User:Mpatel/sandbox/Electromagnetic stress-energy tensor

In physics, the electromagnetic stress-energy tensor is the portion of the stress-energy tensor due to the electromagnetic field. In free space (vacuum), it is given in SI units by:


 * $$T_{ab} = \, \frac{1}{\mu_o}( F_{a}{}^{s} F_{sb} + {1 \over 4} F_{st} F^{st} g_{ab})$$

where $$F_{ab}$$ is the electromagnetic field tensor, $$g_{ab}$$ is the metric tensor and $$\mu_o$$ is the permeability of free space

And in explicit matrix form:
 * $$T^{\alpha\beta} =\begin{bmatrix} \frac{1}{2}(\epsilon_o E^2+\frac{1}{\mu_0}B^2) & S_x & S_y & S_z \\

S_x & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\ S_y & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\ S_z & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix}$$,

with
 * Poynting vector $$\vec{S}=\frac{1}{\mu_o}\vec{E}\times\vec{B}$$,
 * electromagnetic field tensor $$F_{\alpha\beta}\!$$,
 * metric tensor $$g_{\alpha\beta}\!$$, and
 * Maxwell stress tensor $$\sigma_{ij} = \epsilon _o E_i E_j  + \frac{1}

B_i B_j - \frac{1} {2}\left( {\epsilon _o E^2 + \frac{1} B^2 } \right)\delta _{ij} $$. Note that $$c^2=\frac{1}{\epsilon_o \mu_0}$$ where c is light speed.

In cgs units, we simply substitute $$\epsilon_o\,$$ with $$\frac{1}{4\pi}$$ and $$\mu_o\,$$ with $$4\pi\,$$ :
 * $$T^{\alpha\beta} = \frac{1}{4\pi} [ -F^{\alpha \gamma}F_{\gamma}{}^{\beta} - \frac{1}{4}g^{\alpha\beta}F_{\gamma\delta}F^{\gamma\delta}]$$.

And in explicit matrix form:
 * $$T^{\alpha\beta} =\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 the form:
 * $$\vec{S}=\frac{c}{4\pi}\vec{E}\times\vec{H}$$.

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.

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