The electromagnetic stress-energy tensor and covariant energy-momentum conservation

Intuition Beginner

The electromagnetic field carries both energy and momentum. In the non-relativistic treatment, these are described by separate quantities: the energy density $u$, the Poynting vector $\mathbf{S}$ (energy flow), the momentum density $\mathbf{g}$, and the Maxwell stress tensor $\overleftrightarrow{\mathbf{T}}$.

Special relativity unifies all of these into a single object: the stress-energy tensor $T^{\mu \nu }$. This is a 4-by-4 matrix whose 16 components encode everything about the energy and momentum stored in and transported by the field. The component $T^{00}$ is the energy density, $T^{0i}$ is the energy flux (Poynting vector), $T^{i0}$ is the momentum density, and $T^{ij}$ is the stress (pressure and shear).

The key advantage of the tensor formulation is that it transforms properly under Lorentz transformations. If you change to a moving frame, the energy and momentum mix, just as space and time mix. The tensor captures this mixing automatically.

The conservation law for free fields (no charges) combines Poynting's theorem (energy conservation) and the momentum conservation law into a single covariant equation.

Visual Beginner

Component	Physical meaning
$T^{00}$	Energy density $u$
$T^{0i}$	Energy flux (Poynting vector $S_i/c$)
$T^{i0}$	Momentum density $c{g}_i$
$T^{ij}$	Maxwell stress tensor (pressure)

Worked example Beginner

For a plane wave travelling in the $z$-direction with electric field amplitude $E_0$, the energy density is $u={ϵ}_0{E}_0^2$ (time-averaged: $⟨u⟩={ϵ}_0{E}_0^2/2$). The Poynting vector is $S=uc$.

In the stress-energy tensor, $T^{00}=u$, $T^{03}=T^{30}=u$ (for propagation along $z$), and $T^{33}=u$ (radiation pressure). The other components are zero for a plane wave.

The trace $T_{\mu }^{\mu }=T^{00}-T^{11}-T^{22}-T^{33}=u-0-0-u=0$. The electromagnetic stress-energy tensor is traceless — a consequence of the masslessness of the photon.

Check your understanding Beginner

Formal definition Intermediate+

Definition. The electromagnetic stress-energy tensor is:

$$T^{\mu \nu }=F^{\mu \alpha }{F}_{\alpha }^{\nu }-\frac{1}{4}{\eta }^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }$$

where $F^{\mu \nu }$ is the Faraday tensor 10.06.01 and ${\eta }^{\mu \nu }=\text{diag}(+1,-1,-1,-1)$ is the Minkowski metric.

Component identification. In terms of the fields $\mathbf{E}$ and $\mathbf{B}$:

$$T^{00}=\frac{1}{2}\left({ϵ}_0{E}^2+\frac{B^2}{{\mu }_0}\right)=u$$ $$T^{0i}=T^{i0}=\frac{1}{{\mu }_0}(\mathbf{E}\times \mathbf{B}{)}_i=\frac{S_i}{c}=c{g}_i$$ $$T^{ij}=-{ϵ}_0\left(E_i{E}_j-\frac{1}{2}{\delta }_{ij}{E}^2\right)-\frac{1}{{\mu }_0}\left(B_i{B}_j-\frac{1}{2}{\delta }_{ij}{B}^2\right)$$

Covariant conservation law. In the presence of sources:

$${\partial }_{\mu }{T}^{\mu \nu }=-F^{\nu \alpha }{J}_{\alpha }$$

For the free field ($J_{\alpha }=0$): ${\partial }_{\mu }{T}^{\mu \nu }=0$, which gives four conservation laws ($\nu =0$ for energy, $\nu =i$ for momentum).

Properties.

1. Symmetric: $T^{\mu \nu }=T^{\nu \mu }$
2. Traceless: $T_{\mu }^{\mu }={\eta }_{\mu \nu }{T}^{\mu \nu }=0$ (for $D=4$ spacetime dimensions)
3. Conserved: ${\partial }_{\mu }{T}^{\mu \nu }=0$ for free fields
4. Gauge invariant: $T^{\mu \nu }$ depends only on $F^{\mu \nu }$, not on the potentials

Key derivation Intermediate+

Derivation (Covariant energy-momentum conservation).

Theorem. The electromagnetic stress-energy tensor satisfies ${\partial }_{\mu }{T}^{\mu \nu }=-F^{\nu \alpha }{J}_{\alpha }$.

Proof. Compute ${\partial }_{\mu }{T}^{\mu \nu }={\partial }_{\mu }(F^{\mu \alpha }{F}_{\alpha }^{\nu })-\frac{1}{4}{\eta }^{\nu \sigma }{\partial }_{\mu }(F_{\alpha \beta }{F}^{\alpha \beta }{\delta }_{\sigma }^{\mu })$.

Wait — let me be more careful. ${\partial }_{\mu }{T}^{\mu \nu }=({\partial }_{\mu }{F}^{\mu \alpha })F_{\alpha }^{\nu }+F^{\mu \alpha }({\partial }_{\mu }{F}_{\alpha }^{\nu })-\frac{1}{4}{\eta }^{\mu \nu }{\partial }_{\sigma }(F_{\alpha \beta }{F}^{\alpha \beta })$.

The first term: ${\partial }_{\mu }{F}^{\mu \alpha }={\mu }_0{J}^{\alpha }$ (from the inhomogeneous Maxwell equation).

The second and third terms: using the Bianchi identity ${\partial }_{\mu }{F}_{\nu \alpha }+{\partial }_{\nu }{F}_{\alpha \mu }+{\partial }_{\alpha }{F}_{\mu \nu }=0$, one can show $F^{\mu \alpha }{\partial }_{\mu }{F}_{\alpha }^{\nu }=\frac{1}{4}{\partial }^{\nu }(F_{\alpha \beta }{F}^{\alpha \beta })$.

Combining: ${\partial }_{\mu }{T}^{\mu \nu }={\mu }_0{J}^{\alpha }{F}_{\alpha }^{\nu }+\frac{1}{4}{\partial }^{\nu }(F^2)-\frac{1}{4}{\partial }^{\nu }(F^2)={\mu }_0{J}^{\alpha }{F}_{\alpha }^{\nu }$.

In Gaussian units (absorbing ${\mu }_0$): ${\partial }_{\mu }{T}^{\mu \nu }=-F^{\nu \alpha }{J}_{\alpha }$. $\square$

The $\nu =0$ component gives $\partial u/\partial t+\nabla \cdot \mathbf{S}=-\mathbf{J}\cdot \mathbf{E}$ (Poynting's theorem 10.03.05). The $\nu =i$ components give $\partial g_i/\partial t+{\partial }_j{T}_{ji}=-(\rho E_i+(\mathbf{J}\times \mathbf{B}{)}_i)$ (momentum conservation).

Bridge. The stress-energy tensor builds toward the full relativistic formulation of energy-momentum conservation in all of physics. The foundational insight is that energy and momentum are unified into a single rank-2 tensor, just as space and time are unified into spacetime. The central message is that the conservation law ${\partial }_{\mu }{T}^{\mu \nu }=0$ for free fields is a direct consequence of Maxwell's equations in covariant form. This is exactly the Noether current associated with translational symmetry of the action. Putting these together, the stress-energy tensor generalises the Poynting vector and Maxwell stress tensor 10.03.05, the field transformation under Lorentz boosts follows from the tensor transformation law 10.06.04, and the covariant Larmor formula for synchrotron radiation 10.07.05 is derived from the radiated four-momentum $d{P}^{\mu }/d\tau$.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has Minkowski space and tensor products but does not contain the stress-energy tensor, the covariant conservation law, the trace-free property, or the identification of components with energy density, Poynting vector, and Maxwell stress tensor. lean_status: none.

Advanced results Master

The stress-energy tensor and general relativity. In general relativity, the electromagnetic field contributes to the spacetime curvature through $T^{\mu \nu }$ in Einstein's equation $G_{\mu \nu }=8\pi G{T}_{\mu \nu }/c^4$. The Reissner-Nordstrom metric describes a charged black hole whose electromagnetic field modifies the Schwarzschild geometry. The trace-free property $T_{\mu }^{\mu }=0$ means the electromagnetic field is conformally invariant and does not contribute to the Ricci scalar curvature.

The Wilson loop and stress-energy. In quantum electrodynamics, the expectation value of the stress-energy tensor near a Wilson loop (the path-ordered exponential of the vector potential around a closed curve) measures the energy of the field configuration created by the charge sources. This connects the classical stress-energy tensor to the quantum theory.

Stress-energy of electromagnetic waves in media. In a material medium with $ϵ$ and $\mu$, the stress-energy tensor acquires additional terms from the Minkowski and Abraham forms (see the discussion in 10.03.05). The correct form is still debated and depends on how one partitions the total momentum between the field and the medium.

Synthesis. The electromagnetic stress-energy tensor is the covariant formulation of energy-momentum conservation, unifying four separate conservation laws into a single tensor equation. The foundational insight is that the symmetry $T^{\mu \nu }=T^{\nu \mu }$ is related to the conservation of angular momentum, and the tracelessness is related to the conformal invariance (masslessness) of the photon. The central message is that the stress-energy tensor is the source of spacetime curvature in general relativity, making the electromagnetic field a contributor to the geometry of the universe. Putting these together, the stress-energy tensor generalises the Poynting vector 10.03.05, determines the E and B field transformations 10.06.04, and provides the basis for the covariant radiation formulae used in synchrotron physics 10.07.05.

Full proof set Master

Proposition (Tracelessness). The electromagnetic stress-energy tensor in four-dimensional Minkowski space satisfies $T_{\mu }^{\mu }=0$.

Proof. $T_{\mu }^{\mu }={\eta }_{\mu \nu }{T}^{\mu \nu }={\eta }_{\mu \nu }{F}^{\mu \alpha }{F}_{\alpha }^{\nu }-\frac{1}{4}{\eta }_{\mu \nu }{\eta }^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }=F^{\mu \alpha }{F}_{\mu \alpha }-\frac{1}{4}\cdot 4\cdot F^{\alpha \beta }{F}_{\alpha \beta }=F^{\mu \alpha }{F}_{\mu \alpha }-F^{\alpha \beta }{F}_{\alpha \beta }=0$ since both terms are $F^{\mu \alpha }{F}_{\mu \alpha }$ with relabelled dummy indices. $\square$

Connections Master

• Faraday tensor 10.06.01 is the fundamental object from which $T^{\mu \nu }$ is built.
• Four-vectors 10.05.03 provide the framework; $T^{\mu \nu }$ is a rank-2 tensor generalisation.
• Poynting vector and Maxwell stress tensor 10.03.05 are the non-covariant components of $T^{\mu \nu }$.
• Lorentz transformations 10.06.04 of E and B follow from the tensor nature of $F^{\mu \nu }$ and $T^{\mu \nu }$.
• Synchrotron radiation 10.07.05 uses the covariant Larmor formula derived from $T^{\mu \nu }$.

Historical & philosophical context Master

The stress-energy tensor was introduced by Minkowski (1908) in his formulation of relativistic electrodynamics. The covariant form of energy-momentum conservation was developed by Laue (1911) and others who recognised that the separate energy and momentum conservation laws are spacetime components of a single tensor equation.

The Belinfante-Rosenfeld procedure (1939-1940) resolved the long-standing problem of the non-symmetric canonical stress-energy tensor by showing that the symmetrised form (which is the one that appears in general relativity) is obtained by adding a total divergence related to the spin angular momentum of the field.

In general relativity, $T^{\mu \nu }$ serves as the source term in Einstein's field equations, coupling matter and fields to spacetime geometry. The fact that the electromagnetic $T^{\mu \nu }$ is traceless means that the electromagnetic field produces no Ricci scalar curvature — a deep connection between conformal invariance and the geometry of spacetime.

Bibliography Master

• Minkowski, H., "Die Grundgleichungen fuer die elektromagnetischen Vorgaenge in bewegten Koerpern," Nachr. Ges. Wiss. Goettingen, 53-111 (1908).
• Belinfante, F. J., "On the current and the density of the four-momentum, the angular momentum, and the spin of the field," Physica 6, 887-898 (1939).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields, 4th ed. (Butterworth-Heinemann, 1975).