# Electromagnetic four-potential

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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. [1]

General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

An electromagnetic field is a physical field produced by 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. It is one of the four fundamental forces of nature.

In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

## Contents

As measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four-potential is conventionally taken to be the electric scalar potential, and the other three components make up the magnetic vector potential. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant.

In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements.

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

In relativistic physics, Lorentz symmetry, named for Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".

Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge.

This article uses tensor index notation and the Minkowski metric sign convention (+ − − −). See also covariance and contravariance of vectors and raising and lowering indices for more details on notation. Formulae are given in SI units and Gaussian-cgs units.

In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. The choices made may differ between authors. Disagreement about sign conventions is a frequent source of confusion, frustration, misunderstandings, and even outright errors in scientific work. In general, a sign convention is a special case of a choice of coordinate system for the case of one dimension.

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 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.

## Definition

The electromagnetic four-potential can be defined as: [2]

SI unitsGaussian units
${\displaystyle A^{\alpha }=\left(\phi /c,\mathbf {A} \right)\,\!}$${\displaystyle A^{\alpha }=(\phi ,\mathbf {A} )}$

in which ϕ is the electric potential, and A is the magnetic potential (a vector potential). The units of Aα are V·s·m −1 in SI, and Mx·cm −1 in Gaussian-cgs.

An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point beyond the influence of the electric field charge can be used.

The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, or simply vector potential, A; and the magnetic scalar potentialψ. Both quantities can be used in certain circumstances to calculate the magnetic field B.

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 gradient is a given vector field.

The electric and magnetic fields associated with these four-potentials are: [3]

SI unitsGaussian units
${\displaystyle \mathbf {E} =-\mathbf {\nabla } \phi -{\frac {\partial \mathbf {A} }{\partial t}}}$${\displaystyle \mathbf {E} =-\mathbf {\nabla } \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}}$
${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .}$${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .}$

In special relativity, the electric and magnetic fields transform under Lorentz transformations. This can be written in the form of a tensor-the electromagnetic tensor. This is written in terms of the electromagnetic four-potential and the four-gradient as:

${\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}}$

This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.

## In the Lorenz gauge

Often, the Lorenz gauge condition ${\displaystyle \partial _{\alpha }A^{\alpha }=0}$ in an inertial frame of reference is employed to simplify Maxwell's equations as: [2]

SI unitsGaussian units
${\displaystyle \Box A^{\alpha }=\mu _{0}J^{\alpha }}$${\displaystyle \Box A^{\alpha }={\frac {4\pi }{c}}J^{\alpha }}$

where Jα are the components of the four-current, and

${\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}}$

is the d'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes:

SI unitsGaussian units
${\displaystyle \Box \phi ={\frac {\rho }{\epsilon _{0}}}}$${\displaystyle \Box \phi =4\pi \rho }$
${\displaystyle \Box \mathbf {A} =\mu _{0}\mathbf {j} }$${\displaystyle \Box \mathbf {A} ={\frac {4\pi }{c}}\mathbf {j} }$

For a given charge and current distribution, ρ(r, t) and j(r, t), the solutions to these equations in SI units are: [3]

${\displaystyle \phi (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \mathrm {d} ^{3}x^{\prime }{\frac {\rho (\mathbf {r} ^{\prime },t_{r})}{\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}}}$
${\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \mathrm {d} ^{3}x^{\prime }{\frac {\mathbf {j} (\mathbf {r} ^{\prime },t_{r})}{\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}},}$

where

${\displaystyle t_{r}=t-{\frac {\left|\mathbf {r} -\mathbf {r} '\right|}{c}}}$

is the retarded time. This is sometimes also expressed with

${\displaystyle \rho (\mathbf {r} ',t_{r})=[\rho (\mathbf {r} ',t)],}$

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 according to r−2 (the induction field) and a component decreasing as r−1 (the radiation field).[ clarification needed ]

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