View source for Electromagnetic stress–energy tensor

{{Short description|Type of stress-energy tensor}}
{{Electromagnetism|Covariance}}

In [[relativistic physics]], the '''electromagnetic stress–energy tensor''' is the contribution to the [[stress–energy tensor]] due to the [[electromagnetic field]].<ref name="WheelerEtAl">Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, {{ISBN|0-7167-0344-0}}</ref> 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<ref name="WheelerEtAl"/>

:<math>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] \,.</math>

where <math>F^{\mu\nu}</math> is the [[electromagnetic tensor]] and where <math>\eta_{\mu\nu}</math> is the [[Metric tensor (general relativity)#Flat spacetime|Minkowski metric tensor]] of [[metric signature]] {{nowrap|(− + + +)}}. When using the metric with signature {{nowrap|(+ − − −)}}, the expression on the right of the equals sign will have opposite sign.  

Explicitly in matrix form:

:<math>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},</math>

where

:<math>\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B},</math>

is the [[Poynting vector]], 

:<math>\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} </math>

is the [[Maxwell stress tensor]], and ''c'' is the [[speed of light]]. Thus, <math>T^{\mu\nu}</math> is expressed and measured in SI pressure units ([[Pascal (unit)|pascals]]).

=== CGS unit conventions ===

The [[permittivity of free space]] and [[permeability of free space]] in [[Gaussian units|cgs-Gaussian units]] are

:<math>\epsilon_0 = \frac{1}{4\pi},\quad \mu_0 = 4\pi\,</math>

then:

:<math>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] \,.</math>

and in explicit matrix form:

:<math>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}</math>

where [[Poynting vector]] becomes: 

:<math>\mathbf{S} = \frac{c}{4\pi}\mathbf{E}\times\mathbf{B}.</math>

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]].<ref>however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)</ref>

The element <math>T^{\mu\nu}\!</math> of the stress–energy tensor represents the flux of the ''μ''th-component of the [[four-momentum]] of the electromagnetic field, <math>P^{\mu}\!</math>, going through a [[hyperplane]] (<math> x^{\nu}</math> 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:
{{unordered list
| It is a [[symmetric tensor]]:
<math display="block">T^{\mu\nu} = T^{\nu\mu}</math> 
| The tensor <math>T^{\nu}{}_{\alpha}</math> is [[Trace (linear algebra)|traceless]]:
<math display="block">T^{\alpha}{}_{\alpha} = 0.</math>

{{math proof
|proof = Starting with
<math display="block">T^{\mu}_{\mu} = \eta_{\mu\nu}T^{\mu\nu}</math>

Using the explicit form of the tensor,
<math display="block">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]</math>

Lowering the indices and using the fact that <math>\eta^{\mu\nu}\eta_{\mu\nu} = \delta^{\mu}_{\mu}</math>
<math display="block">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]</math>

Then, using <math>\delta^{\mu}_{\mu} = 4</math>,
<math display="block">T^{\mu}_{\mu} = \frac{1}{4\pi}\left[F^{\mu\alpha} F_{\mu\alpha} - F^{\alpha\beta} F_{\alpha\beta}\right]</math>

Note that in the first term, μ and α and just dummy indices, so we relabel them as α and β respectively.
<math display="block">T^{\alpha}_{\alpha} = \frac{1}{4\pi}\left[F^{\alpha\beta} F_{\alpha\beta} - F^{\alpha\beta} F_{\alpha\beta}\right] = 0</math>
}}
| The energy density is [[Positive-definite function|positive-definite]]:
<math display="block">T^{00} \ge 0</math>
}}

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 transformation|Lorentz-invariant]] energy scale, so its energy–momentum tensor must have a vanishing trace. This tracelessness eventually relates to the masslessness of the [[photon]].<ref>Garg, Anupam. ''Classical Electromagnetism in a Nutshell'', p. 564 (Princeton University Press, 2012).</ref>

== Conservation laws ==

{{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:

:<math>\partial_\nu T^{\mu \nu} + \eta^{\mu \rho} \, f_\rho = 0 \,</math>

where <math>f_\rho</math> is the (4D) [[Lorentz force]] per unit volume on [[matter]].

This equation is equivalent to the following 3D conservation laws

:<math>\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}</math>

respectively describing the flux of electromagnetic energy density

:<math>u_\mathrm{em} = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2 \,</math>

and electromagnetic momentum density 

:<math>\mathbf{p}_\mathrm{em} = {\mathbf{S} \over {c^2}} </math>

where '''J''' is the [[electric current density]], ''ρ'' the [[electric charge density]], and <math>\mathbf{f} </math> is the Lorentz force density.

==See also==
*[[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]]

==References==

{{reflist}}

{{DEFAULTSORT:Electromagnetic stress-energy tensor}}
[[Category:Tensor physical quantities]]
[[Category:Electromagnetism]]