Electromagnetic stress-energy tensor

From Wikipedia, the free encyclopedia

Electrostatics
Electric charge Coulomb's law Electric field Gauss's law Electric potential Electric dipole moment

Magnetostatics
Ampère's circuital law Magnetic field Magnetic flux Biot-Savart law Magnetic dipole moment

Electrodynamics
Electric current Lorentz force law Electromotive force Electromagnetic induction Faraday's law of induction Displacement current Maxwell's equations Electromagnetic field Electromagnetic radiation

Electrical Network
Conduction Resistance Capacitance Inductance Impedance Resonant cavities Waveguides

Tensors in Relativity
Electromagnetic tensor Electromagnetic stress-energy tensor

This box: view • talk • edit

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 given by in SI units:

<math>T^{\alpha\beta} = \frac{1}{\mu_0}[ -F^{\alpha \gamma}F_{\gamma}{}^{\beta} - \frac{1}{4}g^{\alpha\beta}F_{\gamma\delta}F^{\gamma\delta}]</math>.

And in explicit matrix form:

<math>T^{\alpha\beta} =\begin{bmatrix} \frac{1}{2}(\epsilon_0 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}</math>,

with

Poynting vector <math>\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}</math>,

electromagnetic field tensor <math>F_{\alpha\beta}\!</math>,

metric tensor <math>g_{\alpha\beta}\!</math>, and

Maxwell stress tensor <math>\sigma_{ij} = \epsilon_0 E_i E_j + \frac{1}

Template:\mu 0B_i B_j - \frac{1} {2}\left( {\epsilon_0 E^2 + \frac{1} Template:\mu 0B^2 } \right)\delta _{ij} </math>. Note that <math>c^2=\frac{1}{\epsilon_0 \mu_0}</math> where c is light speed.

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

<math>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}]</math>.

And in explicit matrix form:

<math>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}</math>

where Poynting vector becomes the form:

<math>\vec{S}=\frac{c}{4\pi}\vec{E}\times\vec{H}</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.

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