Electromagnetic Tensor - Definition

Definition

The electromagnetic tensor can be defined using the electromagnetic four-potential:

and its covariant form is found by multiplying by the Minkowski metric η of signature (+,−,−,−) :

where A is the vector potential, ϕ is the scalar potential and c is the speed of light.

The Electric and magnetic fields can be expressed in terms of A and ϕ by:

By definition, the electromagnetic tensor is the exterior derivative of the differential 1-form :

therefore F is a differential 2-form on spacetime. In an inertial frame, the matrices of F read:

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}

and by lowering indices

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}

Read more about this topic:  Electromagnetic Tensor

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