Electromagnetic stress–energy tensor

In physics, the electromagnetic stress–energy tensor is the portion of the stress–energy tensor due to the electromagnetic field.[1]

Definition

SI units

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

$T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right] \,.$

where $F^{\mu\nu}$ is the electromagnetic tensor. This expression is when using a metric of signature (-,+,+,+). If using the metric with signature (+,-,-,-), the expression for $T^{\mu \nu}$ will have opposite sign.

Explicitly in matrix form:

$T^{\mu\nu} =\begin{bmatrix} \frac{1}{2}\left(\epsilon_0 E^2+\frac{1}{\mu_0}B^2\right) & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix},$

where $\eta_{\mu\nu}$ is the Minkowski metric tensor of metric signature (−+++),

$\bold{S}=\frac{1}{\mu_0}\bold{E}\times\bold{B},$

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:

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

and in explicit matrix form:

$T^{\mu\nu} =\begin{bmatrix} \frac{1}{8\pi}(E^2+B^2) & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix}$

where Poynting vector becomes:

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

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 $T^{\mu\nu}\!$ of the stress–energy tensor represents the flux of the μth-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

This tensor has several noteworthy algebraic properties. First, it is a symmetric tensor:

$T^{\mu\nu}=T^{\nu\mu}$

Second, the tensor $T^{\nu}_{\ \alpha}$ is traceless:

$T^{\alpha}_{\ \alpha}= 0$.

Third, the energy density is positive-definite:

$T^{00}>0$

These three algebraic properties have varying importance in the context of modern physics, and they remove or reduce ambiguity of the definition of the electromagnetic stress-energy tensor. The symmetry of the tensor is important in General Relativity, because the Einstein tensor is symmetric. The tracelessness is regarded as important for the masslessness of the photon.[4]

Conservation laws

Main article: 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 (3D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 3D conservation laws

$\frac{\partial u_\mathrm{em}}{\partial t} + \bold{\nabla} \cdot \bold{S} + \bold{J} \cdot \bold{E} = 0 \,$
$\frac{\partial \bold{p}_\mathrm{em}}{\partial t} - \bold{\nabla}\cdot \sigma + \rho \bold{E} + \bold{J} \times \bold{B} = 0 \,$

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

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

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