Four-current
Encyclopedia
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:
where
This can also be expressed in terms of four-velocity
as
where
In this article the metric
is used.
conservation
(also called the continuity equation
) is that the Lorentz invariant divergence of J is zero :
where D is an operator called the four-gradient
and given by (1/c ∂/∂t, ∇). The summation convention
has been used, so that the space-time dimensions are implicitly summed over. i.e.
Sometimes, the above relation is written as
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative
.
as [ref. 1, p519]
where
is the D'Alembert operator
, defined as
Also, two of the Maxwell's Equations
can be written in terms of the electromagnetic tensor
as [ref. 1, p520]
, and
Using this metric, the four-current is rewritten as
where i is the imaginary unit
.
Thus,
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.
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, the four-current is the Lorentz covariant four-vector
Four-vector
In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
that replaces the electromagnetic
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
, or indeed any conventional charge
Charge (physics)
In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...
current density. Its four components are given by:
where
- c is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
- ρ the charge densityCharge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
- j the conventional current densityCurrent densityCurrent density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
. - a labels the space-timeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s
This can also be expressed in terms of four-velocity
Four-velocity
In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector that replaces classicalvelocity...
as
where
In this article the metric
is used.Continuity equation
In special relativity, the statement of chargeCharge (physics)
In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...
conservation
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
(also called the continuity equation
Continuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
) is that the Lorentz invariant divergence of J is zero :
where D is an operator called the four-gradient
Four-gradient
The four-gradient is the four-vector generalization of the gradient:\partial_\alpha \ = \left...
and given by (1/c ∂/∂t, ∇). The summation convention
Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...
has been used, so that the space-time dimensions are implicitly summed over. i.e.
Sometimes, the above relation is written as
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative
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 differential operator, to be contrasted with the approach given...
.
Other uses
This four-vector is related to four-potentialElectromagnetic four-potential
The electromagnetic four-potential is a potential from which the electromagnetic field can be derived. It combines both the electric scalar potential and the magnetic vector potential into a single space-time four-vector. In a given reference frame, the first component is the scalar potential and...
as [ref. 1, p519]
where
is the D'Alembert operatorD'Alembert operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator , also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named for French mathematician and physicist Jean le Rond d'Alembert...
, defined as
Also, two of the Maxwell's Equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
can be written in terms of the electromagnetic tensor
Electromagnetic tensor
The electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism...
as [ref. 1, p520]
Euclidean metric
Although Minkowski's space is not euclidean, some authors prefer to use a euclidean metric, where, andUsing this metric, the four-current is rewritten as
where i is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
.
General Relativity
In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined asThus,
Physical interpretation
This four-vectorFour-vector
In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
unifies charge density (related to electricity) and current density (related to magnetism) in one electromagnetic entity.
As studied in electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
, 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.