Editing Four-vector (section)

{{Short description|4-dimensional vector in relativity}}
{{distinguish|p-vector}}
{{Use American English|date = March 2019}}
{{spacetime|cTopic=Mathematics}}
In [[special relativity]], a '''four-vector''' (or '''4-vector''', sometimes '''Lorentz vector''')<ref>Rindler, W. ''Introduction to Special Relativity (2nd edn.)'' (1991) Clarendon Press Oxford {{ISBN|0-19-853952-5}}</ref> is an object with four components, which transform in a specific way under [[Lorentz transformation]]s. Specifically, a four-vector is an element of a four-dimensional [[vector space]] considered as a [[representation space]] of the [[Representation theory of the Lorentz group|standard representation]] of the [[Lorentz group]], the ({{sfrac|1|2}},{{sfrac|1|2}}) representation. It differs from a [[Euclidean vector]] in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include [[Rotation group SO(3)|spatial rotations]] and [[Lorentz transformation#Physical formulation of Lorentz boosts|boosts]] (a change by a constant velocity to another [[inertial reference frame]]).<ref name="BaskalKim2015">{{cite book|author1=Sibel Baskal|author2=Young S Kim|author3=Marilyn E Noz|title=Physics of the Lorentz Group|date=1 November 2015|publisher=Morgan & Claypool Publishers|isbn=978-1-68174-062-1}}</ref>{{rp|ch1}}

Four-vectors describe, for instance, position {{math|''x''{{i sup|''μ''}}}} in spacetime modeled as [[Minkowski space]], a particle's [[four-momentum]] {{math|''p''{{i sup|''μ''}}}}, the amplitude of the [[electromagnetic four-potential]] {{math|''A''{{i sup|''μ''}}(''x'')}} at a point {{mvar|x}} in spacetime, and the elements of the subspace spanned by the [[gamma matrices]] inside the [[Representation theory of the Lorentz group#Reducible representations|Dirac algebra]].

The Lorentz group may be represented by 4×4 matrices {{math|Λ}}. The action of a Lorentz transformation on a general [[contravariant vector|contravariant]] four-vector {{mvar|X}} (like the examples above), regarded as a column vector with [[Cartesian coordinates]] with respect to an [[Inertial frame#Special relativity|inertial frame]] in the entries, is given by

<math display="block">X' = \Lambda X,</math>

(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 vector]]s {{math|''x''<sub>''μ''</sub>}}, {{math|''p''<sub>''μ''</sub>}} and {{math|''A''<sub>''μ''</sub>(''x'')}}. These transform according to the rule

<math display="block">X' = \left(\Lambda^{-1}\right)^\textrm{T} X,</math>

where {{math|<sup>T</sup>}} 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 [[Representation theory of the Lorentz group#Dual representations|equivalent]] to the original representation. 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 [[bispinor]]. 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 {{math|''X''{{′}} {{=}} Π(Λ)''X''}}, where {{math|Π(Λ)}} is a 4×4 matrix other than {{math|Λ}}. Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These include [[scalar field|scalar]]s, [[spinor]]s, [[tensor field|tensor]]s 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.<!-- TO DO: provide a GR section for this article! -->