Editing Four-vector (section)

== 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:<ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (BSA), 2006, {{ISBN|0-07-145545-0}}</ref>

<math display="block"> \begin{align}  
  \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{align}</math>

where ''A<sup>α</sup>'' is the magnitude component and '''E'''<sub>α</sub> is the [[basis vector]] component; note that both are necessary to make a vector, and that when ''A<sup>α</sup>'' is seen alone, it refers strictly to the <em>components</em> of the vector.

The upper indices indicate [[Covariance and contravariance of vectors|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 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]].

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:

<math display="block"> \begin{align}  
  \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{align}</math>

although, of course, any other basis and components may be used, such as [[spherical polar coordinates]]

<math display="block"> \begin{align}  
  \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{align}</math>

or [[cylindrical polar coordinates]],

<math display="block"> \begin{align}  
  \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{align}</math>

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 vector]]s:

<math display="block">
  \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>

so that:

<math display="block"> \mathbf{A} = \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix} </math>

The relation between the [[covariant vector|covariant]] and contravariant coordinates is through the [[Minkowski metric|Minkowski]] [[metric tensor]] (referred to as the metric), ''η'' which [[raising and lowering indices|raises and lowers indices]] as follows:

<math display="block">A_{\mu} =  \eta_{\mu \nu} A^{\nu} \,, </math>

and in various equivalent notations the covariant components are:

<math display="block"> \begin{align}  
  \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{align}</math>

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]].

The bases can be represented by [[row vector]]s:

<math display="block">\begin{align}
  \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{align}</math>
so that:
<math display="block"> \mathbf{A} = \begin{pmatrix} A_0 & A_1 & A_2 & A_3 \end{pmatrix} </math>

The motivation for the above conventions are that the inner product is a scalar, see below for details.

=== Lorentz transformation ===

{{main|Lorentz transformation}}

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&nbsp;'''Λ''':
<math display="block">\mathbf{A}' = \boldsymbol{\Lambda}\mathbf{A}</math>

In index notation, the contravariant and covariant components transform according to, respectively:
<math display="block">{A'}^\mu = \Lambda^\mu {}_\nu A^\nu \,, \quad{A'}_\mu = \Lambda_\mu {}^\nu A_\nu</math>
in which the matrix {{math|'''Λ'''}} has components {{math|Λ''<sup>μ</sup><sub>ν</sub>''}} in row&nbsp;{{math|''μ''}} and column&nbsp;{{math|''ν''}}, and the matrix {{math|('''Λ'''<sup>−1</sup>)<sup>T</sup>}} has components {{math|Λ''<sub>μ</sub><sup>ν</sup>''}} in row&nbsp;{{math|''μ''}} and column&nbsp;{{math|''ν''}}.

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]].

====Pure rotations about an arbitrary axis ====

For two frames rotated by a fixed angle {{math|''θ''}} about an axis defined by the [[unit vector]]:

<math display="block">\hat{\mathbf{n}} = \left(\hat{n}_1, \hat{n}_2, \hat{n}_3\right)\,,</math>

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=[https://archive.org/details/mcgrawhillencycl1993park/page/1333 1333]| year=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park/page/1333}}</ref>

<math display="block">\begin{align}
                 \Lambda_{00} &= 1 \\
  \Lambda_{0i} = \Lambda_{i0} &= 0 \\
                 \Lambda_{ij} &= \left(\delta_{ij} - \hat{n}_i \hat{n}_j\right) \cos\theta - \varepsilon_{ijk} \hat{n}_k \sin\theta + \hat{n}_i \hat{n}_j
\end{align}</math>

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 four-vectors are rotated, while the timelike components remain unchanged.

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:

<math display="block">
  \begin{pmatrix}
    {A'}^0 \\ {A'}^1 \\ {A'}^2 \\ {A'}^3
  \end{pmatrix} =
  \begin{pmatrix}
    1 & 0 & 0 & 0 \\
    0 & \cos\theta & -\sin\theta & 0 \\
    0 & \sin\theta &  \cos\theta & 0 \\
    0 & 0 & 0 & 1 \\
  \end{pmatrix}
  \begin{pmatrix}
    A^0 \\ A^1 \\ A^2 \\ A^3
  \end{pmatrix}\ .
</math>

====Pure boosts in an arbitrary direction====

[[File:Standard conf.png|right|thumb|300px|Standard configuration of coordinate systems; for a Lorentz boost in the ''x''-direction.]]

For two frames moving at constant relative three-velocity '''v''' (not four-velocity, [[#Four-velocity|see below]]), it is convenient to denote and define the relative velocity in units of ''c'' by:

<math display="block"> \boldsymbol{\beta} = (\beta_1,\,\beta_2,\,\beta_3) = \frac{1}{c}(v_1,\,v_2,\,v_3) = \frac{1}{c}\mathbf{v} \,. </math>

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>
<math display="block">\begin{align}
                 \Lambda_{00} &= \gamma, \\
  \Lambda_{0i} = \Lambda_{i0} &= -\gamma \beta_{i}, \\
  \Lambda_{ij} = \Lambda_{ji} &= (\gamma - 1)\frac{\beta_{i}\beta_{j}}{\beta^2} + \delta_{ij} = (\gamma - 1)\frac{v_i v_j}{v^2} + \delta_{ij}, \\
\end{align}</math>
where the [[Lorentz factor]] is defined by:
<math display="block">\gamma = \frac{1}{\sqrt{1 - \boldsymbol{\beta}\cdot\boldsymbol{\beta}}} \,,</math>
and {{math|''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.

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>

<math display="block">
  \begin{pmatrix}
    A'^0 \\ A'^1 \\ A'^2 \\ A'^3
  \end{pmatrix} =
  \begin{pmatrix}
     \cosh\phi &-\sinh\phi & 0 & 0 \\
    -\sinh\phi  & \cosh\phi & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1 \\
  \end{pmatrix}
  \begin{pmatrix}
    A^0 \\ A^1 \\ A^2 \\ A^3
  \end{pmatrix} 
</math>

Where the [[rapidity]] {{math|''ϕ''}} expression has been used, written in terms of the [[hyperbolic function]]s:
<math display="block">\gamma = \cosh \phi</math>

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.

===Properties===

====Linearity====

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:
<math display="block">\begin{align}
\mathbf{A} + \mathbf{B} &= \left(A^0, A^1, A^2, A^3\right) + \left(B^0, B^1, B^2, B^3\right) \\
&= \left(A^0 + B^0, A^1 + B^1, A^2 + B^2, A^3 + B^3\right)
\end{align}</math>
and similarly [[scalar multiplication]] by a [[scalar (mathematics)|scalar]] ''λ'' is defined entrywise by:
<math display="block">\lambda\mathbf{A} = \lambda\left(A^0, A^1, A^2, A^3\right) = \left(\lambda A^0, \lambda A^1, \lambda A^2, \lambda A^3\right)</math>

Then subtraction is the inverse operation of addition, defined entrywise by:
<math display="block">\begin{align}
\mathbf{A} + (-1)\mathbf{B} &= \left(A^0, A^1, A^2, A^3\right) + (-1)\left(B^0, B^1, B^2, B^3\right) \\
&= \left(A^0 - B^0, A^1 - B^1, A^2 - B^2, A^3 - B^3\right)
\end{align}</math>

====Minkowski tensor====

{{See also|spacetime interval}}

Applying the [[Minkowski tensor]] {{math|''η<sub>μν</sub>''}} to two four-vectors {{math|'''A'''}} and {{math|'''B'''}}, writing the result in [[dot product]] notation, we have, using [[Einstein notation]]:
<math display="block">\mathbf{A} \cdot \mathbf{B} = A^{\mu} B^{\nu}  \mathbf{E}_{\mu} \cdot \mathbf{E}_{\nu} = A^{\mu} \eta_{\mu \nu} B^{\nu} </math>

in special relativity. The dot product of the basis vectors is the Minkowski metric, as opposed to the Kronecker delta as in Euclidean space. It is convenient to rewrite the definition in [[matrix (mathematics)|matrix]] form:
<math display="block">\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>
in which case {{math|''η<sub>μν</sub>''}} above is the entry in row {{math|''μ''}} and column {{math|''ν''}} of the Minkowski metric as a square matrix. The Minkowski metric is not a [[Euclidean metric]], because it is indefinite (see [[metric signature]]). A number of other expressions can be used because the metric tensor can raise and lower the components of {{math|'''A'''}} or {{math|'''B'''}}. For contra/co-variant components of {{math|'''A'''}} and co/contra-variant components of {{math|'''B'''}}, we have:
<math display="block">\mathbf{A} \cdot \mathbf{B} = A^{\mu} \eta_{\mu \nu} B^{\nu} = A_{\nu} B^{\nu} = A^{\mu} B_{\mu} </math>
so in the matrix notation:
<math display="block">\begin{align}
\mathbf{A} \cdot \mathbf{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} \\[1ex]
  &= \begin{pmatrix} B_0 & B_1 & B_2 & B_3 \end{pmatrix} \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix}
\end{align}
</math>
while for {{math|'''A'''}} and {{math|'''B'''}} each in covariant components:
<math display="block">\mathbf{A} \cdot \mathbf{B} = A_{\mu} \eta^{\mu \nu} B_{\nu}</math>
with a similar matrix expression to the above.

Applying the Minkowski tensor to a four-vector '''A''' with itself we get:
<math display="block">\mathbf{A \cdot A} = A^\mu \eta_{\mu\nu} A^\nu </math>
which, depending on the case, may be considered the square, or its negative, of the length of the vector.

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.

=====Standard basis, (+−−−) signature=====

The (+−−−) [[metric signature]] is sometimes called the "mostly minus" convention, or the "west coast" convention.

In the (+−−−) [[metric signature]], evaluating the [[Einstein notation|summation over indices]] gives:
<math display="block">\mathbf{A} \cdot \mathbf{B} =  A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3  </math>
while in matrix form:
<math display="block">\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>

It is a recurring theme in special relativity to take the expression
<math display="block"> \mathbf{A}\cdot\mathbf{B} = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = C</math>
in one [[Frame of reference|reference frame]], where ''C'' is the value of the inner product in this frame, and:
<math display="block"> \mathbf{A}'\cdot\mathbf{B}' = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3 {B'}^3 = C' </math>
in another frame, in which ''C''&prime; is the value of the inner product in this frame. Then since the inner product is an invariant, these must be equal:
<math display="block"> \mathbf{A}\cdot\mathbf{B} = \mathbf{A}'\cdot\mathbf{B}' </math>
that is:
<math display="block"> \begin{align}
C &= A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 \\[2pt]
&= {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3{B'}^3
\end{align} </math>

Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a "[[Conservation law (physics)|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'''&prime; are connected by a [[Lorentz transformation]], and similarly for '''B''' and '''B'''&prime;, 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).

In this signature we have:
<math display="block"> \mathbf{A \cdot A} = \left(A^0\right)^2 - \left(A^1\right)^2 - \left(A^2\right)^2 - \left(A^3\right)^2 </math>

With the signature (+−−−), four-vectors may be classified as either [[Minkowski space#Causal structure|spacelike]] if <math>\mathbf{A \cdot A} < 0</math>, [[Minkowski space#Causal structure|timelike]] if <math>\mathbf{A \cdot A} > 0</math>, and [[Minkowski space#Causal structure|null vector]]s if <math>\mathbf{A \cdot A} = 0</math>.

=====Standard basis, (−+++) signature=====
The (-+++) [[metric signature]] is sometimes called the "east coast" convention.

Some authors define ''η'' with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature:

<math display="block">\mathbf{A \cdot B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 </math>

while the matrix form is:

<math display="block">\mathbf{A \cdot B} = \left( \begin{matrix}A^0 & A^1 & A^2 & A^3 \end{matrix} \right) 
\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)
\left( \begin{matrix}B^0 \\ B^1 \\ B^2 \\ B^3 \end{matrix} \right) </math>

Note that in this case, in one frame:

<math display="block"> \mathbf{A}\cdot\mathbf{B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = -C </math>

while in another:

<math display="block"> \mathbf{A}'\cdot\mathbf{B}' = - {A'}^0 {B'}^0 + {A'}^1 {B'}^1 + {A'}^2 {B'}^2 + {A'}^3 {B'}^3 = -C'</math>

so that:

<math display="block"> \begin{align}
-C &= - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 \\[2pt]
&= - {A'}^0 {B'}^0 + {A'}^1 {B'}^1 + {A'}^2 {B'}^2 + {A'}^3 {B'}^3
\end{align}</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.

We have:

<math display="block"> \mathbf{A \cdot A} = - \left(A^0\right)^2 + \left(A^1\right)^2 + \left(A^2\right)^2 + \left(A^3\right)^2 </math>

With the signature (−+++), four-vectors may be classified as either [[Minkowski space#Causal structure|spacelike]] if <math>\mathbf{A \cdot A} > 0</math>, [[Minkowski space#Causal structure|timelike]] if <math>\mathbf{A \cdot A} < 0</math>, and [[Minkowski space#Causal structure|null]] if <math>\mathbf{A \cdot A} = 0</math>.

=====Dual vectors=====

Applying the Minkowski tensor is often expressed as the effect of the [[dual space#Ailinear products and dual spaces|dual vector]] of one vector on the other:

<math display="block">\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.