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Electromagnetism |
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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.

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:^{[2]}^{[3]}

where:

- is the charge density measured by an inertial observer O who sees the electric current moving at speed
*u*(the magnitude of the 3-velocity); - is “the rest charge density”, i.e., the charge density for a comoving observer (an observer moving at the speed
*u*- with respect to the inertial observer O - 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.

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:^{[4]}

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^{[5]} when the Lorenz gauge condition is fulfilled:

where is the D'Alembert operator, or the electromagnetic field tensor:

where *μ*_{0} 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

then

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]}