Fourvector
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In special relativity, a fourvector is an object with four in general complex components that transform in a specific way under Lorentz transformations. Specifically, a fourvector is an element of a 4dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations. They include spatial rotations, boosts (a change by a constant velocity to another inertial reference frame), and temporal and spatial inversions.
Fourvectors describe, for instance, position x^{μ} in spacetime modeled as Minkowski space, a particles 4momentum p^{μ}, the amplitude of the electromagnetic fourpotential A^{μ}(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 fourvector 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
(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
where T 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 fourvectors as well.
For an example of a wellbehaved fourcomponent object in special relativity that is not a fourvector, 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 wellbehaved under Lorentz transformations. These include scalars, spinors, tensors and spinortensors.
The article considers fourvectors in the context of special relativity. Although the concept of fourvectors also extends to general relativity, some of the results stated in this article require modification in general relativity.
Contents
Notation
The notations in this article are: lowercase bold for threedimensional vectors, hats for threedimensional unit vectors, capital bold for four dimensional vectors (except for the fourgradient), and tensor index notation.
Fourvector algebra
Fourvectors in a realvalued basis
A fourvector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations:^{[1]}
The upper indices indicate contravariant components. Here the standard convention 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 E_{1}, E_{2}, E_{3} and components A^{1}, A^{2}, A^{3} 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 fourvector 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, fourvectors 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, η 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 (see line element), 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 inner product is a scalar, see below for details.
Lorentz transformation
Given two inertial or rotated frames of reference, a fourvector is defined as a quantity which transforms according to the Lorentz transformation matrix Λ:
In index notation, the contravariant and covariant components transform according to, respectively:
in which the matrix Λ has components Λ^{μ}_{ν} in row μ and column ν, and the inverse matrix Λ^{−1} has components Λ_{μ}^{ν} in row μ and column ν.
For background on the nature of this transformation definition, see tensor. All fourvectors transform in the same way, and this can be generalized to fourdimensional 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:
without any boosts, the matrix Λ has components given by:^{[2]}
where δ_{ij} is the Kronecker delta, and ε_{ijk} is the threedimensional LeviCivita symbol. The spacelike components of 4vectors are rotated, while the timelike components remain unchanged.
For the case of rotations about the zaxis only, the spacelike part of the Lorentz matrix reduces to the rotation matrix about the zaxis:
Pure boosts in an arbitrary direction
For two frames moving at constant relative 3velocity v (not 4velocity, see below), it is convenient to denote and define the relative velocity in units of c by:
Then without rotations, the matrix Λ has components given by:^{[3]}
where the Lorentz factor is defined by:
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 xdirection only, the matrix reduces to;^{[4]}^{[5]}
Where the rapidity ϕ 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 threedimensional space.
Properties
Linearity
Fourvectors 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 η to two fourvectors A and B, writing the result in dot product notation, we have, using Einstein notation:
It is convenient to rewrite the definition in matrix form:
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 raises and lowers the components of A and B. For contra/covariant components of A and co/contravariant components of B, we have:
so in the matrix notation:
while for A and B each in covariant components:
with a similar matrix expression to the above.
Applying the Minkowski tensor to a fourvector A 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 (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:
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 inner product in this frame, and:
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:
that is:
Considering that physical quantities in relativity are fourvectors, 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 fourvectors, 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 fourvectors 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 energymomentum relation derived from the fourmomentum vector (see also below).
In this signature we have:
With the signature (+−−−), fourvectors may be classified as either spacelike if , timelike if , and null vectors if .
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:
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 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 fourvector components are signs, therefore the signs depend on which sign convention is used.
We have:
With the signature (−+++), fourvectors may be classified as either spacelike if , timelike if , and null vectors if ..
Dual vectors
Applying the Minkowski tensor is often expressed as the effect of the dual vector of one vector on the other:
Here the A_{ν}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 A^{ν} components are called the contravariant coordinates.
Fourvector calculus
Derivatives and differentials
In special relativity (but not general relativity), the derivative of a fourvector with respect to a scalar λ (invariant) is itself a fourvector. It is also useful to take the differential of the fourvector, 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 fourvector and divides by the differential in proper time (see below).
Fundamental fourvectors
Fourposition
A point in Minkowski space is a time and spatial position, called an "event", or sometimes the position 4vector or 4position, described in some reference frame by a set of four coordinates:
where r is the threedimensional space position vector. 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 units (of distance).^{[6]}^{[7]}^{[8]} These coordinates are the components of the position fourvector for the event. The displacement fourvector is defined to be an "arrow" linking two events:
For the differential 4position on a world line we have, using a norm notation:
defining the differential line 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 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 propertimederivative of any fourvector is itself a fourvector. It is then important to find a relation between this propertimederivative 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:
where u = dr/dt is the coordinate 3velocity of an object measured in the same frame as the coordinates x, y, z, and coordinate time t, and
is the Lorentz 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 the Lorentz transformations. Important fourvectors in relativity theory can be defined by dividing by this differential.
Fourgradient
Considering that partial derivatives are linear operators, one can form a fourgradient from the partial time derivative ∂/∂t and the spatial gradient ∇. 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 fourvector. 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 the D'Alembert operator.
Kinematics
Fourvelocity
The fourvelocity of a particle is defined by:
Geometrically, U is a normalized vector tangent to the world line of the particle. Using the differential of the 4position, the magnitude of the 4velocity can be obtained:
in short, the magnitude of the 4velocity for any object is always a fixed constant:
The norm is also:
so that:
which reduces to the definition the Lorentz factor.
Fouracceleration
The fouracceleration is given by:
where a = du/dt is the coordinate 3acceleration. 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 fouracceleration and the fourvelocity is zero:
which is true for all world lines. The geometric meaning of 4acceleration is the curvature vector of the world line in Minkowski space.
Dynamics
Fourmomentum
For a massive particle of rest mass (or invariant mass) m_{0}, the fourmomentum is given by:
where the total energy of the moving particle is:
and the total relativistic momentum is:
Taking the inner product of the fourmomentum with itself:
and also:
which leads to the energy–momentum relation:
This last relation is useful relativistic mechanics, essential in relativistic quantum mechanics and relativistic quantum field theory, all with applications to particle physics.
Fourforce
The fourforce acting on a particle is defined analogously to the 3force as the time derivative of 3momentum in Newton's second law:
where P is the power transferred to move the particle, and f is the 3force acting on the particle. For a particle of constant invariant mass m_{0}, this is equivalent to
An invariant derived from the 4force is:
from the above result.
Thermodynamics
Fourheat flux
The 4heat flux vector field, is essentially similar to the 3d heat flux vector field q, in the local frame of the fluid:^{[9]}
where T is absolute temperature and k is thermal conductivity.
Fourbaryon number flux
The flux of baryons is:^{[10]}
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 4velocity field (of the fluid) as above.
Fourentropy
The 4entropy vector is defined by:^{[11]}
where s is the entropy per baryon, and T the absolute temperature, in the local rest frame of the fluid.^{[12]}
Electromagnetism
Examples of fourvectors in electromagnetism include the following.
Fourcurrent
The electromagnetic fourcurrent is defined by
formed from the current density j and charge density ρ.
Fourpotential
The electromagnetic fourpotential defined by
formed from the vector potential a and the scalar potential ϕ. The fourpotential is not uniquely determined, because it depends on a choice of gauge.
Waves
Fourfrequency
A photonic plane wave can be described by the fourfrequency defined as
where ν is the frequency of the wave and is a unit vector in the travel direction of the wave. Now:
so the 4frequency of a photon is always a null vector.
Fourwavevector
The quantities reciprocal to time t and space r are the angular frequency ω and wave vector k, respectively. They form the components of the 4WaveVector or wave 4vector:
A wave packet of nearly monochromatic light can be described by:
The de Broglie relations then showed that 4WaveVector applied to matter waves as well as to light waves. :
yielding and , where ħ is the Planck constant divided by 2π.
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 case m_{0} = 0, we have:
or k = ω/c. Note this is consistent with the above case; for photons with a 3wavevector of modulus ω/c, in the direction of wave propagation defined by the unit vector .
Quantum theory
4Probability current
In quantum mechanics, the 4probability current or probability 4current is analogous to the electromagnetic 4current:^{[13]}
where ρ is the probability density function corresponding to the time component, and j is the probability current vector. In nonrelativistic 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, 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 fourmomentum, one obtains the fourmomentum operator, used in relativistic wave equations.
4Spin
The fourspin of a particle is defined in the rest frame of a particle to be
where s 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
This value is observable and quantized, with s the spin quantum number (not the magnitude of the spin vector).
Other formulations
Fourvectors in the algebra of physical space
A fourvector A can also be defined in using the Pauli matrices as a basis, again in various equivalent notations:^{[14]}
or explicitly:
and in this formulation, the fourvector is represented as a Hermitian matrix (the matrix transpose and complex conjugate of the matrix leaves it unchanged), rather than a realvalued column or row vector. The determinant of the matrix is the modulus of the fourvector, so the determinant is an invariant:
This idea of using the Pauli matrices as basis vectors is employed in the algebra of physical space, an example of a Clifford algebra.
Fourvectors 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). 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 fourvector A contracted with the gamma matrices:
The fourmomentum contracted with the gamma matrices is an important case in relativistic quantum mechanics and relativistic quantum field theory. In the Dirac equation and other relativistic wave equations, terms of the form:
appear, in which the energy E and momentum components (p_{x}, p_{y}, p_{z}) are replaced by their respective operators.
See also
 Relativistic mechanics
 paravector
 wave vector
 Dust (relativity) for the numberflux fourvector
 Basic introduction to the mathematics of curved spacetime
 Minkowski space
References
 ↑ Relativity DeMystified, D. McMahon, Mc Graw Hill (BSA), 2006, ISBN 0071455450
 ↑ C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. p. 1333. ISBN 0070514003.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Gravitation, J.B. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISAN 0716703440
 ↑ Dynamics and Relativity, J.R. Forshaw, B.G. Smith, Wiley, 2009, ISAN 9780470014608
 ↑ Relativity DeMystified, D. McMahon, Mc Graw Hill (ASB), 2006, ISAN 0071455450
 ↑ JeanBernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5 , ISBN 0070320713
 ↑ Charles W. Misner, Kip S. Thorne & John A. Wheeler,Gravitation, pg 51, ISBN 0716703440
 ↑ George Sterman, An Introduction to Quantum Field Theory, pg 4 , ISBN 0521311322
 ↑ Ali, Y. M.; Zhang, L. C. (2005). "Relativistic heat conduction". Int. J. Heat Mass Trans. 48 (12). doi:10.1016/j.ijheatmasstransfer.2005.02.003.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 558559. ISBN 0716703440.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 567. ISBN 0716703440.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 558. ISBN 0716703440.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Vladimir G. Ivancevic, Tijana T. Ivancevic (2008) Quantum leap: from Dirac and Feynman, across the universe, to human body and mind. World Scientific Publishing Company, ISBN 9789812819277, p. 41
 ↑ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 1142–1143. ISBN 0716703440.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Rindler, W. Introduction to Special Relativity (2nd edn.) (1991) Clarendon Press Oxford ISBN 0198539525