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In electromagnetism, the **electromagnetic tensor** or **electromagnetic field tensor** (sometimes called the **field strength tensor**, **Faraday tensor** or **Maxwell bivector**) 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.

The electromagnetic tensor, conventionally labelled *F*, is defined as the exterior derivative of the electromagnetic four-potential, *A*, a differential 1-form:^{ [1] }^{ [2] }

Therefore, *F* is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

where is the four-gradient and is the four-potential.

SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space (+ − − −), will be used throughout this article.

The Faraday differential 2-form is given by

This is the exterior derivative of its 1-form antiderivative

- ,

where has ( is a scalar potential for the irrotational/conservative vector field ) and has ( is a vector potential for the solenoidal vector field ).

Here is a video series by Michael Penn explaining the Faraday 2-form and its relations to Maxwell's equations.

Note that

where is the exterior derivative, is the Hodge star, (where is the electric current density, and is the electric charge density) is the 4-current density 1-form, is the differential forms version of Maxwell's equations.

The electric and magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in Cartesian coordinates:

where *c* is the speed of light, and

where is the Levi-Civita tensor. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will transform covariantly, and the fields in the new frame will be given by the new components.

In contravariant matrix form,

The covariant form is given by index lowering,

The Faraday tensor's Hodge dual is

From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is assumed, and the electric and magnetic fields are with respect to the coordinate system's reference frame, as in the equations above.

The matrix form of the field tensor yields the following properties:^{ [3] }

**Antisymmetry:****Six independent components:**In Cartesian coordinates, these are simply the three spatial components of the electric field (*E*) and magnetic field (_{x}, E_{y}, E_{z}*B*)._{x}, B_{y}, B_{z}**Inner product:**If one forms an inner product of the field strength tensor a Lorentz invariant is formedmeaning this number does not change from one frame of reference to another.**Pseudoscalar invariant:**The product of the tensor with its Hodge dual gives a Lorentz invariant:where is the rank-4 Levi-Civita symbol. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is .**Determinant:**which is proportional to the square of the above invariant.**Trace:**which is equal to zero.

This tensor simplifies and reduces Maxwell's equations as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively:

and reduce to the inhomogeneous Maxwell equation:

- , where is the four-current.

In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively:

which reduce to Bianchi identity:

or using the index notation with square brackets ^{ [note 1] } for the antisymmetric part of the tensor:

The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law, this general property of physical laws being recognised after the advent of special relativity. This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of tensors. The tensor formalism also leads to a mathematically simpler presentation of physical laws.

The inhomogeneous Maxwell equation leads to the continuity equation:

implying conservation of charge.

Maxwell's laws above can be generalised to curved spacetime by simply replacing partial derivatives with covariant derivatives:

- and

where the semi-colon notation represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime):

Classical electromagnetism and Maxwell's equations can be derived from the action:

where is over space and time.

This means the Lagrangian density is

The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is

Substituting this into the Euler–Lagrange equation of motion for a field:

So the Euler–Lagrange equation becomes:

The quantity in parentheses above is just the field tensor, so this finally simplifies to

That equation is another way of writing the two inhomogeneous Maxwell's equations (namely, Gauss's law and Ampère's circuital law) using the substitutions:

where *i, j, k* take the values 1, 2, and 3.

The Hamiltonian density can be obtained with the usual relation,

- .

The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons):

where the first part in the right hand side, containing the Dirac spinor , represents the Dirac field. In quantum field theory it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.

**^**By definition,So if

then

- ↑ J. A. Wheeler; C. Misner; K. S. Thorne (1973).
*Gravitation*. W.H. Freeman & Co. ISBN 0-7167-0344-0. - ↑ D. J. Griffiths (2007).
*Introduction to Electrodynamics*(3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2. - ↑ J. A. Wheeler; C. Misner; K. S. Thorne (1973).
*Gravitation*. W.H. Freeman & Co. ISBN 0-7167-0344-0.

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 the mathematical field of differential geometry, the **Riemann curvature tensor** or **Riemann–Christoffel tensor** is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is *flat*, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In the general theory of relativity, the **Einstein field equations** relate the geometry of spacetime to the distribution of matter within it.

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.

In differential geometry, the **Einstein tensor** is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum.

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

In differential geometry, a **tensor density** or **relative tensor** is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another, except that it is additionally multiplied or *weighted* by a power *W* of the Jacobian determinant of the coordinate transition function or its absolute value. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight *W* are called **tensor capacity.** A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.

When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.

In general relativity, a **geodesic** generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

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.

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. 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 mathematics, **Ricci calculus** constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the **absolute differential calculus**, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.

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

The **Optical Metric** was defined by German theoretical physicist Walter Gordon in 1923 to study the geometrical optics in curved space-time filled with moving dielectric materials. Let u_{a} be the normalized (covariant) 4-velocity of the arbitrarily-moving dielectric medium filling the space-time, and assume that the fluid’s electromagnetic properties are linear, isotropic, transparent, nondispersive, and can be summarized by two scalar functions: a dielectric permittivity ε and a magnetic permeability μ. Then **optical metric ** tensor is defined as

In theoretical physics, the **dual graviton** is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of supergravity in eleven dimensions.

- Brau, Charles A. (2004).
*Modern Problems in Classical Electrodynamics*. Oxford University Press. ISBN 0-19-514665-4. - Jackson, John D. (1999).
*Classical Electrodynamics*. John Wiley & Sons, Inc. ISBN 0-471-30932-X. - Peskin, Michael E.; Schroeder, Daniel V. (1995).
*An Introduction to Quantum Field Theory*. Perseus Publishing. ISBN 0-201-50397-2.

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