# Covariant formulation of classical electromagnetism
The covariant formulation of classical electromagnetism is the rewriting of [Maxwell's equations](https://www.edgepedia.org/maxwells-equations) and the [Lorentz force](https://www.edgepedia.org/lorentz-force) law in the four-dimensional tensor language of special relativity, so that their invariance under Lorentz transformations is visible in the form of the equations themselves. In the usual three-vector form, the electric field **E** and magnetic field **B** are split apart and the equations refer to a preferred time slicing; under a Lorentz boost, time mixes with space and **E** mixes with **B**, so the three-vector form is not covariant[1](https://vitenti-physics.github.io/courses/electrodynamics/maxwell-equations-forms.html). The covariant form packages the potentials into a four-vector, the fields into an antisymmetric rank-2 tensor, and the sources into a [four-current](https://www.edgepedia.org/four-current), reducing the four Maxwell equations to two tensor equations[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf).
| Key fact | Detail |
|---|---|
| Four-potential | \( A^{\mu} = (\varphi/c,\ \vec{A}) \), combining the scalar and vector potentials into one four-vector[3](https://www.physics2u.com/vault/lorentz-covariance-of-maxwell.html) |
| Four-current | \( J^{\mu} = (c\rho,\ \vec{J}) \); charge conservation becomes \( \partial_{\mu}J^{\mu} = 0 \)[3](https://www.physics2u.com/vault/lorentz-covariance-of-maxwell.html) |
| Field tensor | \( F^{\mu\nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu} \), an antisymmetric 4×4 matrix with six independent components[4](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/16_7420/Lectures/16_7420_13_Covariant_Electromagnetism.pdf) |
| Maxwell equations | \( \partial_{\mu}F^{\mu\nu} = \mu_0 j^{\nu} \) and \( \partial_{\mu}\tilde{F}^{\mu\nu} = 0 \)[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf) |
| Lorenz gauge | \( \partial_{\mu}A^{\mu} = 0 \), a Lorentz-invariant condition under which the potentials obey \( \Box^2 A_{\mu} = j_{\mu}/\varepsilon_0 \)[5](https://www.feynmanlectures.caltech.edu/II_25.html) |
| Field invariants | \( \tfrac{1}{2}F_{\mu\nu}F^{\mu\nu} = c^2B^2 - E^2 \) and \( \tfrac{1}{4}G_{\mu\nu}F^{\mu\nu} = c\,\vec{E}\cdot\vec{B} \), the same in every inertial frame[6](https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node149.html) |
| Metric convention | Signature \(\{+,-,-,-\}\) is the most commonly used in current physics literature; \(\{-,+,+,+\}\) is an equivalent alternative[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf) |
## Why the three-vector form is not enough
[Classical electromagnetism](https://www.edgepedia.org/classical-electromagnetism) is usually written with a preferred time slicing, with the fields split into the spatial vectors \(\mathbf{E}\) and \(\mathbf{B}\). Under Lorentz transformations time mixes with space, and electric and magnetic fields mix with each other: a point charge at rest produces a purely electric field, while in a frame where the charge moves the same field has both electric and magnetic parts, so the E-field alone cannot be a Lorentz vector[1](https://vitenti-physics.github.io/courses/electrodynamics/maxwell-equations-forms.html) • [7](https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html). The correct object combines both fields into a single rank-2 antisymmetric tensor[7](https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html).
**Charge is the anchor.** [Experiment](https://www.edgepedia.org/experiment) shows that electric charge is a Lorentz scalar, invariant under Lorentz transformations, and a four-dimensional volume element is invariant as well; this is what allows the charge density and current density to be assembled into the four-current \( J^{\alpha} = (c\rho, \mathbf{J}) \), whose four-divergence \( \partial_{\alpha}J^{\alpha} = 0 \) is the continuity equation in manifestly covariant form[8](https://webhome.phy.duke.edu/~rgb/Class/Electrodynamics/Electrodynamics/node172.html). Because \( \partial_{\mu}J^{\mu} \) is an invariant scalar, if charge is conserved in one inertial frame it is conserved in all frames[5](https://www.feynmanlectures.caltech.edu/II_25.html).
## The four-potential and the field tensor
The scalar potential \(\varphi\) and vector potential \(\mathbf{A}\) belong together as components of the four-potential \( A^{\mu} = (\varphi/c, \mathbf{A}) \)[3](https://www.physics2u.com/vault/lorentz-covariance-of-maxwell.html). The electromagnetic field tensor is the four-dimensional curl of this object[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf):
\[ F^{\mu\nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}. \]
Because it is built from Lorentz vectors, it necessarily transforms as a tensor, \( F^{\mu\nu} = \Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma}F^{\rho\sigma} \)[4](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/16_7420/Lectures/16_7420_13_Covariant_Electromagnetism.pdf). It is antisymmetric, so its four diagonal elements vanish and the remaining elements occur in pairs with opposite signs, leaving six independent components: three encode the magnetic field (\( F_{23} = B_x \) and cyclic permutations) and three encode the electric field[4](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/16_7420/Lectures/16_7420_13_Covariant_Electromagnetism.pdf). In SI convention the contravariant components carry \( E_x/c \) in the time-space slots and \( B_z, B_y, B_x \) in the space-space slots[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf).
Fields derived from a vector potential automatically satisfy the two source-free Maxwell equations, which is one reason the potential formulation is natural[7](https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html).
## Maxwell's equations as two tensor equations
In vacuum, or for the microscopic equations without macroscopic material descriptions, Maxwell's equations reduce to two tensor statements. The two source equations become[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf)
\[ \partial_{\mu}F^{\mu\nu} = \mu_0 j^{\nu}, \]
whose \(\nu = 0\) component reproduces \( \nabla\cdot\mathbf{E} = \rho/\varepsilon_0 \) and whose spatial components reproduce \( \nabla\times\mathbf{B} = \mu_0\mathbf{j} + \varepsilon_0\mu_0\,\partial_t\mathbf{E} \)[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf). The two homogeneous equations, \( \nabla\times\mathbf{E} = -\partial_t\mathbf{B} \) and \( \nabla\cdot\mathbf{B} = 0 \), are packaged with the dual tensor \( \tilde{F}^{\mu\nu} = \tfrac{1}{2}\epsilon^{\mu\nu\kappa\lambda}F_{\kappa\lambda} \) as[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf)
\[ \partial_{\mu}\tilde{F}^{\mu\nu} = 0, \]
which is equivalent, with no summation, to the cyclic identity \( \partial_{\kappa}F_{\mu\nu} + \partial_{\mu}F_{\nu\kappa} + \partial_{\nu}F_{\kappa\mu} = 0 \)[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf). The dual tensor is obtained from the ordinary one by the duality transformation \( \mathbf{E} \to c^2\mathbf{B} \) and \( \mathbf{B} \to -\mathbf{E} \)[2](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf).
## The Lorenz gauge and the wave equation
The equations take their simplest form in the Lorenz gauge (named for Ludvig Lorenz, not [Hendrik Lorentz](https://www.edgepedia.org/hendrik-lorentz)), the condition (1/c²)∂φ/∂t + ∇·A = 0, which says that the four-divergence of \( A_{\mu} \) is zero[5](https://www.feynmanlectures.caltech.edu/II_25.html) • [12](https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node144.html). This condition is invariant, so Maxwell's equations keep the same form in every frame[5](https://www.feynmanlectures.caltech.edu/II_25.html). In the gauge, the potentials satisfy a single four-vector wave equation[5](https://www.feynmanlectures.caltech.edu/II_25.html) • [9](https://farside.ph.utexas.edu/teaching/em/lectures/node117.html)
\[ \Box^2 A_{\mu} = \frac{j_{\mu}}{\varepsilon_0}, \qquad \text{equivalently} \qquad \Box\,\Phi^{\mu} = \frac{J^{\mu}}{c\,\varepsilon_0}, \]
where \( \Box^2 = \partial^2/\partial t^2 - \nabla^2 \) is the invariant d'Alembertian operator[5](https://www.feynmanlectures.caltech.edu/II_25.html). All the field equations of classical electromagnetism are thus summed up in one four-vector equation[9](https://farside.ph.utexas.edu/teaching/em/lectures/node117.html).
## Invariants and the transformation of fields
Two combinations of the fields have the same value in every inertial frame[6](https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node149.html):
\[ \frac{1}{2}F_{\mu\nu}F^{\mu\nu} = c^2B^2 - E^2, \qquad \frac{1}{4}G_{\mu\nu}F^{\mu\nu} = c\,\vec{E}\cdot\vec{B}, \]
the first a proper scalar and the second a pseudoscalar[6](https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node149.html). The sign of \( E^2 - c^2B^2 \) determines whether a field is electrically dominated, magnetically dominated, or balanced, and this classification holds identically in all frames; consequently, boosting cannot transform a field dominated by one component into one dominated by the other, while a field for which both invariants disappear is a radiation field, characterized by \( \mathbf{E} \perp \mathbf{B} \) and \( E = cB \) regardless of frame[10](https://notes.amittai.studio/relativity/covariant-electrodynamics/the-electromagnetic-field-tensor).
Applying the tensor transformation rule to a boost along \(\mathbf{v}\) yields the standard field transformation laws: components parallel to \(\mathbf{v}\) are unchanged, \( E_{\parallel}' = E_{\parallel} \) and \( B_{\parallel}' = B_{\parallel} \), while \( \mathbf{E}_{\perp}' = \gamma(\mathbf{E}_{\perp} + \mathbf{v}\times\mathbf{B}) \) and \( \mathbf{B}_{\perp}' = \gamma(\mathbf{B}_{\perp} - \mathbf{v}\times\mathbf{E}/c^2) \)[6](https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node149.html).
## Force, energy, and momentum
The Lorentz force law also becomes a tensor equation. The four-force density on a charge distribution is \( f_{\mu} = c^{-1}F_{\mu\nu}J^{\nu} \), whose components are \( f^{\mu} = (\rho\mathbf{E} + \mathbf{j}\times\mathbf{B},\ \mathbf{E}\cdot\mathbf{j}/c) \), the spatial part being the familiar Lorentz force density[11](https://farside.ph.utexas.edu/teaching/em/lectures/node128.html). Combining this with \( \partial_{\mu}F^{\mu\nu} = J^{\nu}/(c\varepsilon_0) \) gives \( f^{\nu} = -\partial_{\mu}T^{\mu\nu} \), where \( T^{\mu\nu} \) is the symmetric electromagnetic energy (stress-energy) tensor[11](https://farside.ph.utexas.edu/teaching/em/lectures/node128.html). Its \( T^{44} \) component is the energy density \( U = \varepsilon_0E^2/2 + B^2/(2\mu_0) \), the mixed components give the Poynting flux \( \mathbf{u} = \mathbf{E}\times\mathbf{B}/\mu_0 \), and the spatial block is the [Maxwell stress tensor](https://www.edgepedia.org/maxwell-stress-tensor); the time and space parts of \( f^{\nu} = -\partial_{\mu}T^{\mu\nu} \) are respectively the energy and momentum conservation equations for the field[11](https://farside.ph.utexas.edu/teaching/em/lectures/node128.html).
## The action
The whole theory follows from a Lorentz-invariant action[3](https://www.physics2u.com/vault/lorentz-covariance-of-maxwell.html)
\[ S = -\frac{1}{4\mu_0}\int F_{\mu\nu}F^{\mu\nu}\,d^4x - \int A_{\mu}J^{\mu}\,d^4x, \]
in which charge conservation drops out as an algebraic identity and gauge invariance is the statement that \( F^{\mu\nu} \) is unchanged by \( A^{\mu} \to A^{\mu} + \partial^{\mu}\chi \)[3](https://www.physics2u.com/vault/lorentz-covariance-of-maxwell.html). This structure, a field strength tensor built from a gauge potential coupled to a conserved current, served as the template for later gauge theories[3](https://www.physics2u.com/vault/lorentz-covariance-of-maxwell.html).
## References
1. [Maxwell Equations in Covariant Form, Physics Lecture Notes](https://vitenti-physics.github.io/courses/electrodynamics/maxwell-equations-forms.html)
2. [Covariant classical electrodynamics (EMFT textbook, Lund)](https://www.purco.qc.ca/ftp/Researchers/EMFT_Book.pdf)
3. [Covariant Formulation of Electrodynamics, physics2u](https://www.physics2u.com/vault/lorentz-covariance-of-maxwell.html)
4. [Lorentz Invariant Formulation of Electromagnetism, University of Virginia lecture notes](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/16_7420/Lectures/16_7420_13_Covariant_Electromagnetism.pdf)
5. [The Feynman Lectures on Physics Vol. II Ch. 25: Electrodynamics in Relativistic Notation](https://www.feynmanlectures.caltech.edu/II_25.html)
6. [Transformation of Fields, R. Fitzpatrick, UT Austin electromagnetism notes](https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node149.html)
7. [The Electromagnetic Field Tensor, UCSD Physics 130 notes](https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html)
8. [Covariant Formulation of Electrodynamics, Duke University Electrodynamics notes](https://webhome.phy.duke.edu/~rgb/Class/Electrodynamics/Electrodynamics/node172.html)
9. [The potential 4-vector, R. Fitzpatrick, UT Austin electromagnetism notes](https://farside.ph.utexas.edu/teaching/em/lectures/node117.html)
10. [The Electromagnetic Field Tensor, Amittai study notes](https://notes.amittai.studio/relativity/covariant-electrodynamics/the-electromagnetic-field-tensor)
11. [The electromagnetic energy tensor, R. Fitzpatrick, UT Austin electromagnetism notes](https://farside.ph.utexas.edu/teaching/em/lectures/node128.html)
12. [Gauge Invariance](https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node144.html)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Maxwell's equations and field formulations › Covariant formulation of electromagnetism*
*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026; Oct 25, 2026 · Last review: Oct 15, 2026*
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