * * * *

Books

* * * * * * * * * * * * * * * * * * * * * * * * * * *Further reading

* * *External links

Derivation of the Lorentz transformations

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

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