Tensor Form of the Electromagnetic Field Energy-Momentum Tensor

In classical electrodynamics the conservation of energy and momentum is most elegantly expressed through a single rank‑two tensor, the electromagnetic energy‑momentum tensor (often called the stress‑energy tensor). This object encapsulates the density and flux of energy, the density and flux of momentum, and the stresses exerted by the field on matter, all within the language of four‑dimensional spacetime. Its tensorial form is the cornerstone for extending Maxwell’s theory to special relativity, and it provides the bridge to quantum field theory where the same tensor appears as the generator of spacetime translations.

Below we present a concise yet complete exposition of the tensor form of the electromagnetic field, covering its definition, derivation from a Lagrangian, symmetrization, connection to the conservation laws, and a concrete example involving a plane wave. The discussion is written in a self‑contained style, suitable for readers already familiar with basic differential geometry and Maxwell’s equations.

1. Tensorial definition

In a flat Minkowski spacetime with coordinates (x^{\mu}=(ct,\mathbf{x})) ((\mu=0,1,2,3)), the electromagnetic field is encoded in the antisymmetric field strength tensor

[
F^{\mu\nu}= \partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu},
]

where (A^{\mu}=(\phi/c,\mathbf{A})) is the four‑potential. The symmetric energy‑momentum tensor for the free electromagnetic field is

[
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).
]

Here (\eta^{\mu\nu}=\text{diag}(1,-1,-1,-1)) is the Minkowski metric and (\mu_{0}) is the vacuum permeability. The components of (T^{\mu\nu}) have clear physical interpretations:

The symmetry (T^{\mu\nu}=T^{\nu\mu}) guarantees that the angular‑momentum current is conserved, a requirement that will be justified in the next section.

2. From the Lagrangian to the tensor

The dynamics of the electromagnetic field follow from the Lagrangian density

[
\mathcal{L}= -\frac{1}{4\mu_{0}}F^{\mu\nu}F_{\mu\nu}.
]

Because the theory is invariant under spacetime translations, Noether’s theorem yields a conserved current. Explicitly,

[
T^{\mu\nu}= \frac{\partial\mathcal{L}}{\partial(\partial_{\mu}A_{\alpha})}\partial^{\nu}A_{\alpha}
-\eta^{\mu\nu}\mathcal{L}.
]

Using (\partial\mathcal{L}/\partial(\partial_{\mu}A_{\alpha})= -\frac{1}{\mu_{0}}F^{\mu\alpha}) and substituting into the expression above gives the canonical tensor

[
T_{\text{can}}^{\mu\nu}= -\frac{1}{\mu_{0}}F^{\mu\alpha}\partial^{\nu}A_{\alpha}
-\eta^{\mu\nu}\mathcal{L}.
]

This canonical tensor is not symmetric. The antisymmetric part would spoil the conservation of angular momentum. To restore symmetry we add a superpotential term, following the Belinfante–Rosenfeld prescription:

[
\Theta^{\mu\nu}=T_{\text{can}}^{\mu\nu}
+\partial_{\lambda}X^{\lambda\mu\nu},
\qquad
X^{\lambda\mu\nu}= \frac{1}{\mu_{0}}\bigl(F^{\mu\lambda}A^{\nu}-F^{\nu\lambda}A^{\mu}\bigr).
]

Because (\partial_{\lambda}X^{\lambda\mu\nu}) vanishes identically in vacuum, the resulting tensor (\Theta^{\mu\nu}) coincides with the symmetric expression given in Section 1. Thus the Belinfante‑Rosenfeld symmetrization guarantees that the energy‑momentum tensor is both conserved and symmetric.

3. Conservation laws

The divergence of the energy‑momentum tensor vanishes identically:

[
\partial_{\mu}T^{\mu\nu}=0.
]

When expanded in 3‑vector notation, the four equations (\nu=0,1,2,3) reproduce the familiar conservation laws:

Here (\mathbf{g}=\mathbf{S}/c^{2}) is the electromagnetic momentum density. In the absence of sources ((\rho=\mathbf{J}=0)), the right‑hand sides vanish, and the equations reduce to the standard conservation of energy and momentum in free space.

4. Plane‑wave illustration

Consider a linearly polarized plane wave propagating along the (z)-axis in vacuum:

[
\mathbf{E}=E_{0}\cos(kz-\omega t),\hat{\mathbf{x}},\qquad
\mathbf{B}=\frac{E_{0}}{c}\cos(kz-\omega t),\hat{\mathbf{y}}.
]

From these fields we compute:

• Energy density
  [
  u=\varepsilon_{0}E_{0}^{2}\cos^{2}(kz-\omega t).
  ]
• Poynting vector
  [
  \mathbf{S}= \frac{E_{0}^{2}}{\mu_{0}c}\cos^{2}(kz-\omega t),\hat{\mathbf{z}}.
  ]
• Stress tensor components
  [
  T^{zz}=u,\qquad T^{xx}=T^{yy}=-\frac{1}{2}u,\qquad T^{xy}=T^{xz}=T^{yz}=0.
  ]

Collecting these into a matrix gives

[
T^{\mu\nu}=u
\begin{pmatrix}
1 & 0 & 0 & 1\
0 & -\tfrac12 & 0 & 0\
0 & 0 & -\tfrac12 & 0\
1 & 0 & 0 & 1
\end{pmatrix}.
]

A straightforward calculation shows that (\partial_{\mu}T^{\mu\nu}=0) for this configuration, confirming that the tensor correctly encodes the conservation of energy and momentum for a propagating electromagnetic wave. The non‑zero (T^{zz}) component represents the radiation pressure exerted along the direction of propagation, while the negative transverse components (T^{xx}) and (T^{yy}) correspond to a tension perpendicular to the wave.

5. Summary

• The electromagnetic energy‑momentum tensor (T^{\mu\nu}) provides a unified, covariant description of energy density, momentum density, and stresses in the electromagnetic field.
• It is derived from the field Lagrangian via Noether’s theorem and made symmetric through the Belinfante–Rosenfeld procedure.
• The vanishing divergence (\partial_{\mu}T^{\mu\nu}=0) encapsulates both the energy conservation and momentum conservation equations of classical electrodynamics.
• Explicit calculation for a plane wave demonstrates how the tensor’s components translate into familiar physical quantities such as the Poynting vector and radiation pressure.

Mastering the tensorial form of the electromagnetic field not only deepens one’s understanding of classical field theory but also lays the groundwork for exploring relativistic electrodynamics, gravitational coupling, and quantum field theoretic extensions where the same tensor plays a central role.