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In relativistic physics, the **electromagnetic stress–energy tensor** is the contribution to the stress–energy tensor due to the electromagnetic field.^{ [1] } 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 **stress–energy tensor**, sometimes **stress–energy–momentum tensor** or **energy–momentum tensor**, is a tensor quantity in physics 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. The stress–energy tensor is the source 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.

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 physics, **spacetime** is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive *where* and *when* events occur.

In free space and flat space–time, the electromagnetic stress–energy tensor in SI units is^{ [2] }

In mathematics, a **tensor** is a geometric object that maps in a multi-linear manner geometric vectors, scalars, and other tensors to a resulting tensor. Vectors and scalars which are often used in elementary physics and engineering applications, are considered as the simplest tensors. Vectors from the dual space of the vector space, which supplies the geometric vectors, are also included as tensors. *Geometric* in this context is chiefly meant to emphasize independence of any selection of a coordinate system.

where is the electromagnetic tensor and where is the Minkowski metric tensor of metric signature (−+++). When using the metric with signature (+−−−), the expression for will have opposite sign.

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.

The **signature**(*v*, *p*, *r*) of a metric tensor *g* is the number of positive, zero, and negative eigenvalues of the real symmetric matrix *g*_{ab} of the metric tensor with respect to a basis. In physics, the *v* represents for the time or virtual dimension, and the *p* for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (*v*, *p*) implying *r* = 0 or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signature (1, 3, 0), respectively.

Explicitly in matrix form:

where

is the Poynting vector,

is the Maxwell stress tensor, and *c* is the speed of light. Thus, is expressed and measured in SI pressure units (pascals).

The **Maxwell stress tensor** is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

The **speed of light** in vacuum, commonly denoted * c*, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second. It is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity,

The **pascal** is the SI derived unit of pressure used to quantify internal pressure, stress, Young's modulus and ultimate tensile strength. It is defined as one newton per square metre. It is named after the French polymath Blaise Pascal.

The permittivity of free space and permeability of free space in cgs-Gaussian units are

**Gaussian units** constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the **Gaussian unit system**, **Gaussian-cgs units**, or often just **cgs units**. The term "cgs units" is ambiguous and therefore to be avoided if possible: cgs contains within it several conflicting sets of electromagnetism units, not just Gaussian units, as described below.

then:

and in explicit matrix form:

where Poynting vector becomes:

The stress–energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the unresolved Abraham–Minkowski controversy.^{ [3] }

A **dielectric** is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

The **Abraham–Minkowski controversy** is a physics debate concerning electromagnetic momentum within dielectric media. Traditionally, it is argued that in the presence of matter the electromagnetic stress-energy tensor by itself is not conserved (divergenceless). Only the total stress-energy tensor carries unambiguous physical significance, and how one apportions it between an "electromagnetic" part and a "matter" part depends on context and convenience. In other words, the electromagnetic part and the matter part in the total momentum can be arbitrarily distributed as long as the total momentum is kept the same. There are two incompatible equations to describe momentum transfer between matter and electromagnetic fields. These two equations were first suggested by Hermann Minkowski (1908) and Max Abraham (1909), from which the controversy's name derives. Both were claimed to be supported by experimental data. Theoretically, it is usually argued that Abraham's version of momentum "does indeed represent the true momentum density of electromagnetic fields" for electromagnetic waves, while Minkowski's version of momentum is "pseudomomentum" or "wave momentum".

The element of the stress–energy tensor represents the flux of the *μ*th-component of the four-momentum of the electromagnetic field, , going through a hyperplane ( is constant). It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space–time) in general relativity.

The electromagnetic stress–energy tensor has several algebraic properties:

- It is a symmetric tensor:

- The tensor is traceless:

- .

- The energy density is positive-definite:

The symmetry of the tensor is as for a general stress–energy tensor in general relativity. The trace of the energy-momentum tensor is a Lorentz scalar; the electromagnetic field (and in particular electromagnetica waves) has no Lorentz-invariant energy scale, so its energy-momentum tensor must have a vanishing trace. This tracelessness eventually relates to the masslessness of the photon.^{ [4] }

The electromagnetic stress–energy tensor allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism. The divergence of the stress–energy tensor is:

where is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 4D conservation laws

- (or equivalently with being the Lorentz force density),

respectively describing the flux of electromagnetic energy density

and electromagnetic momentum density

where **J** is the electric current density and *ρ* the electric charge density.

**Noether's** (**first**) **theorem** states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat & F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

**Linear elasticity** is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.

The **Einstein field equations** comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy. First published by Einstein in 1915 as a tensor equation, the EFE relate local *spacetime curvature* with the local energy and momentum within that spacetime.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

The **vierbein** or **tetrad** theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like **triad**, **pentad**, **zweibein**, **fünfbein**, **elfbein** etc. have been used. **Vielbein** covers all dimensions.

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 consistent with energy and momentum conservation.

In physics, the **Polyakov action** is an action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by L. Brink, P. Di Vecchia and P. S. Howe, and has become associated with Alexander Polyakov after he made use of it in quantizing the string. The action reads

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

The **mathematics of general relativity** refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity. 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.

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.

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.

In the theory of general relativity, a **stress–energy–momentum pseudotensor**, such as the **Landau–Lifshitz pseudotensor**, is an extension of the non-gravitational stress–energy tensor which incorporates the energy–momentum of gravity. It allows the energy–momentum of a system of gravitating matter to be defined. In particular it allows the total of matter plus the gravitating energy–momentum to form a conserved current within the framework of general relativity, so that the *total* energy–momentum crossing the hypersurface of *any* compact space–time hypervolume vanishes.

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 mathematical physics, the **Belinfante–Rosenfeld tensor** is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved.

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 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.

In physics, **zilch** is a conserved quantity of the electromagnetic field.

- ↑ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
- ↑ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
- ↑ however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)
- ↑ Garg, Anupam.
*Classical Electromagnetism in a Nutshell*, p. 564 (Princeton University Press, 2012).

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