 Electromagnetic Stress-energy Tensor
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Electromagnetic Stress%E2%80%93energy Tensor

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.

## Definition

### SI units

In free space and flat space-time, the electromagnetic stress-energy tensor in SI units is

$T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,.$ where $F^{\mu \nu }$ is the electromagnetic tensor and where $\eta _{\mu \nu }$ is the Minkowski metric tensor of metric signature . When using the metric with signature , the expression on the right of the equation will have opposite sign.

Explicitly in matrix form:

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

$\mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} ,$ is the Poynting vector,

$\sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}\left(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2}\right)\delta _{ij}$ is the Maxwell stress tensor, and c is the speed of light. Thus, $T^{\mu \nu }$ is expressed and measured in SI pressure units (pascals).

### CGS units

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

$T^{\mu \nu }={\frac {1}{4\pi }}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,.$ and in explicit matrix form:

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

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

The element $T^{\mu \nu }\!$ of the stress-energy tensor represents the flux of the ?th-component of the four-momentum of the electromagnetic field, $P^{\mu }\!$ , going through a hyperplane ($x^{\nu }$ 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:
$T^{\mu \nu }=T^{\nu \mu }$ • The tensor $T^{\nu }{}_{\alpha }$ is traceless:
$T^{\alpha }{}_{\alpha }=0$ .
Proof

Starting with

$T_{\mu }^{\mu }=\eta _{\mu \nu }T^{\mu \nu }$ Using the explicit form of the tensor,

$T_{\mu }^{\mu }={\frac {1}{4\pi }}\left[\eta _{\mu \nu }F^{\mu \alpha }F^{\nu }{}_{\alpha }-\eta _{\mu \nu }\eta ^{\mu \nu }{\frac {1}{4}}F^{\alpha \beta }F_{\alpha \beta }\right]$ Lowering the indices and using the fact that $\eta ^{\mu \nu }\eta _{\mu \nu }=\delta _{\mu }^{\mu }$ $T_{\mu }^{\mu }={\frac {1}{4\pi }}\left[F^{\mu \alpha }F_{\mu \alpha }-\delta _{\mu }^{\mu }{\frac {1}{4}}F^{\alpha \beta }F_{\alpha \beta }\right]$ Then, using $\delta _{\mu }^{\mu }=4$ ,

$T_{\mu }^{\mu }={\frac {1}{4\pi }}\left[F^{\mu \alpha }F_{\mu \alpha }-F^{\alpha \beta }F_{\alpha \beta }\right]$ Note that in the first term, ? and ? and just dummy indices, so we relabel them as ? and ? respectively.

$T_{\alpha }^{\alpha }={\frac {1}{4\pi }}\left[F^{\alpha \beta }F_{\alpha \beta }-F^{\alpha \beta }F_{\alpha \beta }\right]=0$ • The energy density is positive-definite:
$T^{00}\geq 0$ 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.

## 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:

$\partial _{\nu }T^{\mu \nu }+\eta ^{\mu \rho }\,f_{\rho }=0\,$ where $f_{\rho }$ is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 3D conservation laws

{\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

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

$\mathbf {p} _{\mathrm {em} }={\mathbf {S} \over {c^{2}}}$ where J is the electric current density, ? the electric charge density, and $\mathbf {f}$ is the Lorentz force density.