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Inspecial relativity, afour-vector (or4-vector, sometimesLorentz vector)[1] is an object with four components, which transform in a specific way underLorentz transformations. Specifically, a four-vector is an element of a four-dimensionalvector space considered as arepresentation space of thestandard representation of theLorentz group, the (1/2,1/2) representation. It differs from aEuclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which includespatial rotations andboosts (a change by a constant velocity to anotherinertial reference frame).[2]: ch1
Four-vectors describe, for instance, positionxμ in spacetime modeled asMinkowski space, a particle'sfour-momentumpμ, the amplitude of theelectromagnetic four-potentialAμ(x) at a pointx in spacetime, and the elements of the subspace spanned by thegamma matrices inside theDirac algebra.
The Lorentz group may be represented by 4×4 matricesΛ. The action of a Lorentz transformation on a generalcontravariant four-vectorX (like the examples above), regarded as a column vector withCartesian coordinates with respect to aninertial frame in the entries, is given by
(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 correspondingcovariant vectorsxμ,pμ andAμ(x). These transform according to the rule
whereT denotes thematrix transpose. This rule is different from the above rule. It corresponds to thedual representation of the standard representation. However, for the Lorentz group the dual of any representation isequivalent 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 isnot a four-vector, seebispinor. 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 readsX′ = Π(Λ)X, whereΠ(Λ) is a 4×4 matrix other thanΛ. Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These includescalars,spinors,tensors and spinor-tensors.
The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends togeneral relativity, some of the results stated in this article require modification in general relativity.
The notations in this article are: lowercase bold forthree-dimensional vectors, hats for three-dimensionalunit vectors, capital bold forfour dimensional vectors (except for the four-gradient), andtensor index notation.
Afour-vectorA is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations:[3]
whereAα is the magnitude component andEα is thebasis vector component; note that both are necessary to make a vector, and that whenAα is seen alone, it refers strictly to thecomponents of the vector.
The upper indices indicatecontravariant components. Here the standard convention is that Latin indices take values for spatial components, so thati = 1, 2, 3, and Greek indices take values for spaceand time components, soα = 0, 1, 2, 3, used with thesummation 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), orraising and lowering indices.
In special relativity, the spacelike basisE1,E2,E3 and componentsA1,A2,A3 are oftenCartesian basis and components:
although, of course, any other basis and components may be used, such asspherical polar coordinates
orcylindrical polar coordinates,
or any otherorthogonal coordinates, or even generalcurvilinear 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 ofMinkowski diagram (also calledspacetime diagram). In this article, four-vectors will be referred to simply as vectors.
It is also customary to represent the bases bycolumn vectors:
so that:
The relation between thecovariant and contravariant coordinates is through theMinkowskimetric tensor (referred to as the metric),η whichraises and lowers indices as follows:
and in various equivalent notations the covariant components are:
where the lowered index indicates it to becovariant. Often the metric is diagonal, as is the case fororthogonal coordinates (seeline element), but not in generalcurvilinear coordinates.
The bases can be represented byrow vectors:
so that:
The motivation for the above conventions are that the inner product is a scalar, see below for details.
Given two inertial or rotatedframes of reference, a four-vector is defined as a quantity which transforms according to theLorentz transformation matrix Λ:
In index notation, the contravariant and covariant components transform according to, respectively:in which the matrixΛ has componentsΛμν in row μ and column ν, and the matrix(Λ−1)T has componentsΛμν in row μ and column ν.
For background on the nature of this transformation definition, seetensor. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; seespecial relativity.
For two frames rotated by a fixed angleθ about an axis defined by theunit vector:
without any boosts, the matrixΛ has components given by:[4]
whereδij is theKronecker delta, andεijk is thethree-dimensionalLevi-Civita symbol. The spacelike components of four-vectors are rotated, while the timelike components remain unchanged.
For the case of rotations about thez-axis only, the spacelike part of the Lorentz matrix reduces to therotation matrix about thez-axis:
For two frames moving at constant relative three-velocityv (not four-velocity,see below), it is convenient to denote and define the relative velocity in units ofc by:
Then without rotations, the matrixΛ has components given by:[5]where theLorentz factor is defined by:andδij is theKronecker 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 thex-direction only, the matrix reduces to;[6][7]
Where therapidityϕ expression has been used, written in terms of thehyperbolic functions:
This Lorentz matrix illustrates the boost to be ahyperbolic rotation in four dimensional spacetime, analogous to the circular rotation above in three-dimensional space.
Four-vectors have the samelinearity properties asEuclidean vectors inthree dimensions. They can be added in the usual entrywise way:and similarlyscalar multiplication by ascalarλ is defined entrywise by:
Then subtraction is the inverse operation of addition, defined entrywise by:
Applying theMinkowski tensorημν to two four-vectorsA andB, writing the result indot product notation, we have, usingEinstein 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 inmatrix form:in which caseημν above is the entry in rowμ and columnν of the Minkowski metric as a square matrix. The Minkowski metric is not aEuclidean metric, because it is indefinite (seemetric signature). A number of other expressions can be used because the metric tensor can raise and lower the components ofA orB. For contra/co-variant components ofA and co/contra-variant components ofB, we have:so in the matrix notation:while forA andB each in covariant components:with a similar matrix expression to the above.
Applying the Minkowski tensor to a four-vectorA 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 thestandard 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.
The (+−−−)metric signature is sometimes called the "mostly minus" convention, or the "west coast" convention.
In the (+−−−)metric signature, evaluating thesummation over indices gives:while in matrix form:
It is a recurring theme in special relativity to take the expressionin onereference frame, whereC is the value of the inner product in this frame, and:in another frame, in whichC′ is the value of the inner product in this frame. Then since the inner 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 inner product is that for any two four-vectors, its value isinvariant 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 andA′ are connected by aLorentz transformation, and similarly forB andB′, 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 theenergy-momentum relation derived from thefour-momentum vector (see also below).
In this signature we have:
With the signature (+−−−), four-vectors may be classified as eitherspacelike if,timelike if, andnull vectors if.
The (-+++)metric signature is sometimes called the "east coast" convention.
Some authors defineη with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature:
while the matrix form is:
Note that in this case, in one frame:
while in another:
so that:
which is equivalent to the above expression forC in terms ofA andB. 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:
With the signature (−+++), four-vectors may be classified as eitherspacelike if,timelike if, andnull if.
Applying the Minkowski tensor is often expressed as the effect of thedual vector of one vector on the other:
Here theAνs are the components of the dual vectorA* ofA in thedual basis and called thecovariant coordinates ofA, while the originalAν components are called thecontravariant coordinates.
In special relativity (but not general relativity), thederivative of a four-vector with respect to a scalarλ (invariant) is itself a four-vector. It is also useful to take thedifferential of the four-vector,dA and divide it by the differential of the scalar,dλ:
where the contravariant components are:
while the covariant components are:
In relativistic mechanics, one often takes the differential of a four-vector and divides by the differential inproper time (see below).
A point inMinkowski space is a time and spatial position, called an "event", or sometimes theposition four-vector orfour-position or4-position, described in some reference frame by a set of four coordinates:
wherer is thethree-dimensional spaceposition vector. Ifr is a function of coordinate timet in the same frame, i.e.r =r(t), this corresponds to a sequence of events ast varies. The definitionR0 =ct ensures that all the coordinates have the samedimension (oflength) and units (in theSI, meters).[8][9][10][11] These coordinates are the components of theposition four-vector for the event.
Thedisplacement four-vector is defined to be an "arrow" linking two events:
For thedifferential four-position on a world line we have, usinga norm notation:
defining the differentialline element ds and differential proper time increment dτ, but this "norm" is also:
so that:
When considering physical phenomena, differential equations arise naturally; however, when considering space andtime 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 theproper time. 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 thecoordinate timet of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by (cdt)2 to obtain:
whereu =dr/dt is the coordinate 3-velocity of an object measured in the same frame as the coordinatesx,y,z, andcoordinate timet, and
is theLorentz factor. This provides a useful relation between the differentials in coordinate time and proper time:
This relation can also be found from the time transformation in theLorentz transformations.
Important four-vectors in relativity theory can be defined by applying this differential.
Considering thatpartial derivatives arelinear operators, one can form afour-gradient from the partialtime derivative∂/∂t and the spatialgradient ∇. Using the standard basis, in index and abbreviated notations, the contravariant components are:
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:
Since this is an operator, it doesn't have a "length", but evaluating the inner product of the operator with itself gives another operator:
called theD'Alembert operator.
Thefour-velocity of a particle is defined by:
Geometrically,U is a normalized vector tangent to theworld line of the particle. Using the differential of the four-position, the magnitude of the four-velocity can be obtained:
in short, the magnitude of the four-velocity for any object is always a fixed constant:
The norm is also:
so that:
which reduces to the definition of theLorentz factor.
Units of four-velocity are m/s inSI and 1 in thegeometrized unit system. Four-velocity is a contravariant vector.
Thefour-acceleration is given by:
wherea =du/dt is the coordinate 3-acceleration. Since the magnitude ofU 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:
which is true for all world lines. The geometric meaning of four-acceleration is thecurvature vector of the world line in Minkowski space.
For a massive particle ofrest mass (orinvariant mass)m0, thefour-momentum is given by:
where the total energy of the moving particle is:
and the totalrelativistic momentum is:
Taking the inner product of the four-momentum with itself:
and also:
which leads to theenergy–momentum relation:
This last relation is useful inrelativistic mechanics, essential inrelativistic quantum mechanics andrelativistic quantum field theory, all with applications toparticle physics.
Thefour-force acting on a particle is defined analogously to the 3-force as the time derivative of 3-momentum inNewton's second law:
whereP is thepower transferred to move the particle, andf is the 3-force acting on the particle. For a particle of constant invariant massm0, this is equivalent to
An invariant derived from the four-force is:
from the above result.
The four-heat flux vector field, is essentially similar to the 3dheat flux vector fieldq, in the local frame of the fluid:[12]
whereT isabsolute temperature andk isthermal conductivity.
The flux of baryons is:[13]wheren is thenumber density ofbaryons in the localrest frame of the baryon fluid (positive values for baryons, negative forantibaryons), andU thefour-velocity field (of the fluid) as above.
The four-entropy vector is defined by:[14]wheres is the entropy per baryon, andT theabsolute temperature, in the local rest frame of the fluid.[15]
Examples of four-vectors inelectromagnetism include the following.
The electromagneticfour-current (or more correctly a four-current density)[16] is defined byformed from thecurrent densityj andcharge densityρ.
Theelectromagnetic four-potential (or more correctly a four-EM vector potential) defined byformed from thevector potentiala and the scalar potentialϕ.
The four-potential is not uniquely determined, because it depends on a choice ofgauge.
In thewave equation for the electromagnetic field:
A photonicplane wave can be described by thefour-frequency, defined as
whereν is the frequency of the wave and is aunit vector in the travel direction of the wave. Now:
so the four-frequency of a photon is always a null vector.
The quantities reciprocal to timet and spacer are theangular frequencyω andangular wave vectork, respectively. They form the components of thefour-wavevector orwave four-vector:
The wave four-vector hascoherent derived unit ofreciprocal meters in the SI.[17]
A wave packet of nearlymonochromatic light can be described by:
The de Broglie relations then showed that four-wavevector applied tomatter waves as well as to light waves:yielding and, whereħ is thePlanck constant divided by2π .
The square of the norm is:and by the de Broglie relation:we have the matter wave analogue of the energy–momentum relation:
Note that for massless particles, in which casem0 = 0, we have:or‖k‖ =ω/c . Note this is consistent with the above case; for photons with a 3-wavevector of modulusω / c , in the direction of wave propagation defined by the unit vector
Inquantum mechanics, the four-probability current or probability four-current is analogous to theelectromagnetic four-current:[18]whereρ is theprobability density function corresponding to the time component, andj is theprobability 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. Inrelativistic quantum mechanics andquantum field theory, it is not always possible to find a current, particularly when interactions are involved.
Replacing the energy by theenergy operator and the momentum by themomentum operator in the four-momentum, one obtains thefour-momentum operator, used inrelativistic wave equations.
Thefour-spin of a particle is defined in the rest frame of a particle to bewheres is thespin 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
This value is observable and quantized, withs thespin quantum number (not the magnitude of the spin vector).
A four-vectorA can also be defined in using thePauli matrices as abasis, again in various equivalent notations:[19]or explicitly:and in this formulation, the four-vector is represented as aHermitian matrix (thematrix transpose andcomplex conjugate of the matrix leaves it unchanged), rather than a real-valued column or row vector. Thedeterminant of the matrix is the modulus of the four-vector, so the determinant is an invariant:
This idea of using the Pauli matrices asbasis vectors is employed in thealgebra of physical space, an example of aClifford algebra.
Inspacetime algebra, another example of Clifford algebra, thegamma matrices can also form abasis. (They are also called the Dirac matrices, owing to their appearance in theDirac equation). There is more than one way to express the gamma matrices, detailed in that main article.
TheFeynman slash notation is a shorthand for a four-vectorA contracted with the gamma matrices:
The four-momentum contracted with the gamma matrices is an important case inrelativistic quantum mechanics andrelativistic quantum field theory. In the Dirac equation and otherrelativistic wave equations, terms of the form:appear, in which the energyE and momentum components(px,py,pz) are replaced by their respectiveoperators.
