In
special
Special or specials may refer to:
Policing
* Specials, Ulster Special Constabulary, the Northern Ireland police force
* Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer
Literature
* ''Specia ...
and
general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the
electric current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
. 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
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
for more on this quantity.
This article uses the
summation convention for indices. See
covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation ...
for background on raised and lowered indices, and
raising and lowering indices In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Mat ...
on how to switch between them.
Definition
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
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
, and j the conventional
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
. 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:
:
where:
*
is the
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
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.
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
In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is al ...
is that the
Lorentz invariant divergence of ''J'' is zero:
:
where
is the
four-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus.
In special relativity and in quantum mechanics, the four-gradient is used to define the properties ...
. This is the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
.
In general relativity, the continuity equation is written as:
:
where the semi-colon represents a
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differen ...
.
Maxwell's equations
The four-current appears in two equivalent formulations of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, in terms of the
four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
when the
Lorenz gauge condition
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ...
is fulfilled:
:
where
is the
D'Alembert operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of M ...
, or the
electromagnetic field tensor:
:
where ''μ''
0 is the
permeability of free space
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
and ∇
''β'' is the
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differen ...
.
General relativity
In
general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as
:
then
:
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
Yakov Borisovich Zeldovich ( be, Я́каў Бары́савіч Зяльдо́віч, russian: Я́ков Бори́сович Зельдо́вич; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet physicist of Bel ...
considered the conserved vector current (CVC) hypothesis for electroweak interactions.
See also
*
Four-vector
*
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
*
Covariant formulation of classical electromagnetism
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformatio ...
*
Ricci calculus
References
{{DEFAULTSORT:Four-Current
Electromagnetism
Four-vectors