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special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
, a four-vector (or 4-vector, sometimes Lorentz 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 vel ...
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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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. Euclidean vectors can be added and scal ...
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, 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 i ...
, 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. 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 In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
with respect to an inertial frame 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 In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
. 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 , 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, spinors,
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
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 as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, 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 In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
vectors, hats for three-dimensional
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s, capital bold for four dimensional 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 \end where ''Aα'' is the magnitude component and Eα is the basis vector component; note that both are necessary to make a vector, and that when ''Aα'' 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 ''i'' = 1, 2, 3, and Greek indices take values for space ''and time'' components, so ''α'' = 0, 1, 2, 3, used with the summation convention. The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in inner products (examples are given below), or raising and lowering indices. In special relativity, the spacelike basis 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 In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
, or even general
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...
. Note the coordinate labels are always subscripted as labels and are not indices taking numerical values. In general relativity, local curvilinear coordinates in a local basis must be used. Geometrically, a four-vector can still be interpreted as an arrow, but in spacetime - not just space. In relativity, the arrows are drawn as part of Minkowski diagram (also called ''spacetime diagram''). In this article, four-vectors will be referred to simply as vectors. It is also customary to represent the bases by
column vector In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of 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 , 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 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 allows ...
(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 In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
(see line element), but not in general
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...
. The bases can be represented by row vectors: \begin \mathbf^0 &= \begin 1 & 0 & 0 & 0 \end \,, & \mathbf^1 &= \begin 0 & 1 & 0 & 0 \end \,, \\ ex \mathbf^2 &= \begin 0 & 0 & 1 & 0 \end \,, & \mathbf^3 &= \begin 0 & 0 & 0 & 1 \end, \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 vel ...
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 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 associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
. 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 of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
.


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 (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
: \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 In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
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 (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \t ...
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 (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
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 special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velo ...
expression has been used, written in terms of the hyperbolic functions: \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. Euclidean vectors can be added and scal ...
s in three dimensions. They can be added in the usual entrywise way: \begin \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) \end 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 ...
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: \begin \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) \end


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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
notation, we have, using
Einstein notation In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
: \mathbf \cdot \mathbf = A^ B^ \mathbf_ \cdot \mathbf_ = A^ \eta_ B^ 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 Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
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: \begin \mathbf \cdot \mathbf &= \begin A_0 & A_1 & A_2 & A_3 \end \begin B^0 \\ B^1 \\ B^2 \\ B^3 \end \\ ex &= \begin B_0 & B_1 & B_2 & B_3 \end \begin A^0 \\ A^1 \\ A^2 \\ A^3 \end \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, each of whose components are all zero, except one that equals 1. For exampl ...
(essentially Cartesian coordinates). If orthogonal coordinates are used, there would be scale factors along the diagonal part of the spacelike part of the metric, while for general curvilinear coordinates the entire spacelike part of the metric would have components dependent on the curvilinear basis used.


=Standard basis, (+−−−) signature

= The (+−−−) metric signature is sometimes called the "mostly minus" convention, or the "west coast" convention. In the (+−−−) metric signature, evaluating the 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, 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: \begin C &= A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 \\ pt&= ^0 ^0 - ^1 ^1 - ^2 ^2 - ^3^3 \end Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a " conservation law", but there is no "conservation" involved. The primary significance of the Minkowski inner product is that for any two four-vectors, its value is invariant 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 vel ...
, 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 i ...
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 vectors if \mathbf = 0.


=Standard basis, (−+++) signature

= The (-+++) metric signature is sometimes called the "east coast" convention. Some authors define ''η'' with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature: \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: \begin -C &= - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 \\ pt&= - ^0 ^0 + ^1 ^1 + ^2 ^2 + ^3 ^3 \end 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 Null may refer to: Science, technology, and mathematics Astronomy *Nuller, an optical tool using interferometry to block certain sources of light Computing *Null (SQL) (or NULL), a special marker and keyword in SQL indicating that a data value do ...
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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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. The proper time interval between two events on a world line is the change 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: \mathbf = \left(ct, \mathbf\right) where r is the
three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
. 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
dimension 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 coo ...
(of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
) and units (in the SI, meters). 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 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. The proper time interval between two events on a world line is the change in proper time ...
\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 ''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 a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
of an object measured in the same frame as the coordinates ''x'', ''y'', ''z'', and coordinate time ''t'', and \gamma(\mathbf) = \frac is the
Lorentz factor The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
. 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 vel ...
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). P ...
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 mapping V \to W between two vector spaces that pr ...
s, one can form a four-gradient from the partial time derivative /''t'' and the spatial
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
∇. 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 of 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 (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
. 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: \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, ...
(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, ...
) ''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 i ...
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 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 in
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 o ...
, essential in relativistic quantum mechanics and
relativistic quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of suba ...
, all with applications to
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
.


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: \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 In physics and engineering, 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 (physics), time. Its SI units are watts per sq ...
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, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
and ''k'' is
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...
.


Four-baryon number flux

The flux of baryons is: \mathbf = n\mathbf where is the number density of
baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...
s in the local rest frame of the baryon fluid (positive values for baryons, negative for antibaryons), and the four-velocity field (of the fluid) as above.


Four-entropy

The four-
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
vector is defined by: \mathbf = s\mathbf + \frac where is the entropy per baryon, and the
absolute temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
, 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 via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
include the following.


Four-current

The electromagnetic four-current (or more correctly a four-current density) is defined by \mathbf = \left( \rho c, \mathbf \right) formed from the current density j and charge density ''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 and the scalar potential . 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, (\boldsymbol \cdot \boldsymbol) \mathbf = 0 * With a four-current source and using the Lorenz gauge condition (\boldsymbol \cdot \mathbf) = 0, (\boldsymbol \cdot \boldsymbol) \mathbf = \mu_0 \mathbf


Waves


Four-frequency

A photonic plane wave can be described by the '' four-frequency'', defined as \mathbf = \nu\left(1 , \hat \right) where 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 (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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 and space are the
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
and angular wave vector , respectively. They form the components of the four-wavevector or wave four-vector: \mathbf = \left(\frac, \vec\right) = \left(\frac, \frac \hat\mathbf\right) \,. The wave four-vector has coherent derived unit of reciprocal meters in the SI. 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, mon ...
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 is the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
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 the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, 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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
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 Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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 \mathbf = (0, \mathbf) where 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 \, \mathbf\, ^2 = -, \mathbf, ^2 = -\hbar^2 s(s + 1) This value is observable and quantized, with 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 In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
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 In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
(the
matrix transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
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 numbers, then the complex conjugate of a + bi is 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 (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
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 \\ ex & = \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) \\ ex & = \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 with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
.


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 In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
). 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: \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 field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of suba ...
. In the Dirac equation and other relativistic wave equations, terms of the form: \begin \mathbf\!\!\!\!/ = P_\alpha \gamma^\alpha &= P_0 \gamma^0 + P_1 \gamma^1 + P_2 \gamma^2 + P_3 \gamma^3 \\ pt&= \dfrac \gamma^0 - p_x \gamma^1 - p_y \gamma^2 - p_z \gamma^3 \\ \end 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 *
Paravector The name paravector is used for the combination of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists. This name was given by J. G. Maks in a doctoral dissertation at Technische Universiteit Delft, Nethe ...
*
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 o ...
*
Wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength) ...


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)