Electromagnetic four-potential
Encyclopedia
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 the other three components make up the magnetic vector potential as measured by that frame. While both the scalar and vector potential are frame dependent, the electromagnetic four-potential is Lorentz covariant. However as it is a gauge potential, it is not gauge covariant. That is, there are many gauge equivalent electromagnetic four-potentials corresponding to the same physical situation.
Mathematical note: In this article, the abstract index notation
will be used. Formulas are given in SI units
, and in parentheses in Gaussian-cgs units
.
in which
is the electrical potential, and 
is the magnetic potential
, a vector potential
.
The units of
are volt·seconds/meter in SI, and maxwell
/centimeter in Gaussian-cgs
.
The electric and magnetic fields associated with these four-potentials are:
In order for the electric and magnetic fields to satisfy special relativity
, they must be written in the form of a tensor
that transforms correctly under Lorentz transformations (see electromagnetic tensor
). This is written in terms of the electromagnetic four-potential as:
This essentially defines the four-potential in terms of physically observable quantities.
in an inertial frame of reference
to simplify Maxwell's equations
as:
where
are the components of the four-current
,
and
is the d'Alembertian operator.
In terms of the scalar and vector potentials, this last equation becomes:
For a given charge and current distribution,
and 
, the solutions to these equations in SI units are
,
where
is the retarded time. This is sometimes also expressed with 
, where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous
differential equation
, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.
When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying as
(the induction field) and a component decreasing as 
(the radiation field).
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
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 the other three components make up the magnetic vector potential as measured by that frame. While both the scalar and vector potential are frame dependent, the electromagnetic four-potential is Lorentz covariant. However as it is a gauge potential, it is not gauge covariant. That is, there are many gauge equivalent electromagnetic four-potentials corresponding to the same physical situation.
Mathematical note: In this article, the abstract index notation
Abstract index notation
Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical...
will be used. Formulas are given in SI units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
, and in parentheses in Gaussian-cgs units
Gaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...
.
Definition
The electromagnetic four-potential can be defined as:in which
is the electrical potential, and is the magnetic potentialMagnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
, a vector potential
Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
.
The units of
are volt·seconds/meter in SI, and maxwellMaxwell (unit)
The maxwell, abbreviated as Mx, is the compound derived CGS unit of magnetic flux. The unit was previously called a line. The unit name honours James Clerk Maxwell, who presented the unified theory of electromagnetism, and was established by the IEC in 1930.In a magnetic field of strength one...
/centimeter in Gaussian-cgs
Gaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...
.
The electric and magnetic fields associated with these four-potentials are:
In order for the electric and magnetic fields to satisfy special relativity
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...
, they must be written in the form of a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
that transforms correctly under Lorentz transformations (see 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...
). This is written in terms of the electromagnetic four-potential as:
This essentially defines the four-potential in terms of physically observable quantities.
In the Lorenz Gauge
Often, physicists employ the Lorenz gauge conditionLorenz gauge condition
In electromagnetism, the Lorenz gauge or Lorenz gauge condition is a partial gauge fixing of the electromagnetic vector potential. The condition is that \partial_\mu A^\mu=0...
in an inertial frame of referenceInertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...
to simplify 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...
as:
where
are the components of the four-currentFour-current
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...
,
and
is the d'Alembertian operator.In terms of the scalar and vector potentials, this last equation becomes:
For a given charge and current distribution,
and , the solutions to these equations in SI units are,where
is the retarded time. This is sometimes also expressed with , where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneousHomogeneous differential equation
The term homogeneous differential equation has several distinct meanings.One meaning is that a first-order ordinary differential equation is homogeneous if it has the formwhere F is a homogeneous function of degree zero; that is to say, that F = F.In a related, but distinct, usage, the term linear...
differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.
When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying as
(the induction field) and a component decreasing as (the radiation field).