Four-vector

In special relativity, a four-vector is an element of a four-dimensional vector space object with four components, which transform under Lorentz transformations with respect to a change of basis. Its magnitude is determined by an indefinite quadratic form, the preservation of which defines the Lorentz transformations, which include spatial rotations and boosts.
Four-vectors describe, for instance, position in spacetime modeled as Minkowski space, a particle's four-momentum, 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 a set of matrices. The action of a Lorentz transformation on a general contravariant four-vector , regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by
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
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. 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, tensors and spinor-tensors.
The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity.
In the standard configuration, where the primed frame has speed along the positive x-axis, the transformation of four-vectors is:
or
depending on convention.

Notation

The notations in this article are: lowercase bold for three-dimensional vectors, hats for three-dimensional unit vectors, capital bold for four dimensional vectors, 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:
where is the component multiplier and is the basis vector; note that both are necessary to make a vector, and that when 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, and Greek indices take values for time and space components, so, used with the summation convention. The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in scalar products, or raising and lowering indices.
In special relativity, the spacelike basis,, and components,, are often Cartesian basis and components:
although, of course, any other basis and components may be used, such as spherical polar coordinates
or cylindrical polar coordinates,
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. In this article, four-vectors will be referred to simply as vectors.
It is also customary to represent the bases by column vectors:
so that:
The relation between the covariant and contravariant coordinates is through the Minkowski metric tensor, which raises and lowers indices as follows:
and in various equivalent notations the covariant components are:
where the lowered index indicates it to be covariant. Often the metric is diagonal, as is the case for orthogonal coordinates, but not in general curvilinear coordinates.
The bases can be represented by row vectors:
so that:
The motivation for the above conventions are that the scalar product is a scalar, [|see below] for details.

Lorentz transformation

Given two inertial or rotated frames of reference, a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix :
In index notation, the contravariant and covariant components transform according to, respectively:
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. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see special relativity.

Pure rotations about an arbitrary axis

For two frames rotated by a fixed angle about an axis defined by the unit vector:
without any boosts, the matrix has components given by:
where is the Kronecker delta, and is the three-dimensional Levi-Civita symbol. The spacelike components of four-vectors are rotated, while the timelike components remain unchanged.
For the case of rotations about the z-axis only, the spacelike part of the Lorentz matrix reduces to the rotation matrix about the z-axis:

Pure boosts in any direction

For two frames moving at constant relative three-velocity , it is convenient to denote and define the relative velocity in units of by:
Then without rotations, the matrix has components given by:
where the Lorentz factor is defined by:
and is the Kronecker delta. Contrary to the case for pure rotations, the spacelike and timelike components are mixed together under boosts.
For the case of a boost in the x-direction only, the matrix reduces to;
Where the rapidity expression has been used, written in terms of the hyperbolic functions:
This Lorentz matrix illustrates the boost to be a hyperbolic rotation in four dimensional spacetime, analogous to the circular rotation above in three-dimensional space.

Properties

Linearity

Four-vectors have the same linearity properties as Euclidean vectors in three dimensions. They can be added in the usual entrywise way:
and similarly scalar multiplication by a scalar λ is defined entrywise by:
Then subtraction is the inverse operation of addition, defined entrywise by:

Minkowski tensor

Applying the Minkowski tensor to two four-vectors and, writing the result in dot product notation, we have, using Einstein notation:
in special relativity. The dot product of the basis vectors is the Minkowski metric, as opposed to the Kronecker delta as in Euclidean space. It is convenient to rewrite the definition in matrix form:
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. 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:
so in the matrix notation:
while for and each in covariant components:
with a similar matrix expression to the above.
Applying the Minkowski tensor to a four-vector with itself we get:
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. 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:
while in matrix form:
It is a recurring theme in special relativity to take the expression
in one reference frame, where C is the value of the scalar product in this frame, and:
in another frame, in which C′ is the value of the scalar product in this frame. Then since the scalar product is an invariant, these must be equal:
that is:
Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a "conservation law", but there is no "conservation" involved. The primary significance of the Minkowski scalar product is that for any two four-vectors, its value is invariant for all observers; a change of coordinates does not result in a change in value of the scalar product. The components of the four-vectors change from one frame to another; A and A′ are connected by a Lorentz transformation, and similarly for B and B′, although the scalar products are the same in all frames. Nevertheless, this type of expression is exploited in relativistic calculations on a par with conservation laws, since the magnitudes of components can be determined without explicitly performing any Lorentz transformations. A particular example is with energy and momentum in the energy-momentum relation derived from the four-momentum vector.
In this signature we have:
With the signature, four-vectors may be classified as either spacelike if, timelike if, and null vectors if.