Electromagnetism Electricity · Magnetism
In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density, or indeed any conventional charge current density. Its four components are given by:
- c is the speed of light
- ρ the charge density
- j the conventional current density.
- a labels the space-time dimensions
In this article the metric ημν = diag[1, − 1, − 1, − 1] is used.
Sometimes, the above relation is written as
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative.
This four-vector is related to four-potential as [ref. 1, p519]
where is the D'Alembert operator, defined as
Although Minkowski's space is not euclidean, some authors prefer to use a euclidean metric, where , and
Using this metric, the four-current is rewritten as
where i is the imaginary unit.
In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as
This four-vector 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.
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