Electromagnetic stress–energy tensor | Wikipedia audio article

In relativistic physics, the electromagnetic
stress-energy tensor is the contribution to the stress-energy tensor due to the electromagnetic
field. The stress-energy tensor describes the flow of energy and momentum in spacetime.
The electromagnetic stress-energy tensor contains the negative of the classical Maxwell
stress tensor that governs the electromagnetic interactions. == Definition == === SI units ===
In free space and flat space-time, the electromagnetic stress-energy tensor in SI units is T μ
ν = 1 μ 0 [ F μ
α F ν α − 1
4 η μ
ν F α
β F α
β ].

{displaystyle T^{mu nu }={frac {1}{mu
_{0}}}left[F^{mu alpha }F^{nu }{}_{alpha }-{frac {1}{4}}eta ^{mu nu }F_{alpha
beta }F^{alpha beta }right],.} where F μ
ν {displaystyle F^{mu nu }}
is the electromagnetic tensor and where η μ
ν {displaystyle eta _{mu nu }}
is the Minkowski metric tensor of metric signature (−+++). When using the metric with a signature
(+−−−), the expression for T μ
ν {displaystyle T^{mu nu }}
will have the opposite sign. Explicitly in matrix form: T μ
ν = [ 1
2 ( ϵ 0 E 2 + 1 μ 0 B 2 ) S x / c S y / c S z / c S x / c − σ xx − σ xy − σ xz S y / c − σ yx − σ yy − σ yz S z / c − σ zx − σ zy − σ ZZ ] , {displaystyle T^{mu nu }={begin{bmatrix}{frac
{1}{2}}left(epsilon _{0}E^{2}+{frac {1}{mu _{0}}}B^{2}right)&S_{text{x}}/c&S_{text{y}}/c&S_{text{z}}/c\S_{text{x}}/c&-sigma
_{text{xx}}&-sigma _{text{xy}}&-sigma _{text{xz}}\S_{text{y}}/c&-sigma _{text{yx}}&-sigma
_{text{yy}}&-sigma _{text{yz}}\S_{text{z}}/c&-sigma _{text{zx}}&-sigma _{text{zy}}&-sigma
_{text{zz}}end{bmatrix}},} where S = 1 μ 0 E × B , {displaystyle mathbf {S} ={frac {1}{mu
_{0}}}mathbf {E} times mathbf {B} ,} is the Poynting vector, σ i
j = ϵ 0 E i E j + 1 μ 0 B i B j − 1
2 ( ϵ 0 E 2 + 1 μ 0 B 2 ) δ i
j {displaystyle 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}} is the Maxwell stress tensor, and c is the
speed of light.

Thus, T μ
ν {displaystyle T^{mu nu }}
is expressed and measured in SI pressure units (pascals). === CGS units ===
The permittivity of free space and permeability of free space in cgs-Gaussian units are ϵ 0 = 1 4
π, μ 0 =
4 π {displaystyle epsilon _{0}={frac {1}{4pi
}},quad mu _{0}=4pi ,} then: T μ
ν = 1 4
π [ F μ
α F ν α − 1
4 η μ
ν F α
β F α
β ]. {displaystyle T^{mu nu }={frac {1}{4pi
}}[F^{mu alpha }F^{nu }{}_{alpha }-{frac {1}{4}}eta ^{mu nu }F_{alpha beta }F^{alpha
beta }],.} and in explicit matrix form: T μ
ν = [ 1 8
π ( E 2 + B 2 ) S x / c S y / c S z / c S x / c − σ xx − σ xy − σ xz S y / c − σ yx − σ yy − σ yz S z / c − σ zx − σ zy − σ ZZ ] {displaystyle T^{mu nu }={begin{bmatrix}{frac
{1}{8pi }}(E^{2}+B^{2})&S_{text{x}}/c&S_{text{y}}/c&S_{text{z}}/c\S_{text{x}}/c&-sigma _{text{xx}}&-sigma _{text{xy}}&-sigma
_{text{xz}}\S_{text{y}}/c&-sigma _{text{yx}}&-sigma _{text{yy}}&-sigma _{text{yz}}\S_{text{z}}/c&-sigma
_{text{zx}}&-sigma _{text{zy}}&-sigma _{text{zz}}end{bmatrix}}}
where Poynting vector becomes: S = c 4
π E × B .

{displaystyle mathbf {S} ={frac {c}{4pi
}}mathbf {E} times mathbf {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.The element T μ
ν {displaystyle T^{mu nu }!}
of the stress-energy tensor represents the flux of the μth-component of the four-momentum
of the electromagnetic field, P μ {displaystyle P^{mu }!}
, going through a hyperplane ( x ν {displaystyle x^{nu }}
is constant). It represents the contribution of electromagnetism to the source of the gravitational
field (curvature of space-time) in general relativity.

== Algebraic properties ==
The electromagnetic stress-energy tensor has several algebraic properties: It is a symmetric tensor: T μ
ν = T ν
μ {displaystyle T^{mu nu }=T^{nu mu }}
The tensor T ν α {displaystyle T^{nu }{}_{alpha }}
is traceless: T α α =
0 {displaystyle T^{alpha }{}_{alpha }=0}
.The energy density is positive-definite: T 00 ≥
0 {displaystyle T^{00}geq 0}
The symmetry of the tensor is as for a general stress-energy tensor in general relativity.
The trace of the energy-momentum tensor is a Lorentz scalar; the electromagnetic field
(and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so
its energy-momentum tensor must have a vanishing trace. This trace-lessness eventually relates
to the masslessness of the photon. == Conservation laws == The electromagnetic stress-energy tensor
allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism.
The divergence of the stress-energy tensor is: ∂ ν T μ
ν + η μ
ρ f ρ =
0 {displaystyle partial _{nu }T^{mu nu
}+eta ^{mu rho },f_{rho }=0,} where f ρ {displaystyle f_{rho }}
is the (4D) Lorentz force per unit volume on matter.
This equation is equivalent to the following 4D conservation laws ∂ u e
m ∂
t + ∇ ⋅ S + J ⋅ E =
0 {displaystyle {frac {partial u_{mathrm
{em} }}{partial t}}+mathbf {nabla } not mathbf {S} +mathbf {J} not mathbf {E}
=0,} ∂ p e
m ∂
t − ∇ ⋅
σ +
ρ E + J × B =
0 {displaystyle {frac {partial mathbf {p}
_{mathrm {em} }}{partial t}}-mathbf {nabla } dot sigma +rho mathbf {E} +mathbf
{J} times mathbf {B} =0,} (or equivalently f + ϵ 0 μ 0 ∂ S ∂
t =
∇ ⋅ σ {displaystyle mathbf {f} +epsilon _{0}mu
_{0}{frac {partial mathbf {S} }{partial t}},=nabla not mathbf {sigma } }
with f {displaystyle mathbf {f} }
being the Lorentz force density),respectively describing the flux of electromagnetic energy
density u e
m = ϵ 0 2 E 2 + 1 2 μ 0 B 2 {displaystyle u_{mathrm {em} }={frac {epsilon
_{0}}{2}}E^{2}+{frac {1}{2mu _{0}}}B^{2},} and electromagnetic momentum density p e
m = S c 2 {displaystyle mathbf {p} _{mathrm {em}
}={mathbf {S} over {c^{2}}}} where J is the electric current density and
ρ the electric charge density.

== See also ==
Ricci calculus Covariant formulation of classical electromagnetism
Mathematical descriptions of the electromagnetic field
Maxwell’s equations Maxwell’s equations in curved spacetime
General relativity Einstein field equations
Magnetohydrodynamics Vector calculus.

 

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