Four-vector
Template:Short description Template:Distinguish Template:Use American English Template:Spacetime In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector)<ref>Template:Citation</ref> 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. It 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).<ref name="BaskalKim2015">Template:Cite book</ref>Template:Rp
Four-vectors describe, for instance, position Template:Math in spacetime modeled as Minkowski space, a particle's four-momentum Template:Math, the amplitude of the electromagnetic four-potential Template:Math at a point Template:Mvar 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 Template:Nowrap matrices Template:Math. The action of a Lorentz transformation on a general contravariant four-vector Template:Mvar (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an 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 vectors Template:Math, Template:Math and Template:Math. These transform according to the rule <math display="block">X' = \left(\Lambda^{-1}\right)^\textrm{T} X,</math> where Template:Math 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. 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 Template:Math, where Template:Math is a 4×4 matrix other than Template:Math. 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 Template:Math along the positive x-axis, the transformation of four-vectors is:<ref>Template:Cite web</ref> <math display="block">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,</math> or <math display="block">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,</math> depending on convention (viz. whether events are written Template:Math or Template:Math, 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:<ref>Template:Citation</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 Template:Math is the component multiplier and Template:Math is the basis vector; note that both are necessary to make a vector, and that when Template:Math 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 Template:Math, and Greek indices take values for time and space components, so Template:Math, 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 Template:Math, Template:Math, Template:Math and components Template:Math, Template:Math, Template:Math are often 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 vectors: <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 and contravariant coordinates is through the Minkowski metric tensor (referred to as the metric), Template:Math which 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 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: <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 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 Template:Math: <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 Template:Math has components Template:Math in row Template:Math and column Template:Math, and the matrix Template:Math has components Template:Math in row Template:Math and column Template:Math.
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 Template:Math 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 Template:Math has components given by:<ref>Template:Cite book</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 Template:Math is the Kronecker delta, and Template:Math 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 any direction
For two frames moving at constant relative three-velocity Template:Math (not four-velocity, see below), it is convenient to denote and define the relative velocity in units of Template:Math 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 Template:Math 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 Template:Math 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 Template:Math expression has been used, written in terms of the hyperbolic functions: <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 linearity properties as Euclidean vectors 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 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
Applying the Minkowski tensor Template:Math to two four-vectors Template:Math and Template:Math, 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 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 Template:Math above is the entry in row Template:Math and column Template:Math 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 Template:Math or Template:Math. For contra/co-variant components of Template:Math and co/contra-variant components of Template:Math, 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 Template:Math and Template:Math 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 Template:Math 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 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 Template:Math metric signature is sometimes called the "mostly minus" convention, or the "west coast" convention.
In the Template:Math metric signature, evaluating the 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 reference frame, where C is the value of the scalar 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′ is the value of the scalar product in this frame. Then since the scalar 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", but there is no "conservation" involved. The primary significance of the Minkowski scalar product is that for any two four-vectors, its value is invariant for all observers; a change of coordinates does not result in a change in value of the scalar 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 scalar 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 Template:Math, four-vectors may be classified as either spacelike if Template:Math, timelike if Template:Math, and null vectors if Template:Math.
Standard basis, (−+++) signature
The Template:Math metric signature is sometimes called the "east coast" convention.
Some authors define Template:Math with the opposite sign, in which case we have the Template:Math 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 Template:Math in terms of Template:Math and Template:Math. 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 Template:Math, four-vectors may be classified as either spacelike if Template:Math, timelike if Template:Math, and null if Template:Math.
Dual vectors
Applying the Minkowski tensor is often expressed as the effect of the 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 Template:Math are the components of the dual Template:Math of vector Template:Math in the dual basis and called the covariant coordinates of Template:Math, while the original Template:Math components are called the 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 Template:Math (invariant) is itself a four-vector. It is also useful to take the differential of the four-vector, Template:Math and divide it by the differential of the scalar, Template:Math: <math display="block">\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 display="block"> d\mathbf{A} = \left(dA^0, dA^1, dA^2, dA^3\right) </math> while the covariant components are: <math display="block"> d\mathbf{A} = \left(dA_0, dA_1, dA_2, dA_3\right) </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 four-vector or four-position or 4-position, described in some reference frame by a set of four coordinates: <math display="block"> \mathbf{R} = \left(ct, \mathbf{r}\right) </math> where Template:Math is the three-dimensional space position vector. If Template:Math is a function of coordinate time Template:Math in the same frame, i.e. Template:Math, this corresponds to a sequence of events as Template:Math varies. The definition Template:Math ensures that all the coordinates have the same dimension (of length) and units.<ref name="e561">Template:Cite web</ref><ref>Template:Citation</ref><ref>Template:Cite book</ref><ref>Template:Citation</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 display="block"> \Delta \mathbf{R} = \left(c\Delta t, \Delta \mathbf{r} \right) </math>
For the differential four-position on a world line we have, using a norm notation: <math display="block">\|d\mathbf{R}\|^2 = \mathbf{dR \cdot dR} = dR^\mu dR_\mu = c^2d\tau^2 = ds^2 \,,</math> defining the differential line element ds and differential proper time increment dτ, but this "norm" is also: <math display="block">\|d\mathbf{R}\|^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,,</math> so that: <math display="block">(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 derivatives 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 <math>\tau</math>. 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 Template:Math of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by Template:Math to obtain <math display="block">\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 Template:Math is the coordinate 3-velocity of an object measured in the same frame as the coordinates Template:Math, Template:Math, Template:Math, and coordinate time Template:Math, and <math display="block">\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 display="block">dt = \gamma(\mathbf{u})d\tau \,.</math>
This relation can also be found from the time transformation in the Lorentz transformations.
Important four-vectors in relativity theory can be defined by applying this differential Template:Math.
Four-gradient
Considering that partial derivatives are linear operators, one can form a four-gradient from the partial time derivative Template:Math and the spatial gradient operator Template:Math. Using the standard basis, in index and abbreviated notations, the contravariant components are: <math display="block">\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) \\
& = \left(\frac{\partial_t}{c},- \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 display="block">\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) \\
& = \left(\frac{\partial_t}{c}, \nabla \right) \\
& = \mathbf{E}^0\frac{1}{c}\frac{\partial}{\partial t} + \nabla \\
\end{align}</math>
Since this is an operator, it does not have a "length", but evaluating the scalar product of the operator with itself gives another operator: <math display="block">\partial^\mu \partial_\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 = \frac{{\partial_t}^2}{c^2} - \nabla^2</math> called the D'Alembert operator.
Kinematics
Four-velocity
The four-velocity of a particle is defined by: <math display="block">\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 normalized vector tangent to the world line of the particle. Using the differential of the four-position, the magnitude of the four-velocity can be obtained: <math display="block">\|\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 four-velocity for any object is always a fixed constant: <math display="block">\| \mathbf{U} \|^2 = c^2 </math>
The norm is also: <math display="block">\|\mathbf{U}\|^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,</math> so that: <math display="block">c^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,</math> which reduces to the definition of the Lorentz factor.
Units of four-velocity are m/s in SI and 1 in the geometrized unit system. Four-velocity is a contravariant vector.
Four-acceleration
The four-acceleration is given by: <math display="block">\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 Template:Math is the coordinate 3-acceleration. Since the magnitude of U is a constant, the four acceleration is orthogonal to the four velocity, i.e. the Minkowski scalar product of the four-acceleration and the four-velocity is zero: <math display="block">\mathbf{A}\cdot\mathbf{U} = A^\mu U_\mu = \frac{dU^\mu}{d\tau} U_\mu = \frac{1}{2} \, \frac{d}{d\tau} \left(U^\mu U_\mu\right) = 0 \,</math> which is true for all world lines. The geometric meaning of four-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) Template:Math, the four-momentum is given by: <math display="block">\mathbf{P} = m_0 \mathbf{U} = m_0\gamma(\mathbf{u})(c, \mathbf{u}) = \left(\frac{E}{c}, \mathbf{p}\right)</math> where the total energy of the moving particle is: <math display="block">E = \gamma(\mathbf{u}) m_0 c^2 </math> and the total relativistic momentum is: <math display="block">\mathbf{p} = \gamma(\mathbf{u}) m_0 \mathbf{u} </math>
Taking the scalar product of the four-momentum with itself: <math display="block">\|\mathbf{P}\|^2 = P^\mu P_\mu = m_0^2 U^\mu U_\mu = m_0^2 c^2</math> and also: <math display="block">\|\mathbf{P}\|^2 = \frac{E^2}{c^2} - \mathbf{p}\cdot\mathbf{p}</math> which leads to the energy–momentum relation: <math display="block">E^2 = c^2 \mathbf{p}\cdot\mathbf{p} + \left(m_0 c^2\right)^2 \,.</math>
This last relation is useful in 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 display="block">\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})\left(\frac{P}{c}, \mathbf{f}\right)</math> where Template:Math is the power transferred to move the particle, and Template:Math is the 3-force acting on the particle. For a particle of constant invariant mass Template:Math, this is equivalent to <math display="block">\mathbf{F} = m_0 \mathbf{A} = m_0\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 four-force is: <math display="block">\mathbf{F}\cdot\mathbf{U} = F^\mu U_\mu = m_0 A^\mu U_\mu = 0</math> from the above result.
Thermodynamics
Four-heat flux
The four-heat flux vector field, is essentially similar to the 3-d heat flux vector field Template:Math, in the local frame of the fluid:<ref>Template:Cite journal</ref> <math display="block">\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>Template:Cite book</ref> <math display="block">\mathbf{S} = n\mathbf{U}</math> where Template:Math is the number density of baryons in the local rest frame of the baryon fluid (positive values for baryons, negative for antibaryons), and Template:Math the four-velocity field (of the fluid) as above.
Four-entropy
The four-entropy vector is defined by:<ref>Template:Cite book</ref> <math display="block">\mathbf{s} = s\mathbf{S} + \frac{\mathbf{Q}}{T}</math> where Template:Math is the entropy per baryon, and Template:Mvar the absolute temperature, in the local rest frame of the fluid.<ref>Template:Cite book</ref>
Electromagnetism
Examples of four-vectors in electromagnetism include the following.
Four-current
The electromagnetic four-current (or more correctly a four-current density)<ref> Template:Cite book</ref> is defined by <math display="block"> \mathbf{J} = \left( \rho c, \mathbf{j} \right) </math> formed from the current density Template:Math and charge density Template:Math.
Four-potential
The electromagnetic four-potential (or more correctly a four-EM vector potential) defined by <math display="block">\mathbf{A} = \left( \frac{\phi}{c}, \mathbf{a} \right)</math> formed from the vector potential Template:Math and the scalar potential Template:Math.
The four-potential is not uniquely determined, because it depends on a choice of gauge.
In the wave equation for the electromagnetic field:
- In vacuum, <math display="block">(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = 0</math>
- With a four-current source and using the Lorenz gauge condition <math>(\boldsymbol{\partial} \cdot \mathbf{A}) = 0</math>, <math display="block">(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = \mu_0 \mathbf{J}</math>
Waves
Four-frequency
A photonic plane wave can be described by the four-frequency, defined as <math display="block">\mathbf{N} = \nu\left(1 , \hat{\mathbf{n}} \right)</math> where Template:Mvar 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 display="block">\|\mathbf{N}\| = N^\mu N_\mu = \nu ^2 \left(1 - \hat{\mathbf{n}}\cdot\hat{\mathbf{n}}\right) = 0</math> so the four-frequency of a photon is always a null vector.
Four-wavevector
The quantities reciprocal to time Template:Mvar and space Template:Math are the angular frequency Template:Mvar and angular wave vector Template:Math, respectively. They form the components of the four-wavevector or wave four-vector: <math display="block">\mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \left(\frac{\omega}{c}, \frac{\omega}{v_p} \hat\mathbf{n}\right) \,.</math>
The wave four-vector has coherent derived unit of reciprocal meters in the SI.<ref name="o144">Template:Cite web</ref>
A wave packet of nearly monochromatic light can be described by: <math display="block">\mathbf{K} = \frac{2\pi}{c}\mathbf{N} = \frac{2\pi}{c} \nu\left(1,\hat{\mathbf{n}}\right) = \frac{\omega}{c} \left(1, \hat{\mathbf{n}}\right) ~.</math>
The de Broglie relations then showed that four-wavevector applied to matter waves as well as to light waves: <math display="block">\mathbf{P} = \hbar \mathbf{K} = \left(\frac{E}{c},\vec{p}\right) = \hbar \left(\frac{\omega}{c},\vec{k} \right) ,</math> yielding <math>E = \hbar \omega</math> and <math>\vec{p} = \hbar \vec{k}</math>, where Template:Mvar is the Planck constant divided by Template:Math.
The square of the norm is: <math display="block">\| \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 display="block">\| \mathbf{K} \|^2 = \frac{1}{\hbar^2} \| \mathbf{P} \|^2 = \left(\frac{m_0 c}{\hbar}\right)^2 \,,</math> we have the matter wave analogue of the energy–momentum relation: <math display="block">\left(\frac{\omega}{c}\right)^2 - \mathbf{k}\cdot\mathbf{k} = \left(\frac{m_0 c}{\hbar}\right)^2 ~.</math>
Note that for massless particles, in which case Template:Math, we have: <math display="block">\left(\frac{\omega}{c}\right)^2 = \mathbf{k}\cdot\mathbf{k} \,,</math> or Template:Math. Note this is consistent with the above case; for photons with a 3-wavevector of modulus Template:Nobr in the direction of wave propagation defined by the unit vector <math>\hat{\mathbf{n}} ~.</math>
Quantum theory
Four-probability current
In quantum mechanics, the four-probability current or probability four-current is analogous to the electromagnetic four-current:<ref>Template:Citation, p. 41</ref> <math display="block">\mathbf{J} = (\rho c, \mathbf{j}) </math> where Template:Math is the probability density function corresponding to the time component, and Template:Math 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 equations.
Four-spin
The four-spin of a particle is defined in the rest frame of a particle to be <math display="block">\mathbf{S} = (0, \mathbf{s})</math> where Template:Math is the spin pseudovector. In quantum mechanics, not all three components of this vector are simultaneously measurable, only one component is. The timelike component is zero in the particle's rest frame, but not in any other frame. This component can be found from an appropriate Lorentz transformation.
The norm squared is the (negative of the) magnitude squared of the spin, and according to quantum mechanics we have <math display="block">\|\mathbf{S}\|^2 = -|\mathbf{s}|^2 = -\hbar^2 s(s + 1)</math>
This value is observable and quantized, with Template:Math the spin quantum number (not the magnitude of the spin vector).
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, again in various equivalent notations:<ref>Template:Cite book</ref> <math display="block"> \begin{align}
\mathbf{A} & = \left(A^0, \, A^1, \, A^2, \, A^3\right) \\
& = 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 display="block">\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 represented as a Hermitian 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 display="block"> \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} \\[1ex]
& = \left(A^0 + A^3\right)\left(A^0 - A^3\right) - \left(A^1 -i A^2\right)\left(A^1 + i A^2\right) \\[1ex]
& = \left(A^0\right)^2 - \left(A^1\right)^2 - \left(A^2\right)^2 - \left(A^3\right)^2
\end{align}</math>
This idea of using the Pauli matrices as basis vectors 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. (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.
The Feynman slash notation is a shorthand for a four-vector A contracted with the gamma matrices: <math display="block">\mathbf{A}\!\!\!\!/ = A_\alpha \gamma^\alpha = A_0 \gamma^0 + A_1 \gamma^1 + A_2 \gamma^2 + A_3 \gamma^3 </math>
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 equations, terms of the form: <math display="block">\begin{align} \mathbf{P}\!\!\!\!/ = P_\alpha \gamma^\alpha &= P_0 \gamma^0 + P_1 \gamma^1 + P_2 \gamma^2 + P_3 \gamma^3 \\[4pt] &= \dfrac{E}{c} \gamma^0 - p_x \gamma^1 - p_y \gamma^2 - p_z \gamma^3 \\ \end{align} </math> appear, in which the energy Template:Mvar and momentum components Template:Math are replaced by their respective operators.
See also
- Basic introduction to the mathematics of curved spacetime
- Dust (relativity) for the number-flux four-vector
- Minkowski space
- Paravector
- Relativistic mechanics
- Wave vector