Misplaced Pages

Latest revision as of 13:05, 26 October 2024

Part of a series on
Spacetime
Special relativity General relativity
Spacetime concepts Spacetime manifold Equivalence principle Lorentz transformations Minkowski space
General relativity Introduction to general relativity Mathematics of general relativity Einstein field equations
Classical gravity Introduction to gravitation Newton's law of universal gravitation
Relevant mathematics Four-vector Derivations of relativity Spacetime diagrams Differential geometry Curved space Curved spacetime Mathematics of general relativity Spacetime topology
Physics portal  Category
v t e

In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (⁠1/2⁠,⁠1/2⁠) representation. It differs from a Euclidean vector 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 x in spacetime modeled as Minkowski space, a particle's four-momentum p, the amplitude of the electromagnetic four-potential A(x) at a point x 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 Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by

$${\displaystyle X'=\Lambda X,}$$

(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 x_μ, p_μ and A_μ(x). These transform according to the rule

$${\displaystyle X'=\left(\Lambda ^{-1}\right)^{\textrm {T}}X,}$$

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. 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 X′ = Π(Λ)X, where Π(Λ) is a 4×4 matrix other than Λ. 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.

Notation

The notations in this article are: lowercase bold for three-dimensional vectors, hats for three-dimensional unit vectors, capital bold for four dimensional vectors (except for the four-gradient), 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:

$${\displaystyle {\begin{aligned}\mathbf {A} &=\left(A^{0},\,A^{1},\,A^{2},\,A^{3}\right)\\&=A^{0}\mathbf {E} _{0}+A^{1}\mathbf {E} _{1}+A^{2}\mathbf {E} _{2}+A^{3}\mathbf {E} _{3}\\&=A^{0}\mathbf {E} _{0}+A^{i}\mathbf {E} _{i}\\&=A^{\alpha }\mathbf {E} _{\alpha }\end{aligned}}}$$

where A is the magnitude component and E_α is the basis vector component; note that both are necessary to make a vector, and that when A 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, A, A are often Cartesian basis and components:

$${\displaystyle {\begin{aligned}\mathbf {A} &=\left(A_{t},\,A_{x},\,A_{y},\,A_{z}\right)\\&=A_{t}\mathbf {E} _{t}+A_{x}\mathbf {E} _{x}+A_{y}\mathbf {E} _{y}+A_{z}\mathbf {E} _{z}\\\end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}\mathbf {A} &=\left(A_{t},\,A_{r},\,A_{\theta },\,A_{\phi }\right)\\&=A_{t}\mathbf {E} _{t}+A_{r}\mathbf {E} _{r}+A_{\theta }\mathbf {E} _{\theta }+A_{\phi }\mathbf {E} _{\phi }\\\end{aligned}}}$$

or cylindrical polar coordinates,

$${\displaystyle {\begin{aligned}\mathbf {A} &=(A_{t},\,A_{r},\,A_{\theta },\,A_{z})\\&=A_{t}\mathbf {E} _{t}+A_{r}\mathbf {E} _{r}+A_{\theta }\mathbf {E} _{\theta }+A_{z}\mathbf {E} _{z}\\\end{aligned}}}$$

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:

$${\displaystyle \mathbf {E} _{0}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}\,,\quad \mathbf {E} _{1}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}}\,,\quad \mathbf {E} _{2}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}}\,,\quad \mathbf {E} _{3}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}}$$

so that:

$${\displaystyle \mathbf {A} ={\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}}$$

The relation between the covariant and contravariant coordinates is through the Minkowski metric tensor (referred to as the metric), η which raises and lowers indices as follows:

$${\displaystyle A_{\mu }=\eta _{\mu \nu }A^{\nu }\,,}$$

and in various equivalent notations the covariant components are:

$${\displaystyle {\begin{aligned}\mathbf {A} &=(A_{0},\,A_{1},\,A_{2},\,A_{3})\\&=A_{0}\mathbf {E} ^{0}+A_{1}\mathbf {E} ^{1}+A_{2}\mathbf {E} ^{2}+A_{3}\mathbf {E} ^{3}\\&=A_{0}\mathbf {E} ^{0}+A_{i}\mathbf {E} ^{i}\\&=A_{\alpha }\mathbf {E} ^{\alpha }\\\end{aligned}}}$$

where the lowered index indicates it to be covariant. Often the metric is diagonal, as is the case for orthogonal coordinates (see line element), but not in general curvilinear coordinates.

The bases can be represented by row vectors:

$${\displaystyle \mathbf {E} ^{0}={\begin{pmatrix}1&0&0&0\end{pmatrix}}\,,\quad \mathbf {E} ^{1}={\begin{pmatrix}0&1&0&0\end{pmatrix}}\,,\quad \mathbf {E} ^{2}={\begin{pmatrix}0&0&1&0\end{pmatrix}}\,,\quad \mathbf {E} ^{3}={\begin{pmatrix}0&0&0&1\end{pmatrix}}}$$ so that: $${\displaystyle \mathbf {A} ={\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}}}$$

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 matrix Λ: $${\displaystyle \mathbf {A} '={\boldsymbol {\Lambda }}\mathbf {A} }$$

In index notation, the contravariant and covariant components transform according to, respectively: $${\displaystyle {A'}^{\mu }=\Lambda ^{\mu }{}_{\nu }A^{\nu }\,,\quad {A'}_{\mu }=\Lambda _{\mu }{}^{\nu }A_{\nu }}$$ in which the matrix Λ has components Λ_ν in row μ and column ν, and the matrix (Λ) has components Λ_μ in row μ and column ν.

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 θ about an axis defined by the unit vector:

$${\displaystyle {\hat {\mathbf {n} }}=\left({\hat {n}}_{1},{\hat {n}}_{2},{\hat {n}}_{3}\right)\,,}$$

without any boosts, the matrix Λ has components given by:

$${\displaystyle {\begin{aligned}\Lambda _{00}&=1\\\Lambda _{0i}=\Lambda _{i0}&=0\\\Lambda _{ij}&=\left(\delta _{ij}-{\hat {n}}_{i}{\hat {n}}_{j}\right)\cos \theta -\varepsilon _{ijk}{\hat {n}}_{k}\sin \theta +{\hat {n}}_{i}{\hat {n}}_{j}\end{aligned}}}$$

where δ_ij is the Kronecker delta, and ε_ijk 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:

$${\displaystyle {\begin{pmatrix}{A'}^{0}\\{A'}^{1}\\{A'}^{2}\\{A'}^{3}\end{pmatrix}}={\begin{pmatrix}1&0&0&0\\0&\cos \theta &-\sin \theta &0\\0&\sin \theta &\cos \theta &0\\0&0&0&1\\\end{pmatrix}}{\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}\ .}$$

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:

$${\displaystyle {\boldsymbol {\beta }}=(\beta _{1},\,\beta _{2},\,\beta _{3})={\frac {1}{c}}(v_{1},\,v_{2},\,v_{3})={\frac {1}{c}}\mathbf {v} \,.}$$

Then without rotations, the matrix Λ has components given by: $${\displaystyle {\begin{aligned}\Lambda _{00}&=\gamma ,\\\Lambda _{0i}=\Lambda _{i0}&=-\gamma \beta _{i},\\\Lambda _{ij}=\Lambda _{ji}&=(\gamma -1){\frac {\beta _{i}\beta _{j}}{\beta ^{2}}}+\delta _{ij}=(\gamma -1){\frac {v_{i}v_{j}}{v^{2}}}+\delta _{ij},\\\end{aligned}}}$$ where the Lorentz factor is defined by: $${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\boldsymbol {\beta }}\cdot {\boldsymbol {\beta }}}}}\,,}$$ and δ_ij 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;

$${\displaystyle {\begin{pmatrix}A'^{0}\\A'^{1}\\A'^{2}\\A'^{3}\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi &0&0\\-\sinh \phi &\cosh \phi &0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}{\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}}$$

Where the rapidity ϕ expression has been used, written in terms of the hyperbolic functions: $${\displaystyle \gamma =\cosh \phi }$$

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 vectors in three dimensions. They can be added in the usual entrywise way: $${\displaystyle \mathbf {A} +\mathbf {B} =\left(A^{0},A^{1},A^{2},A^{3}\right)+\left(B^{0},B^{1},B^{2},B^{3}\right)=\left(A^{0}+B^{0},A^{1}+B^{1},A^{2}+B^{2},A^{3}+B^{3}\right)}$$ and similarly scalar multiplication by a scalar λ is defined entrywise by: $${\displaystyle \lambda \mathbf {A} =\lambda \left(A^{0},A^{1},A^{2},A^{3}\right)=\left(\lambda A^{0},\lambda A^{1},\lambda A^{2},\lambda A^{3}\right)}$$

Then subtraction is the inverse operation of addition, defined entrywise by: $${\displaystyle \mathbf {A} +(-1)\mathbf {B} =\left(A^{0},A^{1},A^{2},A^{3}\right)+(-1)\left(B^{0},B^{1},B^{2},B^{3}\right)=\left(A^{0}-B^{0},A^{1}-B^{1},A^{2}-B^{2},A^{3}-B^{3}\right)}$$

Minkowski tensor

Applying the Minkowski tensor η_μν to two four-vectors A and B, writing the result in dot product notation, we have, using Einstein notation: $${\displaystyle \mathbf {A} \cdot \mathbf {B} =A^{\mu }B^{\nu }\mathbf {E} _{\mu }\cdot \mathbf {E} _{\nu }=A^{\mu }\eta _{\mu \nu }B^{\nu }}$$

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: $${\displaystyle \mathbf {A\cdot B} ={\begin{pmatrix}A^{0}&A^{1}&A^{2}&A^{3}\end{pmatrix}}{\begin{pmatrix}\eta _{00}&\eta _{01}&\eta _{02}&\eta _{03}\\\eta _{10}&\eta _{11}&\eta _{12}&\eta _{13}\\\eta _{20}&\eta _{21}&\eta _{22}&\eta _{23}\\\eta _{30}&\eta _{31}&\eta _{32}&\eta _{33}\end{pmatrix}}{\begin{pmatrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{pmatrix}}}$$ in which case η_μν above is the entry in row μ and column ν of the Minkowski metric as a square matrix. The Minkowski metric is not a Euclidean metric, because it is indefinite (see metric signature). A number of other expressions can be used because the metric tensor can raise and lower the components of A or B. For contra/co-variant components of A and co/contra-variant components of B, we have: $${\displaystyle \mathbf {A} \cdot \mathbf {B} =A^{\mu }\eta _{\mu \nu }B^{\nu }=A_{\nu }B^{\nu }=A^{\mu }B_{\mu }}$$ so in the matrix notation: $${\displaystyle \mathbf {A} \cdot \mathbf {B} ={\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}}{\begin{pmatrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{pmatrix}}={\begin{pmatrix}B_{0}&B_{1}&B_{2}&B_{3}\end{pmatrix}}{\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}}$$ while for A and B each in covariant components: $${\displaystyle \mathbf {A} \cdot \mathbf {B} =A_{\mu }\eta ^{\mu \nu }B_{\nu }}$$ with a similar matrix expression to the above.

Applying the Minkowski tensor to a four-vector A with itself we get: $${\displaystyle \mathbf {A\cdot A} =A^{\mu }\eta _{\mu \nu }A^{\nu }}$$ 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 (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: $${\displaystyle \mathbf {A} \cdot \mathbf {B} =A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}}$$ while in matrix form: $${\displaystyle \mathbf {A\cdot B} ={\begin{pmatrix}A^{0}&A^{1}&A^{2}&A^{3}\end{pmatrix}}{\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}{\begin{pmatrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{pmatrix}}}$$

It is a recurring theme in special relativity to take the expression $${\displaystyle \mathbf {A} \cdot \mathbf {B} =A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}=C}$$ in one reference frame, where C is the value of the inner product in this frame, and: $${\displaystyle \mathbf {A} '\cdot \mathbf {B} '={A'}^{0}{B'}^{0}-{A'}^{1}{B'}^{1}-{A'}^{2}{B'}^{2}-{A'}^{3}{B'}^{3}=C'}$$ 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: $${\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {A} '\cdot \mathbf {B} '}$$ that is: $${\displaystyle C=A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}={A'}^{0}{B'}^{0}-{A'}^{1}{B'}^{1}-{A'}^{2}{B'}^{2}-{A'}^{3}{B'}^{3}}$$

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, 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 vector (see also below).

In this signature we have: $${\displaystyle \mathbf {A\cdot A} =\left(A^{0}\right)^{2}-\left(A^{1}\right)^{2}-\left(A^{2}\right)^{2}-\left(A^{3}\right)^{2}}$$

With the signature (+−−−), four-vectors may be classified as either spacelike if ${\displaystyle \mathbf {A\cdot A} <0}$, timelike if ${\displaystyle \mathbf {A\cdot A} >0}$, and null vectors if ${\displaystyle \mathbf {A\cdot A} =0}$.

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:

$${\displaystyle \mathbf {A\cdot B} =-A^{0}B^{0}+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3}}$$

while the matrix form is:

$${\displaystyle \mathbf {A\cdot B} =\left({\begin{matrix}A^{0}&A^{1}&A^{2}&A^{3}\end{matrix}}\right)\left({\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\left({\begin{matrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{matrix}}\right)}$$

Note that in this case, in one frame:

$${\displaystyle \mathbf {A} \cdot \mathbf {B} =-A^{0}B^{0}+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3}=-C}$$

while in another:

$${\displaystyle \mathbf {A} '\cdot \mathbf {B} '=-{A'}^{0}{B'}^{0}+{A'}^{1}{B'}^{1}+{A'}^{2}{B'}^{2}+{A'}^{3}{B'}^{3}=-C'}$$

so that:

$${\displaystyle -C=-A^{0}B^{0}+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3}=-{A'}^{0}{B'}^{0}+{A'}^{1}{B'}^{1}+{A'}^{2}{B'}^{2}+{A'}^{3}{B'}^{3}}$$

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:

$${\displaystyle \mathbf {A\cdot A} =-\left(A^{0}\right)^{2}+\left(A^{1}\right)^{2}+\left(A^{2}\right)^{2}+\left(A^{3}\right)^{2}}$$

With the signature (−+++), four-vectors may be classified as either spacelike if ${\displaystyle \mathbf {A\cdot A} >0}$, timelike if ${\displaystyle \mathbf {A\cdot A} <0}$, and null if ${\displaystyle \mathbf {A\cdot A} =0}$.