In special and general relativity, the four-current (technically the four-current density) 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.
Using the Minkowski metric of metric signature (+ − − −), the four-current components are given by:
where c is the speed of light, ρ is the volume charge density, and j the conventional current density. The dummy index α labels the spacetime dimensions.
See also: Lorentz transformations
This can also be expressed in terms of the four-velocity by the equation:
Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to Lorentz contraction.
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.
Main article: Continuity equation
In special relativity, the statement of charge conservation is that the Lorentz invariant divergence of J is zero:
where is the four-gradient. This is the continuity equation.
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative.
Main article: Maxwell's equations
The four-current appears in two equivalent formulations of Maxwell's equations, in terms of the four-potential when the Lorenz gauge condition is fulfilled:
where is the D'Alembert operator, or the electromagnetic field tensor:
where μ0 is the permeability of free space and ∇β is the covariant derivative.
See also: Maxwell's equations in curved spacetime
In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as
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 (CVC) hypothesis for electroweak interactions.