# 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. 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: 

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

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

$F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}$ 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 $\partial _{\alpha }A^{\alpha }=0$ in an inertial frame of reference is employed to simplify Maxwell's equations as: 

SI unitsGaussian units
$\Box A^{\alpha }=\mu _{0}J^{\alpha }$ $\Box A^{\alpha }={\frac {4\pi }{c}}J^{\alpha }$ where Jα are the components of the four-current, and

$\Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}$ is the d'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes:

SI unitsGaussian units
$\Box \phi ={\frac {\rho }{\epsilon _{0}}}$ $\Box \phi =4\pi \rho$ $\Box \mathbf {A} =\mu _{0}\mathbf {j}$ $\Box \mathbf {A} ={\frac {4\pi }{c}}\mathbf {j}$ For a given charge and current distribution, ρ(r, t) and j(r, t), the solutions to these equations in SI units are: 

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

$t_{r}=t-{\frac {\left|\mathbf {r} -\mathbf {r} '\right|}{c}}$ is the retarded time. This is sometimes also expressed with

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

## Related Research Articles In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

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In physics, chemistry and biology, a potential gradient is the local rate of change of the potential with respect to displacement, i.e. spatial derivative, or gradient. This quantity frequently occurs in equations of physical processes because it leads to some form of flux.

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations. The term 'classical field theory' is commonly reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature. Theories that incorporate quantum mechanics are called quantum field theories.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

In electromagnetism, the Lorenz gauge condition or Lorenz gauge is a partial gauge fixing of the electromagnetic vector potential. The condition is that This does not completely determine the gauge: one can still make a gauge transformation where is a harmonic scalar function. In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely. In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector.

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form: In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions. The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems. In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations makes the partial differential equations inhomogeneous, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

In mathematics and physics, the vector Laplace operator, denoted by , named after Pierre-Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian. Whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component. There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking. The theory of special relativity plays an important role in the modern theory of classical electromagnetism. First of all, it gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

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