Skip to main content

Mathematics for Physics

51 Field Tensor

An antisymmetric spacetime tensor that is used to package the electric and magnetic fields into one Lorentz-covariant object, where an antisymmetric tensor is a two-index object that changes sign when its indices are swapped.

The component identification of \(F^{\mu\nu}\). The six independent components of \(F^{\mu\nu}\) are the three electric and three magnetic field components. This principle is used to treat \(\mathbf{E}\) and \(\mathbf{B}\) as parts of one spacetime field.

The electromagnetic field tensor is

\[ F^{\mu\nu} = \begin{pmatrix} 0 & -E_{x}/c & -E_{y}/c & -E_{z}/c \\ E_{x}/c & 0 & -B_{z} & B_{y} \\ E_{y}/c & B_{z} & 0 & -B_{x} \\ E_{z}/c & -B_{y} & B_{x} & 0 \end{pmatrix} \]

where

  • \(F^{\mu\nu}\) is the electromagnetic field tensor.
  • \(E_{x}\), \(E_{y}\), \(E_{z}\) are the electric-field components.
  • \(B_{x}\), \(B_{y}\), \(B_{z}\) are the magnetic-field components.
  • \(c\) is the speed of light.

The potential formula \(F=dA\). The field tensor is the curl of the four-potential. This principle is used to obtain \(F^{\mu\nu}\) from \(A^{\mu}\).

The field tensor from the four-potential is

\[ F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} \]

where

  • \(F_{\mu\nu}\) is the covariant field tensor.
  • \(A_{\mu}\) is the four-potential.
  • \(\partial_{\mu}\) is the spacetime derivative.

The covariant Maxwell equations. Maxwell’s equations are two tensor equations for \(F^{\mu\nu}\). This principle is used to write electrodynamics in every inertial frame at once.

The inhomogeneous Maxwell equation is

\[ \dfrac{\partial F^{\mu\nu}}{\partial x^{\nu}} = \mu_{0}J^{\mu} \]

The homogeneous Maxwell equation is

\[ \dfrac{\partial F_{\mu\nu}}{\partial x^{\lambda}} + \dfrac{\partial F_{\nu\lambda}}{\partial x^{\mu}} + \dfrac{\partial F_{\lambda\mu}}{\partial x^{\nu}} = 0 \]

where

  • \(F^{\mu\nu}\) is the electromagnetic field tensor.
  • \(J^{\mu}\) is the four-current.
  • \(\mu_{0}\) is the permeability of free space.
  • \(x^{\nu}\) are the spacetime coordinates.

Note: Also called the Faraday tensor. Also called the electromagnetic field tensor.