# Electromagnetic tensor
The electromagnetic tensor, also called the electromagnetic field tensor or field-strength tensor, is the antisymmetric rank-2 tensor \( F^{\mu\nu} \) that combines the three components of the electric field and the three components of the magnetic field into a single Lorentz-covariant object in four-dimensional spacetime[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html). It exists because electric and magnetic fields are not separately Lorentz vectors: a charge at rest produces only an electric field, while the same charge seen from a moving frame produces both an electric and a magnetic field, so the two fields must be packaged together to transform correctly under Lorentz transformations[2](https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html).
| Key fact | Detail |
|---|---|
| Definition | \( F^{\mu\nu} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu} \), built from the four-potential \( A^{\mu} = (\phi/c, \vec{A}) \)[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html) |
| Structure | Antisymmetric 4×4 matrix: \( F_{\mu\nu} = -F_{\nu\mu} \), four zero diagonal elements, six independent components[3](https://farside.ph.utexas.edu/teaching/em/lectures/node121.html) |
| Component content | Space-space components carry the magnetic field; time-space components carry the electric field (as \( E_i/c \) in the common convention)[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html) |
| Transformation | \( F'^{\mu\nu} = \Lambda^{\mu}{}_{\alpha} \Lambda^{\nu}{}_{\beta} F^{\alpha\beta} \); field components parallel to a boost are unchanged, perpendicular components mix[4](https://farside.ph.utexas.edu/teaching/em/lectures/node123.html) |
| Invariants | \( \tfrac{1}{2} F_{\mu\nu} F^{\mu\nu} = c^2 B^2 - E^2 \) (proper scalar) and \( \tfrac{1}{4} G_{\mu\nu} F^{\mu\nu} = c\, \vec{E} \cdot \vec{B} \) (pseudoscalar)[4](https://farside.ph.utexas.edu/teaching/em/lectures/node123.html) |
| Maxwell equations | \( \partial_\mu F^{\mu\nu} = J^{\nu}/(c\,\epsilon_0) \) and \( \partial_\mu G^{\mu\nu} = 0 \), two tensor equations replacing the four vector equations[5](https://farside.ph.utexas.edu/teaching/em/lectures/node122.html) |
| Gauge invariance | A gauge shift of the potential leaves \( F^{\mu\nu} \) unchanged[6](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/EMTensor.html) |
## Definition from the four-potential
The tensor is defined as the antisymmetrized derivative of the [electromagnetic four-potential](https://www.edgepedia.org/electromagnetic-four-potential) \( A^{\mu} \), which combines the scalar potential \( \phi \) and the vector potential \( \vec{A} \):
\[ F^{\mu\nu} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu}. \]
Because it is constructed from Lorentz vectors, it necessarily transforms as a tensor, \( F^{\rho'\sigma'} = \Lambda^{\rho'}{}_{\mu} \Lambda^{\sigma'}{}_{\nu} F^{\mu\nu} \)[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html). Antisymmetry follows immediately from the definition, \( F_{\mu\nu} = -F_{\nu\mu} \), so the four diagonal elements vanish and only six independent components remain[3](https://farside.ph.utexas.edu/teaching/em/lectures/node121.html).
In the convention used, for example, by the [University of Virginia](https://www.edgepedia.org/university-of-virginia) lecture notes, the contravariant matrix reads
\[ F^{\mu\nu} = \begin{pmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & B_z & -B_y \\ -E_y/c & -B_z & 0 & B_x \\ -E_z/c & B_y & -B_x & 0 \end{pmatrix}. \]
The three space-space components \( (F^{23}, F^{31}, F^{12}) \) are the magnetic field components \( (B_x, B_y, B_z) \), while the time-space components carry \( \vec{E}/c \)[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html). Other textbooks place \( cB \) or \( E \) in these slots instead, depending on the metric signature and unit conventions; the physics is the same, but the component layout differs between sources[3](https://farside.ph.utexas.edu/teaching/em/lectures/node121.html).
**Gauge invariance.** The vector potential is not unique: adding the gradient of any scalar function of space and time to it leaves the fields unchanged, and the tensor \( F^{\mu\nu} \), being built from derivatives of \( A^{\mu} \), is unchanged under such a gauge shift[6](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/EMTensor.html). This makes \( F^{\mu\nu} \), rather than \( A^{\mu} \) itself, the directly physical field quantity in the covariant formulation.
## Lorentz transformation of the fields
Applying the tensor transformation law to a Lorentz boost yields the standard rules for how electric and magnetic fields change between inertial frames. With \( \vec{v} \) the relative velocity between the frames, the components parallel to the boost are unchanged,
\[ E'_{\parallel} = E_{\parallel}, \qquad B'_{\parallel} = B_{\parallel}, \]
while the perpendicular components mix,
\[ \vec{E}'_{\perp} = \gamma \left( \vec{E}_{\perp} + \vec{v} \times \vec{B} \right), \qquad \vec{B}'_{\perp} = \gamma \left( \vec{B}_{\perp} - \frac{\vec{v} \times \vec{E}}{c^2} \right), \]
where \( \gamma = 1/\sqrt{1 - v^2/c^2} \)[4](https://farside.ph.utexas.edu/teaching/em/lectures/node123.html). The Feynman Lectures give the same content component by component for a boost along \( x \): \( E_x' = E_x \), \( E_y' = (E_y - v B_z)/\sqrt{1-v^2} \), \( B_z' = (B_z - v E_y)/\sqrt{1-v^2} \), and so on, in units with \( c = 1 \)[7](https://www.feynmanlectures.caltech.edu/II_26.html).
The mixing has a direct physical reading. A point charge at rest has a purely radial electric field; in a frame where the charge moves, the transformed tensor shows a magnetic field circling the line of motion, with \( \vec{B} = \vec{v} \times \vec{E}/c^2 \) for slow motion[7](https://www.feynmanlectures.caltech.edu/II_26.html). Conversely, an observer moving through a pure static magnetic field sees an electric field \( \vec{E}' = \vec{v} \times \vec{B}' \) appear, which is the relativistic account of motional induction, for example the charge separation induced across a wire moving through a field[7](https://www.feynmanlectures.caltech.edu/II_26.html).
## Invariants and the dual tensor
Two scalar combinations of the field are the same in every inertial frame. The contraction of the tensor with itself gives
\[ \frac{1}{2} F_{\mu\nu} F^{\mu\nu} = c^2 B^2 - E^2, \]
a proper scalar, and the contraction with the dual tensor gives
\[ \frac{1}{4} G_{\mu\nu} F^{\mu\nu} = c\, \vec{E} \cdot \vec{B}, \]
a pseudoscalar[4](https://farside.ph.utexas.edu/teaching/em/lectures/node123.html). In the \( (-,+,+,+) \) convention these appear as \( B^2 - E^2/c^2 \) and \( \vec{E} \cdot \vec{B} \)[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html). All other Lorentz-invariant combinations of the fields can be expressed in terms of these two fundamental invariants[6](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/EMTensor.html).
The invariants classify fields frame-independently. If \( \vec{E} \cdot \vec{B} = 0 \), a reference frame always exists in which the field is purely electric or purely magnetic, according to the sign of \( E^2 - H^2 \); and if in some frame the field is purely magnetic, no frame exists in which it is purely electric[8](https://haidinh89.wordpress.com/wp-content/uploads/2015/08/landau-l-d-lifshitz-e-m-course-of-theoretical-physics-vol-02-the-classical-theory-of-fields-3305.pdf) • [1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html).
**The dual tensor.** Contracting \( F_{\mu\nu} \) with the totally antisymmetric [Levi-Civita symbol](https://www.edgepedia.org/levi-civita-symbol) \( \epsilon^{\mu\nu\rho\sigma} \) produces the dual field tensor
\[ G^{\alpha\beta} = \frac{1}{2} \epsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta}, \]
which in matrix form interchanges the roles of \( \vec{E} \) and \( \vec{B} \)[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html). Some books write it as \( {}^{*}F \), the Hodge dual[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html). The Levi-Civita symbol itself is invariant under proper Lorentz transformations but changes sign under reflection, which is why it is called a pseudotensor[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html).
## Maxwell's equations in tensor form
The four vector equations of electromagnetism collapse into two tensor equations. Combining [Gauss's law](https://www.edgepedia.org/gausss-law) and the Ampère-Maxwell law gives
\[ \partial_\mu F^{\mu\nu} = \frac{J^{\nu}}{c\,\epsilon_0}, \]
where \( J^{\mu} = (\vec{j}, \rho c) \) is the four-current[3](https://farside.ph.utexas.edu/teaching/em/lectures/node121.html). Combining the two source-free equations, \( \nabla \cdot \vec{B} = 0 \) and \( \nabla \times \vec{E} = -\partial \vec{B}/\partial t \), gives
\[ \partial_\mu G^{\mu\nu} = 0. \]
Together these two equations are equivalent to Maxwell's four equations, and their tensor form makes the invariance of electromagnetism under translations, rotations, special Lorentz transformations, and parity inversions explicit[5](https://farside.ph.utexas.edu/teaching/em/lectures/node122.html). The first equation is automatically consistent with charge conservation, \( \partial_\mu J^{\mu} = 0 \), because applying \( \partial_\nu \) to both sides gives \( \partial_\nu \partial_\mu F^{\mu\nu} \), which vanishes identically due to the antisymmetry of \( F^{\mu\nu} \)[3](https://farside.ph.utexas.edu/teaching/em/lectures/node121.html).
## Role in the action and the force law
The invariant \( F^{\mu\nu} F_{\mu\nu} \) is, up to sign conventions, the Lagrangian density of the free electromagnetic field, playing a role analogous to the difference of kinetic and potential energy in classical mechanics[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html). The same tensor also encodes the force on a charge: the covariant equation of motion
\[ \frac{d p^{\mu}}{d \tau} = q\, U_{\nu} F^{\mu\nu} \]
reduces in the nonrelativistic limit to the [Lorentz force](https://www.edgepedia.org/lorentz-force) law \( \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \)[1](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html).
## Related tensors
The field-strength tensor generalizes beyond electromagnetism. For fields in material media, Minkowski introduced a separate tensor \( G^{\mu\nu} \) built from the electric displacement \( \vec{D} \) and the magnetic field \( \vec{H} \), parallel to Einstein's construction of \( F^{\mu\nu} \) from \( \vec{E} \) and \( \vec{B} \)[9](https://arxiv.org/abs/2003.05275). In curved spacetime, defining measured electric and magnetic fields from \( F_{ab} \) requires choosing an observer's frame four-vector, since the split of the tensor into electric and magnetic parts depends on that choice[10](https://arxiv.org/abs/2303.07562).
## References
1. [Relativistic Electrodynamics, University of Virginia (2022 lecture)](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_65_Rel_Electrodynamics.html)
2. [The Electromagnetic Field Tensor, UCSD Physics 130a notes](https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html)
3. [The electromagnetic field tensor, R. Fitzpatrick, Electromagnetism lectures, University of Texas](https://farside.ph.utexas.edu/teaching/em/lectures/node121.html)
4. [Transformation of fields, R. Fitzpatrick, Electromagnetism lectures, University of Texas](https://farside.ph.utexas.edu/teaching/em/lectures/node123.html)
5. [The dual electromagnetic field tensor, R. Fitzpatrick, Electromagnetism lectures, University of Texas](https://farside.ph.utexas.edu/teaching/em/lectures/node122.html)
6. [EM Tensor, Fundamentals of Time and Relativity](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/EMTensor.html)
7. [The Feynman Lectures on Physics, Vol. II, Ch. 26: Lorentz Transformations of the Fields](https://www.feynmanlectures.caltech.edu/II_26.html)
8. [L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics Vol. 2](https://haidinh89.wordpress.com/wp-content/uploads/2015/08/landau-l-d-lifshitz-e-m-course-of-theoretical-physics-vol-02-the-classical-theory-of-fields-3305.pdf)
9. [Minkowski tensor in electrodynamics of moving media, arXiv:2003.05275](https://arxiv.org/abs/2003.05275)
10. [Definition of electric and magnetic fields in curved spacetime, arXiv:2303.07562](https://arxiv.org/abs/2303.07562)
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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 › Electromagnetic field tensor*
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