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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] }

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.

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.

The **electromagnetic four-potential** can be defined as:^{ [2] }

SI units Gaussian units

in which *ϕ* is the electric potential, and **A** is the magnetic potential (a vector potential). The units of *A ^{α}* are V·s·m

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

SI units Gaussian units

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:

assuming that the signature of the Minkowski metric is (+ − − −). If the said signature is instead (− + + +) then:

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

Often, the Lorenz gauge condition in an inertial frame of reference is employed to simplify Maxwell's equations as:^{ [2] }

SI units Gaussian units

where *J ^{α}* 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:

SI units Gaussian units

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

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

When flattened to a one-form, *A* can be decomposed via the Hodge decomposition theorem as the sum of an exact, a coexact, and a harmonic form,

- .

There is gauge freedom in *A* in that of the three forms in this decomposition, only the coexact form has any effect on the electromagnetic tensor

- .

Exact forms are closed, as are harmonic forms over an appropriate domain, so and , always. So regardless of what and are, we are left with simply

- .

In physics, the **Lorentz transformations** are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

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.

The **stress–energy tensor**, sometimes called the **stress–energy–momentum tensor** or the **energy–momentum tensor**, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

In special relativity, **four-momentum** is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy *E* and three-momentum **p** = = *γm***v**, where **v** is the particle's three-velocity and γ the Lorentz factor, is

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 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.

In physics, the **Hamilton–Jacobi equation**, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

In physics, specifically field theory and particle physics, the **Proca action** describes a massive spin-1 field of mass *m* in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the **Proca equation**. The Proca action and equation are named after Romanian physicist Alexandru Proca.

In special and general relativity, the **four-current** is the four-dimensional analogue of the electric current density. 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.

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

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.

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

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 physics, **Maxwell's equations in curved spacetime** govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

**Paraboloidal coordinates** are three-dimensional orthogonal coordinates that generalize two-dimensional parabolic coordinates. They possess elliptic paraboloids as one-coordinate surfaces. As such, they should be distinguished from parabolic cylindrical coordinates and parabolic rotational coordinates, both of which are also generalizations of two-dimensional parabolic coordinates. The coordinate surfaces of the former are parabolic cylinders, and the coordinate surfaces of the latter are *circular* paraboloids.

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.

In quantum mechanics, the **Pauli equation** or **Schrödinger–Pauli equation** is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. 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. It sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. It motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

In theoretical physics, **relativistic Lagrangian mechanics** is Lagrangian mechanics applied in the context of special relativity and general relativity.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of 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.

- ↑ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
- 1 2 D.J. Griffiths (2007).
*Introduction to Electrodynamics*(3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2. - 1 2 I.S. Grant, W.R. Phillips (2008).
*Electromagnetism*(2nd ed.). Manchester Physics, John Wiley & Sons. ISBN 978-0-471-92712-9.

- Rindler, Wolfgang (1991).
*Introduction to Special Relativity (2nd)*. Oxford: Oxford University Press. ISBN 0-19-853952-5. - Jackson, J D (1999).
*Classical Electrodynamics (3rd)*. New York: Wiley. ISBN 0-471-30932-X.

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