Electromagnetic stress-energy tensor


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:T^{mu u} = -frac{1}{mu_0} [ F^{mu alpha}F_{alpha}{}^{ u} + frac{1}{4} eta^{mu u}F_{alphaeta}F^{alphaeta}] ,.And in explicit matrix form::T^{mu u} =egin{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} 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, ::T^{mu u} = -frac{1}{4pi} [ F^{mualpha}F_{alpha}{}^{ u} + frac{1}{4} eta^{mu u}F_{alphaeta}F^{alphaeta}] ,.And in explicit matrix form::T^{mu u} =egin{bmatrix} frac{1}{8pi}(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}{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 μth-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


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