---
title: Covariant Maxwell and the Stress–Energy Tensor
module: Covariant Electromagnetism
moduleNumber: 4
lessonNumber: 4
order: 404
summary: >
  Maxwell's four equations collapse into two tensor equations, one sourced by the
  four-current and one an identity on the field strength, with charge conservation
  automatic. The Lorentz force becomes a four-vector law, and the field's energy,
  momentum, and stress assemble into a symmetric, conserved stress–energy tensor —
  the object that will source gravity.
topics: [Covariant Electromagnetism]
draft: false
sources:
  - book: Schutz
    ref: "Ch. 4 — Maxwell's equations in covariant form; the electromagnetic stress–energy tensor"
  - book: Carroll
    ref: "Lecture Notes on General Relativity §1 — Special Relativity and Flat Spacetime (Maxwell's equations, energy–momentum tensor), arXiv:gr-qc/9712019"
---

The [field-strength
tensor](/relativity/covariant-electrodynamics/the-electromagnetic-field-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.[^carroll4]

Conventions as before: signature $(-,+,+,+)$, $F^{\mu\nu} = \partial^\mu A^\nu -
\partial^\nu A^\mu$ with $F^{0i} = E^i/c$ and $F^{ij} = \epsilon^{ijk}B_k$, 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,

$$
\partial_\mu F^{\nu\mu} = \mu_0 J^\nu.
$$

Check the components. For $\nu = 0$, $\partial_\mu F^{0\mu} = \partial_i F^{0i} =
\tfrac{1}{c}\nabla \cdot \vec E$, and $\mu_0 J^0 = \mu_0 c\rho = \tfrac{1}{c}
\rho/\epsilon_0$, so the equation is Gauss's law $\nabla \cdot \vec E =
\rho/\epsilon_0$. For $\nu = i$, the equation is the Ampère–Maxwell law $\nabla
\times \vec B = \mu_0 \vec J + \tfrac{1}{c^2}\partial_t \vec E$. 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

$$
\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu
F_{\lambda\mu} = 0,
$$

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.

> **Result.** Maxwell's four equations are the two tensor equations
> $$
> \partial_\mu F^{\nu\mu} = \mu_0 J^\nu, \qquad \partial_{[\lambda} F_{\mu\nu]} =
> 0.
> $$
> The first is sourced by the four-current; the second is the integrability
> condition that lets $F$ derive from a potential. Both are manifestly Lorentz
> covariant: they equate tensors, so if they hold in one inertial frame they hold
> in all.

$$
% caption: The two covariant equations reproduce the familiar four: the sourced
% equation carries Gauss's law and the Ampère–Maxwell law, and the identity on the
% field strength carries the no-monopole law and Faraday's law.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % sourced equation box
  \node[draw, acc, very thick, fill=acc!12, minimum width=3.2cm, minimum height=0.9cm]
    (src) at (0,2.4) {sourced tensor equation};
  \node[draw, black, minimum width=2.6cm] (g1) at (5.0,3.1) {Gauss law};
  \node[draw, black, minimum width=2.6cm] (g2) at (5.0,1.7) {Ampere and Maxwell};
  \draw[->, black] (src) -- (g1);
  \draw[->, black] (src) -- (g2);
  % identity box
  \node[draw, acc, very thick, fill=acc!12, minimum width=3.2cm, minimum height=0.9cm]
    (id) at (0,0.0) {strength-tensor identity};
  \node[draw, black, minimum width=2.6cm] (h1) at (5.0,0.7) {no monopoles};
  \node[draw, black, minimum width=2.6cm] (h2) at (5.0,-0.7) {Faraday law};
  \draw[->, black] (id) -- (h1);
  \draw[->, black] (id) -- (h2);
\end{tikzpicture}
$$

## Charge conservation for free

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

$$
\partial_\nu \partial_\mu F^{\nu\mu} = \mu_0\, \partial_\nu J^\nu.
$$

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

$$
\frac{\d p^\mu}{\d \tau} = q\, F^{\mu\nu} U_\nu,
$$

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

$$
\frac{\d p^i}{\d \tau} = q\big(F^{i0}U_0 + F^{ij}U_j\big) = q\gamma\big(\vec E +
\vec v \times \vec B\big)_i,
$$

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

$$
\frac{\d p^0}{\d \tau} = q F^{0i} U_i = \frac{q\gamma}{c}\,\vec v \cdot \vec E
\quad\Longrightarrow\quad \frac{\d \mathcal E}{\d t} = q\,\vec v \cdot \vec E,
$$

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, $U_\mu (\d p^\mu/\d\tau) = q F^{\mu\nu} U_\mu U_\nu = 0$ 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

$$
T^{\mu\nu} = \frac{1}{\mu_0}\left(F^{\mu\alpha} F^\nu{}_\alpha - \frac{1}{4}
\eta^{\mu\nu} F_{\alpha\beta} F^{\alpha\beta}\right).
$$

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**,
  $$
  u = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2.
  $$
- **$T^{0i} = T^{i0}$ is the energy flux over $c$, equal to $c$ times the
  momentum density**,
  $$
  T^{0i} = \frac{S_i}{c}, \qquad \vec S = \frac{1}{\mu_0}\,\vec E \times \vec B,
  $$
  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 $j$,
  $$
  T^{ij} = -\epsilon_0\!\left(E_i E_j - \tfrac{1}{2}\delta_{ij} E^2\right) -
  \frac{1}{\mu_0}\!\left(B_i B_j - \tfrac{1}{2}\delta_{ij} B^2\right).
  $$

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.

$$
% caption: The block structure of the stress–energy tensor: energy density in the
% corner, energy flux and momentum density along the border, and the momentum-flux
% (stress) block in the spatial part.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black] (0,0) rectangle (4.2,4.2);
  \draw[black] (1.2,0) -- (1.2,4.2);
  \draw[black] (0,3.0) -- (4.2,3.0);
  % energy density corner
  \fill[acc!16] (0,3.0) rectangle (1.2,4.2);
  \node[acc, align=center] at (0.6,3.6) {energy\\density};
  % energy flux / momentum density border
  \fill[black!8] (1.2,3.0) rectangle (4.2,4.2);
  \fill[black!8] (0,0) rectangle (1.2,3.0);
  \node[black!70, align=center] at (2.7,3.6) {energy transport /\\momentum density};
  \node[black!70, align=center, rotate=90] at (0.6,1.5) {momentum density};
  % stress block
  \node[black!70, align=center] at (2.7,1.5) {momentum transport\\(Maxwell stress)};
  % labels
  \node[black, anchor=south] at (0.6,4.3) {t};
  \node[black, anchor=south] at (2.7,4.3) {x y z};
\end{tikzpicture}
$$

## Conservation and momentum flow

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

$$
\partial_\mu T^{\mu\nu} = -F^\nu{}_\lambda J^\lambda.
$$

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,

$$
\partial_\mu T^{\mu\nu} = 0.
$$

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

$$
\frac{\partial u}{\partial t} + \nabla \cdot \vec S = -\vec J \cdot \vec E,
$$

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.

$$
% caption: Conservation as a flux balance on a box: the energy inside changes by
% the Poynting flux through the walls, and the momentum inside changes by the
% Maxwell stress across them, with any imbalance going to the charges as work and
% force.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, fill=acc!10] (0,0) rectangle (3.4,2.6);
  \node[black!70, align=center] at (1.7,1.3) {stored energy\\and momentum};
  % Poynting flux out
  \draw[->, very thick] (3.45,1.9) -- (4.7,1.9) node[right, black!70] {energy transport};
  \draw[->, very thick] (-0.05,1.9) -- (-1.3,1.9) node[left, black!70] {energy transport};
  % stress on walls
  \draw[->, black, thick] (3.45,0.6) -- (4.5,0.6) node[right, black!70] {stress};
  \draw[->, black, thick] (1.7,2.65) -- (1.7,3.6) node[above, black!70] {momentum transport};
\end{tikzpicture}
$$

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](/relativity/foundations/relativistic-momentum-energy), here derived from
the field.

$$
% caption: In a light wave the electric and magnetic fields are perpendicular and
% equal in the sense E equals c B; their cross product, the Poynting vector, points
% along the propagation direction and carries momentum density equal to the energy
% density over c.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, very thick] (0,0) -- (0,2.0) node[above] {E};
  \draw[->, black, very thick] (0,0) -- (1.7,-0.9) node[below, black!70] {B};
  \draw[->, acc, thick] (0,0) -- (3.0,0.6) node[right, acc] {S: energy and momentum};
  \node[black, anchor=west] at (0.2,1.2) {perpendicular, E = cB};
\end{tikzpicture}
$$

## 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](/relativity/curved-spacetime/the-einstein-field-equations) 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 $\d p^\mu/\d\tau = q F^{\mu\nu} U_\nu$,
  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 $T^{\mu\nu} = \mu_0^{-1}(F^{\mu\alpha}F^\nu{}_\alpha -
  \tfrac14 \eta^{\mu\nu} F_{\alpha\beta}F^{\alpha\beta})$ 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.

[^carroll4]: Carroll, _Lecture Notes on General Relativity_, §1 (Maxwell's
equations and the energy–momentum tensor), arXiv:gr-qc/9712019; Schutz, _A First
Course in General Relativity_, Ch. 4 (covariant Maxwell equations and the
electromagnetic stress–energy tensor). The inhomogeneous equation is written
$\partial_\mu F^{\nu\mu} = \mu_0 J^\nu$ following Carroll's index order.