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.
The electromagnetic stress–energy tensor in the International System of Quantities (ISQ), which underlies the SI, is[1]
is the Minkowski metric tensor of metric signature (− + + +) and the Einstein summation convention over repeated indices is used.
Explicitly in matrix form:
{\displaystyle T^{\mu \nu }={\begin{bmatrix}u&{\frac {1}{c}}S_{\text{x}}&{\frac {1}{c}}S_{\text{y}}&{\frac {1}{c}}S_{\text{z}}\\{\frac {1}{c}}S_{\text{x}}&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\{\frac {1}{c}}S_{\text{y}}&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\{\frac {1}{c}}S_{\text{z}}&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}},}
is the volumetric energy density,
is the Maxwell stress tensor, and
is dimensionally equivalent to pressure (with SI unit pascal).
The in the Gaussian system (shown here with a prime) that correspond to the permittivity of free space and permeability of free space are
and in explicit matrix form:
{\displaystyle T^{\mu \nu }={\begin{bmatrix}u&{\frac {1}{c}}S_{\text{x}}&{\frac {1}{c}}S_{\text{y}}&{\frac {1}{c}}S_{\text{z}}\\{\frac {1}{c}}S_{\text{x}}&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\{\frac {1}{c}}S_{\text{y}}&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\{\frac {1}{c}}S_{\text{z}}&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}}}
The stress–energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the Abraham–Minkowski controversy.
of the stress–energy tensor represents the flux of the component with index
of the four-momentum of the electromagnetic field,
It represents the contribution of electromagnetism to the source of the gravitational field (curvature of spacetime) in general relativity.
The electromagnetic stress–energy tensor has several algebraic properties: Starting with
Using the explicit form of the tensor,
Lowering the indices and using the fact that
are dummy indices, so we relabel them as
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.
is the (4D) Lorentz force per unit volume on matter.
This equation is equivalent to the following 3D conservation laws
respectively describing the electromagnetic energy density
and electromagnetic momentum density
is the electric current density,
the electric charge density, and