 Fourcurrent

Electromagnetism Electricity · Magnetism Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy currentIn special and general relativity, the fourcurrent is the Lorentz covariant fourvector that replaces the electromagnetic current density, or indeed any conventional charge current density. Its four components are given by:
where
 c is the speed of light
 ρ the charge density
 j the conventional current density.
 a labels the spacetime dimensions
This can also be expressed in terms of fourvelocity as ^{[1]} ^{[2]}
where
In this article the metric η_{μν} = diag[1, − 1, − 1, − 1] is used.
Contents
Continuity equation
In special relativity, the statement of charge conservation (also called the continuity equation) is that the Lorentz invariant divergence of J is zero ^{[3]}:
where D is an operator called the fourgradient and given by (1/c ∂/∂t, ∇). The summation convention has been used, so that the spacetime dimensions are implicitly summed over. i.e.
Sometimes, the above relation is written as
In general relativity, the continuity equation is written as:
where the semicolon represents a covariant derivative.
Other uses
This fourvector is related to fourpotential as [ref. 1, p519]
where is the D'Alembert operator, defined as
Also, two of the Maxwell's Equations can be written in terms of the electromagnetic tensor as [ref. 1, p520]
Euclidean metric
Although Minkowski's space is not euclidean, some authors prefer to use a euclidean metric, where , and
Using this metric, the fourcurrent is rewritten as
where i is the imaginary unit.
General Relativity
In general relativity, the fourcurrent is defined as the divergence of the electromagnetic displacement, defined as
Thus,
Physical interpretation
This fourvector unifies charge density (related to electricity) and current density (related to magnetism) in one electromagnetic entity. As studied in electrostatics, charges or distributions of charge moving only through time, thus having no velocity, have only charge density, while if they are moving through space too, thus having some velocity v, they have current density too. This means that charge density is related to time, while current density is related to space.
See also
References
Categories: Electromagnetism
 Minkowski spacetime
 Relativity
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