Electromagnetic stress-energy tensor

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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, it is (SI units):

T^{\mu\nu} = -\frac{1}{\mu_0}[ F^{\mu \alpha}F_{\alpha}{}^{\nu} + \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}].

And in explicit matrix form:

T^{\mu\nu} =\begin{bmatrix} \frac{1}{2}(\epsilon_0 E^2+\frac{1}{\mu_0}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},

with

Poynting vector \vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B},
electromagnetic field tensor F_{\mu\nu}\!,
Lorentzian metric tensor g_{\mu\nu}\!, 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 } \right)\delta _{ij} .

Note that c^2=\frac{1}{\epsilon_0 \mu_0} where c is light speed.

In cgs units, we simply substitute \epsilon_0\, with \frac{1}{4\pi} and \mu_0\, with 4\pi\, :

T^{\mu\nu} = -\frac{1}{4\pi} [ F^{\mu\alpha}F_{\alpha}{}^{\nu} + \frac{1}{4}g^{\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 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 (however see Pfeifer et. al, Rev. Mod. Phys. 79, 1197 (2007)).

The element, T^{\mu\nu}\!, of the energy momentum tensor represents the flux of the muth-component of the four-momentum of the electromagnetic field, P^{\mu}\!, going through a hyperplane xν = C. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space-time) in general relativity.

The electromagnetic stress-energy tensor allows a compact way of writing the conservation laws of momentum and energy of electromagnetic fields.

\partial_{\nu}T^{\mu\nu}= - k^{\mu}

where kμ is the four-force density.

From this equation, the following conservation laws can be derived:

\frac{\partial }{\partial t} (u_{em}+u_{mech}) + \vec{\nabla}\cdot \vec{S}= 0
\frac{\partial }{\partial t} (\vec{p}_{em}+\vec{p}_{mech})-\vec{\nabla}\cdot \sigma= 0

with

Poynting vector \vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B},
Electromagnetic energy density u_{em} = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2
Mechanical energy density u_{mech}=\vec{E}\cdot\vec{J}
Electromagnetic momentum density \vec{p}_{em} = \mu_0\epsilon_0 \vec{S}
Mechanical momentum density \vec{p}_{mech}
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 } \right)\delta _{ij} .


[edit] See also

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