Fourcurrent
In special and general relativity, the fourcurrent (technically the fourcurrent density)[1] is the fourdimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of fourdimensional spacetime, rather than threedimensional space and time separately. Mathematically it is a fourvector, and is Lorentz covariant.
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Electromagnetism 


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 fourcurrent 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.
Motion of charges in spacetime
This can also be expressed in terms of the fourvelocity 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 3velocity);
 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.
Physical interpretation
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 fourcurrent unifies charge density (related to electricity) and current density (related to magnetism) in one electromagnetic entity.
Continuity equation
In special relativity, the statement of charge conservation is that the Lorentz invariant divergence of J is zero:[4]
where is the fourgradient. This is the continuity equation.
In general relativity, the continuity equation is written as:
where the semicolon represents a covariant derivative.
Maxwell's equations
The fourcurrent appears in two equivalent formulations of Maxwell's equations, in terms of the fourpotential[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.
General relativity
In general relativity, the fourcurrent is defined as the divergence of the electromagnetic displacement, defined as
then
Quantum field theory
The fourcurrent 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]
See also
References
 Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 103–107. ISBN 9780198539520.
 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]
 Cottingham, W. Noel; Greenwood, Derek A. (2003). An introduction to the standard model of particle physics. Cambridge University Press. p. 67. ISBN 9780521588324.
 Marshak, Robert E. (1993). Conceptual foundations of modern particle physics. World Scientific Publishing Company. p. 20. ISBN 9789813103368.
 Gershtein, S. S.; Zeldovich, Y. B. (1956), Soviet Phys. JETP, 2 576.
 Thomas, Anthony W. (1996). "CVC in particle physics". arXiv:nuclth/9609052.