View source for Lorentz transformation

{{Short description|Family of linear transformations}}
[[File:Hendrik Antoon Lorentz.jpg|thumb|[[Hendrik Lorentz]] in 1916.]]
{{spacetime|cTopic=Mathematics}}

In [[physics]], the '''Lorentz transformations''' are a six-parameter family of [[Linear transformation|linear]] [[coordinate transformation|transformations]] from a [[Frame of Reference|coordinate frame]] in [[spacetime]] 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]] [[Hendrik Lorentz]].

The most common form of the transformation, parametrized by the real constant <math>v,</math> representing a velocity confined to the {{mvar|x}}-direction, is expressed as<ref>{{cite book |title=The Rotation and Lorentz Groups and Their Representations for Physicists |edition=illustrated |last1= Rao|first1= K. N. Srinivasa |publisher=John Wiley & Sons |year=1988 |isbn=978-0-470-21044-4 |page=213 |url=https://books.google.com/books?id=XRJIsf5zoM0C}} [https://books.google.com/books?id=XRJIsf5zoM0C&pg=PA210 Equation 6-3.24, page 210]</ref><ref>{{harvnb|Forshaw|Smith|2009}}</ref>
<math display="block">\begin{align}
t' &= \gamma \left( t  - \frac{vx}{c^2} \right)  \\
x' &= \gamma \left( x - v t \right)\\
y' &= y \\
z' &= z
\end{align}</math>
where {{math|(''t'', ''x'', ''y'', ''z'')}} and {{math|(''t''′, ''x''′, ''y''′, ''z''′)}} are the coordinates of an event in two frames with the spatial origins coinciding at {{math|1=''t'' = ''t''′ = 0}}, where the primed frame is seen from the unprimed frame as moving with speed {{mvar|v}} along the {{mvar|x}}-axis, where {{mvar|c}} is the [[speed of light]], and 
<math display="block"> \gamma = \frac{1}{\sqrt{1 - v^2 / c^2 }}</math>
is the [[Lorentz factor]]. When speed {{mvar|v}} is much smaller than {{mvar|c}}, the Lorentz factor is negligibly different from 1, but as {{mvar|v}} approaches {{mvar|c}}, <math>\gamma</math> grows without bound. The value of {{mvar|v}} must be smaller than {{mvar|c}} for the transformation to make sense.

Expressing the speed as a fraction of the speed of light, <math display="inline"> \beta = v/c,</math> an equivalent form of the transformation is<ref>{{harvnb|Cottingham|Greenwood|2007|p=[https://books.google.com/books?id=Dm36BYq9iu0C&pg=PA21 21]}}</ref>
<math display="block">\begin{align}
ct' &= \gamma \left( c t - \beta x \right)  \\
x' &= \gamma \left( x - \beta ct \right)  \\
y' &= y \\
z' &= z.
\end{align}</math>

Frames of reference can be divided into two groups: [[Inertial frame of reference|inertial]] (relative motion with constant velocity) and [[Non-inertial reference frame|non-inertial]] (accelerating, moving in curved paths, rotational motion with constant [[angular velocity]], etc.). The term "Lorentz transformations" only refers to transformations between ''inertial'' frames, usually in the context of special relativity.

In each [[frame of reference|reference frame]], an observer can use a local coordinate system (usually [[Cartesian coordinates]] in this context) to measure lengths, and a clock to measure time intervals. An [[Event (relativity)|event]] is something that happens at a point in space at an instant of time, or more formally a point in [[spacetime]]. The transformations connect the space and time coordinates of an [[Event (relativity)|event]] as measured by an observer in each frame.<ref group=nb>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., {{harvtxt|Sard|1970}}.</ref>

They supersede the [[Galilean transformation]] of [[Newtonian physics]], which assumes an [[absolute space and time]] (see ''[[Galilean relativity]]''). The Galilean transformation is a good approximation only at relative speeds much 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 [[velocity|velocities]] may measure different [[Length contraction|distances]], [[time dilation|elapsed times]], and even different [[Relativity of simultaneity|orderings of events]], but always such that the [[speed of light]] is the same in all inertial reference frames. The invariance of light speed is one of the [[postulates of special relativity]].

Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of [[light]] was observed to be independent of the [[frame of reference|reference frame]], and to understand the symmetries of the laws of [[electromagnetism]]. The transformations later became a cornerstone for [[special relativity]].

The Lorentz transformation is a [[linear transformation]]. It may include a rotation of space; a rotation-free Lorentz transformation is called a '''Lorentz boost'''. In [[Minkowski space]]—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the [[spacetime interval]] between any two events. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a [[hyperbolic rotation]] of Minkowski space. The more general set of transformations that also includes translations is known as the [[Poincaré group]].

== History ==
{{main|History of Lorentz transformations}}

Many physicists—including [[Woldemar Voigt]], [[George Francis FitzGerald|George FitzGerald]], [[Joseph Larmor]], and [[Hendrik Lorentz]]<ref>{{harvnb|Lorentz|1904}}</ref> himself—had been discussing the physics implied by these equations since 1887.<ref>{{harvnb|O'Connor|Robertson|1996}}</ref> Early in 1889, [[Oliver Heaviside]] had shown from [[Maxwell's equations]] that the [[electric field]] surrounding a spherical distribution of charge should cease to have [[spherical symmetry]] once the charge is in motion relative to the [[luminiferous aether]]. 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 [[Michelson–Morley experiment|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 [[Length contraction|FitzGerald–Lorentz contraction hypothesis]].<ref>{{harvnb|Brown|2003}}</ref> Their explanation was widely known before 1905.<ref>{{harvnb|Rothman|2006|pages = 112f}}</ref>

Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous aether hypothesis, also looked for the transformation under which [[Maxwell's equations]] are invariant when transformed from the aether to a moving frame. They extended the [[Fitzgerald–Lorentz contraction|FitzGerald–Lorentz contraction]] hypothesis and found out that the time coordinate has to be modified as well ("[[relativity of simultaneity|local time]]"). [[Henri Poincaré]] gave a physical interpretation to local time (to first order in {{math|''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.<ref>{{harvnb|Darrigol|2005|pages=1–22}}</ref> Larmor is credited to have been the first to understand the crucial [[time dilation]] property inherent in his equations.<ref>
{{harvnb|Macrossan|1986|pages=232–34}}</ref>

In 1905, Poincaré was the first to recognize that the transformation has the properties of a [[group (mathematics)|mathematical group]],
and he named it after Lorentz.<ref>The reference is within the following paper:{{harvnb|Poincaré|1905|pages = 1504–1508}}</ref>
Later in the same year [[Albert Einstein]] published what is now called [[special relativity]], by deriving the Lorentz transformation under the assumptions of the [[principle of relativity]] and the constancy of the speed of light in any [[inertial reference frame]], and by abandoning the mechanistic aether as unnecessary.<ref>{{harvnb|Einstein|1905|pages=891–921}}</ref>

== Derivation of the group of Lorentz transformations ==
{{Main|Derivations of the Lorentz transformations|Lorentz group}}

An ''[[Event (relativity)|event]]'' 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 {{math|''ct''}} and a set of [[Cartesian coordinate]]s {{mvar|x}}, {{mvar|y}}, {{mvar|z}} to specify position in space in that frame. Subscripts label individual events.

From Einstein's [[Postulates of special relativity|second postulate of relativity]] (invariance of [[Speed of light|{{mvar|c}}]]) it follows that:
{{NumBlk||<math display="block">c^2(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 = 0 \quad \text{(lightlike separated events 1, 2)}</math>|{{EquationRef|D1}}}}
in all inertial frames for events connected by ''light signals''. The quantity on the left is called the [[spacetime interval]] between events {{math|1=''a''{{sub|1}} = (''t''{{sub|1}}, ''x''{{sub|1}}, ''y''{{sub|1}}, ''z''{{sub|1}})}} and {{math|1=''a''{{sub|2}} = (''t''{{sub|2}}, ''x''{{sub|2}}, ''y''{{sub|2}}, ''z''{{sub|2}})}}. 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 [[Derivations of the Lorentz transformations#Invariance of interval|shown using homogeneity and isotropy of space]]. The transformation sought after thus must possess the property that:
{{NumBlk||<math display="block"> \begin{align}
c^2(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 & = & \\[6pt]
c^2(t_2' - t_1')^2 - (x_2' - x_1')^2 - (y_2' - y_1')^2 - (z_2' - z_1')^2 & & \quad \text{(all events 1, 2)}.
\end{align}
</math>|{{EquationRef|D2}}}}
where {{math|(''t'', ''x'', ''y'', ''z'')}} are the spacetime coordinates used to define events in one frame, and {{math|(''t''′, ''x''′, ''y''′, ''z''′)}} are the coordinates in another frame. First one observes that ({{EquationNote|D2}}) is satisfied if an arbitrary {{math|4}}-tuple {{mvar|b}} of numbers are added to events {{math|''a''{{sub|1}}}} and {{math|''a''{{sub|2}}}}. 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:
{{NumBlk||<math display="block">\begin{align}
c^2t^2 - x^2 - y^2 - z^2 &= c^2t'^2 - x'^2 - y'^2 - z'^2 & \quad \text{or} \\[6pt]
c^2t_1t_2 - x_1x_2 - y_1y_2 - z_1z_2 &= c^2t'_1t'_2 - x'_1x'_2 - y'_1y'_2 - z'_1z'_2 &
\end{align}</math>|{{EquationRef|D3}}}}
(a solution satisfying the first formula automatically satisfies the second one as well; see ''[[Polarization identity]]''). Finding the solution to the simpler problem is just a matter of look-up in the theory of [[classical group]]s that preserve [[bilinear form]]s of various signature.<ref group=nb>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]] or the ''inhomogeneous Lorentz group''. The first equation (or the second restricted to lightlike separation) leads to a yet larger group, the [[conformal group]] of spacetime.</ref> First equation in ({{EquationNote|D3}}) can be written more compactly as:
{{NumBlk||<math display="block">(a, a) = (a', a') \quad \text{or} \quad a \cdot a = a' \cdot a',</math>|{{EquationRef|D4}}}}
where {{math|(·, ·)}} refers to the bilinear form of [[Signature (quadratic form)|signature]] {{math|(1, 3)}} on {{math|'''R'''{{sup|4}}}} exposed by the right hand side formula in ({{EquationNote|D3}}). The alternative notation defined on the right is referred to as the ''relativistic dot product''. Spacetime mathematically viewed as {{math|'''R'''{{sup|4}}}} endowed with this bilinear form is known as [[Minkowski space]] {{mvar|M}}. The Lorentz transformation is thus an element of the group {{math|O(1, 3)}}, the [[Lorentz group]] or, for those that prefer the other [[metric signature]], {{math|O(3, 1)}} (also called the Lorentz group).<ref group=nb>The groups {{math|O(3, 1)}} and {{math|O(1, 3)}} are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related to {{math|O(3, 1)}} and {{math|O(1, 3)}} respectively, e.g., the [[Clifford algebra]]s corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.</ref> One has:
{{NumBlk||<math display="block">(a, a) = (\Lambda a,\Lambda a) = (a', a'), \quad \Lambda \in \mathrm{O}(1, 3), \quad a, a' \in M,</math>|{{EquationRef|D5}}}}
which is precisely preservation of the bilinear form ({{EquationNote|D3}}) which implies (by linearity of {{math|Λ}} and bilinearity of the form) that ({{EquationNote|D2}}) is satisfied. The elements of the Lorentz group are [[Rotation group SO(3)|rotations]] and ''boosts'' and mixes thereof. If the spacetime translations are included, then one obtains the ''inhomogeneous Lorentz group'' or the [[Poincaré group]].

== Generalities ==
The relations between the primed and unprimed spacetime coordinates are the '''Lorentz transformations''', each coordinate in one frame is a [[linear function]] of all the coordinates in the other frame, and the [[inverse function]]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.

{{anchor|boost}}Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are called '''Lorentz boosts''' or simply ''boosts'', and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformation is only a rotation in the spatial coordinates. Unlike boosts, these 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]], or [[Euler angle]]s, etc.). A combination of a rotation and a boost is a ''homogeneous transformation'', which transforms the origin back to the origin.

The full Lorentz group {{math|O(3, 1)}} also contains special transformations that are neither rotations nor boosts, but rather [[Reflection (mathematics)|reflection]]s in a plane through the origin. Two of these can be singled out; [[P-symmetry|spatial inversion]] in which the spatial coordinates of all events are reversed in sign and [[T-symmetry|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 ==
{{further|Derivations of the Lorentz transformations}}

=== Coordinate transformation ===
{{anchor|Coordinate transformation}} <!-- "Spacetime" links here -->
[[File:Lorentz boost x direction standard configuration.svg|thumb|right|upright=1.75|The spacetime coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles.<br />'''Top:''' frame {{math|''F''{{′}}}} moves at velocity {{mvar|v}} along the {{mvar|x}}-axis of frame {{mvar|F}}.<br />'''Bottom:''' frame {{mvar|F}} moves at velocity −{{mvar|v}} along the {{math|''x''′}}-axis of frame {{math|''F''{{′}}}}.<ref>{{harvnb|Young|Freedman|2008}}</ref>]]

A "stationary" observer in frame {{mvar|F}} defines events with coordinates {{mvar|t}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}}. Another frame {{math|''F''{{′}}}} moves with velocity {{mvar|v}} relative to {{mvar|F}}, and an observer in this "moving" frame {{math|''F''{{′}}}} defines events using the coordinates {{math|''t''′}}, {{math|''x''′}}, {{math|''y''′}}, {{math|''z''′}}.

The coordinate axes in each frame are parallel (the {{mvar|x}} and {{math|''x''′}} axes are parallel, the {{mvar|y}} and {{math|''y''′}} axes are parallel, and the {{mvar|z}} and {{math|''z''′}} axes are parallel), remain mutually perpendicular, and relative motion is along the coincident {{math|''xx''′}} axes. At {{math|1=''t'' = ''t''′ = 0}}, the origins of both coordinate systems are the same, {{math|1=(''x'', ''y'', ''z'') = (''x''′, ''y''′, ''z''′) = (0, 0, 0)}}. 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 {{mvar|F}} records an event {{mvar|t}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}}, then an observer in {{math|''F''{{′}}}} records the ''same'' event with coordinates<ref>{{harvnb|Forshaw|Smith|2009}}</ref>

{{Equation box 1
|title='''Lorentz boost''' ({{mvar|x}} '' direction'')
|indent =:
|equation =
<math>\begin{align}
  t' &= \gamma \left( t - \frac{v x}{c^2} \right)  \\
  x' &= \gamma \left( x - v t \right)\\
  y' &= y \\
  z' &= z
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

where {{mvar|v}} is the relative velocity between frames in the {{mvar|x}}-direction, {{mvar|c}} is the [[speed of light]], and
<math display="block"> \gamma = \frac{1}{ \sqrt{1 - \frac{v^2}{c^2}}}</math>
(lowercase [[gamma]]) is the [[Lorentz factor]].

Here, {{mvar|v}} is the ''[[parameter]]'' 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 {{math|''v'' > 0}} is motion along the positive directions of the {{math|''xx''′}} axes, zero relative velocity {{math|''v'' {{=}} 0}} is no relative motion, while negative relative velocity {{math|''v'' < 0}} is relative motion along the negative directions of the {{math|''xx''′}} axes. The magnitude of relative velocity {{mvar|v}} cannot equal or exceed {{mvar|c}}, so only subluminal speeds {{math|−''c'' < ''v'' < ''c''}} are allowed. The corresponding range of {{mvar|γ}} is {{math|1 ≤ ''γ'' < ∞}}.

The transformations are not defined if {{mvar|v}} is outside these limits. At the speed of light ({{math|''v'' {{=}} ''c''}}) {{mvar|γ}} is infinite, and [[faster than light]] ({{math|''v'' > ''c''}}) {{mvar|γ}} is a [[complex number]], 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]], an observer in {{math|''F''{{′}}}} notices the coordinates of the event to be "boosted" in the negative directions of the {{math|''xx''′}} axes, because of the {{math|−''v''}} in the transformations. This has the equivalent effect of the ''coordinate system'' {{math|''F''′}} boosted in the positive directions of the {{math|''xx''′}} axes, while the event does not change and is simply represented in another coordinate system, a [[passive transformation]].

The inverse relations ({{mvar|t}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}} in terms of {{math|''t''′}}, {{math|''x''′}}, {{math|''y''′}}, {{math|''z''′}}) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here {{math|''F''{{′}}}} is the "stationary" frame while {{mvar|F}} is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from {{math|''F''′}} to {{mvar|F}} must take exactly the same form as the transformations from {{mvar|F}} to {{math|''F''{{′}}}}. The only difference is {{mvar|F}} moves with velocity {{math|−''v''}} relative to {{math|''F''′}} (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in {{math|''F''′}} notes an event {{math|''t''′}}, {{math|''x''′}}, {{math|''y''′}}, {{math|''z''′}}, then an observer in {{mvar|F}} notes the ''same'' event with coordinates

{{Equation box 1
|title='''Inverse Lorentz boost''' ({{mvar|x}} '' direction'')
|indent =:
|equation =
<math>\begin{align}
  t &= \gamma \left( t' + \frac{v x'}{c^2} \right)  \\
  x &= \gamma \left( x' + v t' \right)\\
  y &= y' \\
  z &= z',
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

and the value of {{mvar|γ}} 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.<ref>{{cite book |title=Special Relativity and How it Works |author1=Moses Fayngold |edition=illustrated |publisher=John Wiley & Sons |year=2008 |isbn=978-3-527-40607-4 |page=102 |url=https://books.google.com/books?id=Q3egk8Ds6ogC}} [https://books.google.com/books?id=Q3egk8Ds6ogC&pg=PA102 Extract of page 102]</ref><ref>{{cite book |title=Advanced University Physics |author1=Mircea S. Rogalski |author2=Stuart B. Palmer |edition=2nd, revised |publisher=CRC Press |year=2018 |isbn=978-1-4200-5712-6 |page=70 |url=https://books.google.com/books?id=cmYIEQAAQBAJ}} [https://books.google.com/books?id=cmYIEQAAQBAJ&pg=PA70 Extract of page 70]</ref>

Sometimes it is more convenient to use {{math|''β'' {{=}} ''v''/''c''}} (lowercase [[beta]]) instead of {{mvar|v}}, so that
<math display="block">\begin{align}
  ct' &= \gamma \left( ct - \beta x \right) \,, \\
   x' &= \gamma \left( x - \beta ct \right) \,, \\
\end{align}</math>
which shows much more clearly the symmetry in the transformation. From the allowed ranges of {{mvar|v}} and the definition of {{mvar|β}}, it follows {{math|−1 < ''β'' < 1}}. The use of {{mvar|β}} and {{mvar|γ}} is standard throughout the literature. In the case of three spatial dimensions {{math|[''ct'', ''x'', ''y'', ''z'']}}, where the boost <math>\beta</math> is in the ''x'' direction, the [[Eigenvalues and eigenvectors|eigenstates]] of the  transformation are {{math|[1, 1, 0, 0]}} with eigenvalue <math>\sqrt{(1-\beta)/(1+\beta)}</math>, {{math|[1, −1, 0, 0]}} with eigenvalue <math>\sqrt{(1+\beta)/(1-\beta)}</math>, and {{math|[0, 0, 1, 0]}} and {{math|[0, 0, 0, 1]}}, the latter two with eigenvalue 1. 

When the boost velocity <math>\boldsymbol{v}</math> is in an arbitrary vector direction with the boost vector <math>\boldsymbol{\beta}=\boldsymbol{v}/c</math>, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by<ref>{{cite book |title=Relativity Made Relatively Easy |last1=Steane |first1=Andrew M. |edition=illustrated |publisher=OUP Oxford |year=2012 |isbn=978-0-19-966286-9 |page=124 |url=https://books.google.com/books?id=75rCErZkh7EC}} [https://books.google.com/books?id=75rCErZkh7EC&pg=PA124 Extract of page 124]</ref><ref>{{cite web |last1=Steane |first1=Andrew |title=The Lorentz transformation |url=https://users.physics.ox.ac.uk/~Steane/teaching/rel_A.pdf}}</ref>

<math display="block">\begin{bmatrix} ct' \vphantom{-\gamma\beta _\text{x}} \\ x' \vphantom{1+\frac{\gamma^2}{1+\gamma}\beta _\text{x}^2} \\ y' \vphantom{ \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y} } \\ z' \vphantom{\frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z}} \end{bmatrix} =
\begin{bmatrix}
  \gamma & -\gamma\beta _\text{x}  & -\gamma\beta _\text{y}  & -\gamma\beta _\text{z} \\
  -\gamma\beta _\text{x}   & 1+\frac{\gamma^2}{1+\gamma}\beta _\text{x}^2 & \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y}& \frac{\gamma^2}{1+\gamma}\beta _\text{x}\beta _\text{z} \\
  -\gamma\beta _\text{y} &  \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y} & 1+\frac{\gamma^2}{1+\gamma}\beta _\text{y}^2 & \frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z} \\
  -\gamma\beta _\text{z} &  \frac{\gamma^2}{1+\gamma}\beta _\text{x}\beta _\text{z} & \frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z}  & 1+\frac{\gamma^2}{1+\gamma}\beta _\text{z}^2 \\
\end{bmatrix}
\begin{bmatrix} ct \vphantom{-\gamma\beta _\text{x}} \\ x \vphantom{1+\frac{\gamma^2}{1+\gamma}\beta _\text{x}^2} \\ y \vphantom{ \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y} } \\ z \vphantom{\frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z}} \end{bmatrix}, </math> 

where the [[Lorentz factor]] is <math>\gamma =1/\sqrt{1 - \boldsymbol{\beta}^2} </math>. The [[determinant]] of the transformation matrix is +1 and its [[Trace (linear algebra)|trace]] is <math>2(1+\gamma)</math>. The inverse of the transformation is given by reversing the sign of <math>\boldsymbol{\beta}</math>. The quantity <math> c^2t^2-x^2-y^2-z^2</math> is invariant under the transformation: namely <math>(ct'^2-x'^2-y'^2-z'^2)=(ct^2-x^2-y^2-z^2)</math>.

The Lorentz transformations can also be derived in a way that resembles circular rotations in 3-dimensional space using the [[hyperbolic function]]s. For the boost in the {{mvar|x}} direction, the results are

{{Equation box 1
|title='''Lorentz boost''' ({{mvar|x}} '' direction with rapidity '' {{mvar|ζ}})
|indent =:
|equation =
<math>\begin{align}
  ct' &=  ct \cosh\zeta - x \sinh\zeta \\
   x' &= x \cosh\zeta - ct \sinh\zeta \\
   y' &= y \\
   z' &= z
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

where {{mvar|ζ}} (lowercase [[zeta]]) is a parameter called ''[[rapidity]]'' (many other symbols are used, including {{mvar|θ}}, {{mvar|ϕ}}, {{mvar|φ}}, {{mvar|η}}, {{mvar|ψ}}, {{mvar|ξ}}). Given the strong resemblance to rotations of spatial coordinates in 3-dimensional space in the Cartesian {{math|''xy''}}, {{math|''yz''}}, and {{math|''zx''}} planes, a Lorentz boost can be thought of as a [[hyperbolic rotation]] of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4-dimensional [[Minkowski space]]. The parameter {{mvar|ζ}} is the [[hyperbolic angle]] of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a [[Minkowski diagram]].

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 {{math|1=''x'' = 0}} or {{math|1=''ct'' = 0}} in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying {{mvar|ζ}}, which parametrizes the curves according to the identity
<math display="block"> \cosh^2\zeta - \sinh^2\zeta = 1 \,. </math>

Conversely the {{math|''ct''}} and {{mvar|x}} axes can be constructed for varying coordinates but constant {{mvar|ζ}}. The definition
<math display="block"> \tanh\zeta = \frac{\sinh\zeta}{\cosh\zeta} \,, </math>
provides the link between a constant value of rapidity, and the [[slope]] of the {{math|''ct''}} axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor
<math display="block"> \cosh\zeta = \frac{1}{\sqrt{1 - \tanh^2\zeta}} \,. </math>

Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between {{mvar|β}}, {{mvar|γ}}, and {{mvar|ζ}} are
<math display="block">\begin{align}
       \beta &= \tanh\zeta  \,, \\
      \gamma &= \cosh\zeta  \,, \\
\beta \gamma &= \sinh\zeta  \,.
\end{align}</math>

Taking the inverse hyperbolic tangent gives the rapidity <math display="block"> \zeta = \tanh^{-1}\beta  \,.</math>

Since {{math|−1 < ''β'' < 1}}, it follows {{math|−∞ < ''ζ'' < ∞}}. From the relation between {{mvar|ζ}} and {{mvar|β}}, positive rapidity {{math|''ζ'' > 0}} is motion along the positive directions of the {{math|''xx''′}} axes, zero rapidity {{math|1=''ζ'' = 0}} is no relative motion, while negative rapidity {{math|''ζ'' < 0}} is relative motion along the negative directions of the {{math|''xx''′}} axes.

The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity {{math|''ζ'' → −''ζ''}} since this is equivalent to negating the relative velocity. Therefore,

{{Equation box 1
|title='''Inverse Lorentz boost''' ({{mvar|x}} '' direction with rapidity '' {{mvar|ζ}})
|indent =:
|equation =
<math>\begin{align}
  ct & = ct' \cosh\zeta + x' \sinh\zeta \\
   x &= x' \cosh\zeta + ct' \sinh\zeta \\
   y &= y' \\
   z &= z'
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The inverse transformations can be similarly visualized by considering the cases when {{math|1=''x''′ = 0}} and {{math|1=''ct''′ = 0}}.

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 [[linear transformation|linearity]] 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:
<math display="block">\begin{align}
\Delta t' &= \gamma \left( \Delta t - \frac{v \, \Delta x}{c^2} \right) \,, \\
\Delta x' &= \gamma \left( \Delta x - v \, \Delta t \right) \,,
\end{align}</math>
with inverse relations
<math display="block">\begin{align}
\Delta t &= \gamma \left( \Delta t' + \frac{v \, \Delta x'}{c^2} \right) \,, \\
\Delta x &= \gamma \left( \Delta x' + v \, \Delta t' \right) \,.
\end{align}</math>
where {{math|Δ}} (uppercase [[delta (letter)|delta]]) indicates a difference of quantities; e.g., {{math|1=Δ''x'' = ''x''{{sub|2}} − ''x''{{sub|1}}}} for two values of {{mvar|x}} 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 {{math|''t''{{sub|0}}, ''x''{{sub|0}}, ''y''{{sub|0}}, ''z''{{sub|0}}}} in {{mvar|F}} and {{math|''t''′{{sub|0}}, ''x''′{{sub|0}}, ''y''′{{sub|0}}, ''z''′{{sub|0}}}} in {{math|''F''′}}, 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., {{math|1=Δ''x'' = ''x'' − ''x''{{sub|0}}}}, {{math|1=Δ''x''′ = ''x''′ − ''x''′{{sub|0}}}}, 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 {{mvar|F}} the equation for a pulse of light along the {{mvar|x}} direction is {{math|1=''x'' = ''ct''}}, then in {{mvar|F′}} the Lorentz transformations give {{math|1=''x''′ = ''ct''′}}, and vice versa, for any {{math|−''c'' < ''v'' < ''c''}}.

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the [[Galilean transformation]]:<ref>{{cite book |title=International Edition University Physics |author1=George Arfken |edition= |publisher=Elsevier |year=2012 |isbn=978-0-323-14203-8 |page=367 |url=https://books.google.com/books?id=XzQK42x6uaEC}} [https://books.google.com/books?id=XzQK42x6uaEC&pg=PA367 Extract of page 367]</ref><ref>{{cite book |title=Basic Electromagnetism |author1=E.R. Dobbs |edition=illustrated |publisher=Springer Science & Business Media |year=2013 |isbn=978-94-011-2112-5 |page=113 |url=https://books.google.com/books?id=v3fsCAAAQBAJ}} [https://books.google.com/books?id=v3fsCAAAQBAJ&pg=PA113 Extract of page 113]</ref>
<math display="block">\begin{align}
t' &\approx t \\
x' &\approx x - vt
\end{align}</math>
in accordance with the [[correspondence principle]]. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".<ref>{{harvnb|Einstein|1916}}</ref>

Three counterintuitive, but correct, predictions of the transformations are:
; [[Relativity of simultaneity]] : Suppose two events occur along the x axis simultaneously ({{math|1=Δ''t'' = 0}}) in {{mvar|F}}, but separated by a nonzero displacement {{math|Δ''x''}}.  Then in {{mvar|F′}}, we find that <math>\Delta t' = -\gamma {v\,\Delta x}/{c^2} </math>, so the events are no longer simultaneous according to a moving observer. 
; [[Time dilation]] : Suppose there is a clock at rest in {{mvar|F}}. If a time interval is measured at the same point in that frame, so that {{math|1=Δ''x'' = 0}}, then the transformations give this interval in {{math|''F''′}} by {{math|1=Δ''t''′ = ''γ''Δ''t''}}. Conversely, suppose there is a clock at rest in {{math|''F''′}}. If an interval is measured at the same point in that frame, so that {{math|1=Δ''x''′ = 0}}, then the transformations give this interval in {{mvar|F}} by {{math|1=Δ''t'' = ''γ''Δ''t''′}}. Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor {{mvar|γ}} than the time interval between ticks of his own clock.
; [[Length contraction]] : Suppose there is a rod at rest in {{mvar|F}} aligned along the {{mvar|x}} axis, with length {{math|Δ''x''}}.  In {{math|''F''′}}, the rod moves with velocity {{math|−''v''}}, so its length must be measured by taking two simultaneous ({{math|1=Δ''t''′ = 0}}) measurements at opposite ends.  Under these conditions, the inverse Lorentz transform shows that {{math|1=Δ''x'' = ''γ''Δ''x''′}}. In {{mvar|F}} the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in {{mvar|F}}. So each observer measures the distance between the end points of a moving rod to be shorter by a factor {{math|1/''γ''}} 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 ===
{{further|Euclidean vector|vector projection}}

[[File:Lorentz boost any direction standard configuration.svg|upright=1.75|thumb|An observer in frame {{mvar|F}} observes {{math|''F''′}} to move with velocity {{math|'''v'''}}, while {{math|''F''′}} observes {{mvar|F}} to move with velocity {{math|−'''v'''}}. {{According to whom|The coordinate axes of each frame are still parallel|date=November 2020}} and orthogonal. The position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vector {{math|'''v'''}}.<br />'''Left:''' Standard configuration. '''Right:''' Inverse configuration.]]

The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative [[velocity vector]] {{math|'''v'''}} with a magnitude {{math|1={{abs|'''v'''}} = ''v''}} that cannot equal or exceed {{mvar|c}}, so that {{math|0 ≤ ''v'' < ''c''}}.

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]] {{math|'''r'''}} as measured in {{mvar|F}}, and {{math|'''r'''′}} as measured in {{math|''F''′}}, each into components perpendicular ({{math|⊥}}) and parallel ({{math| {{!!}} }}) to {{math|'''v'''}},
<math display="block">\mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|\,,\quad \mathbf{r}' = \mathbf{r}_\perp' + \mathbf{r}_\|' \,, </math>
then the transformations are
<math display="block">\begin{align}
              t'  &= \gamma \left(t - \frac{\mathbf{r}_\parallel \cdot \mathbf{v}}{c^2} \right) \\
   \mathbf{r}_\|' &= \gamma (\mathbf{r}_\| - \mathbf{v} t) \\
\mathbf{r}_\perp' &= \mathbf{r}_\perp
\end{align}</math>
where {{math|·}} is the [[dot product]]. The Lorentz factor {{mvar|γ}} retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition {{math|1='''β''' = '''v'''/''c''}} with magnitude {{math|0 ≤ ''β'' < 1}} is also used by some authors.

Introducing a [[unit vector]] {{math|1='''n''' = '''v'''/''v'' = '''β'''/''β''}} in the direction of relative motion, the relative velocity is {{math|1='''v''' = ''v'''''n'''}} with magnitude {{mvar|v}} and direction {{math|'''n'''}}, and [[vector projection]] and rejection give respectively
<math display="block">\mathbf{r}_\parallel = (\mathbf{r}\cdot\mathbf{n})\mathbf{n}\,,\quad \mathbf{r}_\perp = \mathbf{r} - (\mathbf{r}\cdot\mathbf{n})\mathbf{n}</math>

Accumulating the results gives the full transformations,

{{Equation box 1
|title='''Lorentz boost''' (''in direction '' {{math|'''n'''}} '' with magnitude '' {{mvar|v}})
|indent =:
|equation =
<math>\begin{align}
          t'  &= \gamma \left(t - \frac{v\mathbf{n}\cdot \mathbf{r}}{c^2} \right) \,, \\
  \mathbf{r}' &= \mathbf{r} + (\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma t v\mathbf{n} \,.
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The projection and rejection also applies to {{math|'''r'''′}}. For the inverse transformations, exchange {{math|'''r'''}} and {{math|'''r'''′}} to switch observed coordinates, and negate the relative velocity {{math|'''v''' → −'''v'''}} (or simply the unit vector {{math|'''n''' → −'''n'''}} since the magnitude {{mvar|v}} is always positive) to obtain

{{Equation box 1
|title='''Inverse Lorentz boost''' (''in direction '' {{math|'''n'''}} '' with magnitude '' {{mvar|v}})
|indent =:
|equation =
<math>\begin{align}
           t &= \gamma \left(t' + \frac{\mathbf{r}' \cdot v\mathbf{n}}{c^2} \right) \,, \\
  \mathbf{r} &= \mathbf{r}' + (\gamma-1)(\mathbf{r}'\cdot\mathbf{n})\mathbf{n} + \gamma t' v\mathbf{n} \,,
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The unit vector has the advantage of simplifying equations for a single boost, allows either {{math|'''v'''}} or {{math|'''β'''}} to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing {{mvar|β}} and {{math|''βγ''}}. It is not convenient for multiple boosts.

The vectorial relation between relative velocity and rapidity is<ref>{{harvnb|Barut|1964|page=18–19}}</ref>
<math display="block"> \boldsymbol{\beta} = \beta \mathbf{n} = \mathbf{n} \tanh\zeta \,,</math>
and the "rapidity vector" can be defined as
<math display="block"> \boldsymbol{\zeta} = \zeta\mathbf{n} = \mathbf{n}\tanh^{-1}\beta \,, </math>
each of which serves as a useful abbreviation in some contexts. The magnitude of {{math|'''ζ'''}} is the absolute value of the rapidity scalar confined to {{math|0 ≤ ''ζ'' < ∞}}, which agrees with the range {{math|0 ≤ ''β'' < 1}}.

=== Transformation of velocities ===
{{further|differential of a function|velocity addition formula}}

[[File:Lorentz transformation of velocity including velocity addition.svg|upright=1.75|thumb|The transformation of velocities provides the definition [[velocity addition formula|relativistic velocity addition]] {{math|⊕}}, the ordering of vectors is chosen to reflect the ordering of the addition of velocities; first {{math|'''v'''}} (the velocity of {{math|''F''′}} relative to {{mvar|F}}) then {{math|'''u'''′}} (the velocity of {{mvar|X}} relative to {{math|''F''′}}) to obtain {{math|'''u''' {{=}} '''v''' ⊕ '''u'''′}} (the velocity of {{mvar|X}} relative to {{mvar|F}}).]]

Defining the coordinate velocities and Lorentz factor by
<math display="block">\mathbf{u} = \frac{d\mathbf{r}}{dt} \,,\quad \mathbf{u}' = \frac{d\mathbf{r}'}{dt'} \,,\quad \gamma_\mathbf{v} = \frac{1}{\sqrt{1-\dfrac{\mathbf{v}\cdot\mathbf{v}}{c^2}}}</math>
taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to
<math display="block">\mathbf{u}' = \frac{1}{ 1 - \frac{\mathbf{v}\cdot\mathbf{u}}{c^2} }\left[\frac{\mathbf{u}}{\gamma_\mathbf{v}} - \mathbf{v} + \frac{1}{c^2}\frac{\gamma_\mathbf{v}}{\gamma_\mathbf{v} + 1}\left(\mathbf{u}\cdot\mathbf{v}\right)\mathbf{v}\right] </math>

The velocities {{math|'''u'''}} and {{math|'''u'''′}} are the velocity of some massive object. They can also be for a third inertial frame (say {{mvar|F′′}}), in which case they must be ''constant''. Denote either entity by {{mvar|X}}. Then {{mvar|X}} moves with velocity {{math|'''u'''}} relative to {{mvar|F}}, or equivalently with velocity {{math|'''u'''′}} relative to {{mvar|F′}}, in turn {{mvar|F′}} moves with velocity {{math|'''v'''}} relative to {{mvar|F}}. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange {{math|'''u'''}} and {{math|'''u'''′}}, and change {{math|'''v'''}} to {{math|−'''v'''}}.

The transformation of velocity is useful in [[stellar aberration]], the [[Fizeau experiment]], and the [[relativistic Doppler effect]].

The [[Acceleration (special relativity)#Three-acceleration|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 {{mvar|A}} and {{math|1='''Z''' = (''Z''{{sub|''x''}}, ''Z''{{sub|''y''}}, ''Z''{{sub|''z''}})}} and their Lorentz-boosted counterparts {{mvar|A′}} and {{math|1='''Z'''′ = (''Z''′{{sub|''x''}}, ''Z''′{{sub|''y''}}, ''Z''′{{sub|''z''}})}}, a relation of the form
<math display="block">A^2 - \mathbf{Z}\cdot\mathbf{Z} = {A'}^2 - \mathbf{Z}'\cdot\mathbf{Z}'</math>
implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;
<math display="block">\begin{align}
           A' &= \gamma \left(A - \frac{v\mathbf{n}\cdot \mathbf{Z}}{c} \right) \,, \\
  \mathbf{Z}' &= \mathbf{Z} + (\gamma-1)(\mathbf{Z}\cdot\mathbf{n})\mathbf{n} - \frac{\gamma A v\mathbf{n}}{c} \,.
\end{align}</math>

The decomposition of {{math|'''Z'''}} (and {{math|'''Z'''′}}) into components perpendicular and parallel to {{math|'''v'''}} is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange {{math|(''A'', '''Z''')}} and {{math|(''A''′, '''Z'''′)}} to switch observed quantities, and reverse the direction of relative motion by the substitution {{math|'''n''' ↦ −'''n'''}}).

The quantities {{math|(''A'', '''Z''')}} collectively make up a ''[[four-vector]]'', where {{mvar|A}} is the "timelike component", and {{math|'''Z'''}} the "spacelike component". Examples of {{mvar|A}} and {{math|'''Z'''}} are the following:

{| class="wikitable"
|-
! Four-vector
! {{mvar|A}}
! {{math|'''Z'''}}
|-
| position [[four-vector]]
| [[time]] (multiplied by {{mvar|c}}), {{math|''ct''}}
| [[position vector]], {{math|'''r'''}}
|-
| [[four-momentum]]
| [[energy]] (divided by {{mvar|c}}), {{math|''E''/''c''}}
| [[momentum]], {{math|'''p'''}}
|-
| [[four-vector|Four-wave vector]]
| [[angular frequency]] (divided by {{mvar|c}}), {{math|''ω''/''c''}}
| [[wave vector]], {{math|'''k'''}}
|-
| [[four-spin]]
| (No name), {{math|''s''{{sub|''t''}}}}
| [[Spin (physics)|spin]], {{math|'''s'''}}
|-
| [[four-current]]
| [[charge density]] (multiplied by {{mvar|c}}), {{math|''ρc''}}
| [[current density]], {{math|'''j'''}}
|-
| [[electromagnetic four-potential]]
| [[electric potential]] (divided by {{mvar|c}}), {{math|''φ''/''c''}}
| [[magnetic vector potential]], {{math|'''A'''}}
|}

For a given object (e.g., particle, fluid, field, material), if {{mvar|A}} or {{math|'''Z'''}} correspond to properties specific to the object like its [[charge density]], [[mass density]], [[Spin (physics)|spin]], 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 {{mvar|E}} 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]] 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]] spin {{math|'''s'''}} 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 {{math|''s''{{sub|''t''}}}}, however a boosted observer will perceive a nonzero timelike component and an altered spin.<ref>{{harvnb|Chaichian|Hagedorn|1997|page=239}}</ref>

Not all quantities are invariant in the form as shown above, for example orbital [[angular momentum]] {{math|'''L'''}} does not have a timelike quantity, and neither does the [[electric field]] {{math|'''E'''}} nor the [[magnetic field]] {{math|'''B'''}}. The definition of angular momentum is {{math|1='''L''' = '''r''' × '''p'''}}, and in a boosted frame the altered angular momentum is {{math|1='''L'''′ = '''r'''′ × '''p'''′}}. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out {{math|'''L'''}} transforms with another vector quantity {{math|1='''N''' = (''E''/''c''{{sup|2}})'''r''' − ''t'''''p'''}} related to boosts, see ''[[Relativistic angular momentum]]'' for details. For the case of the {{math|'''E'''}} and {{math|'''B'''}} fields, the transformations cannot be obtained as directly using vector algebra. The [[Lorentz force]] is the definition of these fields, and in {{mvar|F}} it is {{math|1='''F''' = ''q''('''E''' + '''v''' × '''B''')}} while in {{mvar|F′}} it is {{math|1='''F'''′ = ''q''('''E'''′ + '''v'''′ × '''B'''′)}}. A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra, [[Lorentz transformation#Transformation of the electromagnetic field|given below]].

== Mathematical formulation ==
{{main|Lorentz group}}
{{further|Matrix (mathematics)|matrix product|linear algebra|rotation formalisms in three dimensions}}

Throughout, italic non-bold capital letters are {{math|4 × 4}} matrices, while non-italic bold letters are {{math|3 × 3}} matrices.

=== Homogeneous Lorentz group ===

Writing the coordinates in column vectors and the [[Minkowski metric]] {{mvar|η}} as a square matrix
<math display="block"> X' = \begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix} \,, \quad \eta = \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} \,, \quad X = \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}  </math>
the spacetime interval takes the form (superscript {{math|T}} denotes [[transpose]])
<math display="block"> X \cdot X = X^\mathrm{T} \eta X = {X'}^\mathrm{T} \eta {X'} </math>
and is [[Invariant (physics)|invariant]] under a Lorentz transformation
<math display="block">X' = \Lambda X  </math>
where {{math|Λ}} is a square matrix which can depend on parameters.

The [[set (mathematics)|set]] of all Lorentz transformations <math>\Lambda  </math> in this article is denoted <math>\mathcal{L}</math>. This set together with matrix multiplication forms a [[group (mathematics)|group]], in this context known as the ''[[Lorentz group]]''. Also, the above expression {{math|''X''·''X''}} is a [[quadratic form]] of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the [[indefinite orthogonal group]] O(3,1), a [[Lie group]]. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are [[matrix Lie group]]s. In this context the operation of composition amounts to [[matrix multiplication]].

From the invariance of the spacetime interval it follows
<math display="block">\eta = \Lambda^\mathrm{T} \eta \Lambda </math>
and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the [[determinant]] of the equation using the product rule<ref group=nb>For two square matrices {{mvar|A}} and {{mvar|B}}, {{math|1=det(''AB'') = det(''A'')det(''B'')}}</ref> gives immediately
<math display="block">\left[\det (\Lambda)\right]^2 = 1 \quad \Rightarrow \quad \det(\Lambda) = \pm 1 </math>

Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,
<math display="block">\eta = \begin{bmatrix}-1 & 0 \\ 0 & \mathbf{I}\end{bmatrix} \,, \quad \Lambda=\begin{bmatrix}\Gamma & -\mathbf{a}^\mathrm{T}\\-\mathbf{b} & \mathbf{M}\end{bmatrix} \,, </math>
carrying out the block matrix multiplications obtains general conditions on {{math|Γ, '''a''', '''b''', '''M'''}} to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results
<math display="block">\Gamma^2 = 1 + \mathbf{b}^\mathrm{T}\mathbf{b}</math>
is useful; {{math|'''b'''{{sup|T}}'''b''' ≥ 0}} always so it follows that
<math display="block"> \Gamma^2 \geq 1 \quad \Rightarrow \quad \Gamma \leq - 1 \,,\quad \Gamma \geq  1 </math>

The negative inequality may be unexpected, because {{math|Γ}} multiplies the time coordinate and this has an effect on [[Time translation symmetry|time symmetry]]. If the positive equality holds, then {{math|Γ}} is the Lorentz factor.

The determinant and inequality provide four ways to classify '''L'''orentz '''T'''ransformations (''herein '''LT'''s 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 (set theory)|intersection]]s ("n"-shaped symbol meaning "and") of these classifying sets.

{| class="wikitable"
|-
! Intersection, ∩
! '''Antichronous''' (or non-orthochronous) LTs
: <math> \mathcal{L}^\downarrow = \{ \Lambda : \Gamma \leq -1 \} </math>
! '''Orthochronous''' LTs
: <math> \mathcal{L}^\uparrow = \{ \Lambda : \Gamma \geq 1 \} </math>
|-
! '''Proper''' LTs
: <math> \mathcal{L}_{+} = \{ \Lambda : \det(\Lambda) = +1 \} </math>
| '''Proper antichronous''' LTs
: <math>\mathcal{L}_+^\downarrow = \mathcal{L}_+ \cap \mathcal{L}^\downarrow </math>
|'''Proper orthochronous''' LTs
: <math>\mathcal{L}_+^\uparrow = \mathcal{L}_+ \cap \mathcal{L}^\uparrow </math>
|-
! '''Improper''' LTs
: <math> \mathcal{L}_{-} = \{ \Lambda : \det(\Lambda) = -1 \} </math>
|'''Improper antichronous''' LTs
: <math>\mathcal{L}_{-}^\downarrow = \mathcal{L}_{-} \cap \mathcal{L}^\downarrow </math>
|'''Improper orthochronous''' LTs
: <math>\mathcal{L}_{-}^\uparrow = \mathcal{L}_{-} \cap \mathcal{L}^\uparrow </math>
|-
|}

where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.

The full Lorentz group splits into the [[Union (set theory)|union]] ("u"-shaped symbol meaning "or") of four [[disjoint set]]s
<math display="block"> \mathcal{L} = \mathcal{L}_{+}^\uparrow \cup \mathcal{L}_{-}^\uparrow \cup \mathcal{L}_{+}^\downarrow \cup \mathcal{L}_{-}^\downarrow </math>

A [[subgroup]] of a group must be [[Closure (mathematics)|closed]] under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations {{math|Λ}} and {{mvar|L}} from a particular subgroup, the composite Lorentz transformations {{math|Λ''L''}} and {{math|''L''Λ}} must be in the same subgroup as {{math|Λ}} and {{mvar|L}}. 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 <math>\mathcal{L}_+^\uparrow </math>, <math>\mathcal{L}_+</math>, <math>\mathcal{L}^\uparrow</math>, and <math>\mathcal{L}_0 = \mathcal{L}_+^\uparrow \cup \mathcal{L}_{-}^\downarrow</math> all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g. <math>\mathcal{L}_+^\downarrow </math>, <math>\mathcal{L}_{-}^\downarrow </math>, <math>\mathcal{L}_{-}^\uparrow </math>) do not form subgroups.

=== Proper transformations ===

If a Lorentz covariant 4-vector is measured in one inertial frame with result <math>X</math>, and the same measurement made in another inertial frame (with the same orientation and origin) gives result <math>X'</math>, the two results will be related by
<math display="block">X' = B(\mathbf{v})X</math>
where the boost matrix <math>B(\mathbf{v})</math> represents the rotation-free Lorentz transformation between the unprimed and primed frames and <math>\mathbf{v}</math> is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by<ref>{{Cite journal|last=Furry|first=W. H.|date=1955-11-01|title=Lorentz Transformation and the Thomas Precession|url=https://aapt.scitation.org/doi/10.1119/1.1934085|journal=American Journal of Physics|volume=23|issue=8|pages=517–525|doi=10.1119/1.1934085| bibcode=1955AmJPh..23..517F| issn=0002-9505|url-access=subscription}}</ref>
<math display="block">B(\mathbf{v}) =
\begin{bmatrix}
 \gamma      &-\gamma v_\text{x}/c                   &-\gamma v_\text{y}/c                   &-\gamma v_\text{z}/c                    \\
-\gamma v_\text{x}/c&1+(\gamma-1)\dfrac{v_\text{x}^2}  {v^2}&  (\gamma-1)\dfrac{v_\text{x} v_\text{y}}{v^2}&  (\gamma-1)\dfrac{v_\text{x} v_\text{z}}{v^2} \\
-\gamma v_\text{y}/c&  (\gamma-1)\dfrac{v_\text{y} v_\text{x}}{v^2}&1+(\gamma-1)\dfrac{v_\text{y}^2}  {v^2}&  (\gamma-1)\dfrac{v_\text{y} v_\text{z}}{v^2} \\
-\gamma v_\text{z}/c&  (\gamma-1)\dfrac{v_\text{z} v_\text{x}}{v^2}&  (\gamma-1)\dfrac{v_\text{z} v_\text{y}}{v^2}&1+(\gamma-1)\dfrac{v_\text{z}^2}  {v^2}
\end{bmatrix} =
\begin{bmatrix}
 \gamma             & -\gamma \vec{\beta}^\text{T} \\
-\gamma \vec{\beta} & I + (\gamma-1)\dfrac{\vec{\beta}\vec{\beta}^\text{T}}{\beta^2}
\end{bmatrix},</math>

where <math display="inline">v=\sqrt{v_\text{x}^2+v_\text{y}^2+v_\text{z}^2}</math> is the magnitude of the velocity and <math display="inline">\gamma=\frac{1}{\sqrt{1-{v^2}/{c^2}}}</math> 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 <math>B(-\mathbf{v})</math>.

If a frame {{math|''F''{{′}}}} is boosted with velocity {{math|'''u'''}} relative to frame {{mvar|F}}, and another frame {{mvar|F′′}} is boosted with velocity {{math|'''v'''}} relative to {{math|''F''{{′}}}}, the separate boosts are
<math display="block">X'' = B(\mathbf{v})X' \,, \quad X' = B(\mathbf{u})X </math>
and the composition of the two boosts connects the coordinates in {{math|''F''′′}} and {{mvar|F}},
<math display="block">X'' = B(\mathbf{v})B(\mathbf{u})X \,. </math>
Successive transformations act on the left. If {{math|'''u'''}} and {{math|'''v'''}} are [[collinear]] (parallel or antiparallel along the same line of relative motion), the boost matrices [[Commutative property|commute]]: {{math|''B''('''v''')''B''('''u''') {{=}} ''B''('''u''')''B''('''v''')}}. This composite transformation happens to be another boost, {{math|''B''('''w''')}}, where {{math|'''w'''}} is collinear with {{math|'''u'''}} and {{math|'''v'''}}.

If {{math|'''u'''}} and {{math|'''v'''}} are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute: {{math|''B''('''v''')''B''('''u''')}} and {{math|''B''('''u''')''B''('''v''')}} are not equal. Although each of these compositions is ''not'' a single boost, each composition is still a Lorentz transformation as it preserves 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 {{math|''R''('''ρ''')''B''('''w''')}} or {{math|''B''({{overline|'''w'''}})''R''({{overline|'''ρ'''}})}}. The {{math|'''w'''}} and {{math|{{overline|'''w'''}}}} are [[velocity addition formula|composite velocities]], while {{math|'''ρ'''}} and {{math|{{overline|'''ρ'''}}}} are rotation parameters (e.g. [[axis-angle representation|axis-angle]] variables, [[Euler angles]], etc.). The rotation in [[block matrix]] form is simply
<math display="block">\quad R(\boldsymbol{\rho}) = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{R}(\boldsymbol{\rho}) \end{bmatrix} \,, </math>
where {{math|'''R'''('''ρ''')}} is a {{math|3 × 3}} [[rotation matrix]], which rotates any 3-dimensional vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is ''not'' simple to connect {{math|'''w'''}} and {{math|'''ρ'''}} (or {{math|{{overline|'''w'''}}}} and {{math|{{overline|'''ρ'''}}}}) to the original boost parameters {{math|'''u'''}} and {{math|'''v'''}}. In a composition of boosts, the {{mvar|R}} matrix is named the [[Wigner rotation]], and gives rise to the [[Thomas precession]]. These articles give the explicit formulae for the composite transformation matrices, including expressions for {{math|'''w''', '''ρ''', {{overline|'''w'''}}, {{overline|'''ρ'''}}}}.

In this article the [[axis-angle representation]] is used for {{math|'''ρ'''}}. The rotation is about an axis in the direction of a [[unit vector]] {{math|'''e'''}}, through angle {{mvar|θ}} (positive anticlockwise, negative clockwise, according to the [[right-hand rule]]). The "axis-angle vector"
<math display="block">\boldsymbol{\theta} = \theta \mathbf{e}</math>
will serve as a useful abbreviation.

Spatial rotations alone are also Lorentz transformations since 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:
* [[matrix inverse|inverse]]s: {{math|1=''B''('''v'''){{sup|−1}} = ''B''(−'''v''')}} (relative motion in the opposite direction), and {{math|1=''R''('''θ'''){{sup|−1}} = ''R''(−'''θ''')}} (rotation in the opposite sense about the same axis)
* [[identity transformation]] for no relative motion/rotation: {{math|1=''B''('''0''') = ''R''('''0''') = ''I''}}
* unit [[determinant]]: {{math|1=det(''B'') = det(''R'') = +1}}. This property makes them proper transformations.
* [[symmetric matrix|matrix symmetry]]: {{mvar|B}} is symmetric (equals [[transpose]]), while {{mvar|R}} is nonsymmetric but [[orthogonal matrix|orthogonal]] (transpose equals [[matrix inverse|inverse]], {{math|1=''R''{{sup|T}} = ''R''{{sup|−1}}}}).

The most general proper Lorentz transformation {{math|Λ('''v''', '''θ''')}} includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, {{math|1=Λ('''0''', '''θ''') = ''R''('''θ''')}} and {{math|1=Λ('''v''', '''0''') = ''B''('''v''')}}. 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 {{math|Λ('''ζ''', '''θ''')}} and {{math|''B''('''ζ''')}}.

==== Lie group SO{{sup|+}}(3,1) ====

The set of transformations
<math display="block"> \{ B(\boldsymbol{\zeta}), R(\boldsymbol{\theta}), \Lambda(\boldsymbol{\zeta}, \boldsymbol{\theta}) \} </math>
with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the [[special indefinite orthogonal group]] SO{{sup|+}}(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 {{mvar|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 {{math|1=''ζ'' = 0}},
<math display="block"> B_\text{x} = I + \zeta \left. \frac{\partial B_\text{x}}{\partial \zeta } \right|_{\zeta=0} + \cdots </math>
where the higher order terms not shown are negligible because {{mvar|ζ}} is small, and {{math|''B''{{sub|''x''}}}} 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 {{math|1=''ζ'' = 0}},
<math display="block"> \left. \frac{\partial B_\text{x} }{\partial \zeta } \right|_{\zeta=0} = - K_\text{x} \,. </math>

For now, {{math|''K''{{sub|''x''}}}} 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
<math display="block"> B_\text{x} =\lim_{N\to\infty}\left(I-\frac{\zeta }{N}K_\text{x}\right)^{\!N} = e^{-\zeta K_\text{x}} </math>
where the [[Exponential function#Formal definition|limit definition of the exponential]] has been used (see also ''[[Characterizations of the exponential function]]''). More generally<ref group="nb">Explicitly,
<math display="block"> \boldsymbol{\zeta} \cdot\mathbf{K} = \zeta_\text{x} K_\text{x} + \zeta_\text{y} K_\text{y} + \zeta_\text{z} K_\text{z} </math>
<math display="block"> \boldsymbol{\theta} \cdot\mathbf{J} = \theta_\text{x} J_\text{x} + \theta_\text{y} J_\text{y} + \theta_\text{z} J_\text{z} </math>
</ref>
<math display="block">B(\boldsymbol{\zeta}) = e^{-\boldsymbol{\zeta}\cdot\mathbf{K}} \, , \quad R(\boldsymbol{\theta}) = e^{\boldsymbol{\theta}\cdot\mathbf{J}} \,. </math>

The axis-angle vector {{math|'''θ'''}} and rapidity vector {{math|'''ζ'''}} are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are {{math|1='''K''' = (''K''{{sub|''x''}}, ''K''{{sub|''y''}}, ''K''{{sub|''z''}})}} and {{math|1='''J''' = (''J''{{sub|''x''}}, ''J''{{sub|''y''}}, ''J''{{sub|''z''}})}}, each vectors of matrices with the explicit forms<ref group=nb>In [[quantum mechanics]], [[relativistic quantum mechanics]], 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 {{math|''i'' {{=}} {{sqrt|−1}}}}.</ref>
<math display="block">\begin{alignat}{3}

K_\text{x} &= \begin{bmatrix}
  0 & 1 & 0 & 0 \\
  1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  \end{bmatrix}\,,\quad &
K_\text{y} &= \begin{bmatrix}
  0 & 0 & 1 & 0\\
  0 & 0 & 0 & 0\\
  1 & 0 & 0 & 0\\
  0 & 0 & 0 & 0
  \end{bmatrix}\,,\quad &
K_\text{z} &= \begin{bmatrix}
  0 & 0 & 0 & 1\\
  0 & 0 & 0 & 0\\
  0 & 0 & 0 & 0\\
  1 & 0 & 0 & 0
\end{bmatrix}

\\[10mu]

J_\text{x} &= \begin{bmatrix}
  0 & 0 & 0 &  0 \\
  0 & 0 & 0 &  0 \\
  0 & 0 & 0 & -1 \\
  0 & 0 & 1 &  0 \\
  \end{bmatrix}\,,\quad &
J_\text{y} &= \begin{bmatrix}
  0 &  0 & 0 & 0 \\
  0 &  0 & 0 & 1 \\
  0 &  0 & 0 & 0 \\
  0 & -1 & 0 & 0
  \end{bmatrix}\,,\quad &
J_\text{z} &= \begin{bmatrix}
  0 & 0 &  0 & 0 \\
  0 & 0 & -1 & 0 \\
  0 & 1 &  0 & 0 \\
  0 & 0 &  0 & 0
  \end{bmatrix}

\end{alignat}</math>

These are all defined in an analogous way to {{math|''K''{{sub|''x''}}}} above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime: {{math|'''J'''}} are the ''rotation generators'' which correspond to [[angular momentum]], and {{math|'''K'''}} are the ''boost generators'' which correspond to the motion of the system in spacetime. The derivative of any smooth curve {{math|''C''(''t'')}} with {{math|1=''C''(0) = ''I''}} in the group depending on some group parameter {{mvar|t}} with respect to that group parameter, evaluated at {{math|1=''t'' = 0}}, serves as a definition of a corresponding group generator {{mvar|G}}, 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 {{mvar|G}} smoothly back into the group via {{math|''t'' → exp(''tG'')}} for all {{mvar|t}}; this curve will yield {{mvar|G}} again when differentiated at {{math|1=''t'' = 0}}.

Expanding the exponentials in their Taylor series obtains
<math display="block"> B({\boldsymbol {\zeta }})=I-\sinh \zeta (\mathbf {n} \cdot \mathbf {K} )+(\cosh \zeta -1)(\mathbf {n} \cdot \mathbf {K} )^2</math>
<math display="block">R(\boldsymbol {\theta })=I+\sin \theta (\mathbf {e} \cdot \mathbf {J} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {J} )^2\,.</math>
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
<math display="block"> \begin{align}
\Lambda
&= (I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots )(I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots ) \\
&= (I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots )(I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots ) \\
&= I - \boldsymbol {\zeta } \cdot \mathbf {K}  + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots  
\end{align} </math>
is commutative because only linear terms are required (products like {{math|('''θ'''·'''J''')('''ζ'''·'''K''')}} and {{math|('''ζ'''·'''K''')('''θ'''·'''J''')}} count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential
<math display="block">\Lambda (\boldsymbol{\zeta}, \boldsymbol{\theta}) = e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} }.</math>

The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,
<math display="block">e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} } \ne e^{-\boldsymbol{\zeta} \cdot\mathbf{K}} e^{\boldsymbol{\theta} \cdot\mathbf{J}},</math>
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 {{math|'''J'''}} and {{math|'''K'''}}), 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.

==== 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 (mathematics)|set]] of all Lorentz generators
<math display="block">V = \{ \boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J}  \} </math>
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.<ref group=nb>Until now the term "vector" has exclusively referred to "[[Euclidean vector]]", examples are position {{math|'''r'''}}, velocity {{math|'''v'''}}, 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 number]]s, [[complex number]]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 3-dimensional space.</ref> The generators {{math|''J''{{sub|''x''}}, ''J''{{sub|''y''}}, ''J''{{sub|''z''}}, ''K''{{sub|''x''}}, ''K''{{sub|''y''}}, ''K''{{sub|''z''}}}} form a [[basis (linear algebra)|basis]] set of ''V'', and the components of the axis-angle and rapidity vectors, {{math|''θ''{{sub|''x''}}, ''θ''{{sub|''y''}}, ''θ''{{sub|''z''}}, ''ζ''{{sub|''x''}}, ''ζ''{{sub|''y''}}, ''ζ''{{sub|''z''}}}}, are the [[coordinate vector|coordinate]]s of a Lorentz generator with respect to this basis.<ref group=nb>In ordinary 3-dimensional [[position space]], the position vector {{math|'''r''' {{=}} ''x'''''e'''{{sub|''x''}} + ''y'''''e'''{{sub|''y''}} + ''z'''''e'''{{sub|''z''}}}} is expressed as a linear combination of the Cartesian unit vectors {{math|'''e'''{{sub|''x''}}, '''e'''{{sub|''y''}}, '''e'''{{sub|''z''}}}} which form a basis, and the Cartesian coordinates {{math|''x, y, z''}} are coordinates with respect to this basis.</ref>

Three of the [[commutation relation]]s of the Lorentz generators are
<math display="block">[ J_\text{x}, J_\text{y} ] =  J_\text{z} \,,\quad [ K_\text{x}, K_\text{y} ] = -J_\text{z} \,,\quad [ J_\text{x}, K_\text{y} ] =  K_\text{z} \,, </math>
where the bracket {{math|1=[''A'', ''B''] = ''AB'' − ''BA''}} is known as the ''[[commutator]]'', and the other relations can be found by taking [[cyclic permutation]]s of {{mvar|x}}, {{mvar|y}}, {{mvar|z}} components (i.e. change {{mvar|x}} to {{mvar|y}}, {{mvar|y}} to {{mvar|z}}, and {{mvar|z}} to {{mvar|x}}, repeat).

These commutation relations, and the vector space of generators, fulfill the definition of the [[Lie algebra]] <math>\mathfrak{so}(3, 1)</math>. In summary, a Lie algebra is defined as a [[vector space]] ''V'' over a [[field (mathematics)|field]] of numbers, and with a [[binary operation]] {{math|[ , ]}} (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 {{math|[ , ]}} 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,
<math display="block">\exp : \mathfrak{so}(3,1) \to \mathrm{SO}(3,1),</math>
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. In 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]]
<math display="block"> P = \begin{bmatrix} 1 & 0 \\ 0 & - \mathbf{I} \end{bmatrix} </math>
which negates all the spatial coordinates only, and [[T-symmetry|time reversal]]
<math display="block"> T = \begin{bmatrix} - 1 & 0 \\ 0 & \mathbf{I} \end{bmatrix}</math>
which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here {{math|'''I'''}} is the {{math|3 × 3}} [[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 {{math|Λ}} is a proper orthochronous Lorentz transformation, then {{math|''T''Λ}} is improper antichronous, {{math|''P''Λ}} is improper orthochronous, and {{math|1=''TP''Λ = ''PT''Λ}} is proper antichronous.

=== Inhomogeneous Lorentz group ===

Two other spacetime symmetries have not been accounted for. In order for the spacetime interval to be invariant, it can be shown<ref>{{harvnb|Weinberg|1972}}</ref> that it is necessary and sufficient for the coordinate transformation to be of the form
<math display="block">X' = \Lambda X + C </math>
where ''C'' is a constant column containing translations in time and space. If ''C'' ≠ 0, this is an '''inhomogeneous Lorentz transformation''' or '''[[Poincaré transformation]]'''.<ref>{{harvnb|Weinberg|2005|pages=55–58}}</ref><ref>{{harvnb|Ohlsson|2011|page=3–9}}</ref> If ''C'' = 0, this is a '''homogeneous Lorentz transformation'''. Poincaré transformations are not dealt further in this article.

== Tensor formulation ==

{{main|Representation theory of the Lorentz group}}
{{For|the notation used|Ricci calculus}}

=== Contravariant vectors ===
Writing the general matrix transformation of coordinates as the matrix equation
<math display="block">\begin{bmatrix} {x'}^0 \\ {x'}^1 \\ {x'}^2 \\ {x'}^3 \end{bmatrix} =
  \begin{bmatrix}
    {\Lambda^0}_0 & {\Lambda^0}_1 & {\Lambda^0}_2 & {\Lambda^0}_3 \vphantom{{x'}^0} \\
    {\Lambda^1}_0 & {\Lambda^1}_1 & {\Lambda^1}_2 & {\Lambda^1}_3 \vphantom{{x'}^0} \\
    {\Lambda^2}_0 & {\Lambda^2}_1 & {\Lambda^2}_2 & {\Lambda^2}_3 \vphantom{{x'}^0} \\
    {\Lambda^3}_0 & {\Lambda^3}_1 & {\Lambda^3}_2 & {\Lambda^3}_3 \vphantom{{x'}^0} \\
  \end{bmatrix}
  \begin{bmatrix} x^0 \vphantom{{x'}^0} \\ x^1 \vphantom{{x'}^0} \\ x^2 \vphantom{{x'}^0} \\ x^3 \vphantom{{x'}^0} \end{bmatrix}</math>
allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., [[tensor]]s or [[spinor]]s of any order in 4-dimensional spacetime, to be defined. In the corresponding [[tensor index notation]], the above matrix expression is
<math display="block">{x'}^\nu = {\Lambda^\nu}_\mu x^\mu,</math>

where lower and upper indices label [[covariance and contravariance of vectors|covariant and contravariant components]] respectively,<ref>{{harvnb|Dennery|Krzywicki|2012|p=[https://books.google.com/books?id=ogHCAgAAQBAJ&pg=PA138 138]}}</ref> 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 (the opposite for Landau and Lifshitz). Note that the first index (reading left to right) corresponds in the matrix notation to a ''row index''. The second index corresponds to the column index.

The transformation matrix is universal for all [[four-vector]]s, not just 4-dimensional spacetime coordinates. If {{mvar|A}} is any four-vector, then in [[tensor index notation]] <math display="block"> {A'}^\nu = {\Lambda^\nu}_\mu A^\mu \,.</math>

Alternatively, one writes <math display="block"> A^{\nu'} = {\Lambda^{\nu'}}_\mu A^\mu \,.</math> in which the primed indices denote the indices of {{math|''A''}} in the primed frame. For a general {{mvar|n}}-component object one may write <math display="block">{X'}^\alpha = {\Pi(\Lambda)^\alpha}_\beta X^\beta \,, </math> where {{math|Π}} is the appropriate [[Representation theory of the Lorentz group|representation of the Lorentz group]], an {{math|''n'' × ''n''}} matrix for every {{math|Λ}}. In this case, the indices should ''not'' be thought of as spacetime indices (sometimes called Lorentz indices), and they run from {{math|1}} to {{mvar|n}}. E.g., if {{mvar|X}} 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.,
<math display="block">x_\nu = \eta_{\mu\nu}x^\mu,</math>
where {{mvar|η}} 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
<math display="block">x^\mu = \eta^{\mu\nu}x_\nu,</math>
where, when viewed as matrices, {{math|''η''{{sup|''μν''}}}} is the inverse of {{math|''η''{{sub|''μν''}}}}. As it happens, {{math|1=''η''{{sup|''μν''}} = {{math|''η''{{sub|''μν''}}}}}}. This is referred to as ''raising an index''. To transform a covariant vector {{math|''A''{{sub|''μ''}}}}, first raise its index, then transform it according to the same rule as for contravariant {{math|4}}-vectors, then finally lower the index;
<math display="block">{A'}_\nu = \eta_{\rho\nu} {\Lambda^\rho}_\sigma \eta^{\mu\sigma}A_\mu.</math>

But <math display="block">\eta_{\rho\nu} {\Lambda^\rho}_\sigma \eta^{\mu\sigma} = {\left(\Lambda^{-1}\right)^\mu}_\nu,</math>

That is, it is the {{math|(''μ'', ''ν'')}}-component of the ''inverse'' Lorentz transformation. One defines (as a matter of notation),
<math display="block">{\Lambda_\nu}^\mu \equiv {\left(\Lambda^{-1}\right)^\mu}_\nu,</math>
and may in this notation write
<math display="block">{A'}_\nu = {\Lambda_\nu}^\mu A_\mu.</math>

Now for a subtlety. The implied summation on the right hand side of
<math display="block">{A'}_\nu = {\Lambda_\nu}^\mu A_\mu = {\left(\Lambda^{-1}\right)^\mu}_\nu A_\mu</math>
is running over ''a row index'' of the matrix representing {{math|Λ{{sup|−1}}}}. Thus, in terms of matrices, this transformation should be thought of as the ''inverse transpose'' of {{math|Λ}} acting on the column vector {{math|''A''{{sub|''μ''}}}}. That is, in pure matrix notation,
<math display="block">A' = \left(\Lambda^{-1}\right)^\mathrm{T} A.</math>

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 {{math|Λ}} with {{math|Π(Λ)}}.

=== Tensors ===
If {{mvar|A}} and {{mvar|B}} are linear operators on vector spaces {{mvar|U}} and {{mvar|V}}, then a linear operator {{math|''A'' ⊗ ''B''}} may be defined on the [[tensor product]] of {{mvar|U}} and {{mvar|V}}, denoted {{math|''U'' ⊗ ''V''}} according to<ref>{{harvnb|Hall|2003|loc=Chapter 4}}</ref>

{{Equation box 1
 |indent =:
 |equation = <math>(A \otimes B)(u \otimes v) = Au \otimes Bv, \qquad u \in U, v \in V, u \otimes v \in U \otimes V.</math> {{spaces|13}} {{EquationRef|(T1)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

From this it is immediately clear that if {{mvar|u}} and {{mvar|v}} are a four-vectors in {{mvar|V}}, then {{math|''u'' ⊗ ''v'' ∈ ''T''{{sub|2}}''V'' ≡ ''V'' ⊗ ''V''}} transforms as

{{Equation box 1
 |indent =:
 |equation =
<math>
  u \otimes v \rightarrow \Lambda u \otimes \Lambda v =
    {\Lambda^\mu}_\nu u^\nu \otimes {\Lambda^\rho}_\sigma v^\sigma =
    {\Lambda^\mu}_\nu {\Lambda^\rho}_\sigma u^\nu \otimes v^\sigma \equiv
    {\Lambda^\mu}_\nu {\Lambda^\rho}_\sigma w^{\nu\sigma}.
</math> {{spaces|13}} {{EquationRef|(T2)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

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 {{math|''u'' ⊗ ''v''}}.

These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space {{mvar|V}} 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 {{mvar|T}}. It is given by<ref>{{harvnb|Carroll|2004|page=22}}</ref>

{{Equation box 1
 |indent =:
 |equation =
<math>
  T^{\alpha'\beta' \cdots \zeta'}_{\theta'\iota' \cdots \kappa'} =
    {\Lambda^{\alpha'}}_\mu {\Lambda^{\beta'}}_\nu \cdots {\Lambda^{\zeta'}}_\rho
    {\Lambda_{\theta'}}^\sigma {\Lambda_{\iota'}}^\upsilon \cdots {\Lambda_{\kappa'}}^\zeta
    T^{\mu\nu \cdots \rho}_{\sigma\upsilon \cdots \zeta},
</math> {{spaces|13}} {{EquationRef|(T3)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

where {{math|''Λ''{{sub|''χ''′}}{{sup|''ψ''}}}} is defined above. This form can generally be reduced to the form for general {{mvar|n}}-component objects given above with a single matrix ({{math|Π(Λ)}}) operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.

==== Transformation of the electromagnetic field ====
[[File:Lorentz boost electric charge.svg|upright=1.75|thumb|Lorentz boost of an electric charge; the charge is at rest in one frame or the other.]]
{{main|Electromagnetic tensor}}
{{Further|classical electromagnetism and special relativity}}

Lorentz transformations can also be used to illustrate that the [[magnetic field]] {{math|'''B'''}} and [[electric field]] {{math|'''E'''}} are simply different aspects of the same force — the [[electromagnetic force]], as a consequence of relative motion between [[electric charge]]s and observers.<ref>{{harvnb|Grant|Phillips|2008}}</ref> The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.<ref>{{harvnb|Griffiths|2007}}</ref>
* An observer measures a charge at rest in frame {{mvar|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 {{mvar|F′}} moves at velocity {{math|'''v'''}} relative to {{mvar|F}} and the charge. ''This'' observer sees a different electric field because the charge moves at velocity {{math|−'''v'''}} in their rest frame. The motion of the charge corresponds to an [[electric current]], and thus the observer in frame {{mvar|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
<math display="block">
  F^{\mu\nu} = \begin{bmatrix}
    0              & -\frac{1}{c}E_\text{x} & -\frac{1}{c}E_\text{y} & -\frac{1}{c}E_\text{z} \\
    \frac{1}{c}E_\text{x} &  0              & -B_\text{z}            &  B_\text{y}   \\
    \frac{1}{c}E_\text{y} &  B_\text{z}            &  0              & -B_\text{x}   \\
    \frac{1}{c}E_\text{z} & -B_\text{y}            &  B_\text{x}            &  0
  \end{bmatrix}
</math>
in with signature {{math|(+, −, −, −)}}. In relativity, the factor {{math|''c''}} may be absorbed into the tensor components to eliminate its explicit appearance in expressions.<ref>{{harvnb|Jackson|1975|p={{page needed|date=November 2023}}}}</ref> Consider a Lorentz boost in the {{mvar|x}}-direction. It is given by<ref>{{harvnb|Misner|Thorne|Wheeler|1973}}</ref>
<math display="block">
  {\Lambda^\mu}_\nu = \begin{bmatrix}
     \gamma      & -\gamma\beta & 0 & 0\\
    -\gamma\beta &  \gamma      & 0 & 0\\
     0           &  0           & 1 & 0\\
     0           &  0           & 0 & 1\\
  \end{bmatrix}, \qquad

  F^{\mu\nu} = \begin{bmatrix}
     0   &  E_\text{x} &  E_\text{y} &  E_\text{z} \\
    -E_\text{x} &  0   &  B_\text{z} & -B_\text{y} \\
    -E_\text{y} & -B_\text{z} &  0   &  B_\text{x} \\
    -E_\text{z} &  B_\text{y} & -B_\text{x} &  0
  \end{bmatrix},
</math>
where the signature is {{math|(−, +, +, +)}} and the field tensor is displayed side by side for easiest possible reference in the manipulations below.

The general transformation law {{EquationNote|(T3)}} becomes
<math display="block">F^{\mu'\nu'} = {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}.</math>

For the magnetic field one obtains
<math display="block">\begin{align}
  B_{x'} &= F^{2'3'}
            = {\Lambda^2}_\mu {\Lambda^3}_\nu F^{\mu\nu}
            = {\Lambda^2}_2 {\Lambda^3}_3 F^{23}
            = 1 \times 1 \times B_\text{x} \\
         &= B_\text{x}, \\
  B_{y'} &= F^{3'1'}
            = {\Lambda^3}_\mu {\Lambda^1}_\nu F^{\mu \nu}
            = {\Lambda^3}_3 {\Lambda^1}_\nu F^{3\nu}
            = {\Lambda^3}_3 {\Lambda^1}_0 F^{30} + {\Lambda^3}_3 {\Lambda^1}_1 F^{31} \\
         &= 1 \times (-\beta\gamma) (-E_\text{z}) + 1 \times \gamma B_\text{y}
            = \gamma B_\text{y} + \beta\gamma E_\text{z} \\
         &= \gamma\left(\mathbf{B} - \boldsymbol{\beta} \times \mathbf{E}\right)_\text{y} \\
  B_{z'} &= F^{1'2'}
            = {\Lambda^1}_\mu {\Lambda^2}_\nu F^{\mu\nu}
            = {\Lambda^1}_\mu {\Lambda^2}_2 F^{\mu 2}
            = {\Lambda^1}_0 {\Lambda^2}_2 F^{02} + {\Lambda^1}_1 {\Lambda^2}_2 F^{12} \\
         &= (-\gamma\beta) \times 1\times E_\text{y} + \gamma \times 1 \times B_\text{z}
            = \gamma B_\text{z} - \beta\gamma E_\text{y} \\
         &= \gamma\left(\mathbf{B} - \boldsymbol{\beta} \times \mathbf{E}\right)_\text{z}
\end{align}</math>

For the electric field results
<math display="block">\begin{align}
  E_{x'} &= F^{0'1'}
            = {\Lambda^0}_\mu {\Lambda^1}_\nu F^{\mu\nu}
            = {\Lambda^0}_1 {\Lambda^1}_0 F^{10} + {\Lambda^0}_0 {\Lambda^1}_1 F^{01} \\
         &= (-\gamma\beta)(-\gamma\beta)(-E_\text{x}) + \gamma\gamma E_\text{x}
            = -\gamma^2\beta^2(E_\text{x}) + \gamma^2 E_\text{x}
            = E_\text{x}(1 - \beta^2)\gamma^2 \\
         &= E_\text{x}, \\
  E_{y'} &= F^{0'2'}
            = {\Lambda^0}_\mu {\Lambda^2}_\nu F^{\mu\nu}
            = {\Lambda^0}_\mu {\Lambda^2}_2 F^{\mu 2}
            = {\Lambda^0}_0 {\Lambda^2}_2 F^{02} + {\Lambda^0}_1 {\Lambda^2}_2 F^{12} \\
         &= \gamma \times 1 \times E_\text{y} + (-\beta\gamma) \times 1 \times B_\text{z}
            = \gamma E_\text{y} - \beta\gamma B_\text{z} \\
         &= \gamma\left(\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B}\right)_\text{y} \\
  E_{z'} &= F^{0'3'}
            = {\Lambda^0}_\mu {\Lambda^3}_\nu F^{\mu\nu}
            = {\Lambda^0}_\mu {\Lambda^3}_3 F^{\mu 3}
            = {\Lambda^0}_0 {\Lambda^3}_3 F^{03} + {\Lambda^0}_1 {\Lambda^3}_3 F^{13} \\
         &= \gamma \times 1 \times E_\text{z} - \beta\gamma \times 1 \times (-B_\text{y})
            = \gamma E_\text{z} + \beta\gamma B_\text{y} \\
         &= \gamma\left(\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B}\right)_\text{z}.
\end{align}</math>

Here, {{math|1='''''β''''' = (''β'', 0, 0)}} is used. These results can be summarized by
<math display="block">\begin{align}
  \mathbf{E}_{\parallel'} &= \mathbf{E}_\parallel \\
  \mathbf{B}_{\parallel'} &= \mathbf{B}_\parallel \\
       \mathbf{E}_{\bot'} &= \gamma \left( \mathbf{E}_\bot + \boldsymbol{\beta} \times \mathbf{B}_\bot \right) = \gamma \left( \mathbf{E} + \boldsymbol{\beta} \times \mathbf{B} \right)_\bot,\\
       \mathbf{B}_{\bot'} &= \gamma \left( \mathbf{B}_\bot - \boldsymbol{\beta} \times \mathbf{E}_\bot \right) = \gamma \left( \mathbf{B} - \boldsymbol{\beta} \times \mathbf{E} \right)_\bot,
\end{align}</math>
and are independent of the metric signature. For SI units, substitute {{math|''E'' → ''E''/''c''}}. {{harvtxt|Misner|Thorne|Wheeler|1973}} refer to this last form as the {{math|3 + 1}} view as opposed to the ''geometric view'' represented by the tensor expression
<math display="block">F^{\mu'\nu'} = {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu},</math>
and make a strong point of the ease with which results that are difficult to achieve using the {{math|3 + 1}} 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 {{math|'''E'''}} (alone) and {{math|'''B'''}} (alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations {{EquationNote|(T1)}} and {{EquationNote|(T2)}} that immediately yield {{EquationNote|(T3)}}. One should note that the primed and unprimed tensors refer to the ''same event in spacetime''. Thus the complete equation with spacetime dependence is
<math display="block">
  F^{\mu' \nu'}\left(x'\right) =
    {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}\left(\Lambda^{-1} x'\right) =
    {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}(x).
</math>

Length contraction has an effect on [[charge density]] {{mvar|ρ}} and [[current density]] {{math|'''J'''}}, 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,
<math display="block">\begin{align}
  \mathbf{j}' &= \mathbf{j} - \gamma\rho v\mathbf{n} + \left( \gamma - 1 \right)(\mathbf{j} \cdot \mathbf{n})\mathbf{n} \\
        \rho' &= \gamma \left(\rho - \mathbf{j} \cdot \frac{v\mathbf{n}}{c^2}\right),
\end{align}</math>
or, in the simpler geometric view,
<math display="block">j^{\mu'} = {\Lambda^{\mu'}}_\mu j^\mu.</math>

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 {{EquationNote|(T1)}} hold unmodified for any representation of the Lorentz group, including the [[bispinor]] representation. In {{EquationNote|(T2)}} one simply replaces all occurrences of {{math|Λ}} by the bispinor representation {{math|Π(Λ)}},

{{Equation box 1
 |indent =:
 |equation =
<math>\begin{align}
u \otimes v \rightarrow \Pi(\Lambda) u \otimes \Pi(\Lambda) v &=
  {\Pi(\Lambda)^\alpha}_\beta u^\beta \otimes {\Pi(\Lambda)^\rho}_\sigma v^\sigma\\ &=
  {\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma u^\beta \otimes v^\sigma\\ &\equiv
  {\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma w^{\beta\sigma}
\end{align}</math> {{spaces|13}} {{EquationRef|(T4)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

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<ref>{{harvnb|Weinberg|2002|loc=Chapter 3}}</ref>
{{NumBlk||<math display="block">\begin{align}
         &U(\Lambda, a) \Psi_{p_1\sigma_1 n_1; p_2\sigma_2 n_2; \cdots} \\
    = {} &e^{-ia_\mu \left[(\Lambda p_1)^\mu + (\Lambda p_2)^\mu + \cdots\right]}
            \sqrt{\frac{(\Lambda p_1)^0(\Lambda p_2)^0\cdots}{p_1^0 p_2^0 \cdots}}
            \left( \sum_{\sigma_1'\sigma_2' \cdots} D_{\sigma_1'\sigma_1}^{(j_1)}\left[W(\Lambda, p_1)\right] D_{\sigma_2'\sigma_2}^{(j_2)}\left[W(\Lambda, p_2)\right] \cdots \right)
            \Psi_{\Lambda p_1 \sigma_1' n_1; \Lambda p_2 \sigma_2' n_2; \cdots},
  \end{align}</math>
 | {{EquationRef|1}}
}}
where {{math|''W''(Λ, ''p'')}} is the [[Wigner's classification|Wigner's little group]]<ref>{{Cite web |title=INSPIRE |url=https://inspirehep.net/literature/26312 |access-date=2024-09-04 |website=inspirehep.net}}</ref> and {{math|''D''{{sup|(''j'')}}}} is the {{nowrap|{{math|(2''j'' + 1)}}-dimensional}} representation of {{math|SO(3)}}.

== See also ==

{{columns-list|colwidth=22em|
* [[Algebra of physical space]]
* [[Electromagnetic field]]
* [[Galilean transformation]]
* [[Gyrovector space]]
* [[Hyperbolic rotation]]
* [[Lorentz group]]
* [[Prandtl–Glauert transformation]]
* [[Principle of relativity]]
* [[Quaternion Lorentz Transformations]]
* [[Relativistic aberration]]
* [[Representation theory of the Lorentz group]]
* [[Ricci calculus]]
* [[Split-complex number]]
}}

== Footnotes ==

{{reflist|group=nb}}

== Notes ==

{{reflist|30em}}

== References ==

=== Websites ===
* {{citation |first1 = John J. |last1 = O'Connor |first2 =  Edmund F. |last2 = Robertson|title = A History of Special Relativity|url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Special_relativity.html|year=1996}}
* {{citation |first = Harvey R. |last = Brown
|title = Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited
|url =  http://philsci-archive.pitt.edu/id/eprint/987|year=2003}}

=== Papers ===

{{refbegin|30em}}
* {{cite journal | first = J. T. | last = Cushing  | title = Vector Lorentz transformations | journal = [[American Journal of Physics]] | year = 1967 |volume = 35 | issue = 9  | doi = 10.1119/1.1974267 | pages = 858–862 |url=https://www.deepdyve.com/browse/journals/the-american-journal-of-physics/1967/v35/i9?page=4|bibcode = 1967AmJPh..35..858C }}
* {{cite journal | first = A. J. | last = Macfarlane| title = On the Restricted Lorentz Group and Groups Homomorphically Related to It | journal = [[Journal of Mathematical Physics]] | year = 1962 | volume = 3| issue=6 | pages = 1116–1129| doi=10.1063/1.1703854 | bibcode=1962JMP.....3.1116M| hdl = 2027/mdp.39015095220474 | hdl-access = free }}
* {{citation |first = Tony |last = Rothman
|title = Lost in Einstein's Shadow
|url = http://www.americanscientist.org/libraries/documents/200622102452_866.pdf
|journal = American Scientist |volume = 94 |issue = 2 |pages = 112f |year = 2006}}
* {{citation
|last=Darrigol |first=Olivier
|title=The Genesis of the theory of relativity
|year=2005
|journal=Séminaire Poincaré
|volume=1
|pages=1–22
|url=http://www.bourbaphy.fr/darrigol2.pdf
|doi=10.1007/3-7643-7436-5_1|isbn=978-3-7643-7435-8
|bibcode=2006eins.book....1D
}}
* {{citation
 |first        = Michael N. |last         = Macrossan
 |title        = A Note on Relativity Before Einstein
 |url          = http://espace.library.uq.edu.au/view.php?pid=UQ:9560
 |journal      = Br. J. Philos. Sci.
 |volume       = 37
 |issue        = 2
 |year         = 1986
 |pages        = 232–34
 |doi          = 10.1093/bjps/37.2.232
 |citeseerx    = 10.1.1.679.5898
 |access-date  = 2007-04-02
 |archive-url  = https://web.archive.org/web/20131029203003/http://espace.library.uq.edu.au/view.php?pid=UQ:9560
 |archive-date = 2013-10-29
 |url-status   = dead
}}
* {{citation |first = Henri |last = Poincaré  |author-link = Henri Poincaré
|title = On the Dynamics of the Electron
|journal = Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences
|volume = 140 |pages = 1504–1508 |year = 1905|title-link = s:Translation:On the Dynamics of the Electron (June)  }}
* {{citation
|last=Einstein |first=Albert
|year=1905
|title=Zur Elektrodynamik bewegter Körper
|journal=Annalen der Physik
|volume=322
|issue=10
|pages=891–921
|url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf
|doi=10.1002/andp.19053221004|bibcode = 1905AnP...322..891E |doi-access=free
}}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].
* {{cite journal |last=Lorentz |first=Hendrik Antoon |year=1904 |title=Electromagnetic phenomena in a system moving with any velocity smaller than that of light |journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences |volume=6 |pages=809–831 |title-link=s:Electromagnetic phenomena}}
* {{cite book|last=Einstein |first=A. |year=1916 |author-link=Albert Einstein|title=Relativity: The Special and General Theory |url=https://archive.org/stream/cu31924011804774#page/n35/mode/2up |access-date=2012-01-23}} {{Cite book |ref=none |first=A. |last=Einstein |author-link=Albert Einstein |title=Relativity: The Special and General Theory |place=New York |url=http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ |publisher=Three Rivers Press |year=1916 |publication-date=1995 |isbn=978-0-517-88441-6 |via=Albert Einstein Reference Archive}}
* {{cite journal|title=Thomas rotation and the parameterization of the Lorentz transformation group|first=A. A. |last=Ungar|journal=Foundations of Physics Letters|year=1988|volume=1|issue=1|pages=55–89|doi=10.1007/BF00661317|issn=0894-9875|bibcode=1988FoPhL...1...57U|s2cid=121240925 }} eqn (55).
* {{cite journal | first = A. A. | last = Ungar | title = The relativistic velocity composition paradox and the Thomas rotation| journal = [[Foundations of Physics]] | volume = 19 | issue = 11 | pages = 1385–1396 | year = 1989 |bibcode = 1989FoPh...19.1385U |doi = 10.1007/BF00732759 | s2cid = 55561589 }}
* {{cite journal | first = A. A. | last = Ungar | title = The relativistic composite-velocity reciprocity principle | citeseerx = 10.1.1.35.1131 | journal = [[Foundations of Physics]] | year = 2000 | volume = 30 | issue = 2 | pages = 331–342 | doi = 10.1023/A:1003653302643 | bibcode = 2000FoPh...30..331U | s2cid = 118634052 }}
* {{cite journal | first = C. I. | last = Mocanu | title = Some difficulties within the framework of relativistic electrodynamics| journal = Archiv für Elektrotechnik | year = 1986 | volume = 69 | issue = 2 | pages = 97–110 | doi=10.1007/bf01574845| s2cid = 123543303 }}
* {{cite journal | first = C. I. | last = Mocanu | title = On the relativistic velocity composition paradox and the Thomas rotation| journal = [[Foundations of Physics]] | year = 1992 | volume = 5 | issue = 5 | pages = 443–456 | doi=10.1007/bf00690425|bibcode = 1992FoPhL...5..443M | s2cid = 122472788 }}
* {{cite book|last=Weinberg|first=S.|year=2002|title=The Quantum Theory of Fields, vol I|isbn=978-0-521-55001-7|author-link=Steven Weinberg|publisher=[[Cambridge University Press]] |url=https://archive.org/details/quantumtheoryoff00stev}}
{{refend}}

=== Books ===
{{refbegin|30em}}
* {{cite book |last1=Barut |first1=Asim Orhan |title=Electrodynamics and Classical Theory of Fields and Particles |date=1964 |publisher=Macmillan |isbn=978-0-486-64038-9 |url=https://books.google.com/books?id=B6_vAAAAMAAJ |language=en}}
* {{cite book |title=Mathematics for Physicists |first1=Philippe |last1=Dennery |first2=André |last2=Krzywicki |publisher=Courier Corporation |year=2012 |isbn=978-0-486-15712-2 |url=https://books.google.com/books?id=ogHCAgAAQBAJ}}
* {{cite book |title=An Introduction to the Standard Model of Particle Physics |edition=2nd |first1=W. N. |last1=Cottingham |first2=D. A. |last2=Greenwood |publisher=Cambridge University Press |year=2007 |isbn=978-1-139-46221-1 |url=https://books.google.com/books?id=Dm36BYq9iu0C}}
* {{cite book|title=University Physics – With Modern Physics|edition=12th|first1=H. D.|last1=Young|first2=R. A.|last2=Freedman|year=2008|publisher=Pearson Addison-Wesley |isbn=978-0-321-50130-1}}
* {{cite book|title=3000 Solved Problems in Physics|series=Schaum Series|first=A.|last=Halpern|publisher=Mc Graw Hill|year=1988|isbn=978-0-07-025734-4|page=688}}
* {{cite book|title=Dynamics and Relativity|first1=J. R.|last1=Forshaw|first2=A. G.|last2=Smith|series=Manchester Physics Series|publisher=John Wiley & Sons Ltd|year=2009|isbn=978-0-470-01460-8|pages=124–126}}
* {{cite book|title=Spacetime Physics|first1=J. A.|last1=Wheeler|first2=E. F|last2=Taylor|author-link1=John Archibald Wheeler|author-link2=Edwin F. Taylor|year=1971|publisher=Freeman|isbn=978-0-7167-0336-5}}
* {{cite book|title=Gravitation|first1=J. A.|last1=Wheeler|first2=K. S.|last2=Thorne|first3=C. W. |last3=Misner|author-link1=John Archibald Wheeler|author-link2=Kip Thorne|author-link3=Charles W. Misner|year=1973|publisher=Freeman|isbn=978-0-7167-0344-0}}
* {{cite book|title=Spacetime and Geometry: An Introduction to General Relativity |edition=illustrated |first1=S. M. |last1=Carroll |author-link=Sean M. Carroll|publisher=Addison Wesley |year=2004 |isbn=978-0-8053-8732-2 |page=22 |url=https://books.google.com/books?id=1SKFQgAACAAJ}}
* {{cite book|title=Electromagnetism |edition=2nd |first1=I. S.|last1=Grant|first2=W. R.|last2=Phillips|series=Manchester Physics|publisher=John Wiley & Sons|year=2008|isbn=978-0-471-92712-9|chapter=14}}
* {{cite book|title=Introduction to Electrodynamics|edition=3rd|first1= D. J.|last1=Griffiths|author-link=David Griffiths (physicist)|publisher=Pearson Education, Dorling Kindersley|year=2007|isbn=978-81-7758-293-2}}
* {{cite book|year=2003|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations An Elementary Introduction|publisher=[[Springer Nature|Springer]]|isbn=978-0-387-40122-5}}
* {{cite book |last1=Weinberg |first1=Steven |title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity |date=1972 |publisher=Wiley |isbn=978-0-471-92567-5 |url=https://books.google.com/books?id=XLbvAAAAMAAJ |language=en}}
* {{citation |first = S. |last = Weinberg |title = Cosmology |author-link = Steven Weinberg |publisher = Wiley |year = 2008|isbn = 978-0-19-852682-7}}
* {{citation |first = S. |last = Weinberg |author-link = Steven Weinberg |title = The quantum theory of fields (3 vol.)|publisher = Cambridge University Press | year = 2005 |isbn = 978-0-521-67053-1|volume=1}}
* {{citation |first = T. |last = Ohlsson |author-link = Tommy Ohlsson |title = Relativistic Quantum Physics|publisher = Cambridge University Press | year = 2011 |isbn = 978-0-521-76726-2}}
* {{cite book|last=Goldstein|first=H.|author-link=Herbert Goldstein|title=Classical Mechanics|edition=2nd|publisher=[[Addison-Wesley Publishing Company|Addison-Wesley]]|location=Reading MA|isbn=978-0-201-02918-5|year=1980|orig-year=1950}}
* {{cite book|first=J. D.|last=Jackson|author-link=John David Jackson (physicist)|title=Classical Electrodynamics|pages=[https://archive.org/details/classicalelectro00jack_0/page/542 542–545]|edition=2nd|year=1975|orig-year=1962|isbn=978-0-471-43132-9|publisher=[[John Wiley & Sons]]|chapter=Chapter 11|chapter-url=https://archive.org/details/classicalelectro00jack_0/page/542}}
* {{cite book|last1=Landau|first1=L. D.|author-link1=Lev Landau|last2=Lifshitz|first2=E. M.|author-link2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=[[Course of Theoretical Physics]]|volume=2|edition=4th|publisher=[[Butterworth–Heinemann]]|isbn=0-7506-2768-9|year=2002|orig-year=1939|pages=9–12}}
* {{cite book|last1=Feynman|first1=R. P.|author-link1=Richard Feynman|last2=Leighton|first2= R. B.|author-link2=Robert B. Leighton|last3=Sands|first3=M. |author-link3=Matthew Sands|title=The Feynman Lectures on Physics|volume=1|publisher=Addison Wesley|isbn=978-0-201-02117-2|year=1977|orig-year=1963|chapter=15}}
* {{cite book|last1=Feynman|first1=R. P.|author-link1=Richard Feynman|last2=Leighton|first2= R. B.|author-link2=Robert B. Leighton|last3=Sands|first3=M. |author-link3=Matthew Sands|title=The Feynman Lectures on Physics|volume=2|publisher=Addison Wesley|isbn=978-0-201-02117-2|year=1977|orig-year=1964|chapter=13}}
* {{cite book|last1=Misner |first1=Charles W. |author-link1=Charles W. Misner|last2=Thorne |first2=Kip S. |author-link2=Kip Thorne |last3=Wheeler |first3=John Archibald |author-link3=John Archibald Wheeler |year=1973 |title=Gravitation |publisher=[[W. H. Freeman]]|location=San Francisco|isbn=978-0-7167-0344-0}}
* {{cite book|first=W.|last=Rindler|author-link=Wolfgang Rindler|year=2006|orig-year=2001|title=Relativity Special, General and Cosmological|chapter = Chapter 9|edition=2nd|publisher=[[Oxford University Press]]|location=Dallas|isbn=978-0-19-856732-5}}
* {{cite book|first=L. H.|last=Ryder|author-link=Lewis Ryder|title=Quantum Field Theory|year=1996|orig-year=1985|isbn=978-0521478144|edition=2nd|publisher=[[Cambridge University Press]]|location=Cambridge}}
* {{cite book|last=Sard|first=R. D.|title=Relativistic Mechanics - Special Relativity and Classical Particle Dynamics|url=https://archive.org/details/relativisticmech0000sard|url-access=registration|year=1970|publisher=W. A. Benjamin|location=New York|isbn=978-0805384918}}
* {{cite book|first1=R. U. |last1=Sexl |first2=H. K. |last2=Urbantke|title=Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics|year=2001|orig-year=1992|publisher=Springer|isbn=978-3211834435|url=https://books.google.com/books?id=iyj0CAAAQBAJ&q=sexl+relativity}}
* {{cite book|last=Gourgoulhon|first=Eric|title=Special Relativity in General Frames: From Particles to Astrophysics|year=2013|publisher=Springer|isbn=978-3-642-37276-6|page=213|url=https://books.google.com/books?id=N4HBBAAAQBAJ&q=mcfarlane+1962+lorentz+transformation&pg=PA215}}
* {{cite book|last1=Chaichian|first1=Masud|last2=Hagedorn|first2=Rolf|title=Symmetry in quantum mechanics:From angular momentum to supersymmetry|year=1997|publisher=IoP|isbn=978-0-7503-0408-5|page=239|url=https://books.google.com/books?id=pEhjQgAACAAJ&q=Symmetry+in+quantum+mechanics}}
* {{cite book|last1=Landau|first1=L.D.|author-link1=Lev Landau|last2=Lifshitz|first2=E.M.|author-link2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=Course of Theoretical Physics|volume=2|edition=4th|publisher=[[Butterworth–Heinemann]]|isbn=0-7506-2768-9|year=2002|orig-year=1939}}
{{refend}}

== Further reading ==
* {{citation
 |first1=A.
 |last1=Ernst
 |first2=J.-P.
 |last2=Hsu
 |title=First proposal of the universal speed of light by Voigt 1887
 |journal=Chinese Journal of Physics
 |volume=39
 |issue=3
 |url=http://psroc.phys.ntu.edu.tw/cjp/v39/211.pdf
 |pages=211–230
 |year=2001
 |bibcode=2001ChJPh..39..211E
 |url-status=dead
 |archive-url=https://web.archive.org/web/20110716083015/http://psroc.phys.ntu.edu.tw/cjp/v39/211.pdf
 |archive-date=2011-07-16
}}
* {{citation |first1 = Stephen T. |last1 = Thornton |first2 = Jerry B. |last2 = Marion |title =  Classical dynamics of particles and systems |edition = 5th |place = Belmont, [CA.] |publisher = Brooks/Cole |year = 2004 |pages = 546–579 |isbn = 978-0-534-40896-1}}
* {{citation |first = Woldemar |last = Voigt |author-link = Woldemar Voigt |title = Über das Doppler'sche princip |journal = Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen |volume = 2 |pages =  41–51 |year = 1887}}

== External links ==
{{Wikisource portal|Relativity}}
{{Wikibooks|special relativity}}
{{Wikiversity|Lorentz transformations}}
* [https://www.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf Derivation of the Lorentz transformations]. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
* [https://web.archive.org/web/20061206092114/http://casa.colorado.edu/~ajsh/sr/paradox.html The Paradox of Special Relativity].  This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
* [http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html Relativity] {{Webarchive|url=https://web.archive.org/web/20110829032009/http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html |date=2011-08-29 }} – a chapter from an online textbook
* [http://www.adamauton.com/warp/ Warp Special Relativity Simulator]. A computer program demonstrating the Lorentz transformations on everyday objects.
* {{YouTube|C2VMO7pcWhg|Animation clip}} visualizing the Lorentz transformation.
* [https://www.youtube.com/watch?v=Rh0pYtQG5wI MinutePhysics video] on [[YouTube]] explaining and visualizing the Lorentz transformation with a mechanical Minkowski diagram
* [https://www.desmos.com/calculator/uxu4h22ya4 Interactive graph] on [[Desmos (graphing)]] showing Lorentz transformations with a virtual Minkowski diagram
* [https://www.desmos.com/calculator/pc7azsxteh Interactive graph] on Desmos showing Lorentz transformations with points and hyperbolas
* [http://math.ucr.edu/~jdp/Relativity/Lorentz_Frames.html Lorentz Frames Animated] ''from John de Pillis.''  Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, ''etc''.

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[[Category:Special relativity]]
[[Category:Mathematical physics]]
[[Category:Spacetime]]
[[Category:Coordinate systems]]
[[Category:Hendrik Lorentz]]