# Electromagnetic stress–energy tensor

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[1]

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

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 on the right of the equals sign will have opposite sign.

Explicitly in matrix form:

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

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 unit conventions

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

then:

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

and in explicit matrix form:

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

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.[2]

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:

• It is a symmetric tensor:
${\displaystyle T^{\mu \nu }=T^{\nu \mu }}$
• The tensor ${\displaystyle T^{\nu }{}_{\alpha }}$ is traceless:
${\displaystyle T^{\alpha }{}_{\alpha }=0.}$
Proof

Starting with

${\displaystyle T_{\mu }^{\mu }=\eta _{\mu \nu }T^{\mu \nu }}$

Using the explicit form of the tensor,

${\displaystyle T_{\mu }^{\mu }={\frac {1}{4\pi }}\left[\eta _{\mu \nu }F^{\mu \alpha }F^{\nu }{}_{\alpha }-\eta _{\mu \nu }\eta ^{\mu \nu }{\frac {1}{4}}F^{\alpha \beta }F_{\alpha \beta }\right]}$

Lowering the indices and using the fact that ${\displaystyle \eta ^{\mu \nu }\eta _{\mu \nu }=\delta _{\mu }^{\mu }}$

${\displaystyle T_{\mu }^{\mu }={\frac {1}{4\pi }}\left[F^{\mu \alpha }F_{\mu \alpha }-\delta _{\mu }^{\mu }{\frac {1}{4}}F^{\alpha \beta }F_{\alpha \beta }\right]}$

Then, using ${\displaystyle \delta _{\mu }^{\mu }=4}$,

${\displaystyle T_{\mu }^{\mu }={\frac {1}{4\pi }}\left[F^{\mu \alpha }F_{\mu \alpha }-F^{\alpha \beta }F_{\alpha \beta }\right]}$

Note that in the first term, μ and α and just dummy indices, so we relabel them as α and β respectively.

${\displaystyle T_{\alpha }^{\alpha }={\frac {1}{4\pi }}\left[F^{\alpha \beta }F_{\alpha \beta }-F^{\alpha \beta }F_{\alpha \beta }\right]=0}$

• The energy density is positive-definite:
${\displaystyle T^{00}\geq 0}$

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 electromagnetic 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.[3]

## 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 3D conservation laws

{\displaystyle {\begin{aligned}{\frac {\partial u_{\mathrm {em} }}{\partial t}}+\mathbf {\nabla } \cdot \mathbf {S} +\mathbf {J} \cdot \mathbf {E} &=0\\{\frac {\partial \mathbf {p} _{\mathrm {em} }}{\partial t}}-\mathbf {\nabla } \cdot \sigma +\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} &=0\ \Leftrightarrow \ \epsilon _{0}\mu _{0}{\frac {\partial \mathbf {S} }{\partial t}}-\nabla \cdot \mathbf {\sigma } +\mathbf {f} =0\end{aligned}}}

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, ρ the electric charge density, and ${\displaystyle \mathbf {f} }$ is the Lorentz force density.