In electromagnetism, the **electromagnetic tensor** or **electromagnetic field tensor** (sometimes called the **field strength tensor**, **Faraday tensor** or **Maxwell bivector**) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely.

## Definition

The electromagnetic tensor, conventionally labelled *F*, is defined as the exterior derivative of the electromagnetic four-potential, *A*, a differential 1-form:^{[1]}^{[2]}

- $F\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} A.$

Therefore, *F* is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

- $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }.$

where $\partial$ is the four-gradient and $A$ is the four-potential.

SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space (+ − − −), will be used throughout this article.

### Relationship with the classical fields

The electric and magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in Cartesian coordinates:

- $E_{i}=cF_{0i},$

where *c* is the speed of light, and

- $B_{i}=-{\frac {1}{2}}\epsilon _{ijk}F^{jk},$

where $\epsilon _{ijk}$ is the Levi-Civita tensor. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will transform covariantly, and the fields in the new frame will be given by the new components.

In contravariant matrix form,

- $F^{\mu \nu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}.$

The covariant form is given by index loweringThe electromagnetic tensor, conventionally labelled *F*, is defined as the exterior derivative of the electromagnetic four-potential, *A*, a differential 1-form:^{[1]}^{[2]}

- $F\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} A.$

Therefore, *F* is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

- $F}_{\mu \nu}={\mathrm{\partial}}_{$
- $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }.$$\partial$ is the four-gradient and $A$ is the four-potential.
SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space (+ − − −), will be used throughout this article.

### Relationship with the classical fields

The electric and magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in Cartesian coordinates:

- $E}_{i}=c{F}_{$(+ − − −), will be used throughout this article.
The electric and magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in Cartesian coordinates:

- $E}_{<$
where *c* is the speed of light, and

- ${B}_{i}=-\frac{1}{2}{\u03f5}_{ijk}{F}^{jk},$$\epsilon _{ijk}$ is the Levi-Civita tensor. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will transform covariantly, and the fields in the new frame will be given by the new components.
In contravariant matrix form,

- $F}^{\mu \nu}=[\begin{array}{ccc}0& -{E}_{x}/c& -\end{array$
In contravariant matrix form,

The covariant form is given by index lowering,

- $F_{\mu \nu }=\$
The Faraday tensor's Hodge dual is

- ${G^{\alpha \beta }={\frac {1}{2}}\epsilon ^{\alpha \beta \gamma \delta }F_{\gamma \delta }={\begin{bmatrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\end{bmatrix}}}$Properties
The matrix form of the field tensor yields the following properties:^{[3]}

**Antisymmetry:**
- $F^{\mu \nu }=-F^{\nu \mu }$

**Six independent components:** In Cartesian coordinates, these are simply the three spatial components of the electric field (*E*_{x}, E_{y}, E_{z}) and magnetic field (*B*_{x}, B_{y}, B_{z}).**Inner product:** If one forms an inner product of the field strength tensor a Lorentz invariant is formed
- $F_{\mu \nu }F^{\mu \nu }=2\left(B^{2}-{\frac {E^{2}}{c^{2}}}\right)$

meaning this number does not change from one frame of reference to another.**Pseudoscalar invariant:** The product of the tensor $F^{\mu \nu }$ with its Hodge dual $G^{\mu \nu }$ gives a Lorentz invariant:
- $\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}},\quad \nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {J}$