# 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.

## Definition

### SI units

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

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

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.

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).

### CGS units

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

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]

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.

## 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).