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.
== Definition ==
Using the Minkowski metric
η
μ
ν
{\displaystyle \eta _{\mu \nu }}
of metric signature (+−−−), the four-current
components are given by:
J
α
=
(
c
ρ
,
j
1
,
j
2
,
j
3
)
=
(
c
ρ
,
j
)
{\displaystyle J^{\alpha }=\left(c\rho ,j^{1},j^{2},j^{3}\right)=\left(c\rho
,\mathbf {j} \right)}
where c is the speed of light, ρ is the 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
four-velocity by the equation:
J
α
=
ρ
0
U
α
=
ρ
1
−
u
2
c
2
U
α
{\displaystyle J^{\alpha }=\rho _{0}U^{\alpha
}=\rho {\sqrt {1-{\frac {u^{2}}{c^{2}}}}}U^{\alpha
}}
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.
=== 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 four-current 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:
∂
J
α
∂
x
α
=
∂
ρ
∂
t
+
∇
⋅
j
=
0
{\displaystyle {\dfrac {\partial J^{\alpha
}}{\partial x^{\alpha }}}={\frac {\partial
\rho }{\partial t}}+\nabla \cdot \mathbf {j}
=0}
where
∂
/
∂
x
α
{\displaystyle \partial /\partial x^{\alpha
}}
is the 4-gradient.
This is the continuity equation.
In general relativity, the continuity equation
is written as:
J
α
;
α
=
0
{\displaystyle J^{\alpha }{}_{;\alpha }=0\,}
where the semi-colon represents a covariant
derivative.
== Maxwell's equations ==
The four-current appears in two equivalent
formulations of Maxwell's equations, in terms
of the four-potential:
◻
A
α
=
μ
0
J
α
{\displaystyle \Box A^{\alpha }=\mu _{0}J^{\alpha
}}
where
◻
{\displaystyle \Box }
is the D'Alembert operator, or the electromagnetic
field tensor:
∂
β
F
α
β
=
μ
0
J
α
{\displaystyle \partial _{\beta }F^{\alpha
\beta }=\mu _{0}J^{\alpha }}
where μ0 is the permeability of free space.
== General relativity ==
In general relativity, the four-current is
defined as the divergence of the electromagnetic
displacement, defined as
D
μ
ν
=
1
μ
0
g
μ
α
F
α
β
g
β
ν
−
g
{\displaystyle {\mathcal {D}}^{\mu \nu }\,=\,{\frac
{1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha
\beta }\,g^{\beta \nu }\,{\sqrt {-g}}\,}
then
J
μ
=
∂
ν
D
μ
ν
{\displaystyle J^{\mu }=\partial _{\nu }{\mathcal
{D}}^{\mu \nu }}
== Quantum field theory ==
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.
== See also ==
Four-vector
Noether's theorem
Covariant formulation of classical electromagnetism
Ricci calculus
