In special and general relativity, the four-current is the 4-dimensional analogue of the electric current density, which is used in the geometric context of 4d spacetime, rather than 3d space and time separately. Mathematically it is a four-vector, and is Lorentz covariant.
Analogously, it's 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 (+−−−), its four components are given by:
Some authors[who?] prefer to work under a Euclidean spacetime rather than Minkowski's, in which case the Euclidean metric of signature (++++) is used, so that:
so the four-current is rewritten as
where i is the imaginary unit.
Motion of charges in spacetime
where ρ is the charge density measured by an observer at rest observing the electric current, and ρ0 the charge density for an observer moving at the speed u (the magnitude of the 3-velocity) 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 electricity magnetism) in one electromagnetic entity.
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative.
where μ0 is the permeability of free space.
In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as
- Roald K. Wangsness, Electromagnetic Fields, 2nd edition (1986), p. 518, 519
- Melvin Schwartz, Principles of Electrodynamics, Dover edition (1987), p. 122, 123
- J. D. Jackson, Classical Electrodynamics, 3rd Edition (1999), p. 554
- as [ref. 1, p519]