electromagnetic stress–energy tensor

The electromagnetic stress–energy tensor in the International System of Quantities (ISQ), which underlies the SI, is

$$T^\{\backslash mu\backslash nu\}\; =\; \backslash frac\{1\}\{\backslash mu\_0\}\; \backslash left[\; F^\{\backslash mu\; \backslash alpha\}F^\backslash nu\{\}\_\{\backslash alpha\}\; -\; \backslash frac\{1\}\{4\}\; \backslash eta^\{\backslash mu\backslash nu\}F\_\{\backslash alpha\backslash beta\}\; F^\{\backslash alpha\backslash beta\}\backslash right]\; \backslash ,,$$

where $F^\{\backslash mu\backslash nu\}$ is the electromagnetic tensor and where $\backslash eta\_\{\backslash mu\backslash nu\}$ is the Minkowski metric tensor of metric signature {{nowrap|(− + + +)}} and the Einstein summation convention over repeated indices is used.

Explicitly in matrix form:

$$T^\{\backslash mu\backslash nu\}\; =\; \backslash begin\{bmatrix\}$$

u & \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

$$u\; =\; \backslash frac\{1\}\{2\}\backslash left(\backslash epsilon\_0\; \backslash mathbf\{E\}^2+\backslash frac\{1\}\{\backslash mu\_0\}\backslash mathbf\{B\}^2\backslash right)$$

is the volumetric energy density,

$$\backslash mathbf\{S\}\; =\; \backslash frac\{1\}\{\backslash mu\_0\}\backslash mathbf\{E\}\backslash times\backslash mathbf\{B\}$$

is the Poynting vector,

$$\backslash sigma\_\{ij\}\; =\; \backslash epsilon\_0\; E\_i\; E\_j\; +\; \backslash frac\{1\}\{\{\backslash mu\; \_0\}\}B\_i\; B\_j\; -\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash epsilon\_0\; \backslash mathbf\{E\}^2\; +\; \backslash frac\{1\}\{\backslash mu\; \_0\}\backslash mathbf\{B\}^2\; \backslash right)\backslash delta\; \_\{ij\}$$

is the Maxwell stress tensor, and $c$ is the speed of light. Thus, each component of $T^\{\backslash mu\backslash nu\}$ is dimensionally equivalent to pressure (with SI unit pascal).

The in the Gaussian system (shown here with a prime) that correspond to the permittivity of free space and permeability of free space are

$$\backslash epsilon\_0\text{'}\; =\; \backslash frac\{1\}\{4\backslash pi\},\backslash quad\; \backslash mu\_0\text{'}\; =\; 4\backslash pi$$

then:

$$T^\{\backslash mu\backslash nu\}\; =\; \backslash frac\{1\}\{4\backslash pi\}\; \backslash left[F\text{'}^\{\backslash mu\backslash alpha\}F\text{'}^\{\backslash nu\}\{\}\_\{\backslash alpha\}\; -\; \backslash frac\{1\}\{4\}\; \backslash eta^\{\backslash mu\backslash nu\}F\text{'}\_\{\backslash alpha\backslash beta\}F\text{'}^\{\backslash alpha\backslash beta\}\backslash right]$$

and in explicit matrix form:

$$T^\{\backslash mu\backslash nu\}\; =\; \backslash begin\{bmatrix\}$$

u & \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 the energy density becomes

$$u\; =\; \backslash frac\{1\}\{8\backslash pi\}\backslash left(\backslash mathbf\{E\}\text{'}^2\; +\; \backslash mathbf\{B\}\text{'}^2\backslash right)$$

and the Poynting vector becomes

$$\backslash mathbf\{S\}\; =\; \backslash frac\{c\}\{4\backslash pi\}\backslash mathbf\{E\}\text{'}\backslash times\backslash mathbf\{B\}\text{'}.$$

The stress–energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the Abraham–Minkowski controversy.however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)

The element $T^\{\backslash mu\backslash nu\}$ of the stress–energy tensor represents the flux of the component with index $\backslash mu$ of the four-momentum of the electromagnetic field, {{tmath|1= P^{\mu} }}, going through a hyperplane. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of spacetime) in general relativity.

The electromagnetic stress–energy tensor has several algebraic properties:

{{unordered list

| It is a symmetric tensor:

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

| The tensor $T^\{\backslash nu\}\{\}\_\{\backslash alpha\}$ is traceless (in 4D):

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

{{math proof

|proof = Starting with

$$T^\{\backslash mu\}\{\}\_\{\backslash mu\}\; =\; \backslash eta\_\{\backslash mu\backslash nu\}T^\{\backslash mu\backslash nu\}$$

Using the explicit form of the tensor,

$$T^\{\backslash mu\}\{\}\_\{\backslash mu\}\; =\; \backslash frac\{1\}\{\backslash mu\_0\}\backslash left[\backslash eta\_\{\backslash mu\backslash nu\}F^\{\backslash mu\backslash alpha\}F^\{\backslash nu\}\{\}\_\{\backslash alpha\}\; -\; \backslash eta\_\{\backslash mu\backslash nu\}\backslash eta^\{\backslash mu\backslash nu\}\backslash frac\{1\}\{4\}F^\{\backslash alpha\backslash beta\}F\_\{\backslash alpha\backslash beta\}\backslash right]$$

Lowering the indices and using the fact that {{tmath|1= \eta^{\mu\nu}\eta_{\mu\nu} = \delta^{\mu}_{\mu} }},

$$T^\{\backslash mu\}\{\}\_\{\backslash mu\}\; =\; \backslash frac\{1\}\{\backslash mu\_0\}\; \backslash left[F^\{\backslash mu\backslash alpha\}\; F\_\{\backslash mu\backslash alpha\}\; -\; \backslash delta^\{\backslash mu\}\_\{\backslash mu\}\; \backslash frac\{1\}\{4\}\; F^\{\backslash alpha\backslash beta\}\; F\_\{\backslash alpha\backslash beta\}\backslash right]$$

Then, using {{tmath|1= \delta^{\mu}_{\mu} = 4 }},

$$T^\{\backslash mu\}\{\}\_\{\backslash mu\}\; =\; \backslash frac\{1\}\{\backslash mu\_0\}\backslash left[F^\{\backslash mu\backslash alpha\}\; F\_\{\backslash mu\backslash alpha\}\; -\; F^\{\backslash alpha\backslash beta\}\; F\_\{\backslash alpha\backslash beta\}\backslash right]$$

Note that in the first term, $\backslash mu$ and $\backslash alpha$ are dummy indices, so we relabel them as $\backslash alpha$ and $\backslash beta$ respectively.

$$T^\{\backslash alpha\}\{\}\_\{\backslash alpha\}\; =\; \backslash frac\{1\}\{\backslash mu\_0\}\backslash left[F^\{\backslash alpha\backslash beta\}\; F\_\{\backslash alpha\backslash beta\}\; -\; F^\{\backslash alpha\backslash beta\}\; F\_\{\backslash alpha\backslash beta\}\backslash right]\; =\; 0$$

}}

| The energy density is positive-definite:

$$T^\{00\}\; \backslash ge\; 0$$

}}

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.Garg, Anupam. Classical Electromagnetism in a Nutshell, p. 564 (Princeton University Press, 2012).

{{further|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:

$$\backslash partial\_\backslash nu\; T^\{\backslash mu\; \backslash nu\}\; +\; \backslash eta^\{\backslash mu\; \backslash rho\}\; \backslash ,\; f\_\backslash rho\; =\; 0\; \backslash ,$$

where $f\_\backslash rho$ is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 3D conservation laws

$$\backslash begin\{align\}$$

\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{align}

respectively describing the electromagnetic energy density

$$u\_\backslash mathrm\{em\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash epsilon\_0\backslash mathbf\{E\}^2\; +\; \backslash frac\{1\}\{\backslash mu\_0\}\backslash mathbf\{B\}^2\; \backslash right)$$

and electromagnetic momentum density

$$\backslash mathbf\{p\}\_\backslash mathrm\{em\}\; =\; \{\backslash mathbf\{S\}\; \backslash over\; \{c^2\}\}\; ,$$

where $\backslash mathbf\{J\}$ is the electric current density, $\backslash rho$ the electric charge density, and $\backslash mathbf\{f\}$ is the Lorentz force density.

• Ricci calculus
• Covariant formulation of classical electromagnetism
• Mathematical descriptions of the electromagnetic field
• Maxwell's equations
• Maxwell's equations in curved spacetime
• General relativity
• Einstein field equations
• Magnetohydrodynamics
• Vector calculus

{{reflist}}

{{DEFAULTSORT:Electromagnetic stress-energy tensor}}

Category:Tensor physical quantities

Category:Electromagnetism