Four-current


In special and general relativity, the four-current 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.
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.

Definition

Using the Minkowski metric of metric signature, the four-current 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 four-velocity by the equation:
where:
- is the charge density measured by an inertial observer O who sees the electric current moving at speed u ;
- is “the rest charge density”, i.e., the charge density for a comoving observer.
Qualitatively, the change in charge density is due to the contracted volume of charge due to Lorentz contraction.

Physical interpretation

Charges at rest will appear to remain at the same spatial position for some interval of time. 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 and current density in one electromagnetic entity.

Continuity equation

In special relativity, the statement of charge conservation is that the Lorentz invariant divergence of J is zero:
where is the 4-gradient. This is the continuity equation.
In general relativity, the continuity equation is written as:
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:
where is the D'Alembert operator, or the electromagnetic field tensor:
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
then

Quantum field theory

The four-current density of charge is an essential component of the Lagrangian density used in quantum electrodynamics. In 1956 Gershtein and Zeldovich considered the conserved vector current hypothesis for electroweak interactions.