Covariant Maxwell and the Stress–Energy Tensor

The field-strength tensor and its dual turn Maxwell's four vector equations into two tensor equations, in which Lorentz covariance is manifest and charge conservation is automatic. The force law and the field's energy and momentum follow the same compression. The end product is the electromagnetic stress–energy tensor, the symmetric conserved object that measures the energy, momentum, and stress carried by the field — and, in general relativity, the source term that tells spacetime how to curve.¹

Conventions as before: signature $(-,+,+,+)$, $F^{\mu \nu }={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$ with $F^{0i}=E^i/c$ and $F^{ij}={ϵ}^{ijk}{B}_k$, and $J^{\mu }=(c\rho ,\vec{J})$.

The two tensor equations

Maxwell's equations split into two groups. The two with sources — Gauss's law and the Ampère–Maxwell law — combine into a single four-vector equation,

Check the components. For $\nu =0$, ${\partial }_{\mu }{F}^{0\mu }={\partial }_i{F}^{0i}=\frac{1}{c}\nabla \cdot \vec{E}$, and ${\mu }_0{J}^0={\mu }_{0}c\rho =\frac{1}{c}\rho /{ϵ}_0$, so the equation is Gauss's law $\nabla \cdot \vec{E}=\rho /{ϵ}_0$. For $\nu =i$, the equation is the Ampère–Maxwell law $\nabla \times \vec{B}={\mu }_0\vec{J}+\frac{1}{c^2}{\partial }_t\vec{E}$. Using the antisymmetry $F^{\nu \mu }=-F^{\mu \nu }$, the same content is written ${\partial }_{\mu }{F}^{\mu \nu }=-{\mu }_0{J}^{\nu }$; the index order is a sign convention with no physical weight.

The two without sources — the no-monopole law $\nabla \cdot \vec{B}=0$ and Faraday's law $\nabla \times \vec{E}=-{\partial }_t\vec{B}$ — combine into the identity

the totally antisymmetric derivative ${\partial }_{[\lambda }{F}_{\mu \nu ]}=0$. Equivalently, in terms of the dual tensor, ${\partial }_{\mu }{\tilde{F}}^{\mu \nu }=0$. This group is an identity rather than a dynamical law: it holds automatically whenever $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ comes from a potential, because it is a statement about second derivatives of $A$ commuting. The existence of the four-potential is equivalent to the source-free half of Maxwell's equations.

Charge conservation for free

The sourced equation forces charge conservation with no extra assumption. Take its four-divergence:

The left side contracts the symmetric operator ${\partial }_{\nu }{\partial }_{\mu }$ (symmetric under $\mu \leftrightarrow \nu$ because derivatives commute) with the antisymmetric $F^{\nu \mu }$. A symmetric object contracted with an antisymmetric one vanishes identically. So ${\partial }_{\nu }{J}^{\nu }=0$: charge conservation is a consequence of Maxwell's equations, not an independent postulate. The structure of the field equations guarantees that their source is conserved, a pattern that recurs when the Einstein equation forces conservation of the stress–energy tensor.

The Lorentz force as a four-vector

The force on a charge $q$ with four-velocity $U^{\mu }$ is

with $\tau$ the proper time along the particle's worldline. This one equation holds the entire Lorentz force. Its spatial part, using $U_{\nu }=\gamma (-c,\vec{v})$ lowered, is

and dividing by $\gamma =\mathbf{d}t/\mathbf{d}\tau$ recovers $\mathbf{d}\vec{p}/\mathbf{d}t=q(\vec{E}+\vec{v}\times \vec{B})$. Its time part is the power delivered to the charge,

where $\mathcal{E}=c{p}^0$ is the particle's energy. The force and the power are the space and time parts of one four-force, and only the electric field does work, since $\vec{v}\times \vec{B}\perp \vec{v}$. The four-force is orthogonal to the four-velocity, $U_{\mu }(\mathbf{d}{p}^{\mu }/\mathbf{d}\tau )=q{F}^{\mu \nu }{U}_{\mu }{U}_{\nu }=0$ by the antisymmetry of $F$, which is the covariant statement that the magnetic force changes direction but not speed.

The stress–energy tensor

The energy and momentum carried by the field assemble into a symmetric rank-two tensor. The electromagnetic stress–energy tensor is

It is symmetric, $T^{\mu \nu }=T^{\nu \mu }$, and traceless, $T^{\mu }{}_{\mu }=0$ (the two terms cancel because ${\eta }^{\mu }{}_{\mu }=4$). Its components are the familiar energy and momentum densities of the field:

• $T^{00}$ is the energy density, $$u=\frac{1}{2}{ϵ}_0{E}^2+\frac{1}{2{\mu }_0}{B}^2.$$
• $T^{0i}=T^{i0}$ is the energy flux over $c$, equal to $c$ times the momentum density, $$T^{0i}=\frac{S_i}{c},\vec{S}=\frac{1}{{\mu }_0}\vec{E}\times \vec{B},$$ with $\vec{S}$ the Poynting vector. The field's momentum density is $\vec{g}=\vec{S}/c^2$.
• $T^{ij}$ is the Maxwell stress tensor, the flux of $i$-momentum across a surface with normal $j$, $$T^{ij}=-{ϵ}_0\left(E_i{E}_j-\frac{1}{2}{\delta }_{ij}{E}^2\right)-\frac{1}{{\mu }_0}\left(B_i{B}_j-\frac{1}{2}{\delta }_{ij}{B}^2\right).$$

The block structure is uniform across relativistic field theories: the time–time corner is energy density, the time–space border is energy flux and momentum density, and the space–space block is the flux of momentum, which is stress.

Conservation and momentum flow

Taking the four-divergence of $T^{\mu \nu }$ and using the field equations gives

The right side is the density of four-force the field exerts on the charges. Where there are no charges, $J^{\lambda }=0$ and the stress–energy tensor is conserved,

The two components of this balance are the theorems of classical electromagnetism. The $\nu =0$ component is Poynting's theorem,

energy conservation: the field energy in a region changes by the Poynting flux through its boundary and the work $\vec{J}\cdot \vec{E}$ done on charges. The $\nu =i$ component is momentum conservation: field momentum changes by the Maxwell stress across the boundary plus the Lorentz force on the charges inside. When charges gain energy and momentum, the field loses exactly as much; the tensor $T^{\mu \nu }$ is the ledger.

That electromagnetic fields carry momentum is a measurable fact: radiation pressure is $T^{ij}$ acting on a surface, and the recoil of an antenna is field momentum leaving through $T^{0i}$. A light wave, with $\vec{E}\perp \vec{B}$ and $E=cB$, has energy density $u$ and momentum density $u/c$ along its direction of travel, so it carries momentum $\mathcal{E}/c$ — the massless energy–momentum relation from the foundations module, here derived from the field.

The bridge to gravity

The symmetry, tracelessness, and conservation of $T^{\mu \nu }$ are the properties that make it a source for gravity. General relativity replaces the Newtonian mass density, a single scalar, with the full stress–energy tensor: energy density, momentum density, and stress all gravitate. The Einstein field equation sets the curvature of spacetime proportional to $T^{\mu \nu }$, and the conservation law ${\partial }_{\mu }{T}^{\mu \nu }=0$ — sharpened to a covariant divergence ${\nabla }_{\mu }{T}^{\mu \nu }=0$ in curved spacetime — is forced by the geometry, exactly as charge conservation was forced by the antisymmetry of $F$. The electromagnetic $T^{\mu \nu }$ built here is one instance of the object that the geometry modules place on the right-hand side of Einstein's equation.

Summary

• Maxwell's four equations are two tensor equations: the sourced ${\partial }_{\mu }{F}^{\nu \mu }={\mu }_0{J}^{\nu }$ (Gauss and Ampère–Maxwell) and the identity ${\partial }_{[\lambda }{F}_{\mu \nu ]}=0$ (no monopoles and Faraday), the latter automatic once $F$ comes from a potential.
• Charge conservation ${\partial }_{\nu }{J}^{\nu }=0$ follows from the sourced equation because a symmetric derivative contracts an antisymmetric field to zero.
• The Lorentz force is the four-vector law $\mathbf{d}{p}^{\mu }/\mathbf{d}\tau =q{F}^{\mu \nu }{U}_{\nu }$, holding both the force $q(\vec{E}+\vec{v}\times \vec{B})$ and the power $q\vec{v}\cdot \vec{E}$, with the four-force orthogonal to the four-velocity.
• The stress–energy tensor $T^{\mu \nu }={\mu }_0^{-1}(F^{\mu \alpha }{F}^{\nu }{}_{\alpha }-\frac{1}{4}{\eta }^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta })$ is symmetric and traceless; its entries are energy density, Poynting flux and momentum density, and Maxwell stress. Its conservation ${\partial }_{\mu }{T}^{\mu \nu }=0$ in vacuum is Poynting's theorem and momentum conservation, and it is the source that will curve spacetime.