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special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s. Specifically, a four-vector is an element of a four-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
considered as a representation space of the standard representation of the Lorentz group, the (,) representation. It differs from a
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). Four-vectors describe, for instance, position in spacetime modeled as
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, a particle's
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
, the amplitude of the electromagnetic four-potential at a point in spacetime, and the elements of the subspace spanned by the gamma matrices inside the
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of th ...
. The Lorentz group may be represented by 4×4 matrices . The action of a Lorentz transformation on a general contravariant four-vector (like the examples above), regarded as a column vector with
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
with respect to an
inertial frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
in the entries, is given by X' = \Lambda X, (matrix multiplication) where the components of the primed object refer to the new frame. Related to the examples above that are given as contravariant vectors, there are also the corresponding covariant vectors , and . These transform according to the rule X' = \left(\Lambda^\right)^\textrm X, where 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 In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, spe ...
. 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 , where is a 4×4 matrix other than . Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These include scalars,
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s,
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s and spinor-tensors. The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, some of the results stated in this article require modification in general relativity.


Notation

The notations in this article are: lowercase bold for
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
vectors, hats for three-dimensional
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
s, capital bold for
four dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
vectors (except for the four-gradient), 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: \begin \mathbf & = \left(A^0, \, A^1, \, A^2, \, A^3\right) \\ & = A^0\mathbf_0 + A^1 \mathbf_1 + A^2 \mathbf_2 + A^3 \mathbf_3 \\ & = A^0\mathbf_0 + A^i \mathbf_i \\ & = A^\alpha\mathbf_\alpha\\ & = A^\mu \end where in the last form the magnitude component and basis vector have been combined to a single element. The upper indices indicate contravariant components. Here the standard convention is that Latin indices take values for spatial components, so that ''i'' = 1, 2, 3, and Greek indices take values for 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 mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions. Vectors, covectors and the metric Mat ...
. In special relativity, the spacelike basis E1, E2, E3 and components ''A''1, ''A''2, ''A''3 are often Cartesian basis and components: \begin \mathbf & = \left(A_t, \, A_x, \, A_y, \, A_z\right) \\ & = A_t \mathbf_t + A_x \mathbf_x + A_y \mathbf_y + A_z \mathbf_z \\ \end although, of course, any other basis and components may be used, such as spherical polar coordinates \begin \mathbf & = \left(A_t, \, A_r, \, A_\theta, \, A_\phi\right) \\ & = A_t \mathbf_t + A_r \mathbf_r + A_\theta \mathbf_\theta + A_\phi \mathbf_\phi \\ \end or cylindrical polar coordinates, \begin \mathbf & = (A_t, \, A_r, \, A_\theta, \, A_z) \\ & = A_t \mathbf_t + A_r \mathbf_r + A_\theta \mathbf_\theta + A_z \mathbf_z \\ \end 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: \mathbf_0 = \begin 1 \\ 0 \\ 0 \\ 0 \end \,,\quad \mathbf_1 = \begin 0 \\ 1 \\ 0 \\ 0 \end \,,\quad \mathbf_2 = \begin 0 \\ 0 \\ 1 \\ 0 \end \,,\quad \mathbf_3 = \begin 0 \\ 0 \\ 0 \\ 1 \end so that: \mathbf = \begin A^0 \\ A^1 \\ A^2 \\ A^3 \end The relation between the covariant and contravariant coordinates is through the Minkowski
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
(referred to as the metric), ''η'' which raises and lowers indices as follows: A_ = \eta_ A^ \,, and in various equivalent notations the covariant components are: \begin \mathbf & = (A_0, \, A_1, \, A_2, \, A_3) \\ & = A_0\mathbf^0 + A_1 \mathbf^1 + A_2 \mathbf^2 + A_3 \mathbf^3 \\ & = A_0\mathbf^0 + A_i \mathbf^i \\ & = A_\alpha\mathbf^\alpha\\ \end where the lowered index indicates it to be covariant. Often the metric is diagonal, as is the case for orthogonal coordinates (see
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
), but not in general curvilinear coordinates. The bases can be represented by
row vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s: \mathbf^0 = \begin 1 & 0 & 0 & 0 \end \,,\quad \mathbf^1 = \begin 0 & 1 & 0 & 0 \end \,,\quad \mathbf^2 = \begin 0 & 0 & 1 & 0 \end \,,\quad \mathbf^3 = \begin 0 & 0 & 0 & 1 \end so that: \mathbf = \begin A_0 & A_1 & A_2 & A_3 \end The motivation for the above conventions are that the inner 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 In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
matrix Λ: \mathbf' = \boldsymbol\mathbf In index notation, the contravariant and covariant components transform according to, respectively: ^\mu = \Lambda^\mu _\nu A^\nu \,, \quad_\mu = \Lambda_\mu ^\nu A_\nu in which the matrix has components in row  and column , and the inverse matrix has components in row  and column . For background on the nature of this transformation definition, see
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
.


Pure rotations about an arbitrary axis

For two frames rotated by a fixed angle about an axis defined by the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
: \hat = \left(\hat_1, \hat_2, \hat_3\right)\,, without any boosts, the matrix Λ has components given by: \begin \Lambda_ &= 1 \\ \Lambda_ = \Lambda_ &= 0 \\ \Lambda_ &= \left(\delta_ - \hat_i \hat_j\right) \cos\theta - \varepsilon_ \hat_k \sin\theta + \hat_i \hat_j \end where ''δij'' is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
, and ''εijk'' is the
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
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 In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \ ...
about the ''z''-axis: \begin ^0 \\ ^1 \\ ^2 \\ ^3 \end = \begin 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \\ \end \begin A^0 \\ A^1 \\ A^2 \\ A^3 \end\ .


Pure boosts in an arbitrary direction

For two frames moving at constant relative three-velocity v (not four-velocity, see below), it is convenient to denote and define the relative velocity in units of ''c'' by: \boldsymbol = (\beta_1,\,\beta_2,\,\beta_3) = \frac(v_1,\,v_2,\,v_3) = \frac\mathbf \,. Then without rotations, the matrix Λ has components given by: \begin \Lambda_ &= \gamma, \\ \Lambda_ = \Lambda_ &= -\gamma \beta_, \\ \Lambda_ = \Lambda_ &= (\gamma - 1)\frac + \delta_ = (\gamma - 1)\frac + \delta_, \\ \end where the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
is defined by: \gamma = \frac \,, and is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. 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; \begin A'^0 \\ A'^1 \\ A'^2 \\ A'^3 \end = \begin \cosh\phi &-\sinh\phi & 0 & 0 \\ -\sinh\phi & \cosh\phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end \begin A^0 \\ A^1 \\ A^2 \\ A^3 \end Where the
rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with d ...
expression has been used, written in terms of the
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s: \gamma = \cosh \phi 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 vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
s in
three dimensions Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
. They can be added in the usual entrywise way: \mathbf + \mathbf = \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) and similarly
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
by a scalar ''λ'' is defined entrywise by: \lambda\mathbf = \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) Then subtraction is the inverse operation of addition, defined entrywise by: \mathbf + (-1)\mathbf = \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)


Minkowski tensor

Applying the Minkowski tensor to two four-vectors and , writing the result in
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
notation, we have, using
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
: \mathbf \cdot \mathbf = A^ \eta_ B^ It is convenient to rewrite the definition in
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
form: \mathbf = \begin A^0 & A^1 & A^2 & A^3 \end \begin \eta_ & \eta_ & \eta_ & \eta_ \\ \eta_ & \eta_ & \eta_ & \eta_ \\ \eta_ & \eta_ & \eta_ & \eta_ \\ \eta_ & \eta_ & \eta_ & \eta_ \end \begin B^0 \\ B^1 \\ B^2 \\ B^3 \end in which case above is the entry in row and column 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 or . For contra/co-variant components of and co/contra-variant components of , we have: \mathbf \cdot \mathbf = A^ \eta_ B^ = A_ B^ = A^ B_ so in the matrix notation: \mathbf \cdot \mathbf = \begin A_0 & A_1 & A_2 & A_3 \end \begin B^0 \\ B^1 \\ B^2 \\ B^3 \end = \begin B_0 & B_1 & B_2 & B_3 \end \begin A^0 \\ A^1 \\ A^2 \\ A^3 \end while for and each in covariant components: \mathbf \cdot \mathbf = A_ \eta^ B_ with a similar matrix expression to the above. Applying the Minkowski tensor to a four-vector A with itself we get: \mathbf = A^\mu \eta_ A^\nu 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 In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
(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

= In the (+−−−) metric signature, evaluating the summation over indices gives: \mathbf \cdot \mathbf = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 while in matrix form: \mathbf = \begin A^0 & A^1 & A^2 & A^3 \end \begin 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end \begin B^0 \\ B^1 \\ B^2 \\ B^3 \end It is a recurring theme in special relativity to take the expression \mathbf\cdot\mathbf = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = C in one
reference frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
, where ''C'' is the value of the inner product in this frame, and: \mathbf'\cdot\mathbf' = ^0 ^0 - ^1 ^1 - ^2 ^2 - ^3 ^3 = C' 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: \mathbf\cdot\mathbf = \mathbf'\cdot\mathbf' that is: C = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = ^0 ^0 - ^1 ^1 - ^2 ^2 - ^3^3 Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a "
conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
", 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 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 In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
, 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 In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
vector (see also below). In this signature we have: \mathbf = \left(A^0\right)^2 - \left(A^1\right)^2 - \left(A^2\right)^2 - \left(A^3\right)^2 With the signature (+−−−), four-vectors may be classified as either spacelike if \mathbf < 0, timelike if \mathbf > 0, and
null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms an ...
s if \mathbf = 0.


=Standard basis, (−+++) signature

= Some authors define ''η'' with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature: \mathbf = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 while the matrix form is: \mathbf = \left( \beginA^0 & A^1 & A^2 & A^3 \end \right) \left( \begin -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end \right) \left( \beginB^0 \\ B^1 \\ B^2 \\ B^3 \end \right) Note that in this case, in one frame: \mathbf\cdot\mathbf = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = -C while in another: \mathbf'\cdot\mathbf' = - ^0 ^0 + ^1 ^1 + ^2 ^2 + ^3 ^3 = -C' so that: -C = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = - ^0 ^0 + ^1 ^1 + ^2 ^2 + ^3 ^3 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: \mathbf = - \left(A^0\right)^2 + \left(A^1\right)^2 + \left(A^2\right)^2 + \left(A^3\right)^2 With the signature (−+++), four-vectors may be classified as either spacelike if \mathbf > 0, timelike if \mathbf < 0, and null if \mathbf = 0.


=Dual vectors

= Applying the Minkowski tensor is often expressed as the effect of the dual vector of one vector on the other: \mathbf = A^*(\mathbf) = AB^. Here the ''Aν''s are the components of the dual vector A* of A in the dual basis and called the covariant coordinates of A, while the original ''Aν'' components are called the contravariant coordinates.


Four-vector calculus


Derivatives and differentials

In special relativity (but not general relativity), the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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 the four-vector, ''d''A and divide it by the differential of the scalar, ''dλ'': \underset = \underset \underset where the contravariant components are: d\mathbf = \left(dA^0, dA^1, dA^2, dA^3\right) while the covariant components are: d\mathbf = \left(dA_0, dA_1, dA_2, dA_3\right) In relativistic mechanics, one often takes the differential of a four-vector and divides by the differential in
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
(see below).


Fundamental four-vectors


Four-position

A point in
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
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: \mathbf = \left(ct, \mathbf\right) where r is the
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
. 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 ''R''0 = ''ct'' ensures that all the coordinates have the same units (of distance). 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: \Delta \mathbf = \left(c\Delta t, \Delta \mathbf \right) For the differential four-position on a world line we have, using a norm notation: \, d\mathbf\, ^2 = \mathbf = dR^\mu dR_\mu = c^2d\tau^2 = ds^2 \,, defining the differential
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
d''s'' and differential proper time increment d''τ'', but this "norm" is also: \, d\mathbf\, ^2 = (cdt)^2 - d\mathbf\cdot d\mathbf \,, so that: (c d\tau)^2 = (cdt)^2 - d\mathbf\cdot d\mathbf \,. 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 In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
\tau. 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 In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spat ...
''t'' of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by (''cdt'')2 to obtain: \left(\frac\right)^2 = 1 - \left(\frac\cdot \frac\right) = 1 - \frac = \frac \,, where u = ''d''r/''dt'' is the coordinate 3-
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of an object measured in the same frame as the coordinates ''x'', ''y'', ''z'', and
coordinate time In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spat ...
''t'', and \gamma(\mathbf) = \frac is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
. This provides a useful relation between the differentials in coordinate time and proper time: dt = \gamma(\mathbf)d\tau \,. This relation can also be found from the time transformation in the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s. Important four-vectors in relativity theory can be defined by applying this differential \frac.


Four-gradient

Considering that
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s are
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s, one can form a
four-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties ...
from the partial time derivative /''t'' and the spatial
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
∇. Using the standard basis, in index and abbreviated notations, the contravariant components are: \begin \boldsymbol & = \left(\frac, \, -\frac, \, -\frac, \, -\frac \right) \\ & = (\partial^0, \, - \partial^1, \, - \partial^2, \, - \partial^3) \\ & = \mathbf_0\partial^0 - \mathbf_1\partial^1 - \mathbf_2\partial^2 - \mathbf_3\partial^3 \\ & = \mathbf_0\partial^0 - \mathbf_i\partial^i \\ & = \mathbf_\alpha \partial^\alpha \\ & = \left(\frac\frac , \, - \nabla \right) \\ & = \left(\frac,- \nabla \right) \\ & = \mathbf_0\frac\frac - \nabla \\ \end 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: \begin \boldsymbol & = \left(\frac, \, \frac, \, \frac, \, \frac \right) \\ & = (\partial_0, \, \partial_1, \, \partial_2, \, \partial_3) \\ & = \mathbf^0\partial_0 + \mathbf^1\partial_1 + \mathbf^2\partial_2 + \mathbf^3\partial_3 \\ & = \mathbf^0\partial_0 + \mathbf^i\partial_i \\ & = \mathbf^\alpha \partial_\alpha \\ & = \left(\frac\frac , \, \nabla \right) \\ & = \left(\frac, \nabla \right) \\ & = \mathbf^0\frac\frac + \nabla \\ \end Since this is an operator, it doesn't have a "length", but evaluating the inner product of the operator with itself gives another operator: \partial^\mu \partial_\mu = \frac\frac - \nabla^2 = \frac - \nabla^2 called the D'Alembert operator.


Kinematics


Four-velocity

The four-velocity of a particle is defined by: \mathbf = \frac = \frac\frac = \gamma(\mathbf)\left(c, \mathbf\right), Geometrically, U is a normalized vector tangent to the
world line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
of the particle. Using the differential of the four-position, the magnitude of the four-velocity can be obtained: \, \mathbf\, ^2 = U^\mu U_\mu = \frac \frac = \frac = c^2 \,, in short, the magnitude of the four-velocity for any object is always a fixed constant: \, \mathbf \, ^2 = c^2 The norm is also: \, \mathbf\, ^2 = ^2 \left( c^2 - \mathbf\cdot\mathbf \right) \,, so that: c^2 = ^2 \left( c^2 - \mathbf\cdot\mathbf \right) \,, which reduces to the definition of the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
. Units of four-velocity are m/s in SI and 1 in the
geometrized unit system A geometrized unit system, geometric unit system or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, ''c'', and the gravitational constant, ''G'', are set equ ...
. Four-velocity is a contravariant vector.


Four-acceleration

The
four-acceleration In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has ...
is given by: \mathbf = \frac = \gamma(\mathbf) \left(\frac c, \frac \mathbf + \gamma(\mathbf) \mathbf \right). where a = ''d''u/''dt'' 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 inner product of the four-acceleration and the four-velocity is zero: \mathbf\cdot\mathbf = A^\mu U_\mu = \frac U_\mu = \frac \, \frac \left(U^\mu U_\mu\right) = 0 \, 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 The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
(or
invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
) ''m''0, the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
is given by: \mathbf = m_0 \mathbf = m_0\gamma(\mathbf)(c, \mathbf) = \left(\frac, \mathbf\right) where the total energy of the moving particle is: E = \gamma(\mathbf) m_0 c^2 and the total
relativistic momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
is: \mathbf = \gamma(\mathbf) m_0 \mathbf Taking the inner product of the four-momentum with itself: \, \mathbf\, ^2 = P^\mu P_\mu = m_0^2 U^\mu U_\mu = m_0^2 c^2 and also: \, \mathbf\, ^2 = \frac - \mathbf\cdot\mathbf which leads to the energy–momentum relation: E^2 = c^2 \mathbf\cdot\mathbf + \left(m_0 c^2\right)^2 \,. This last relation is useful
relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
, essential in relativistic quantum mechanics and
relativistic quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, all with applications to
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
.


Four-force

The
four-force In the special theory of relativity, four-force is a four-vector that replaces the classical force. In special relativity The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper ...
acting on a particle is defined analogously to the 3-force as the time derivative of 3-momentum in
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...
: \mathbf = \frac = \gamma(\mathbf)\left(\frac\frac, \frac\right) = \gamma(\mathbf)\left(\frac, \mathbf\right) where ''P'' is the power transferred to move the particle, and f is the 3-force acting on the particle. For a particle of constant invariant mass ''m''0, this is equivalent to \mathbf = m_0 \mathbf = m_0\gamma(\mathbf)\left( \frac c, \left(\frac \mathbf + \gamma(\mathbf) \mathbf\right) \right) An invariant derived from the four-force is: \mathbf\cdot\mathbf = F^\mu U_\mu = m_0 A^\mu U_\mu = 0 from the above result.


Thermodynamics


Four-heat flux

The four-heat flux vector field, is essentially similar to the 3d
heat flux Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy per unit area per unit time. In SI its units are watts per square metre (W/m2). It has both a ...
vector field q, in the local frame of the fluid: \mathbf = -k \boldsymbol T = -k\left( \frac\frac, \nabla T\right) where ''T'' is
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
and ''k'' is
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
.


Four-baryon number flux

The flux of baryons is: \mathbf = n\mathbf where is the
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...
of
baryon In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifie ...
s in the local rest frame of the baryon fluid (positive values for baryons, negative for
anti Anti may refer to: *Anti-, a prefix meaning "against" *Änti, or Antaeus, a half-giant in Greek and Berber mythology *A false reading of ''Nemty'', the name of the ferryman who carried Isis to Set's island in Egyptian mythology * Áńt’į, or ...
baryons), and the four-velocity field (of the fluid) as above.


Four-entropy

The four-
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
vector is defined by: \mathbf = s\mathbf + \frac where is the entropy per baryon, and the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
, in the local rest frame of the fluid.


Electromagnetism

Examples of four-vectors in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
include the following.


Four-current

The electromagnetic
four-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional sp ...
(or more correctly a four-current density) is defined by \mathbf = \left( \rho c, \mathbf \right) formed from the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...
j and
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
''ρ''.


Four-potential

The electromagnetic four-potential (or more correctly a four-EM vector potential) defined by \mathbf = \left( \frac, \mathbf \right) formed from the
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
and the scalar potential . The four-potential is not uniquely determined, because it depends on a choice of
gauge Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, es ...
. In the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
for the electromagnetic field: * In vacuum, (\boldsymbol \cdot \boldsymbol) \mathbf = 0 * With a
four-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional sp ...
source and using the
Lorenz gauge condition In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ...
(\boldsymbol \cdot \mathbf) = 0, (\boldsymbol \cdot \boldsymbol) \mathbf = \mu_0 \mathbf


Waves


Four-frequency

A photonic
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...
can be described by the
four-frequency The four-frequency of a massless particle, such as a photon, is a four-vector defined by :N^a = \left( \nu, \nu \hat \right) where \nu is the photon's frequency and \hat is a unit vector in the direction of the photon's motion. The four-frequency ...
defined as \mathbf = \nu\left(1 , \hat \right) where ''ν'' is the frequency of the wave and \hat is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
in the travel direction of the wave. Now: \, \mathbf\, = N^\mu N_\mu = \nu ^2 \left(1 - \hat\cdot\hat\right) = 0 so the four-frequency of a photon is always a null vector.


Four-wavevector

The quantities reciprocal to time ''t'' and space r are the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
''ω'' and angular wave vector k, respectively. They form the components of the four-wavevector or wave four-vector: \mathbf = \left(\frac, \vec\right) = \left(\frac, \frac \hat\mathbf\right) \,. A wave packet of nearly
monochromatic A monochrome or monochromatic image, object or palette is composed of one color (or values of one color). Images using only shades of grey are called grayscale (typically digital) or black-and-white (typically analog). In physics, monochro ...
light can be described by: \mathbf = \frac\mathbf = \frac \nu\left(1,\hat\right) = \frac \left(1, \hat\right) \,. The de Broglie relations then showed that four-wavevector applied to matter waves as well as to light waves: \mathbf = \hbar \mathbf = \left(\frac,\vec\right) = \hbar \left(\frac,\vec \right)\,. yielding E = \hbar \omega and \vec = \hbar \vec, where ''ħ'' is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
divided by . The square of the norm is: \, \mathbf \, ^2 = K^\mu K_\mu = \left(\frac\right)^2 - \mathbf\cdot\mathbf\,, and by the de Broglie relation: \, \mathbf \, ^2 = \frac \, \mathbf \, ^2 = \left(\frac\right)^2 \,, we have the matter wave analogue of the energy–momentum relation: \left(\frac\right)^2 - \mathbf\cdot\mathbf = \left(\frac\right)^2 \,. Note that for massless particles, in which case , we have: \left(\frac\right)^2 = \mathbf\cdot\mathbf \,, or . Note this is consistent with the above case; for photons with a 3-wavevector of modulus , in the direction of wave propagation defined by the unit vector \hat.


Quantum theory


Four-probability current

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the four- probability current or probability four-current is analogous to the electromagnetic four-current: \mathbf = (\rho c, \mathbf) where is the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
corresponding to the time component, and 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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, it is not always possible to find a current, particularly when interactions are involved. Replacing the energy by the
energy operator In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry. Definition It is given by: \hat = i\hbar\frac It acts on the wave function (the ...
and the momentum by the
momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensi ...
in the four-momentum, one obtains the four-momentum operator, used in relativistic wave equations.


Four-spin

The
four-spin In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thr ...
of a particle is defined in the rest frame of a particle to be \mathbf = (0, \mathbf) where is the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
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 \, \mathbf\, ^2 = -, \mathbf, ^2 = -\hbar^2 s(s + 1) This value is observable and quantized, with the
spin quantum number In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
(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 In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
as a basis, again in various equivalent notations: \begin \mathbf & = \left(A^0, \, A^1, \, A^2, \, A^3\right) \\ & = A^0\boldsymbol_0 + A^1 \boldsymbol_1 + A^2 \boldsymbol_2 + A^3 \boldsymbol_3 \\ & = A^0\boldsymbol_0 + A^i \boldsymbol_i \\ & = A^\alpha\boldsymbol_\alpha\\ \end or explicitly: \begin \mathbf & = A^0\begin 1 & 0 \\ 0 & 1 \end + A^1\begin 0 & 1 \\ 1 & 0 \end + A^2\begin 0 & -i \\ i & 0 \end + A^3\begin 1 & 0 \\ 0 & -1 \end \\ & = \begin A^0 + A^3 & A^1 - i A^2 \\ A^1 + i A^2 & A^0 - A^3 \end \end and in this formulation, the four-vector is represented as a Hermitian matrix (the matrix transpose and
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the matrix leaves it unchanged), rather than a real-valued column or row vector. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the matrix is the modulus of the four-vector, so the determinant is an invariant: \begin , \mathbf, & = \begin A^0 + A^3 & A^1 - i A^2 \\ A^1 + i A^2 & A^0 - A^3 \end \\ & = \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) \\ & = \left(A^0\right)^2 - \left(A^1\right)^2 - \left(A^2\right)^2 - \left(A^3\right)^2 \end This idea of using the Pauli matrices as basis vectors is employed in the algebra of physical space, an example of a
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
.


Four-vectors in spacetime algebra

In
spacetime algebra In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of spec ...
, 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 In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form), : \ \stackrel\ \gamma^1 A_1 + ...
is a shorthand for a four-vector A contracted with the gamma matrices: \mathbf\!\!\!\!/ = A_\alpha \gamma^\alpha = A_0 \gamma^0 + A_1 \gamma^1 + A_2 \gamma^2 + A_3 \gamma^3 The four-momentum contracted with the gamma matrices is an important case in relativistic quantum mechanics and
relativistic quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. In the Dirac equation and other relativistic wave equations, terms of the form: \mathbf\!\!\!\!/ = P_\alpha \gamma^\alpha = P_0 \gamma^0 + P_1 \gamma^1 + P_2 \gamma^2 + P_3 \gamma^3 = \dfrac \gamma^0 - p_x \gamma^1 - p_y \gamma^2 - p_z \gamma^3 appear, in which the energy and momentum components 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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
* Paravector *
Relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
* Wave vector


References

*Rindler, W. ''Introduction to Special Relativity (2nd edn.)'' (1991) Clarendon Press Oxford {{ISBN, 0-19-853952-5 Minkowski spacetime Theory of relativity Concepts in physics Vectors (mathematics and physics)