In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field.[1] 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.
In free space and flat space–time, the electromagnetic stress–energy tensor in SI units is[1]
T\mu\nu=
1 | |
\mu0 |
\left[F\mu
\nu{} | |
F | |
\alpha |
-
1 | |
4 |
η\mu\nuF\alpha\betaF\alpha\beta\right].
where
F\mu\nu
η\mu\nu
Explicitly in matrix form:
T\mu\nu=\begin{bmatrix}
1 | |
2 |
\left(\epsilon0
| ||||
E |
B2\right)&
1 | |
c |
Sx&
1 | |
c |
Sy&
1 | |
c |
Sz\\
1 | |
c |
Sx&-\sigmaxx&-\sigmaxy&-\sigmaxz\\
1 | |
c |
Sy&-\sigmayx&-\sigmayy&-\sigmayz\\
1 | |
c |
Sz&-\sigmazx&-\sigmazy&-\sigmazz \end{bmatrix},
where
S=
1 | |
\mu0 |
E x B,
is the Poynting vector,
\sigmaij=\epsilon0EiEj+
1 | |
{\mu0 |
is the Maxwell stress tensor, and c is the speed of light. Thus,
T\mu\nu
The permittivity of free space and permeability of free space in cgs-Gaussian units are
\epsilon0=
1 | |
4\pi |
, \mu0=4\pi
then:
T\mu\nu=
1 | |
4\pi |
\left[F\mu\alphaF\nu{}\alpha-
1 | |
4 |
η\mu\nuF\alpha\betaF\alpha\beta\right].
and in explicit matrix form:
T\mu\nu=\begin{bmatrix}
1 | |
8\pi |
\left(E2+B2\right)&
1 | |
c |
Sx&
1 | |
c |
Sy&
1 | |
c |
Sz\\
1 | |
c |
Sx&-\sigmaxx&-\sigmaxy&-\sigmaxz\\
1 | |
c |
Sy&-\sigmayx&-\sigmayy&-\sigmayz\\
1 | |
c |
Sz&-\sigmazx&-\sigmazy&-\sigmazz \end{bmatrix}
where Poynting vector becomes:
S=
c | |
4\pi |
E x 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.[2]
The element
T\mu\nu
P\mu
x\nu
The electromagnetic stress–energy tensor has several algebraic properties:
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 tracelessness eventually relates to the masslessness of the photon.[3]
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:
\partial\nuT\mu+η\muf\rho=0
where
f\rho
This equation is equivalent to the following 3D conservation laws
\begin{align}
\partialuem | |
\partialt |
+\nabla ⋅ S+J ⋅ E&=0\\
\partialpem | |
\partialt |
-\nabla ⋅ \sigma+\rhoE+J x B&=0 \Leftrightarrow \epsilon0\mu0
\partialS | |
\partialt |
-\nabla ⋅ \sigma+f=0 \end{align}
respectively describing the flux of electromagnetic energy density
uem=
\epsilon0 | |
2 |
E2+
1 | |
2\mu0 |
B2
and electromagnetic momentum density
pem={S\over{c2}}
where J is the electric current density, ρ the electric charge density, and
f