Fundamentals of Time and Relativity

EM Tensor

As an example of an external force field, we consider a given electromagnetic field

9.1

F_{μ ν} (x) : = \partial_{μ} A_{ν} (x) - \partial_{ν} A_{μ} (x)

derived from the four-vector potential $A_{μ} = (A_{0}, - A)$ , which combines the scalar and vector potentials of electromagnetism. In a given inertial frame, the connection between the electric and magnetic field strengths, on the one hand,

9.2

E : = - \vec{\nabla} A_{0} - \partial_{0} A, B : = \vec{\nabla} \times A

and the electromagnetic field tensor (9.1), on the other, is

9.3

F_{μ ν} = (\begin{matrix} 0 & E_{1} & E_{2} & E_{3} \\ - E_{1} & 0 & - B_{3} & B_{2} \\ - E_{2} & B_{3} & 0 & - B_{1} \\ - E_{3} & - B_{2} & B_{1} & 0 \end{matrix}), F^{μ ν} = (\begin{matrix} 0 & - E_{1} & - E_{2} & - E_{3} \\ E_{1} & 0 & - B_{3} & B_{2} \\ E_{2} & B_{3} & 0 & - B_{1} \\ E_{3} & - B_{2} & B_{1} & 0 \end{matrix})

Here $μ$ and $ν$ refer to the row and column indices, respectively. Because of its anti-symmetry, the electromagnetic field tensor has only six non-vanishing components.

This separation of the electromagnetic field tensor in electric and magnetic contributions has only relative meaning dependent on the chosen inertial frame. The electric and magnetic components become mixed up if one passes to another inertial frame.

There are two exceptions, namely the invariant combinations:

9.4

E^{2} - B^{2} = \frac{1}{2} F_{μ ν} F^{μ ν}, E \cdot B = - \frac{1}{4} F_{μ ν} {\tilde{F}}^{μ ν}

where the dual (wiggled) field tensor is obtained by the substitutions $E \to B$ and $B \to - E$ . The second invariant states that the inner-product of $E$ and $B$ is the same viewed in all inertial frames. All other invariants can be expressed in terms of these two so-called fundamental invariants.

There is no unique choice of the vector potential $A_{μ}$ . Under a gauge shift $A_{μ} (x) \to A_{μ} (x) - \partial_{μ} χ (x)$ , where $χ (x)$ is any function of space and time, the electromagnetic field tensor remains unchanged.
Remarkably, it is possible to construct a Lorentz covariant object consisting of two 3-vectors, by arranging them into an anti-symmetric tensor that transforms under the Lorentz group.