Fourcurrent
Electromagnetism 


In special and general relativity, the fourcurrent is the fourdimensional analogue of the electric current density, which is used in the geometric context of fourdimensional spacetime, rather than threedimensional space and time separately. Mathematically it is a fourvector, and is Lorentz covariant.
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.
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.
Contents
Definition
Using the Minkowski metric of metric signature (+−−−), its four components are given by:
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.
Motion of charges in spacetime
This can also be expressed in terms of the fourvelocity by the equation:^{[1]}^{[2]}
where ρ is the charge density measured by an observer at rest observing the electric current, and ρ_{0} the charge density for an observer moving at the speed u (the magnitude of the 3velocity) 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 fourcurrent unifies charge density (related to electricity) and current density (related to electricity 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:^{[3]}
where is the 4gradient. This is the continuity equation.
In general relativity, the continuity equation is written as:
where the semicolon represents a covariant derivative.
Maxwell's equations
The fourcurrent appears in two equivalent formulations of Maxwell's equations, in terms of the fourpotential:^{[4]}
where is the D'Alembert operator, or the electromagnetic field tensor:
where μ_{0} is the permeability of free space.
General Relativity
In general relativity, the fourcurrent is defined as the divergence of the electromagnetic displacement, defined as
then