The electromagnetic tensor, also known as the Faraday tensor or electromagnetic field tensor, is a rank‑2 antisymmetric tensor that encapsulates the electric and magnetic fields in a covariant formulation of classical electrodynamics. It provides a compact representation of Maxwell’s equations within the framework of special relativity and facilitates the description of electromagnetic phenomena in any inertial reference frame.
Definition and Mathematical Form
In four‑dimensional spacetime with coordinates $x^{\mu} = (ct, \mathbf{x})$ and metric signature $(+,-,-,-)$, the electromagnetic tensor $F_{\mu u}$ is defined as
$$ F_{\mu u} = \partial_{\mu} A_{ u} - \partial_{ u} A_{\mu}, $$
where $A_{\mu} = (\phi/c, -\mathbf{A})$ is the electromagnetic four‑potential, $\phi$ the scalar potential, and $\mathbf{A}$ the vector potential. The antisymmetry $F_{\mu u} = -F_{ u\mu}$ implies six independent components, corresponding to the three components of the electric field $\mathbf{E}$ and the three components of the magnetic field $\mathbf{B}$.
In matrix notation (using Cartesian coordinates),
$$ F_{\mu u} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \ E_x/c & 0 & -B_z & B_y \ E_y/c & B_z & 0 & -B_x \ E_z/c & -B_y & B_x & 0 \end{pmatrix}. $$
The dual tensor $^\ast!F^{\mu u}$ is defined by
$$ ^\ast!F^{\mu u} = \frac{1}{2}\varepsilon^{\mu u\alpha\beta}F_{\alpha\beta}, $$
where $\varepsilon^{\mu u\alpha\beta}$ is the Levi‑Civita symbol.
Relation to Maxwell’s Equations
The homogeneous and inhomogeneous Maxwell equations acquire concise tensorial forms:
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Inhomogeneous equations (Gauss’s law for electricity and Ampère‑Maxwell law): $$ \partial_{\mu} F^{\mu u} = \mu_0 J^{ u}, $$ where $J^{ u} = (c\rho, \mathbf{J})$ is the four‑current density, $\rho$ the charge density, $\mathbf{J}$ the current density, and $\mu_0$ the vacuum permeability.
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Homogeneous equations (Gauss’s law for magnetism and Faraday’s law of induction): $$ \partial_{\mu} ,^\ast!F^{\mu u} = 0. $$
These expressions are manifestly Lorentz‑covariant; they retain their form under Lorentz transformations, reflecting the relativistic invariance of electromagnetic phenomena.
Physical Invariants
Two Lorentz‑invariant scalars can be constructed from the electromagnetic tensor:
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First invariant: $$ \mathcal{I}1 = \frac{1}{2}F{\mu u}F^{\mu u} = \frac{1}{c^{2}}\left( \mathbf{B}^2 - \frac{\mathbf{E}^2}{c^{2}} \right). $$
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Second invariant: $$ \mathcal{I}2 = \frac{1}{4}\varepsilon^{\mu u\alpha\beta}F{\mu u}F_{\alpha\beta} = \frac{\mathbf{E}\cdot\mathbf{B}}{c^{2}}. $$
These invariants characterize the electromagnetic field independent of the observer’s inertial frame.
Applications
- Relativistic particle dynamics: The Lorentz force law can be expressed compactly as $m \frac{d u^{\mu}}{d\tau} = q F^{\mu}{}_{ u} u^{ u}$, where $u^{\mu}$ is the particle’s four‑velocity, $q$ its charge, and $\tau$ proper time.
- Field theory: In the Lagrangian formulation of electromagnetism, the field‑strength term $-\frac{1}{4\mu_0}F_{\mu u}F^{\mu u}$ appears in the action integral.
- General relativity: The electromagnetic tensor couples to spacetime curvature via the Einstein–Maxwell equations, contributing to the stress‑energy tensor $T_{\mu u}$.
Historical Note
The tensor formalism was introduced in the early 20th century as part of the development of special relativity. The name “Faraday tensor” honors Michael Faraday’s experimental work on electromagnetism, while the term “electromagnetic field tensor” reflects its role as a unified representation of $\mathbf{E}$ and $\mathbf{B}$.
References
- Jackson, J. D. Classical Electrodynamics, 3rd ed., Wiley (1998).
- Landau, L. D., Lifshitz, E. M. The Classical Theory of Fields, 4th ed., Butterworth‑Heinemann (1975).
- Misner, C. W., Thorne, K. S., Wheeler, J. A. Gravitation, W. H. Freeman (1973).
