In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components, which transform under Lorentz transformations with respect to a change of basis. Its magnitude is determined by an indefinite quadratic form, the preservation of which defines the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another reference frame). Four-vectors describe, for instance, position xμ in spacetime modeled as Minkowski space, a particle's four-momentum pμ, the amplitude of the electromagnetic four-potential Aμ(x) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra. The Lorentz group may be represented by a set of 4 × 4 matrices Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by
X ′ = Λ X , {\displaystyle X'=\Lambda X,}
(matrix multiplication) where the components of the primed object refer to the new frame. Related to the examples above that are given as contravariant vectors, there are also the corresponding covariant vectors xμ, pμ and Aμ(x). These transform according to the rule
X ′ = ( Λ − 1 ) T X , {\displaystyle X'=\left(\Lambda ^{-1}\right)^{\textrm {T}}X,}
where T denotes the matrix transpose. This rule is different from the above rule. It corresponds to the dual representation of the standard representation. However, for the Lorentz group the dual of any representation is equivalent to the original representation (see Representation theory of the Lorentz group). Thus the objects with covariant indices are four-vectors as well. For an example of a well-behaved four-component object in special relativity that is not a four-vector, see Dirac spinor. It is similarly defined, the difference being that the transformation rule under Lorentz transformations is given by a representation other than the standard representation. In this case, the rule reads X′ = Π(Λ)X, where Π(Λ) is a 4×4 matrix other than Λ. Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These include scalars, spinors, tensors and spinor-tensors. The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity. In the standard configuration, where the primed frame has speed u along the positive x-axis, the transformation of four-vectors is:
X ′ = [ γ ( u ) − γ ( u ) u c 2 0 0 − γ ( u ) u γ ( u ) 0 0 0 0 1 0 0 0 0 1 ] X , {\displaystyle X'={\begin{bmatrix}{\gamma (u)}&{-\gamma (u){\frac {u}{c^{2}}}}&{0}&{0}\\{-\gamma (u)u}&{\gamma (u)}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\end{bmatrix}}X,}
or
X ′ = [ γ ( u ) − γ ( u ) u c 0 0 − γ ( u ) u c γ ( u ) 0 0 0 0 1 0 0 0 0 1 ] X , {\displaystyle X'={\begin{bmatrix}{\gamma (u)}&{-\gamma (u){\frac {u}{c}}}&{0}&{0}\\{-\gamma (u){\frac {u}{c}}}&{\gamma (u)}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\end{bmatrix}}X,}
depending on convention (viz. whether events are written (t, x, y, z) or (ct, x, y, z), respectively).
Notation The notations in this article are: lowercase bold for three-dimensional vectors, hats for three-dimensional unit vectors, capital bold for four dimensional vectors (except for the four-gradient operator), and tensor index notation.
Four-vector algebra
Four-vectors in a real-valued basis A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations:
A = ( A 0 , A 1 , A 2 , A 3 ) = A 0 E 0 + A 1 E 1 + A 2 E 2 + A 3 E 3 = A 0 E 0 + A i E i = A α E α {\displaystyle {\begin{aligned}\mathbf {A} &=\left(A^{0},\,A^{1},\,A^{2},\,A^{3}\right)\\&=A^{0}\mathbf {E} _{0}+A^{1}\mathbf {E} _{1}+A^{2}\mathbf {E} _{2}+A^{3}\mathbf {E} _{3}\\&=A^{0}\mathbf {E} _{0}+A^{i}\mathbf {E} _{i}\\&=A^{\alpha }\mathbf {E} _{\alpha }\end{aligned}}}
where Aα is the component multiplier and Eα is the basis vector; note that both are necessary to make a vector, and that when Aα is seen alone, it refers strictly to the components of the vector. The upper indices indicate contravariant components. Here the standard convention is that Latin indices take values for spatial components, so that i = 1, 2, 3, and Greek indices take values for time and space components, so α = 0, 1, 2, 3, used with the summation convention. The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in scalar products (examples are given below), or raising and lowering indices. In special relativity, the spacelike basis E1, E2, E3 and components A1, A2, A3 are often Cartesian basis and components:
A = ( A t , A x , A y , A z ) = A t E t + A x E x + A y E y + A z E z {\displaystyle {\begin{aligned}\mathbf {A} &=\left(A_{t},\,A_{x},\,A_{y},\,A_{z}\right)\\&=A_{t}\mathbf {E} _{t}+A_{x}\mathbf {E} _{x}+A_{y}\mathbf {E} _{y}+A_{z}\mathbf {E} _{z}\\\end{aligned}}}
although, of course, any other basis and components may be used, such as spherical polar coordinates
A = ( A t , A r , A θ , A ϕ ) = A t E t + A r E r + A θ E θ + A ϕ E ϕ {\displaystyle {\begin{aligned}\mathbf {A} &=\left(A_{t},\,A_{r},\,A_{\theta },\,A_{\phi }\right)\\&=A_{t}\mathbf {E} _{t}+A_{r}\mathbf {E} _{r}+A_{\theta }\mathbf {E} _{\theta }+A_{\phi }\mathbf {E} _{\phi }\\\end{aligned}}}
or cylindrical polar coordinates,
A = ( A t , A r , A θ , A z ) = A t E t + A r E r + A θ E θ + A z E z {\displaystyle {\begin{aligned}\mathbf {A} &=(A_{t},\,A_{r},\,A_{\theta },\,A_{z})\\&=A_{t}\mathbf {E} _{t}+A_{r}\mathbf {E} _{r}+A_{\theta }\mathbf {E} _{\theta }+A_{z}\mathbf {E} _{z}\\\end{aligned}}}
or any other orthogonal coordinates, or even general curvilinear coordinates. Note the coordinate labels are always subscripted as labels and are not indices taking numerical values. In general relativity, local curvilinear coordinates in a local basis must be used. Geometrically, a four-vector can still be interpreted as an arrow, but in spacetime - not just space. In relativity, the arrows are drawn as part of Minkowski diagram (also called spacetime diagram). In this article, four-vectors will be referred to simply as vectors. It is also customary to represent the bases by column vectors:
E 0 = ( 1 0 0 0 ) , E 1 = ( 0 1 0 0 ) , E 2 = ( 0 0 1 0 ) , E 3 = ( 0 0 0 1 ) {\displaystyle \mathbf {E} _{0}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}\,,\quad \mathbf {E} _{1}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}}\,,\quad \mathbf {E} _{2}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}}\,,\quad \mathbf {E} _{3}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}}
so that:
A = ( A 0 A 1 A 2 A 3 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}}
The relation between the covariant and contravariant coordinates is through the Minkowski metric tensor (referred to as the metric), η which raises and lowers indices as follows:
A μ = η μ ν A ν , {\displaystyle A_{\mu }=\eta _{\mu \nu }A^{\nu }\,,}
and in various equivalent notations the covariant components are:
A = ( A 0 , A 1 , A 2 , A 3 ) = A 0 E 0 + A 1 E 1 + A 2 E 2 + A 3 E 3 = A 0 E 0 + A i E i = A α E α {\displaystyle {\begin{aligned}\mathbf {A} &=(A_{0},\,A_{1},\,A_{2},\,A_{3})\\&=A_{0}\mathbf {E} ^{0}+A_{1}\mathbf {E} ^{1}+A_{2}\mathbf {E} ^{2}+A_{3}\mathbf {E} ^{3}\\&=A_{0}\mathbf {E} ^{0}+A_{i}\mathbf {E} ^{i}\\&=A_{\alpha }\mathbf {E} ^{\alpha }\\\end{aligned}}}
where the lowered index indicates it to be covariant. Often the metric is diagonal, as is the case for orthogonal coordinates (see line element), but not in general curvilinear coordinates. The bases can be represented by row vectors:
E 0 = ( 1 0 0 0 ) , E 1 = ( 0 1 0 0 ) , E 2 = ( 0 0 1 0 ) , E 3 = ( 0 0 0 1 ) , {\displaystyle {\begin{aligned}\mathbf {E} ^{0}&={\begin{pmatrix}1&0&0&0\end{pmatrix}}\,,&\mathbf {E} ^{1}&={\begin{pmatrix}0&1&0&0\end{pmatrix}}\,,\\[1ex]\mathbf {E} ^{2}&={\begin{pmatrix}0&0&1&0\end{pmatrix}}\,,&\mathbf {E} ^{3}&={\begin{pmatrix}0&0&0&1\end{pmatrix}},\end{aligned}}}
so that:
A = ( A 0 A 1 A 2 A 3 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}}}
The motivation for the above conventions are that the scalar product is a scalar, see below for details.
Lorentz transformation
Given two inertial or rotated frames of reference, a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix Λ:
A ′ = Λ A {\displaystyle \mathbf {A} '={\boldsymbol {\Lambda }}\mathbf {A} }
In index notation, the contravariant and covariant components transform according to, respectively:
A ′ μ = Λ μ
ν A ν , A ′ μ = Λ μ
ν A ν {\displaystyle {A'}^{\mu }=\Lambda ^{\mu }{}_{\nu }A^{\nu }\,,\quad {A'}_{\mu }=\Lambda _{\mu }{}^{\nu }A_{\nu }}
in which the matrix Λ has components Λμν in row μ and column ν, and the matrix (Λ−1)T has components Λμν in row μ and column ν. For background on the nature of this transformation definition, see tensor. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see special relativity.
Pure rotations about an arbitrary axis For two frames rotated by a fixed angle θ about an axis defined by the unit vector:
n ^ = ( n ^ 1 , n ^ 2 , n ^ 3 ) , {\displaystyle {\hat {\mathbf {n} }}=\left({\hat {n}}_{1},{\hat {n}}_{2},{\hat {n}}_{3}\right)\,,}
without any boosts, the matrix Λ has components given by:
Λ 00 = 1 Λ 0 i = Λ i 0 = 0 Λ i j = ( δ i j − n ^ i n ^ j ) cos θ − ε i j k n ^ k sin θ + n ^ i n ^ j {\displaystyle {\begin{aligned}\Lambda _{00}&=1\\\Lambda _{0i}=\Lambda _{i0}&=0\\\Lambda _{ij}&=\left(\delta _{ij}-{\hat {n}}_{i}{\hat {n}}_{j}\right)\cos \theta -\varepsilon _{ijk}{\h