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 is2

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

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

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.

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).

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