Math 2.0

Optics, electromagnetic theory Definition

Electromagnetic tensor

Builds on Electric fieldMagnetic fieldMaxwell's equationsTensor fieldSkew-symmetric matrix

Definition

Work in Minkowski space with coordinates x0=ct,x1,x2,x3x^0 = ct, x^1, x^2, x^3, where cc is the speed of light, and metric η=diag⁡(1,−1,−1,−1)\eta = \operatorname{diag}(1, -1, -1, -1). Let E=(E1,E2,E3)E = (E_1, E_2, E_3) be the electric field and B=(B1,B2,B3)B = (B_1, B_2, B_3) the magnetic field. The electromagnetic tensor is the antisymmetric tensor field FμνF^{\mu\nu}, μ,ν∈{0,1,2,3}\mu, \nu \in \set{0, 1, 2, 3}, given by the skew-symmetric matrix Fμν=(0−E1/c−E2/c−E3/cE1/c0−B3B2E2/cB30−B1E3/c−B2B10).F^{\mu\nu} = \begin{pmatrix} 0 & -E_1/c & -E_2/c & -E_3/c \\ E_1/c & 0 & -B_3 & B_2 \\ E_2/c & B_3 & 0 & -B_1 \\ E_3/c & -B_2 & B_1 & 0 \end{pmatrix}. Indices are lowered with η\eta, so Fμν=ημαηνβFαβF_{\mu\nu} = \eta_{\mu\alpha} \eta_{\nu\beta} F^{\alpha\beta}; this gives F0i=Ei/cF_{0i} = E_i / c and Fij=FijF_{ij} = F^{ij}.

Remarks

With the four-current Jν=(cρ,J)J^\nu = (c \rho, J) and the vacuum permeability μ0\mu_0, Maxwell's equations become ∂μFμν=μ0Jν,∂λFμν+∂μFνλ+∂νFλμ=0.\partial_\mu F^{\mu\nu} = \mu_0 J^\nu, \qquad \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0. For ν=0\nu = 0 the first equation reads 1c∇⋅E=μ0cρ\frac{1}{c} \nabla \cdot E = \mu_0 c \rho, which is Gauss's law since ε0μ0c2=1\varepsilon_0 \mu_0 c^2 = 1. The second set contains ∇⋅B=0\nabla \cdot B = 0 and Faraday's law.

The two Lorentz invariants of the field are FμνFμν=2(∣B∣2−∣E∣2/c2)F_{\mu\nu} F^{\mu\nu} = 2 \big( \abs{B}^2 - \abs{E}^2 / c^2 \big) and E⋅BE \cdot B, up to a constant factor for the second.

Example: a point charge qq at rest at the origin has B=0B = 0, so only the entries Fi0=−F0i=Ei/cF^{i0} = -F^{0i} = E_i / c with E=qx/(4πε0∣x∣3)E = q x / (4 \pi \varepsilon_0 \abs{x}^3) are nonzero.

Non-example: a matrix with a nonzero diagonal entry, such as F00=1F^{00} = 1, is not an electromagnetic tensor, because it is not antisymmetric.

Sign conventions differ between books, depending on the signature of the metric and the order of the indices.

Reading path

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Electric field, concept, no evidence levelElectric fieldconceptMagnetic field, concept, no evidence levelMagnetic fieldconceptMaxwell's equations, concept, no evidence levelMaxwell's equationsconceptSkew-symmetric matrix, concept, no evidence levelSkew-symmetric matrixconceptTensor field, concept, no evidence levelTensor fieldconceptElectromagnetic tensor, definition, no evidence levelElectromagnetic tensordefinition
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  3. Maxwell's equations concept
  4. Skew-symmetric matrix concept
  5. Tensor field concept
  6. Electromagnetic tensor definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Electromagnetic_tensor
About this page
Tag60K3E80KCV
URIm2://60K3E80KCV
Kinddefinition
WritingW0 stub
MSC78A25, 83A05
DefinesElectromagnetic tensor
UsesElectric field, Magnetic field, Maxwell's equations, Tensor field, Skew-symmetric matrix
Changed3 November 2026

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