# Maxwell's equations in covariant form
Maxwell's equations in covariant form are two tensor equations written with the electromagnetic field tensor F and the four-current J: the inhomogeneous equation \(\partial_{\mu} F^{\mu\nu} = \mu_0 J^{\nu}\) and the homogeneous, Bianchi-type equation \(\partial_{[\alpha} F_{\beta\gamma]} = 0\). Together they replace the four 3-vector Maxwell equations, and they display the theory's Lorentz invariance, whereas the 3-vector forms are valid only in flat spacetime with a [Cartesian coordinate system](https://www.edgepedia.org/cartesian-coordinate-system).[4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations) This article covers the two tensor equations, the four-current and its conservation, and how the constants \(\mu_0\), \(\varepsilon_0\), and c sit in different unit systems; the construction and components of the field tensor, the four-potential and gauge structure, media, and the Lagrangian formulation are treated in sibling articles. (For the overall framework, see [Covariant formulation of classical electromagnetism](https://www.edgepedia.org/covariant-formulation-of-classical-electromagnetism).)
| Key fact | Value | Meaning |
|---|---|---|
| Inhomogeneous equation | \(\partial_{\mu} F^{\mu\nu} = \mu_0 J^{\nu}\) (SI form) | Encodes Gauss's law and Ampère's law in one tensor statement[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) |
| Homogeneous equation | ∂_[α F_βγ] = 0 (cyclic form) | Encodes Gauss's law for magnetism and Faraday's law as an identity, not a sourced equation[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) • [3](https://encyclopediaofmath.org/wiki/Maxwell_equations) |
| Four-current | \(J^{\mu} = \rho_{0} \cdot U^{\mu}\), time component \(c \cdot \gamma \cdot \rho_{0}\) | Four-vector built from rest-frame charge density \(\rho_{0}\) and 4-velocity \(U^{\mu}\)[2](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/18_7420/18_7420_17_Covariant_Electromagnetism.pdf) |
| Charge conservation | \(\partial_{\mu} J^{\mu} = 0\), automatically | Follows from the antisymmetry of F in the inhomogeneous equation[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) |
| Compression | Eight scalar 3-vector equations → two tensor equations | Nothing physical is lost; the split is ν = 4 (Gauss) versus ν = 1,2,3 (Ampère)[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) • [4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations) |
| Covariant extension | \((1/\sqrt{-g})\partial_{\kappa}(\sqrt{-g}\,F^{\kappa\lambda}) = J^{\lambda}\) | The tensor form generalizes to curved spacetime, where 3-vector forms fail[4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations) |
## The four-current and its conservation
The source of the inhomogeneous equation is the electric four-current \(J^{\mu}\), defined as \(J^{\mu} = \rho_{0} \cdot U^{\mu}\), where \(\rho_{0}\) is the charge density in the charge's rest frame and \(U^{\mu}\) the local 4-velocity. Its time component in a generally moving frame is \(c \cdot \gamma \cdot \rho_{0}\), so in the usual real-time convention \(J^{\mu} = (c \cdot \rho, \, \mathbf{j})\) with the factors of c fixed by this rest-frame normalization.[2](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/18_7420/18_7420_17_Covariant_Electromagnetism.pdf) Because \(J^{\mu}\) is a four-vector, current density transforms together with charge density under Lorentz boosts, which is precisely the mixing the 3-vector formulation treats separately.
Charge conservation comes out of the field equation rather than being added to it. Taking the divergence of \(\partial_{\mu} F^{\mu\nu} = \mu_0 J^{\nu}\) gives \(\partial_{\nu} J^{\nu} \propto \partial_{\nu} \partial_{\mu} F^{\mu\nu}\), and this double divergence vanishes identically because \(\partial_{\mu} \partial_{\nu}\) is symmetric in its two indices while \(F^{\mu\nu}\) is antisymmetric; one of the standard references calls this the covariant counterpart of Maxwell's displacement-current argument.[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) In curved spacetime the same statement reads \(\nabla_{\lambda} J^{\lambda} = 0\), equivalently \(\partial_{\lambda}(\sqrt{-g} \cdot J^{\lambda}) = 0\) after multiplying by the metric determinant.[4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations)
## The inhomogeneous equation ∂_μ F^{μν} = μ0 J^ν
The inhomogeneous equation packages [Gauss's law](https://www.edgepedia.org/gausss-law) and the Ampère–Maxwell law. In flat Cartesian coordinates, with the SI equation \(\partial_{\mu} F^{\mu\nu} = \mu_0 J^{\nu}\), its components are \(\nabla \cdot \mathbf{E} = \rho/\varepsilon_0\) and \(\nabla \times \mathbf{B} = \mu_0 (\mathbf{j} + \varepsilon_0 \partial \mathbf{E}/\partial t)\): the time-like component ν = 4 gives Gauss's law, and the three spatial components ν = 1,2,3 give Ampère's law with Maxwell's [displacement current](https://www.edgepedia.org/displacement-current). Nothing is lost in the compression; the single tensor equation contains exactly the same content as the two vector equations.[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) • [4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations)
The constant on the right-hand side depends on the unit system. In SI units the equation reads \(\partial_{\mu} F^{\mu\nu} = \mu_0 J^{\nu}\), with \(\mu_0 \approx 4\pi \cdot 10^{-7}\) H/m, a quantity that, since the 2019 revision of the SI, is no longer fixed by definition but must be determined experimentally (with \(\varepsilon_0 = 1/(\mu_0 c^2)\) tied to its measured value), and the speed of light c = 299 792 458 m/s exact;[3](https://encyclopediaofmath.org/wiki/Maxwell_equations) • [7](https://pmc.ncbi.nlm.nih.gov/articles/PMC5907514/) in vacuum the excitation and field tensors are proportional, \(G_{kl} = F_{kl}/\mu_0\), so a single tensor suffices.[3](https://encyclopediaofmath.org/wiki/Maxwell_equations) Many relativistic treatments instead rewrite the equations in [Gaussian units](https://www.edgepedia.org/gaussian-units), where the constants are absorbed and (in free space) \(\mu_0 = \varepsilon_0 = 1\), so the equation reads \(E^{\mu\nu}_{,\mu} = (4\pi/c) J^{\nu}\).[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) • [5](https://physics.uwo.ca/~mhoude2/courses/phy502b/Covariant_formulation.pdf) The choice is not neutral: one expert reference argues that the SI system is "extremely inconvenient" for the relativistic formulation because it masks the pseudo-symmetry between the electric and magnetic fields, which is why covariant derivations commonly open by switching to Gaussian units.[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism)
## The homogeneous (Bianchi-type) equation
The second tensor equation, ∂_[α F_βγ] = 0, is the cyclic identity
\(\partial F_{kl}/\partial x_{m} + \partial F_{lm}/\partial x_{k} + \partial F_{mk}/\partial x_{l} = 0,\)
with antisymmetrization over the three free indices. Assigning the values 1,2,3 to (μ,ν,σ) yields Gauss's law for magnetism, and the cyclic sets 2,3,4; 3,4,1; 4,1,2 yield the three components of Faraday's law, so all four homogeneous Maxwell equations sit in this one statement.[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism) • [3](https://encyclopediaofmath.org/wiki/Maxwell_equations)
The field tensor F is built from derivatives of a potential, \(A_{\nu}\), as \(F^{\mu\nu} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu}\).[4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations) In Lorenz gauge with vanishing sources the field equations reduce to homogeneous wave equations for the field strengths themselves.[3](https://encyclopediaofmath.org/wiki/Maxwell_equations)
## Lorentz invariance made manifest
The chief practical gain of the tensor form is that Lorentz invariance becomes a property of the notation. If both sides of \(\partial_{\mu} F^{\mu\nu} = \mu_{0} J^{\nu}\) transform as four-vectors, any observer related by a [Lorentz transformation](https://www.edgepedia.org/lorentz-transformation) sees the same equation, whereas the 3-vector version requires checking how **E** and **B** mix and how **ρ** and **j** mix under each boost. The 3-vector forms are not wrong; they are simply valid only in flat spacetime with a Cartesian coordinate system.[4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations)
The tensor form also generalizes. In a generally covariant setting the inhomogeneous equation becomes \((1/\sqrt{-g})\partial_{\kappa}(\sqrt{-g}\, F^{\kappa\lambda}) = J^{\lambda}\) and the homogeneous one \(\partial_{\mu}(\varepsilon^{\mu\nu\kappa\lambda}F_{\kappa\lambda}) = 0\), and neither equation contains any [Christoffel symbols](https://www.edgepedia.org/christoffel-symbols). This is what allows the same equations to be posed on curved spacetimes, something the 3-vector forms cannot express without modification.[4](https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations)
## Conventions and pedagogy compared
Textbooks differ along several axes, and the equations look different before they disagree.
Real-time versus ict coordinates. Modern treatments use a real time coordinate and \(J^{\mu} = \rho_{0} \cdot U^{\mu}\) with time component \(c \gamma \rho_{0}\).[2](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/18_7420/18_7420_17_Covariant_Electromagnetism.pdf) Older references write the time coordinate as ict and the fourth current component as \(j_{4} = i \cdot c \cdot \rho\), which changes factors of i and c throughout.[3](https://encyclopediaofmath.org/wiki/Maxwell_equations)
One tensor or two. In vacuum the excitation tensor is proportional to \(F\) (\(G_{kl} = F_{kl}/\mu_0\) in SI), so a single tensor suffices.[3](https://encyclopediaofmath.org/wiki/Maxwell_equations) Treatments that anticipate media keep two tensors, \(F\) and \(G\), from the start. Gaussian-unit presentations absorb the constants so the inhomogeneous equation reads \(E^{\mu\nu}_{,\mu} = (4\pi/c) \cdot J^{\nu}\).[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism)
Tensor first, units first. Graduate courses typically rewrite the Maxwell equations in Gaussian units before discussing Lorentz invariance, so the unit choice precedes and shapes the covariant notation.[5](https://physics.uwo.ca/~mhoude2/courses/phy502b/Covariant_formulation.pdf) Recent pedagogical publishing retains this approach: a current IOP monograph chapter introduces tensor calculus and then casts the vacuum Maxwell equations in tensorial form.[6](https://iopscience.iop.org/book/mono/978-0-7503-5884-2/chapter/bk978-0-7503-5884-2ch18)
Exact solutions. In the Lorenz gauge the field equations decouple into four-potential wave equations, □Φ_µ = (4π/c) J_µ in Gaussian units, with the retarded Green-function solution Φ_µ(P) = (1/c) ∫ [J_µ] dV / r, where the bracketed current is evaluated at the retarded time. The field is thereby "built up" at the speed of light.[1](http://scholarpedia.org/article/Special_relativity:_electromagnetism)
## References
1. Special relativity: electromagnetism, Scholarpedia. http://scholarpedia.org/article/Special_relativity:_electromagnetism
2. 17. Lorentz Invariant Formulation of Electromagnetism, University of Virginia graduate lecture notes. https://galileoandeinstein.phys.virginia.edu/Elec_Mag/18_7420/18_7420_17_Covariant_Electromagnetism.pdf
3. Maxwell equations, Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Maxwell_equations
4. A covariant form of Maxwell's equations, Viktor T. Toth physics notes. https://vttoth.com/CMS/physics-notes/289-a-covariant-form-of-maxwell-s-equations
5. Chapter 7. Covariant Formulation of Electrodynamics, Western University graduate course. https://physics.uwo.ca/~mhoude2/courses/phy502b/Covariant_formulation.pdf
6. Covariant formulation of Maxwell's equations, IOP Publishing book chapter. https://iopscience.iop.org/book/mono/978-0-7503-5884-2/chapter/bk978-0-7503-5884-2ch18
7. PMC5907514 (pmc.ncbi.nlm.nih.gov). https://pmc.ncbi.nlm.nih.gov/articles/PMC5907514/
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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 › Maxwell's equations in covariant form*
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