Four-vector

In the theory of relativity, a four-vector or 4-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a Euclidean vector in that four-vectors transform by the Lorentz transformations. The term four-vector tacitly assumes that its components refer to a vector basis. In a standard basis, the components transform between these bases as the space and time coordinate differences, (cΔt, Δx, Δy, Δz) under spatial translations, spatial rotations, spatial and time inversions and boosts (a change by a constant velocity to another inertial reference frame). The set of all such translations, rotations, inversions and boosts (called Poincaré transformations) forms the Poincaré group. The set of rotations, inversions and boosts (Lorentz transformations, described by 4×4 matrices) forms the Lorentz group.

This 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.

Mathematics of four-vectors

Four-vectors in a real-valued basis

A four-vector A is defined as, in various equivalent notations, a vector with a "timelike" component and three "spacelike" components:

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

where the upper index denotes it to be contravariant. Here the standard convention of Latin indices take values for spatial components (so 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 given throughout below), or raising and lowering indices.

It is also customary to represent the bases by column vectors:

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

so that:

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

The relation between the covariant and contravariant coordinates is through the Minkowski metric tensor, η:

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

and in various equivalent notations the covariant components are:

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

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.

Again, the bases can be represented by column vectors:

$\mathbf {e} ^{0}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}\,,\quad \mathbf {e} ^{1}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}}\,,\quad \mathbf {e} ^{2}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}}\,,\quad \mathbf {e} ^{3}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}$

so that:

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

Geometrically, a four-vector can still be interpreted as an arrow. In relativity, the arrows are drawn as part of a spacetime diagram or Minkowski diagram. In this article, four-vectors will be referred to simply as vectors.

Properties

Linearity

Four-vectors have the same linearity properties as Euclidean vectors in three dimensions. They can be added in the usual entrywise way:

$\mathbf {A} +\mathbf {B} =(A^{0},A^{1},A^{2},A^{3})+(B^{0},B^{1},B^{2},B^{3})=(A^{0}+B^{0},A^{1}+B^{1},A^{2}+B^{2},A^{3}+B^{3})$

and similarly scalar multiplication by a scalar λ is defined entrywise by:

$\lambda \mathbf {A} +\mathbf {B} =\lambda (A^{0},A^{1},A^{2},A^{3})=(\lambda A^{0},\lambda A^{1},\lambda A^{2},\lambda A^{3})$

Then subtraction is the inverse operation of addition, defined entrywise by:

$\mathbf {A} +(-1)\mathbf {B} =(A^{0},A^{1},A^{2},A^{3})+(-1)(B^{0},B^{1},B^{2},B^{3})=(A^{0}-B^{0},A^{1}-B^{1},A^{2}-B^{2},A^{3}-B^{3})$

Inner product

See also: spacetime interval

The inner product (also called the scalar product) of two four-vectors A and B is defined, using Einstein notation, as

$\mathbf {A} \cdot \mathbf {B} =A^{\mu }\eta _{\mu \nu }B^{\nu }$

where η is the Minkowski metric. The inner product in this context is also called the Minkowski inner product. For visual clarity, it is convenient to rewrite the definition in matrix form:

$\mathbf {A\cdot B} ={\begin{pmatrix}A^{0}&A^{1}&A^{2}&A^{3}\end{pmatrix}}{\begin{pmatrix}\eta _{00}&\eta _{01}&\eta _{02}&\eta _{03}\\\eta _{10}&\eta _{11}&\eta _{12}&\eta _{13}\\\eta _{20}&\eta _{21}&\eta _{22}&\eta _{23}\\\eta _{30}&\eta _{31}&\eta _{32}&\eta _{33}\end{pmatrix}}{\begin{pmatrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{pmatrix}}$

in which case η_μν above is the entry in row μ and column ν of the Minkowski metric as a square matrix.

The inner product of a four-vector A with itself is the square of the norm of the vector, denoted and defined by:

$\|\mathbf {A} \|^{2}=\mathbf {A\cdot A} =A^{\mu }\eta _{\mu \nu }A^{\nu }$

and intuitively represents (the square of) the length or magnitude of the vector. However, the Minkowski metric is not a Euclidean metric, because it is indefinite (see metric signature). In general, four-vectors can have nonpositive length, contrary to three dimensional vectors in Euclidean space.

(+−−−) signature

In the (+−−−) metric signature, evaluating the summation over indices gives:

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

while in matrix form:

$\mathbf {A\cdot B} ={\begin{pmatrix}A^{0}&A^{1}&A^{2}&A^{3}\end{pmatrix}}{\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}{\begin{pmatrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{pmatrix}}$

It is a recurring theme in special relativity to take the expression

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

in one reference frame, where C is the value of the inner product in this frame, and:

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

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:

$\mathbf {A} \cdot \mathbf {B} =\mathbf {A} '\cdot \mathbf {B} '$

that is:

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

This has the appearance of a "conservation law", but there is no "conservation" involved, since 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, the norm of the vector A is:

$\|\mathbf {A} \|^{2}=(A^{0})^{2}-(A^{1})^{2}-(A^{2})^{2}-(A^{3})^{2}$

With the signature (+−−−), four-vectors may be classified as either spacelike if ||A|| < 0, timelike if ||A|| > 0, and null vectors if ||A|| = 0.

(−+++) signature

Some authors define η with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature:

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

while the matrix form is:

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

Note that in this case, in one frame:

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

while in another:

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

so that:

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

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.

The square of the norm in this signature is:

$\|\mathbf {A} \|^{2}=-(A^{0})^{2}+(A^{1})^{2}+(A^{2})^{2}+(A^{3})^{2}$

With the signature (−+++), four-vectors may be classified as either spacelike if ||A|| > 0, timelike if ||A|| < 0, and null vectors if ||A|| = 0.

Dual vectors

The inner product is often expressed as the effect of the dual vector of one vector on the other:

$\mathbf {A\cdot B} =A^{*}(\mathbf {B} )=A{_{\nu }}B^{\nu }.$

Here the A_νs are the components of the dual vector A* of A in the dual basis and called the covariant coordinates of A, while the original A^ν components are called the contravariant coordinates. Lower and upper indices indicate always covariant and contravariant coordinates, respectively.

Derivatives and differentials

In special relativity (but not general relativity), the derivative 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, dA and divide it by the differential of the scalar, dλ:

${\underset {\text{differential}}{d\mathbf {A} }}={\underset {\text{derivative}}{\frac {d\mathbf {A} }{d\lambda }}}{\underset {\text{differential}}{d\lambda }}$

where the contravariant components are:

$d\mathbf {A} =(dA^{0},dA^{1},dA^{2},dA^{3})$

while the covariant components are:

$d\mathbf {A} =(dA_{0},dA_{1},dA_{2},dA_{3})$

In relativistic mechanics, one often takes the differential of a four-vector and divides by the differential in proper time (see below).

Lorentz transformation

Main article: Lorentz transformation

For two inertial or rotated frames of reference, all four-vectors transform in the same way according to the Lorentz transformation matrix Λ:

$\mathbf {A} '={\boldsymbol {\Lambda }}\mathbf {A} \,\!$

or in index notation:

${A'}^{\mu }=\Lambda ^{\mu }{}_{\nu }A^{\nu }$

in which the matrix Λ has components Λ^μ_ν in row μ and column ν.

Pure rotations about an arbitrary axis

For two frames rotated by a fixed angle θ about an axis defined by the unit vector ${\hat {\mathbf {n} }}$, without any boosts, the matrix Λ has components given by:

$\Lambda _{00}=1$

$\Lambda _{0i}=\Lambda _{i0}=0$

$\Lambda _{ij}=(\delta _{ij}-{\hat {n}}_{i}{\hat {n}}_{j})\cos \theta -\varepsilon _{ijk}{\hat {n}}_{k}\sin \theta +{\hat {n}}_{i}{\hat {n}}_{j}$

where δ_ij is the Kronecker delta, and ε_ijk is the three dimensional Levi-Civita symbol. The spacelike components of 4-vectors rotated, while the time-like 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:

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

Pure boosts in an arbitrary direction

Standard configuration of coordinate systems; for a Lorentz boost in the x-direction.

For two frames moving at constant relative 3-velocity (not 4-velocity) v = (v₁, v₂, v₃) without rotations, the matrix Λ has components given by:

${\begin{aligned}\Lambda _{00}&=\gamma ,\\\Lambda _{0i}&=\Lambda _{i0}=-\gamma \beta _{i},\\\Lambda _{ij}&=\Lambda _{ji}=(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}+\delta _{ij}=(\gamma -1){\dfrac {v_{i}v_{j}}{v^{2}}}+\delta _{ij},\\\end{aligned}}\,\!$

where for convenience the Lorentz factor defined by:

$\gamma ={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\,,$

the relative velocity in units of c:

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

and δ_ij is the Kronecker delta, are used for simplicity. 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;

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

Written in terms of the hyperbolic functions of rapidity ϕ, 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.

Four-vectors in the Pauli basis

A four-vector A can also be defined in using the Pauli matrices as a basis, again in various equivalent notations:

${\begin{aligned}\mathbf {A} &=(A^{0},\,A^{1},\,A^{2},\,A^{3})\\&=A^{0}{\boldsymbol {\sigma }}_{0}+A^{1}{\boldsymbol {\sigma }}_{1}+A^{2}{\boldsymbol {\sigma }}_{2}+A^{3}{\boldsymbol {\sigma }}_{3}\\&=A^{0}{\boldsymbol {\sigma }}_{0}+A^{i}{\boldsymbol {\sigma }}_{i}\\&=A^{\alpha }{\boldsymbol {\sigma }}_{\alpha }\\\end{aligned}}$

or explicitly:

${\begin{aligned}\mathbf {A} &=A^{0}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+A^{1}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}+A^{2}{\begin{pmatrix}0&-i\\i&0\end{pmatrix}}+A^{3}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\\&={\begin{pmatrix}A^{0}+A^{3}&A^{1}-iA^{2}\\A^{1}+iA^{2}&A^{0}-A^{3}\end{pmatrix}}\end{aligned}}$

and in this formulation, the four-vector is a represented as a unitary matrix (the matrix transpose and complex conjugate of the matrix leaves it unchanged), rather than a real-valued column or row vector. The determinant of the matrix is the modulus of the four-vector, so the determinant is an invariant:

${\begin{aligned}|\mathbf {A} |&={\begin{vmatrix}A^{0}+A^{3}&A^{1}-iA^{2}\\A^{1}+iA^{2}&A^{0}-A^{3}\end{vmatrix}}\\&=(A^{0}+A^{3})(A^{0}-A^{3})-(A^{1}-iA^{2})(A^{1}+iA^{2})\\&=(A^{0})^{2}-(A^{1})^{2}-(A^{2})^{2}-(A^{3})^{2}\end{aligned}}$

This idea of using matrices as basis vectors is employed in spacetime algebra and geometric algebra, which are examples of Clifford algebra.

Four-position

A point in Minkowski space is called an "event" and is described in a standard basis by a set of four coordinates such as

$\mathbf {X} =X^{\mu }:=\left(X^{0},X^{1},X^{2},X^{3}\right)=\left(ct,x,y,z\right)$

where $\mu$ = 0, 1, 2, 3, labels the spacetime dimensions and where c is the speed of light. The definition $X^{0}=ct$ 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:

$\Delta X^{\mu }:=\left(c\Delta t,\Delta x,\Delta y,\Delta z\right)$

The position vector is the displacement vector when one of the two events is the origin of the coordinate system. Position vectors are relatively trivial; the general theory of four-vectors is concerned with displacement vectors.

The scalar product of the 4-position with itself is;

$\mathbf {X} \cdot \mathbf {X} =\|\mathbf {X} \|^{2}=X^{\mu }X_{\mu }=(c\tau )^{2}=s^{2}\,\!$

which defines the spacetime interval s and proper time τ in Minkowski spacetime, which are invariant. The scalar product of the differential 4-position with itself is:

$d\mathbf {X} \cdot d\mathbf {X} =\|d\mathbf {X} \|^{2}=dX^{\mu }dX_{\mu }=c^{2}d\tau ^{2}=ds^{2}\,\!$

defining the differential line element ds and differential proper time increment dτ.

Dynamics

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 τ. 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 differential invariant spacetime interval;

$(cd\tau )^{2}=(cdt)^{2}-dr^{2}\,,$

where

$dr^{2}=dx^{2}+dy^{2}+dz^{2}$

is the distance differential of Cartesian coordinates x, y, z measured in some frame. Then dividing by (cdt)² gives:

$\left({\frac {cd\tau }{cdt}}\right)^{2}=1-\left({\frac {1}{c}}{\frac {dr}{dt}}\right)^{2}=1-\left({\frac {u}{c}}\right)^{2}={\frac {1}{\gamma (u)^{2}}}\,,$

where u = dr/dt is the 3-velocity of an object measured in the same frame as the coordinates x, y, z, and coordinate time t, and

$\gamma (u)={\frac {1}{\sqrt {1-(u/c)^{2}}}}$

is the Lorentz factor, so

$dt=\gamma (u)d\tau \,.$

It can also be found from the time transformation in the Lorentz transformations. Important four-vectors in relativity theory can now be defined.

Four-velocity

The four-velocity of an $\mathbf {X} (\tau )$ world line is defined by:

$\mathbf {U} :={\frac {d\mathbf {X} }{d\tau }}={\frac {d\mathbf {X} }{dt}}{\frac {dt}{d\tau }}=\left(\gamma c,\gamma \mathbf {u} \right),$

where, using suffix notation,

$\mathbf {u} =u^{i}={\frac {dX^{i}}{dt}}$

for i = 1, 2, 3. Using the differential of the 4-position, the magnitude of the 4-velocity can be obtained;

${\frac {dX^{\mu }dX_{\mu }}{d\tau ^{2}}}={\frac {dX^{\mu }dX_{\mu }}{d\tau d\tau }}=U^{\mu }U_{\mu }=\|\mathbf {U} \|^{2}=\mathbf {U} \cdot \mathbf {U} =c^{2}\,\!$

in short

$\|\mathbf {U} \|^{2}=U^{\mu }U_{\mu }=c^{2}\,$

The geometric meaning of 4-velocity is the unit vector tangent to the world line in Minkowski space.

Four-acceleration

The four-acceleration is given by:

$\mathbf {A} ={\frac {d\mathbf {U} }{d\tau }}=\left(\gamma {\dot {\gamma }}c,\gamma {\dot {\gamma }}\mathbf {u} +\gamma ^{2}{\dot {\mathbf {u} }}\right).$

Since the magnitude ${\sqrt {|U_{\mu }U^{\mu }|}}$ of $\mathbf {U}$ is a constant, the four acceleration is (pseudo-)orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:

$A^{\mu }U_{\mu }={\frac {dU^{\mu }}{d\tau }}U_{\mu }={\frac {1}{2}}\,{\frac {d}{d\tau }}(U^{\mu }U_{\mu })=0\,$

which is true for all world lines.

The geometric meaning of 4-acceleration is the curvature vector of the world line in Minkowski space.

Four-momentum

The four-momentum for a massive particle is given by:

$\mathbf {P} :=m\mathbf {U} =(\gamma mc,\mathbf {p} )=(\gamma mc,\gamma m\mathbf {u} )$

where m is the invariant mass of the particle and p is the relativistic momentum.

Four-force

The four-force is defined by:

$\mathbf {F} :={\frac {d\mathbf {P} }{d\tau }}.$

For a particle of constant mass, this is equivalent to

$\mathbf {F} =m\mathbf {A} =\left(\gamma {\dot {\gamma }}mc,\gamma \mathbf {f} \right)$

where

$\mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\dot {\gamma }}m\mathbf {u} +\gamma m{\dot {\mathbf {u} }}.$

Thermodynamics

See also: Relativistic heat conduction

Four-heat flux

The 4-heat flux vector field, is essentially similar to the 3d heat flux vector field q:

$\mathbf {Q} =-k\left({\frac {1}{c}}{\frac {\partial T}{\partial t}},\nabla T\right)$

where T is absolute temperature and k is thermal conductivity.

The components are:

$Q_{\alpha }=-k{\frac {\partial T}{\partial x^{\alpha }}}$

in the local frame of the fluid, where ∂_α = ∂/∂x^α is the four-gradient.

Four-baryon number flux

The flux of baryons is:

$\mathbf {S} =n\mathbf {U}$

where n is the number density of baryons in the local rest frame of the baryon fluid (positive values for baryons, negative for antibaryons), and U the 4-velocity field (of the fluid) as above.

Four-entropy

The 4-entropy vector is defined by:

$\mathbf {s} =s\mathbf {S} +{\frac {\mathbf {Q} }{T}}$

where s is the entropy per baryon, and T the absolute temperature, in the local rest frame of the fluid.

Electromagnetism

Examples of four-vectors in electromagnetism include the following.

Four-current

The four-current is defined by

$\mathbf {J} :=\left(\rho c,\mathbf {j} \right)$

formed from the current density j and charge density ρ.

Four-potential

The electromagnetic four-potential defined by

$\mathbf {A} :=\left(\phi /c,\mathbf {a} \right)$

formed from the vector potential a and the scalar potential $\phi \,$. The four-potential is not uniquely determined, because it depends on a choice of gauge.

Four-frequency

A plane electromagnetic wave can be described by the four-frequency defined as

$\mathbf {N} :=\left(\nu ,\nu \mathbf {n} \right)$

where $\nu$ is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that

$N^{\mu }N_{\mu }=\nu ^{2}\left(n^{2}-1\right)=0$

so that the four-frequency is always a null vector.

Four-wavevector

See also: De Broglie relation

A wave packet of nearly monochromatic light can be characterized by the wave vector, or four-wavevector

$\mathbf {K} =\left({\frac {\omega }{c}},\mathbf {k} \right)=2\pi {\frac {\mathbf {N} }{c}}.$

Quantum theory

In relativistic quantum mechanics, the 4-probability current is:

$\mathbf {J} =(\rho c,\mathbf {j} )=(c|\Psi |^{2},\mathbf {j} )$

where ρ is the probability density function corresponding to the time component, in turn Ψ is the wavefunction, and j is the probability current vector.

Physics of four-vectors

One advantage of the four-vector formalism, over that of three-vectors, is illuminated by showing that known relations between energy and matter are contained in the four-momentum vector.

Energy of massive particles

Here, an expression for the total energy of a particle

$E=\gamma mc^{2}\,$

will be derived. The kinetic energy (K) of a particle is defined analogously to the classical definition, namely as

${\frac {dK}{dt}}=\mathbf {f} \cdot \mathbf {u}$

with f as above. Noticing that $F^{\mu }U_{\mu }=0$ and expanding this out we get

$\gamma ^{2}\left(\mathbf {f} \cdot \mathbf {u} -{\dot {\gamma }}mc^{2}\right)=0$

Hence

${\frac {dK}{dt}}=c^{2}{\frac {d\gamma m}{dt}}$

which yields

$K=\gamma mc^{2}+S\,$

for some constant S. When the particle is at rest (u = 0), we take its kinetic energy to be zero (K = 0). This gives

$S=-mc^{2}.\,$

Thus, we interpret the total energy E of the particle as composed of its kinetic energy K and its rest energy m c². Thus, we have

$E=\gamma mc^{2}.\,$

Total energy and invariant mass

We can also derive the energy–momentum relation:

$E^{2}=(pc)^{2}+(mc^{2})^{2}\,\!$

using the four-vector formalism. Using the relation

$E=\gamma mc^{2}$

we can write the four-momentum as

$\mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right).$

Taking the inner product of the four-momentum with itself in two different ways, we obtain the relation

${\frac {E^{2}}{c^{2}}}-p^{2}=P^{\mu }P_{\mu }=m^{2}U^{\mu }U_{\mu }=m^{2}c^{2}$

reducing to

${\frac {E^{2}}{c^{2}}}-p^{2}=m^{2}c^{2}.$

Hence

$E^{2}=p^{2}c^{2}+m^{2}c^{4}.\,$

This last relation is useful in many areas of physics.

See also

• Relativistic mechanics
• paravector
• wave vector
• Dust (relativity) for the number-flux four-vector
• Basic introduction to the mathematics of curved spacetime
• Minkowski space

References

• Rindler, W. Introduction to Special Relativity (2nd edn.) (1991) Clarendon Press Oxford ISBN 0-19-853952-5