# Electromagnetic stress–energy tensor

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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.

## Definition

### SI units

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.

${\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,.}$

where ${\displaystyle F^{\mu \nu }}$ is the electromagnetic tensor and where ${\displaystyle \eta _{\mu \nu }}$ is the Minkowski metric tensor of metric signature (−+++). When using the metric with signature (+−−−), the expression for ${\displaystyle T^{\mu \nu }}$ 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 gab 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:

${\displaystyle T^{\mu \nu }={\begin{bmatrix}{\frac {1}{2}}\left(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2}\right)&S_{\text{x}}/c&S_{\text{y}}/c&S_{\text{z}}/c\\S_{\text{x}}/c&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\S_{\text{y}}/c&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\S_{\text{z}}/c&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}},}$

where

${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} ,}$

is the Poynting vector,

${\displaystyle \sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}\left(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2}\right)\delta _{ij}}$

is the Maxwell stress tensor, and c is the speed of light. Thus, ${\displaystyle T^{\mu \nu }}$ 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, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most commonly associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.

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.

### CGS units

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.

${\displaystyle \epsilon _{0}={\frac {1}{4\pi }},\quad \mu _{0}=4\pi \,}$

then:

${\displaystyle T^{\mu \nu }={\frac {1}{4\pi }}[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }]\,.}$

and in explicit matrix form:

${\displaystyle T^{\mu \nu }={\begin{bmatrix}{\frac {1}{8\pi }}(E^{2}+B^{2})&S_{\text{x}}/c&S_{\text{y}}/c&S_{\text{z}}/c\\S_{\text{x}}/c&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\S_{\text{y}}/c&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\S_{\text{z}}/c&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}}}$

where Poynting vector becomes:

${\displaystyle \mathbf {S} ={\frac {c}{4\pi }}\mathbf {E} \times \mathbf {B} .}$

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 ${\displaystyle T^{\mu \nu }\!}$ of the stress–energy tensor represents the flux of the μth-component of the four-momentum of the electromagnetic field, ${\displaystyle P^{\mu }\!}$, going through a hyperplane (${\displaystyle x^{\nu }}$ is constant). It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space–time) in general relativity.

## Algebraic properties

The electromagnetic stress–energy tensor has several algebraic properties:

${\displaystyle T^{\mu \nu }=T^{\nu \mu }}$
• The tensor ${\displaystyle T^{\nu }{}_{\alpha }}$ is traceless:
${\displaystyle T^{\alpha }{}_{\alpha }=0}$.
${\displaystyle T^{00}\geq 0}$

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]

## Conservation laws

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:

${\displaystyle \partial _{\nu }T^{\mu \nu }+\eta ^{\mu \rho }\,f_{\rho }=0\,}$

where ${\displaystyle f_{\rho }}$ is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 4D conservation laws

${\displaystyle {\frac {\partial u_{\mathrm {em} }}{\partial t}}+\mathbf {\nabla } \cdot \mathbf {S} +\mathbf {J} \cdot \mathbf {E} =0\,}$
${\displaystyle {\frac {\partial \mathbf {p} _{\mathrm {em} }}{\partial t}}-\mathbf {\nabla } \cdot \sigma +\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} =0\,}$ (or equivalently ${\displaystyle \mathbf {f} +\epsilon _{0}\mu _{0}{\frac {\partial \mathbf {S} }{\partial t}}\,=\nabla \cdot \mathbf {\sigma } }$ with ${\displaystyle \mathbf {f} }$ being the Lorentz force density),

respectively describing the flux of electromagnetic energy density

${\displaystyle u_{\mathrm {em} }={\frac {\epsilon _{0}}{2}}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\,}$

and electromagnetic momentum density

${\displaystyle \mathbf {p} _{\mathrm {em} }={\mathbf {S} \over {c^{2}}}}$

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

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## References

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