Electromagnetic stress-energy tensor

In physics, the electromagnetic stress-energy tensor is the portion of the stress-energy tensor due to the electromagnetic field.

Definition

In free space in SI units, the electromagnetic stress-energy tensor is:And in explicit matrix form::,

with:Poynting vector $vec\{S\}=frac\{1\}\{mu\_0\}vec\{E\}\; imesvec\{B\}$,:electromagnetic field tensor $F\_\{mu\; u\}!$,:Minkowski metric tensor $eta\_\{mu\; u\}!$, and:Maxwell stress tensor $sigma\_\{ij\}\; =\; epsilon\_0\; E\_i\; E\_j\; +\; frac\{1\}$mu _0 B_i B_j - frac{1}{2}left( {epsilon_0 E^2 + frac{1}mu _0 B^2 } ight)delta _{ij} .Note that $c^2=frac\{1\}\{epsilon\_0\; mu\_0\}$ where "c" is light speed.

CGS

In free space in cgs units, we simply substitute $epsilon\_0,$ with $frac\{1\}\{4pi\}$ and $mu\_0,$ with $4pi,$ ::And in explicit matrix form::

where Poynting vector becomes the form: :$vec\{S\}=frac\{c\}\{4pi\}vec\{E\}\; imesvec\{B\}$.

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 (however see Pfeifer et. al, Rev. Mod. Phys. 79, 1197 (2007)).

The element, $T^\{mu\; u\}!$, of the energy momentum tensor represents the flux of the μ^th-component of the four-momentum of the electromagnetic field, $P^\{mu\}!$, going through a hyperplane $x^\{\; u\}\; =\; constant$. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space-time) in general relativity.

Conservation laws

The electromagnetic stress-energy tensor allows a compact way of writing the conservation laws of linear momentum and energy by electromagnetism.:$partial\_\{\; u\}T^\{mu\; u\}\; +\; eta^\{mu\; ho\}\; ,\; f\_\{\; ho\}\; =\; 0\; ,$

where $f\_\{\; ho\}$ is the density of the (3D) Lorentz force on matter.

This equation is equivalent to the following 3D conservation laws:$frac\{partial\; u\_\{em\{partial\; t\}\; +\; vec\{\; abla\}\; cdot\; vec\{S\}\; +\; vec\{J\}\; cdot\; vec\{E\}\; =\; 0\; ,$:$frac\{partial\; vec\{p\}\_\{em\{partial\; t\}\; -\; vec\{\; abla\}cdot\; sigma\; +\; ho\; vec\{E\}\; +\; vec\{J\}\; imes\; vec\{B\}\; =\; 0\; ,$

where:Electromagnetic energy density (joules/meter³) is $u\_\{em\}\; =\; frac\{epsilon\_0\}\{2\}E^2\; +\; frac\{1\}\{2mu\_0\}B^2\; ,$:Poynting vector (watts/meter²) is $vec\{S\}\; =\; frac\{1\}\{mu\_0\}\; vec\{E\}\; imes\; vec\{B\}\; ,$:Density of electric current (amperes/meter²) is $vec\{J\}\; ,$:Electromagnetic momentum density (newton·seconds/meter³) is $vec\{p\}\_\{em\}\; =\; \{vec\{S\}\; over\; c^2\}\; ,$:Maxwell stress tensor (newtons/meter²) is $sigma\_\{ij\}\; =\; epsilon\_0\; E\_i\; E\_j\; +\; frac\{1\}$mu _0 B_i B_j - frac{1}{2}left( {epsilon_0 E^2 + frac{1}mu _0 B^2 } ight)delta _{ij} ,:Density of electric charge (coulombs) is $ho\; ,.$

ee also

*Stress-energy tensor
*Covariant formulation of classical electromagnetism