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| {{spacetime}}
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| In the [[theory of relativity]], a '''four-vector''' or '''4-vector''' is a [[vector (geometry)|vector]] in a four-dimensional real [[vector space]], called [[Minkowski space]]. It differs from a [[Euclidean vector]] in that four-vectors transform by the [[Lorentz transformations]]. The term ''four-vector'' tacitly assumes that its components refer to a [[vector basis]]. In a [[standard basis]], the components transform between these bases as the [[space]] and [[time]] coordinate differences, (''c''Δ''t'', Δ''x'', Δ''y'', Δ''z'') under [[Translation (geometry)|spatial translations]], [[Rotation group SO(3)|spatial rotations]], spatial and time [[Inversion in a point#Inversion with respect to the origin|inversions]] and ''[[Lorentz transformation#Boost in the x-direction|boosts]]'' (a change by a constant velocity to another [[inertial reference frame]]). The set of all such translations, rotations, inversions and boosts (called [[Poincaré transformation]]s) forms the [[Poincaré group]]. The set of rotations, inversions and boosts ([[Lorentz transformation]]s, described by 4×4 [[matrix (mathematics)|matrices]]) forms the [[Lorentz group]].
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| 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! -->
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| The notations in this article are: lowercase bold for [[three dimensional space|three dimensional]] vectors, hats for three dimensional [[unit vector]]s, capital bold for [[Spacetime|four dimensional]] vectors (except for the four-gradient), and [[tensor index notation]].
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| == Four-vector algebra ==
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| ===Four-vectors in a real-valued basis===
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| A '''four-vector''' ''A'' is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations:<ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (BSA), 2006, ISBN 0-07-145545-0</ref>
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| :<math> \begin{align}
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| \mathbf{A} & = (A^0, \, A^1, \, A^2, \, A^3) \\
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| & = A^0\mathbf{e}_0 + A^1 \mathbf{e}_1 + A^2 \mathbf{e}_2 + A^3 \mathbf{e}_3 \\
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| & = A^0\mathbf{e}_0 + A^i \mathbf{e}_i \\
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| & = A^\alpha\mathbf{e}_\alpha\\
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| \end{align}</math>
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| The upper indices indicate [[Covariance and contravariance of vectors|contravariant]] components. Here the standard convention that Latin indices take values for spatial components, so that ''i'' = 1, 2, 3, and Greek indices take values for space ''and time'' 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 inner products (examples are given below), or [[raising and lowering indices]].
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| In special relativity, the spacelike basis '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> and components ''A''<sup>1</sup>, ''A''<sup>2</sup>, ''A''<sup>3</sup> are often [[Cartesian coordinates|Cartesian]] basis and components:
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| :<math> \begin{align}
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| \mathbf{A} & = (A_t, \, A_x, \, A_y, \, A_z) \\
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| & = A_t \mathbf{e}_t + A_x \mathbf{e}_x + A_y \mathbf{e}_y + A_z \mathbf{e}_z \\
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| \end{align}</math>
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| although, of course, any other basis and components may be used, such as [[spherical polar coordinates]]
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| :<math> \begin{align}
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| \mathbf{A} & = (A_t, \, A_r, \, A_\theta, \, A_\phi) \\
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| & = A_t \mathbf{e}_t + A_r \mathbf{e}_r + A_\theta \mathbf{e}_\theta + A_\phi \mathbf{e}_\phi \\
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| \end{align}</math>
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| or [[cylindrical polar coordinates]],
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| :<math> \begin{align}
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| \mathbf{A} & = (A_t, \, A_r, \, A_\theta, \, A_z) \\
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| & = A_t \mathbf{e}_t + A_r \mathbf{e}_r + A_\theta \mathbf{e}_\theta + A_z \mathbf{e}_z \\
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| \end{align}</math>
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| 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 a [[spacetime diagram]] or [[Minkowski diagram]]. In this article, four-vectors will be referred to simply as vectors.
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| It is also customary to represent the bases by [[column vector]]s:
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| :<math> \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} </math>
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| so that:
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| :<math> \mathbf{A} = \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix} </math>
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| The relation between the [[covariant vector|covariant]] and contravariant coordinates is through the [[Minkowski metric|Minkowski]] [[metric tensor]], ''η'' which [[raising and lowering indices|raises and lowers indices]] as follows:
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| :<math>A_{\mu} = \eta_{\mu \nu} A^{\nu} \,, </math>
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| and in various equivalent notations the covariant components are:
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| :<math> \begin{align}
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| \mathbf{A} & = (A_0, \, A_1, \, A_2, \, A_3) \\
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| & = A_0\mathbf{e}^0 + A_1 \mathbf{e}^1 + A_2 \mathbf{e}^2 + A_3 \mathbf{e}^3 \\
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| & = A_0\mathbf{e}^0 + A_i \mathbf{e}^i \\
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| & = A_\alpha\mathbf{e}^\alpha\\
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| \end{align}</math>
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| where the lowered index indicates it to be [[Covariance and contravariance of vectors|covariant]]. Often the metric is diagonal, as is the case for [[orthogonal coordinates]] (see [[line element]]), but not in general [[curvilinear coordinates]].
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| The bases can be represented by [[row vector]]s:
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| :<math> \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} </math>
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| so that:
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| :<math> \mathbf{A} = \begin{pmatrix} A_0 & A_1 & A_2 & A_3 \end{pmatrix} </math>
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| The motivation for the above conventions are that the inner product is a scalar, see below for details.
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| === Lorentz transformation ===
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| {{main|Lorentz transformation}}
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| Given two inertial or rotated [[frame of reference|frames of reference]], a four-vector is defined as a quantity which transforms according to the [[Lorentz transformation]] matrix '''Λ''':
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| :<math>\mathbf{A}'=\boldsymbol{\Lambda}\mathbf{A}\,\!</math>
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| In index notation, the contravariant and covariant components transform according to, respectively:
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| :<math>{A'}^\mu = \Lambda^\mu {}_\nu A^\nu \,,\quad {A'}_\mu = \Lambda_\mu {}^\nu A_\nu</math>
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| in which the matrix '''Λ''' has components ''Λ<sup>μ</sup><sub>ν</sub>'' in row ''μ'' and column ''ν'', and the [[inverse matrix]] '''Λ'''<sup>−1</sup> has components ''Λ<sub>μ</sub><sup>ν</sup>'' in row ''μ'' and column ''ν''.
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| For background on the nature of this transformation definition, see [[tensor#Definition|tensor]]. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see [[Special relativity#Transformations of physical quantities between reference frames|special relativity]].
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| ====Pure rotations about an arbitrary axis ====
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| For two frames rotated by a fixed angle ''θ'' about an axis defined by the [[unit vector]]:
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| :<math>\hat{\mathbf{n}} = (\hat{n}_1,\hat{n}_2,\hat{n}_3)\,,</math>
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| without any boosts, the matrix '''Λ''' has components given by:<ref>{{cite book| author=C.B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill|edition=2nd|page=1333|year=1994| isbn=0-07-051400-3}}</ref>
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| :<math>\Lambda_{00} = 1 </math>
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| :<math> \Lambda_{0i} = \Lambda_{i0} = 0 </math>
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| :<math>\Lambda_{ij} = (\delta_{ij} - \hat{n}_i \hat{n}_j) \cos\theta - \varepsilon_{ijk} \hat{n}_k \sin\theta + \hat{n}_i \hat{n}_j </math>
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| where ''δ<sub>ij</sub>'' is the [[Kronecker delta]], and ''ε<sub>ijk</sub>'' is the [[three dimensional]] [[Levi-Civita symbol]]. The spacelike components of 4-vectors are rotated, while the time-like components remain unchanged.
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| For the case of rotations about the ''z''-axis only, the spacelike part of the Lorentz matrix reduces to the [[rotation matrix]] about the ''z''-axis:
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| :<math>
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| \begin{pmatrix}
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| {A'}^0 \\ {A'}^1 \\ {A'}^2 \\ {A'}^3
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| \end{pmatrix}
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| =
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| \begin{pmatrix}
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| 1 & 0 & 0 & 0 \\
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| 0 & \cos\theta &-\sin\theta & 0 \\
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| 0 & \sin\theta & \cos\theta & 0 \\
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| 0 & 0 & 0 & 1 \\
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| \end{pmatrix}
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| \begin{pmatrix}
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| A^0 \\ A^1 \\ A^2 \\ A^3
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| \end{pmatrix}\ .
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| </math>
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| ====Pure boosts in an arbitrary direction====
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| [[Image:Standard conf.png|right|thumb|300px|Standard configuration of coordinate systems; for a Lorentz boost in the ''x''-direction.]]
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| For two frames moving at constant relative 3-velocity '''v''' (not 4-velocity, [[#Four-velocity|see below]]), it is convenient to denote and define the relative velocity in units of ''c'' by:
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| :<math> \boldsymbol{\beta} = (\beta_1,\,\beta_2,\,\beta_3) = \frac{1}{c}(v_1,\,v_2,\,v_3) = \frac{1}{c}\mathbf{v} \,. </math>
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| Then without rotations, the matrix '''Λ''' has components given by:<ref>Gravitation, J.B. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISAN 0-7167-0344-0</ref>
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| :<math> \begin{align} \Lambda_{00} & = \gamma, \\
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| \Lambda_{0i} & = \Lambda_{i0} = - \gamma \beta_{i}, \\
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| \Lambda_{ij} & = \Lambda_{ji} = ( \gamma - 1 )\dfrac{\beta_{i}\beta_{j}}{\beta^{2}} + \delta_{ij}= ( \gamma - 1 )\dfrac{v_i v_j}{v^2} + \delta_{ij}, \\
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| \end{align}
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| \,\!</math>
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| where the [[Lorentz factor]] is defined by:
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| :<math> \gamma = \frac{1}{\sqrt{1- \boldsymbol{\beta}\cdot\boldsymbol{\beta}}} \,,</math>
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| and ''δ<sub>ij</sub>'' is the [[Kronecker delta]]. Contrary to the case for pure rotations, the spacelike and timelike components are mixed together under boosts.
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| For the case of a boost in the ''x''-direction only, the matrix reduces to;<ref>Dynamics and Relativity, J.R. Forshaw, B.G. Smith, Wiley, 2009, ISAN 978-0-470-01460-8</ref><ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (ASB), 2006, ISAN 0-07-145545-0</ref>
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| :<math>
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| \begin{pmatrix}
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| A'^0 \\ A'^1 \\ A'^2 \\ A'^3
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| \end{pmatrix}
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| =\begin{pmatrix}
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| \cosh\phi &-\sinh\phi & 0 & 0 \\
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| -\sinh\phi & \cosh\phi & 0 & 0 \\
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| 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 1 \\
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| \end{pmatrix}
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| \begin{pmatrix}
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| A^0 \\ A^1 \\ A^2 \\ A^3
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| \end{pmatrix}
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| </math>
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| Where the [[rapidity]] ''ϕ'' expression has been used, written in terms of the [[hyperbolic function]]s:
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| :<math>
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| \gamma = \cosh \varphi
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| </math>
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| This Lorentz matrix illustrates the boost to be a ''[[hyperbolic rotation]]'' in four dimensional spacetime, analogous to the circular rotation above in three dimensional space.
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| ===Properties===
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| ====Linearity====
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| Four-vectors have the same [[Linear algebra|linearity properties]] as [[Euclidean vector]]s in [[three dimensions]]. They can be added in the usual entrywise way:
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| :<math>\mathbf{A}+\mathbf{B} = (A^0, A^1, A^2,A^3) + (B^0, B^1, B^2,B^3) = (A^0 + B^0, A^1 + B^1, A^2 + B^2, A^3 + B^3) </math>
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| and similarly [[scalar multiplication]] by a [[scalar (mathematics)|scalar]] ''λ'' is defined entrywise by:
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| :<math>\lambda\mathbf{A} = \lambda(A^0, A^1, A^2,A^3) = (\lambda A^0, \lambda A^1, \lambda A^2, \lambda A^3) </math>
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| Then subtraction is the inverse operation of addition, defined entrywise by:
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| :<math>\mathbf{A}+(-1)\mathbf{B} = (A^0, A^1, A^2,A^3) + (-1)(B^0, B^1, B^2,B^3) = (A^0 - B^0, A^1 - B^1, A^2 - B^2, A^3 - B^3) </math>
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| ====Inner product====
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| {{See also|spacetime interval}}
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| The [[inner product]] (also called the scalar product) of two four-vectors '''A''' and '''B''' is defined, using [[Einstein notation]], as
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| :<math>\mathbf{A} \cdot \mathbf{B} = A^{\mu} \eta_{\mu \nu} B^{\nu} </math>
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| where ''η'' is the [[Minkowski metric]]. The inner product in this context is also called the Minkowski inner product. For visual clarity, it is convenient to rewrite the definition in [[matrix (mathematics)|matrix]] form:
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| :<math>\mathbf{A \cdot B} = \begin{pmatrix} A^0 & A^1 & A^2 & A^3 \end{pmatrix} \begin{pmatrix} \eta_{00} & \eta_{01} & \eta_{02} & \eta_{03} \\ \eta_{10} & \eta_{11} & \eta_{12} & \eta_{13} \\ \eta_{20} & \eta_{21} & \eta_{22} & \eta_{23} \\ \eta_{30} & \eta_{31} & \eta_{32} & \eta_{33} \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix} </math>
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| in which case ''η<sub>μν</sub>'' above is the entry in row ''μ'' and column ''ν'' of the Minkowski metric as a square matrix. The Minkowski metric is not a [[Euclidean metric]], because it is indefinite (see [[metric signature]]). The inner product can be rewritten in a number of other ways because the metric tensor raises and lowers the components of '''A''' and '''B'''. For contra/co-variant components of '''A''' and co/contra-variant components of '''B''', we have:
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| :<math>\mathbf{A} \cdot \mathbf{B} = A_{\nu} B^{\nu} = A^{\mu} B_{\mu} </math>
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| so in the matrix notation:
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| :<math>\mathbf{A \cdot B} = \begin{pmatrix} A_0 & A_1 & A_2 & A_3 \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix} = \begin{pmatrix} B_0 & B_1 & B_2 & B_3 \end{pmatrix} \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix} </math>
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| while for '''A''' and '''B''' each in covariant components:
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| :<math>\mathbf{A} \cdot \mathbf{B} = A_{\mu} \eta^{\mu \nu} B_{\nu} </math>
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| with a similar matrix expression to the above.
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| The inner product of a four-vector '''A''' with itself is the square of the [[Norm (mathematics)|norm]] of the vector, denoted and defined by:
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| :<math> \|\mathbf{A}\|^2 = \mathbf{A \cdot A} = A^\mu \eta_{\mu\nu} A^\nu </math>
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| and intuitively represents (the square of) the length or magnitude of the vector. However, in general, four-vectors can have nonpositive length, contrary to [[three dimensional space|three dimensional]] vectors in [[Euclidean space]].
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| Following are two common choices for the metric tensor in the [[Minkowski space#Standard basis|standard basis]] (essentially Cartesian coordinates). If orthogonal coordinates are used, there would be scale factors along the diagonal part of the spacelike part of the metric, while for general curvilinear coordinates the entire spacelike part of the metric would have components dependent on the curvilinear basis used.
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| =====Standard basis, (+−−−) signature=====
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| In the (+−−−) [[metric signature]], evaluating the [[Einstein notation|summation over indices]] gives:
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| :<math>\mathbf{A} \cdot \mathbf{B} = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 </math>
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| while in matrix form:
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| :<math>\mathbf{A \cdot B} = \begin{pmatrix} A^0 & A^1 & A^2 & A^3 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix} </math>
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| It is a recurring theme in special relativity to take the expression
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| :<math> \mathbf{A}\cdot\mathbf{B} = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = C</math>
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| in one [[Frame of reference|reference frame]], where ''C'' is the value of the inner product in this frame, and:
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| :<math> \mathbf{A}'\cdot\mathbf{B}' = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3 {B'}^3 = C' </math>
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| in another frame, in which ''C''′ is the value of the inner product in this frame. Then since the inner product is an invariant, these must be equal:
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| :<math> \mathbf{A}\cdot\mathbf{B} = \mathbf{A}'\cdot\mathbf{B}' </math>
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| that is:
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| :<math> C = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3{B'}^3 </math>
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| Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a "[[conservation law]]", but there is no "conservation" involved. The primary significance of the Minkowski inner product is that for any two four-vectors, its value is [[invariant (physics)|invariant]] for all observers; a change of coordinates does not result in a change in value of the inner product. The components of the four-vectors change from one frame to another; '''A''' and '''A′''' are connected by a [[Lorentz transformation]], and similarly for '''B''' and '''B′''', although the inner products are the same in all frames. Nevertheless, this type of expression is exploited in relativistic calculations on a par with conservation laws, since the magnitudes of components can be determined without explicitly performing any Lorentz transformations. A particular example is with energy and momentum in the [[energy-momentum relation]] derived from the [[four-momentum]] vector (see also below).
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| In this signature, the norm of the vector '''A''' is:
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| :<math> \|\mathbf{A}\|^2 = (A^0)^2 - (A^1)^2 - (A^2)^2 - (A^3)^2 </math>
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| With the signature (+−−−), four-vectors may be classified as either [[Minkowski space#Causal structure|spacelike]] if ||'''A'''|| < 0, [[Minkowski space#Causal structure|timelike]] if ||'''A'''|| > 0, and [[Minkowski space#Causal structure|null vector]]s if ||'''A'''|| = 0.
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| =====Standard basis, (−+++) signature=====
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| Some authors define ''η'' with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature:
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| :<math>\mathbf{A \cdot B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 </math>
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| while the matrix form is:
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| :<math>\mathbf{A \cdot B} = \left( \begin{matrix}A^0 & A^1 & A^2 & A^3 \end{matrix} \right)
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| \left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)
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| \left( \begin{matrix}B^0 \\ B^1 \\ B^2 \\ B^3 \end{matrix} \right) </math>
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| Note that in this case, in one frame:
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| :<math> \mathbf{A}\cdot\mathbf{B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = -C </math>
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| while in another:
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| :<math> \mathbf{A}'\cdot\mathbf{B}' = - {A'}^0 {B'}^0 + {A'}^1 {B'}^1 + {A'}^2 {B'}^2 + {A'}^3 {B'}^3 = -C'</math>
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| so that:
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| :<math> -C = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = - {A'}^0 {B'}^0 + {A'}^1 {B'}^1 + {A'}^2 {B'}^2 + {A'}^3{B'}^3</math>
| |
| | |
| which is equivalent to the above expression for ''C'' in terms of '''A''' and '''B'''. Either convention will work. With the Minkowski metric defined in the two ways above, the only difference between covariant and contravariant four-vector components are signs, therefore the signs depend on which sign convention is used.
| |
| | |
| The square of the norm in this signature is:
| |
| | |
| :<math> \|\mathbf{A}\|^2 = - (A^0)^2 + (A^1)^2 + (A^2)^2 + (A^3)^2 </math>
| |
| | |
| With the signature (−+++), four-vectors may be classified as either [[Minkowski space#Causal structure|spacelike]] if ||'''A'''|| > 0, [[Minkowski space#Causal structure|timelike]] if ||'''A'''|| < 0, and [[Minkowski space#Causal structure|null vector]]s if ||'''A'''|| = 0.
| |
| | |
| =====Dual vectors=====
| |
| | |
| The inner product is often expressed as the effect of the [[dual space#Ailinear products and dual spaces|dual vector]] of one vector on the other:
| |
| | |
| :<math>\mathbf{A \cdot B} = A^*(\mathbf{B}) = A{_\nu}B^{\nu}. </math>
| |
| | |
| Here the ''A<sub>ν</sub>''s are the components of the dual vector ''A''* of '''A''' in the [[dual basis]] and called the [[Covariance and contravariance of vectors|covariant]] coordinates of '''A''', while the original ''A<sup>ν</sup>'' components are called the [[Covariance and contravariance of vectors|contravariant]] coordinates.
| |
| | |
| == Four-vector calculus ==
| |
| | |
| ===Derivatives and differentials===
| |
| | |
| In special relativity (but not general relativity), the [[derivative]] of a four-vector with respect to a scalar ''λ'' (invariant) is itself a four-vector. It is also useful to take the [[differential of a function|differential]] of the four-vector, ''d'''''A''' and divide it by the differential of the scalar, ''dλ'':
| |
| | |
| :<math>\underset{\text{differential}}{d\mathbf{A}} = \underset{\text{derivative}}{\frac{d\mathbf{A}}{d\lambda}} \underset{\text{differential}}{d\lambda} </math>
| |
| | |
| where the contravariant components are:
| |
| | |
| :<math> d\mathbf{A} = (dA^0, dA^1, dA^2, dA^3) </math>
| |
| | |
| while the covariant components are:
| |
| | |
| :<math> d\mathbf{A} = (dA_0, dA_1, dA_2, dA_3) </math>
| |
| | |
| In relativistic mechanics, one often takes the differential of a four-vector and divides by the differential in [[proper time]] (see below).
| |
| | |
| ==Fundamental four-vectors==
| |
| | |
| ===Four-position===
| |
| | |
| A point in [[Minkowski space]] is a time and spatial position, called an "event", or sometimes the position 4-vector or 4-position, described in some reference frame by a set of four coordinates:
| |
| | |
| :<math> \mathbf{X}= \left(ct, \mathbf{r}\right) </math>
| |
| | |
| where '''r''' is the [[three dimensional space]] [[position vector]]. If '''r''' is a function of coordinate time ''t'' in the same frame, i.e. '''r''' = '''r'''(''t''), this corresponds to a sequence of events as ''t'' varies. The definition ''X''<sup>0</sup> = ''ct'' ensures that all the coordinates have the same units (of distance).<ref>Jean-Bernard Zuber & Claude Itzykson, ''Quantum Field Theory'', pg 5 , ISBN 0-07-032071-3</ref><ref>[[Charles W. Misner]], [[Kip S. Thorne]] & [[John A. Wheeler]],''Gravitation'', pg 51, ISBN 0-7167-0344-0</ref><ref>[[George Sterman]], ''An Introduction to Quantum Field Theory'', pg 4 , ISBN 0-521-31132-2</ref> These coordinates are the components of the ''position four-vector'' for the event.
| |
| The ''displacement four-vector'' is defined to be an "arrow" linking two events:
| |
| | |
| :<math> \Delta \mathbf{X} = \left(c\Delta t, \Delta \mathbf{r} \right) </math>
| |
| | |
| The scalar product of the 4-position with itself is;<ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8</ref>
| |
| | |
| :<math>\|\mathbf{X}\|^2 = X^\mu X_\mu=(c\tau)^2 = s^2 \,,</math>
| |
| | |
| which ''defines'' the [[spacetime]] interval ''s'' and [[proper time]] ''τ'' in Minkowski spacetime, which are invariant. The [[scalar product]] of the [[differential (infinitesimal)|differential]] 4-position with itself is:
| |
| | |
| :<math>\|d\mathbf{X}\|^2 = dX^\mu dX_\mu=c^2d\tau^2=ds^2 \,,</math>
| |
| | |
| defining the differential [[line element]] d''s'' and differential proper time increment d''τ'', but this norm is also:
| |
| | |
| :<math>\|d\mathbf{X}\|^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,,</math>
| |
| | |
| so that:
| |
| | |
| :<math> (c d\tau)^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,.</math>
| |
| | |
| When considering physical phenomena, differential equations arise naturally; however, when considering space and [[time derivative]]s of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the [[proper time]] τ. As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the [[coordinate time]] ''t'' of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by (''cdt'')<sup>2</sup> to obtain:
| |
| | |
| :<math> \left(\frac{cd\tau}{cdt}\right)^2 = 1 - \left(\frac{d\mathbf{r}}{cdt}\cdot \frac{d\mathbf{r}}{cdt}\right) = 1 - \frac{\mathbf{u}\cdot\mathbf{u}}{c^2} = \frac{1}{\gamma(\mathbf{u})^2} \,,</math>
| |
| | |
| where '''u''' = ''d'''''r'''/''dt'' is the coordinate 3-[[velocity]] of an object measured in the same frame as the coordinates ''x'', ''y'', ''z'', and [[coordinate time]] ''t'', and
| |
| | |
| :<math> \gamma(\mathbf{u}) = \frac{1}{\sqrt{1- \frac{\mathbf{u}\cdot\mathbf{u}}{c^2}}}</math>
| |
| | |
| is the [[Lorentz factor]]. This provides a useful relation between the differentials in coordinate time and proper time:
| |
| | |
| :<math> dt = \gamma(\mathbf{u})d\tau \,.</math>
| |
| | |
| This relation can also be found from the time transformation in the [[Lorentz transformation]]s. Important four-vectors in relativity theory can be defined by dividing by this differential.
| |
| | |
| ===Four-gradient===
| |
| | |
| Considering that [[partial derivative]]s are [[linear operator]]s, one can form a [[four-gradient]] from the partial [[time derivative]] {{math|∂}}/{{math|∂}}''t'' and the spatial [[gradient]] ∇. Using the standard basis, in index and abbreviated notations, the contravariant components are:
| |
| | |
| :<math> \begin{align}
| |
| \boldsymbol{\partial} & = \left(\frac{\partial }{\partial x_0}, \, -\frac{\partial }{\partial x_1}, \, -\frac{\partial }{\partial x_2}, \, -\frac{\partial }{\partial x_3} \right) \\
| |
| & = (\partial^0, \, - \partial^1, \, - \partial^2, \, - \partial^3) \\
| |
| & = \mathbf{e}_0\partial^0 - \mathbf{e}_1\partial^1 - \mathbf{e}_2\partial^2 - \mathbf{e}_3\partial^3 \\
| |
| & = \mathbf{e}_0\partial^0 - \mathbf{e}_i\partial^i \\
| |
| & = \mathbf{e}_\alpha \partial^\alpha \\
| |
| & = \left(\frac{1}{c}\frac{\partial}{\partial t} , \, - \nabla \right) \\
| |
| & = \mathbf{e}_0\frac{1}{c}\frac{\partial}{\partial t} - \nabla \\
| |
| \end{align}</math>
| |
| | |
| Note the basis vectors are placed in front of the components, to prevent confusion between taking the derivative of the basis vector, or simply indicating the partial derivative is a component of this four-vector. The covariant components are:
| |
| | |
| :<math> \begin{align}
| |
| \boldsymbol{\partial} & = \left(\frac{\partial }{\partial x^0}, \, \frac{\partial }{\partial x^1}, \, \frac{\partial }{\partial x^2}, \, \frac{\partial }{\partial x^3} \right) \\
| |
| & = (\partial_0, \, \partial_1, \, \partial_2, \, \partial_3) \\
| |
| & = \mathbf{e}^0\partial_0 + \mathbf{e}^1\partial_1 + \mathbf{e}^2\partial_2 + \mathbf{e}^3\partial_3 \\
| |
| & = \mathbf{e}^0\partial_0 + \mathbf{e}^i\partial_i \\
| |
| & = \mathbf{e}^\alpha \partial_\alpha \\
| |
| & = \left(\frac{1}{c}\frac{\partial}{\partial t} , \, \nabla \right) \\
| |
| & = \mathbf{e}^0\frac{1}{c}\frac{\partial}{\partial t} + \nabla \\
| |
| \end{align}</math>
| |
| | |
| Since this is an operator, it doesn't have a "length", but evaluating the inner product of the operator with itself gives another operator:
| |
| | |
| :<math>\partial^\mu \partial_\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 </math>
| |
| | |
| called the [[D'Alembert operator]].
| |
| | |
| ==Kinematics==
| |
| | |
| === Four-velocity ===
| |
| | |
| The [[four-velocity]] of a particle is defined by:
| |
| | |
| :<math>\mathbf{U} = \frac{d\mathbf{X}}{d \tau}= \frac{d\mathbf{X}}{dt}\frac{dt}{d \tau} = \gamma(\mathbf{u})\left(c, \mathbf{u} \right),</math>
| |
| | |
| Geometrically '''U''' is a tangent vector along the [[world line]] of the particle. Using the differential of the 4-position, the magnitude of the 4-velocity can be obtained;
| |
| | |
| :<math> \|\mathbf{U}\|^2 = U^\mu U_\mu = \frac{dX^\mu }{d\tau} \frac{dX_\mu }{d\tau}= \frac{dX^\mu dX_\mu }{d\tau^2} = c^2 \,,</math>
| |
| | |
| in short, the magnitude of the 4-velocity for any all objects is always that of ''c'':
| |
| | |
| :<math> \| \mathbf{U} \|^2 = c^2 \,</math>
| |
| | |
| The norm is also:
| |
| | |
| :<math> \|\mathbf{U}\|^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,</math>
| |
| | |
| so that:
| |
| | |
| :<math> c^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,</math>
| |
| | |
| which reduces to the definition the Lorentz factor.
| |
| | |
| === Four-acceleration ===
| |
| | |
| The [[four-acceleration]] is given by:
| |
| | |
| :<math>\mathbf{A} =\frac{d\mathbf{U} }{d \tau} = \gamma(\mathbf{u}) \left(\frac{d{\gamma}(\mathbf{u})}{dt} c, \frac{d{\gamma}(\mathbf{u})}{dt} \mathbf{u} + \gamma(\mathbf{u}) \mathbf{a} \right).</math>
| |
| | |
| where '''a''' = ''d'''''u'''/''dt'' is the coordinate 3-acceleration. Since the magnitude of '''U''' is a constant, the four acceleration is (pseudo-)orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:
| |
| | |
| :<math> \mathbf{A}\cdot\mathbf{U} = A^\mu U_\mu = \frac{dU^\mu}{d\tau} U_\mu = \frac{1}{2} \, \frac{d}{d\tau} (U^\mu U_\mu) = 0 \,</math>
| |
| | |
| which is true for all world lines. The geometric meaning of 4-acceleration is the [[curvature vector]] of the world line in Minkowski space.
| |
| | |
| ==Dynamics==
| |
| | |
| === Four-momentum ===
| |
| | |
| For a massive particle of [[rest mass]] (or [[invariant mass]]) ''m'', the [[four-momentum]] is given by:
| |
| | |
| :<math>\mathbf{P} = m \mathbf{U} = m\gamma(\mathbf{u})(c, \mathbf{u}) = (E/c, \mathbf{p}) </math>
| |
| | |
| where the total energy of the moving particle is:
| |
| | |
| :<math>E = \gamma(\mathbf{u}) mc^2 </math>
| |
| | |
| and the total [[relativistic momentum]] is:
| |
| | |
| :<math>\mathbf{p} = \gamma(\mathbf{u}) m \mathbf{u} </math>
| |
| | |
| Taking the inner product of the four-momentum with itself:
| |
| | |
| :<math> \|\mathbf{P}\|^2 = P^\mu P_\mu = m^2 U^\mu U_\mu = m^2 c^2</math>
| |
| | |
| and also:
| |
| | |
| :<math> \|\mathbf{P}\|^2 = \frac{E^2}{c^2} - \mathbf{p}\cdot\mathbf{p} </math>
| |
| | |
| which leads to the [[energy–momentum relation]]:
| |
| | |
| :<math>E^2 = c^2 \mathbf{p}\cdot\mathbf{p} + (mc^2)^2 \,.</math>
| |
| | |
| This last relation is useful [[relativistic mechanics]], essential in [[relativistic quantum mechanics]] and [[relativistic quantum field theory]], all with applications to [[particle physics]].
| |
| | |
| === Four-force ===
| |
| | |
| The [[four-force]] acting on a particle is defined analogously to the 3-force as the time derivative of 3-momentum in [[Newton's second law]]:
| |
| | |
| :<math> \mathbf{F} = \frac {d \mathbf{P}} {d \tau} = \gamma(\mathbf{u})\left(\frac{1}{c}\frac{dE}{dt},\frac{d\mathbf{p}}{dt}\right) = \gamma(\mathbf{u})(P/c,\mathbf{f}) </math>
| |
| | |
| where ''P'' is the [[power (physics)|power]] transferred to move the particle, and '''f''' is the 3-force acting on the particle. For a particle of constant invariant mass ''m'', this is equivalent to
| |
| | |
| :<math> \mathbf{F} = m \mathbf{A} = m\gamma(\mathbf{u})\left( \frac{d{\gamma}(\mathbf{u})}{dt} c, \left(\frac{d{\gamma}(\mathbf{u})}{dt} \mathbf{u} + \gamma(\mathbf{u}) \mathbf{a}\right) \right) </math>
| |
| | |
| An invariant derived from the 4-force is:
| |
| | |
| :<math> \mathbf{F}\cdot\mathbf{U} = F^\mu U_\mu = m A^\mu U_\mu = 0</math>
| |
| | |
| from the above result.
| |
| | |
| ==Thermodynamics==
| |
| | |
| {{see also|Relativistic heat conduction}}
| |
| | |
| ===Four-heat flux===
| |
| | |
| The 4-heat flux vector field, is essentially similar to the 3d [[heat flux]] vector field '''q''', in the local frame of the fluid:<ref>{{Cite journal |first=Y. M. |last=Ali |first2=L. C. |last2=Zhang |title=Relativistic heat conduction |journal=Int. J. Heat Mass Trans. |volume=48 |year=2005 |issue=12 |pages= |doi=10.1016/j.ijheatmasstransfer.2005.02.003 }}</ref>
| |
| | |
| :<math>\mathbf{Q} = -k \boldsymbol{\partial} T = - k\left( \frac{1}{c}\frac{\partial T}{\partial t}, \nabla T\right) </math>
| |
| | |
| where ''T'' is [[absolute temperature]] and ''k'' is [[thermal conductivity]].
| |
| | |
| ===Four-baryon number flux===
| |
| | |
| The flux of baryons is:<ref>{{Citebook|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|page=558-559|isbn=0-7167-0344-0}}</ref>
| |
| | |
| :<math>\mathbf{S}= n\mathbf{U}</math>
| |
| | |
| where ''n'' is the [[number density]] of [[baryon]]s in the local [[rest frame]] of the baryon fluid (positive values for baryons, negative for [[antiparticle|anti]]baryons), and '''U''' the 4-[[velocity]] field (of the fluid) as above.
| |
| | |
| ===Four-entropy===
| |
| | |
| The 4-[[entropy]] vector is defined by:<ref>{{Citebook|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|page=567|isbn=0-7167-0344-0}}</ref>
| |
| | |
| :<math>\mathbf{s}= s\mathbf{S} + \frac{\mathbf{Q}}{T}</math>
| |
| | |
| where ''s'' is the entropy per baryon, and ''T'' the [[absolute temperature]], in the local rest frame of the fluid.<ref>{{Citebook|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|page=558|isbn=0-7167-0344-0}}</ref>
| |
| | |
| ==Electromagnetism==
| |
| | |
| Examples of four-vectors in [[electromagnetism]] include the following.
| |
| | |
| ===Four-current===
| |
| | |
| The electromagnetic [[four-current]] is defined by
| |
| | |
| :<math> \mathbf{J} = \left( \rho c, \mathbf{j} \right) </math>
| |
| | |
| formed from the [[current density]] '''j''' and [[charge density]] ''ρ''.
| |
| | |
| ===Four-potential===
| |
| | |
| The [[electromagnetic four-potential]] defined by
| |
| | |
| :<math> \mathbf{A} = \left( \phi /c, \mathbf{a} \right) </math>
| |
| | |
| formed from the [[vector potential]] '''a''' and the scalar potential ''ϕ''. The four-potential is not uniquely determined, because it depends on a choice of [[Coulomb gauge#Coulomb gauge|gauge]].
| |
| | |
| ==Waves==
| |
| | |
| ===Four-frequency===
| |
| | |
| A [[plane wave]] can be described by the [[four-frequency]] defined as
| |
| | |
| :<math>\mathbf{N} = \nu\left(1 , \hat{\mathbf{n}} \right)</math>
| |
| | |
| where ''ν'' is the frequency of the wave and <math>\hat{\mathbf{n}}</math> is a [[unit vector]] in the travel direction of the wave. Now:
| |
| | |
| :<math> \|\mathbf{N}\| = N^\mu N_\mu = \nu ^2 \left(1 - \hat{\mathbf{n}}\cdot\hat{\mathbf{n}}\right) = 0 </math>
| |
| | |
| so the 4-frequency is always a null vector.
| |
| | |
| ===Four-wavevector===
| |
| | |
| {{see also|De Broglie relation}}
| |
| | |
| The quantities reciprocal to time ''t'' and space '''r''' are the [[angular frequency]] ''ω'' and [[wave vector]] '''k''', respectively. The form the components of the 4-wavevector or wave 4-vector:
| |
| | |
| :<math>\mathbf{K} = \left(\frac{\omega}{c}, \mathbf{k} \right) \,. </math>
| |
| | |
| A wave packet of nearly [[monochromatic]] light can be described by:
| |
| | |
| :<math>\mathbf{K} = \frac{2\pi}{c}\mathbf{N} = \frac{2\pi}{c} \nu(1,\hat{\mathbf{n}}) = \frac{\omega}{c}\left( 1 , \hat{\mathbf{n}} \right) \,. </math>
| |
| | |
| For [[matter wave]]s, the de Broglie relations become one equation:
| |
| | |
| :<math>\mathbf{P} = \hbar \mathbf{K}\,.</math>
| |
| | |
| where ''ħ'' is the [[Planck constant]] divided by 2''π''. The square of the norm is:
| |
| | |
| :<math>\| \mathbf{K} \|^2 = K^\mu K_\mu = \left(\frac{\omega}{c}\right)^2 - \mathbf{k}\cdot\mathbf{k}\,,</math>
| |
| | |
| and by the de Broglie relation:
| |
| | |
| :<math> \| \mathbf{K} \|^2 = \frac{1}{\hbar^2} \| \mathbf{P} \|^2 = \left(\frac{mc}{\hbar}\right)^2 \,,</math>
| |
| | |
| we have the matter wave analogue of the energy-momentum relation:
| |
| | |
| :<math>\left(\frac{\omega}{c}\right)^2 - \mathbf{k}\cdot\mathbf{k} = \left(\frac{mc}{\hbar}\right)^2 \,.</math>
| |
| | |
| Note that for massless particles, in which case ''m'' = 0, we have:
| |
| | |
| :<math>\left(\frac{\omega}{c}\right)^2 = \mathbf{k}\cdot\mathbf{k} \,,</math>
| |
| | |
| or ||'''k'''|| = ''ω''/''c''. Note this is consistent with the above case; for photons with a 3-wavevector of modulus ''ω''/''c'', in the direction of wave propagation defined by the unit vector <math>\hat{\mathbf{n}}</math>.
| |
| | |
| ==Quantum theory==
| |
| | |
| In [[quantum mechanics]], the 4-[[probability current]] or probability 4-current is analogous to the electromagnetic 4-current:<ref>Vladimir G. Ivancevic, Tijana T. Ivancevic (2008) ''Quantum leap: from Dirac and Feynman, across the universe, to human body and mind''. World Scientific Publishing Company, ISBN 978-981-281-927-7, [http://books.google.com/books?id=qyK95FevVbIC&pg=PA41 p. 41]</ref>
| |
| :<math>\mathbf{J} = (\rho c, \mathbf{j}) </math>
| |
| | |
| where ''ρ'' is the [[probability density function]] corresponding to the time component, and '''j''' is the [[probability current]] vector. In non-relativistic quantum mechanics, this current is always well defined because the expressions for density and current are positive definite and can admit a probability interpretation. In [[relativistic quantum mechanics]] and [[quantum field theory]], it is not always possible to find a current, particularly when interactions are involved.
| |
| | |
| Replacing the energy by the [[energy operator]] and the momentum by the [[momentum operator]] in the four-momentum, one obtains the [[four-momentum operator]], used in [[relativistic wave equation]]s.
| |
| | |
| ==Other formulations==
| |
| | |
| ===Four-vectors in the algebra of physical space===
| |
| | |
| A four-vector ''A'' can also be defined in using the [[Pauli matrices]] as a [[basis (linear algebra)|basis]], again in various equivalent notations:<ref>{{cite book |pages= 1142-1143| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref>
| |
| | |
| :<math> \begin{align}
| |
| \mathbf{A} & = (A^0, \, A^1, \, A^2, \, A^3) \\
| |
| & = A^0\boldsymbol{\sigma}_0 + A^1 \boldsymbol{\sigma}_1 + A^2 \boldsymbol{\sigma}_2 + A^3 \boldsymbol{\sigma}_3 \\
| |
| & = A^0\boldsymbol{\sigma}_0 + A^i \boldsymbol{\sigma}_i \\
| |
| & = A^\alpha\boldsymbol{\sigma}_\alpha\\
| |
| \end{align}</math>
| |
| | |
| or explicitly:
| |
| | |
| :<math> \begin{align}
| |
| \mathbf{A} & = A^0\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + A^1 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + A^2 \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} + A^3 \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\
| |
| & = \begin{pmatrix} A^0 + A^3 & A^1 -i A^2 \\ A^1 + i A^2 & A^0 - A^3 \end{pmatrix}
| |
| \end{align}</math>
| |
| | |
| and in this formulation, the four-vector is a represented as a [[unitary matrix]] (the [[matrix transpose]] and [[complex conjugate]] of the matrix leaves it unchanged), rather than a real-valued column or row vector. The [[determinant]] of the matrix is the modulus of the four-vector, so the determinant is an invariant:
| |
| | |
| :<math> \begin{align}
| |
| |\mathbf{A}| & = \begin{vmatrix} A^0 + A^3 & A^1 -i A^2 \\ A^1 + i A^2 & A^0 - A^3 \end{vmatrix} \\
| |
| & = (A^0 + A^3)(A^0 - A^3) - (A^1 -i A^2)(A^1 + i A^2) \\
| |
| & = (A^0)^2 - (A^1)^2 - (A^2)^2 - (A^3)^2
| |
| \end{align}</math>
| |
| | |
| This idea of using the Pauli matrices as [[basis vector]]s is employed in the [[algebra of physical space]], an example of a [[Clifford algebra]].
| |
| | |
| ===Four-vectors in spacetime algebra===
| |
| | |
| In [[spacetime algebra]], another example of Clifford algebra, the [[gamma matrices]] can also form a [[basis (linear algebra)|basis]]. (They are also called the Dirac matrices, owing to their appearance in the [[Dirac equation]]). There is more than one way to express the gamma matrices, detailed in that main article.
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| The [[Feynman slash notation]] is a shorthand for a four-vector '''A''' contracted with the gamma matrices:
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| :<math>\mathbf{A}\!\!\!\!/ = A_\alpha \gamma^\alpha = A_0 \gamma^0 + A_1 \gamma^1 + A_2 \gamma^2 + A_3 \gamma^3 </math>
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| The four-momentum contracted with the gamma matrices is an important case in [[relativistic quantum mechanics]] and [[relativistic quantum field theory]]. In the Dirac equation and other [[relativistic wave equation]]s, terms of the form:
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| :<math>\mathbf{P}\!\!\!\!/ = P_\alpha \gamma^\alpha = P_0 \gamma^0 + P_1 \gamma^1 + P_2 \gamma^2 + P_3 \gamma^3 = \dfrac{E}{c} \gamma^0 + p_x \gamma^1 + p_y \gamma^2 + p_z \gamma^3 </math>
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| appear, in which the energy ''E'' and momentum components (''p<sub>x</sub>'',''p<sub>y</sub>'', ''p<sub>z</sub>'') are replaced by their respective [[operator (physics)|operator]]s.
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| ==See also==
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| *[[Relativistic mechanics]]
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| *[[paravector]]
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| *[[wave vector]]
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| *[[Dust (relativity)]] for the number-flux four-vector
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| *[[Basic introduction to the mathematics of curved spacetime]]
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| *[[Minkowski space]]
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| == References ==
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| {{reflist}}
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| *Rindler, W. ''Introduction to Special Relativity (2nd edn.)'' (1991) Clarendon Press Oxford ISBN 0-19-853952-5
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| <!--Categories-->
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| [[Category:Minkowski spacetime]]
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| [[Category:Theory of relativity]]
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| [[Category:Concepts in physics]]
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| [[Category:Vectors]]
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