# Four-current

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In special and general relativity, the four-current (technically the four-current density) [1] is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of four-dimensional spacetime, rather than three-dimensional space and time separately. Mathematically it is a four-vector, and is Lorentz covariant.

In physics, special relativity is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates:

1. the laws of physics are invariant in all inertial systems ; and
2. the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformation. 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 (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, which include spatial rotations and boosts.

## Contents

Analogously, it is possible to have any form of "current density", meaning the flow of a quantity per unit time per unit area. see current density for more on this quantity.

In electromagnetism, current density is the electric current per unit area of cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI units, the electric current density is measured in amperes per square metre.

This article uses the summation convention for indices. See covariance and contravariance of vectors for background on raised and lowered indices, and raising and lowering indices on how to switch between them.

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in applications in physics that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.

In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.

## Definition

Using the Minkowski metric ${\displaystyle \eta _{\mu \nu }}$ of metric signature (+−−−), the four-current components are given by:

The signature(v, p, r) of a metric tensor g is the number of positive, zero, and negative eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis. In physics, the v represents for the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (v, p) implying r = 0 or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signature (1, 3, 0), respectively.

${\displaystyle J^{\alpha }=\left(c\rho ,j^{1},j^{2},j^{3}\right)=\left(c\rho ,\mathbf {j} \right)}$

where c is the speed of light, ρ is the charge density, and j the conventional current density. The dummy index α labels the spacetime dimensions.

The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second. It is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most commonly associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.

In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C•m−3), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C•m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C•m−1), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive where and when events occur.

### Motion of charges in spacetime

This can also be expressed in terms of the four-velocity by the equation: [2] [3]

In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.

${\displaystyle J^{\alpha }=\rho _{0}U^{\alpha }=\rho {\sqrt {1-{\frac {u^{2}}{c^{2}}}}}U^{\alpha }}$

where ρ is the charge density measured by an observer at rest observing the electric current, and ρ0 the charge density for an observer moving at the speed u (the magnitude of the 3-velocity) along with the charges.

Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to Lorentz contraction.

### Physical interpretation

Charges (free or as a distribution) at rest will appear to remain at the same spatial position for some interval of time (as long as they're stationary). When they do move, this corresponds to changes in position, therefore the charges have velocity, and the motion of charge constitutes an electric current. This means that charge density is related to time, while current density is related to space.

The four-current unifies charge density (related to electricity) and current density (related to magnetism) in one electromagnetic entity.

## Continuity equation

In special relativity, the statement of charge conservation is that the Lorentz invariant divergence of J is zero: [4]

${\displaystyle {\dfrac {\partial J^{\alpha }}{\partial x^{\alpha }}}={\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =0}$

where ${\displaystyle \partial /\partial x^{\alpha }}$ is the 4-gradient. This is the continuity equation.

In general relativity, the continuity equation is written as:

${\displaystyle J^{\alpha }{}_{;\alpha }=0\,}$

where the semi-colon represents a covariant derivative.

## Maxwell's equations

The four-current appears in two equivalent formulations of Maxwell's equations, in terms of the four-potential: [5]

${\displaystyle \Box A^{\alpha }=\mu _{0}J^{\alpha }}$

where ${\displaystyle \Box }$ is the D'Alembert operator, or the electromagnetic field tensor:

${\displaystyle \partial _{\beta }F^{\alpha \beta }=\mu _{0}J^{\alpha }}$

where μ0 is the permeability of free space.

## General relativity

In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as

${\displaystyle {\mathcal {D}}^{\mu \nu }\,=\,{\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\sqrt {-g}}\,}$

then

${\displaystyle J^{\mu }=\partial _{\nu }{\mathcal {D}}^{\mu \nu }}$

## Quantum field theory

The four-current density of charge is an essential component of the Lagrangian density used in quantum electrodynamics. [6] In 1956 Gershtein and Zeldovich considered the conserved vector current (CVC) hypothesis for electroweak interactions. [7] [8] [9]

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## References

1. Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 103–107. ISBN   0-19-853952-5.
2. Roald K. Wangsness, Electromagnetic Fields, 2nd edition (1986), p. 518, 519
3. Melvin Schwartz, Principles of Electrodynamics, Dover edition (1987), p. 122, 123
4. J. D. Jackson, Classical Electrodynamics, 3rd Edition (1999), p. 554
5. as [ref. 1, p519]
6. Cottingham, W. Noel; Greenwood, Derek A. (2003). An introduction to the standard model of particle physics. Cambridge University Press. p. 67.
7. Marshak, Robert E. (1993). Conceptual foundations of modern particle physics. World Scientific Publishing Company. p. 20.
8. Gershtein, S. S.; Zeldovich, Y. B. (1956), Soviet Phys. JETP, 2 576.
9. Thomas, Anthony W. (1996). "CVC in particle physics". arXiv preprint nucl-th/9609052.