User:Mpatel/sandbox/Electromagnetic stress-energy tensor

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Electromagnetism
Electricity · Magnetism
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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 Fab is the electromagnetic field tensor, gab is the metric tensor and μo 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} {{\mu _0 }}B_i B_j - \frac{1} {2}\left( {\epsilon _o E^2 + \frac{1} {{\mu _0 }}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.

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