Fundamentals of Time and Relativity

EM Bivector

The electromagnetic field in the STA formalism is a bivector field $F$ with the same components as the anti-symmetric field tensor $F_{μ ν}$ .
In a given inertial frame, the bivector $F$ decomposes into a complex pair of three-vectors $E$ and $i B$ .

As an example of the use of STA, we consider electromagnetism. For each spacetime point $x$ , the standard frame ${e_{μ}}$ as defined in STA, determines a set of ‘rectangular coordinates’ ${x^{μ}}$ given by $x^{μ} = e^{μ} \cdot x$ and $x = x^{μ} e_{μ}$ . In terms of these coordinates the derivative with respect to a spacetime point $x$ is the operator $\nabla : = e^{μ} \partial_{μ}$ , where $\partial_{μ} = e_{μ} \cdot \nabla = (\partial_{0}, \vec{\nabla})$ is the coordinate derivative.

Writing the electromagnetic field $F_{μ ν}$ in terms of the 4-potential $A (x) = A_{μ} (x) e^{μ}$ we have

9.15

\nabla \land A= \partial_{μ} A_{ν} (e^{μ} \land e^{ν}) = \frac{1}{2} F_{μ ν} (e^{μ} \land e^{ν}) = : F

where $F$ is the so-called electromagnetic field bivector with its six components given by:

9.16

F_{μ ν} = e_{μ} \cdot F \cdot e_{ν} = (e_{ν} \land e_{μ}) \cdot F

Like in Minkowski space, in STA a given inertial system is completely characterized by the single future-pointing, timelike unit vector $e_{0}$ . The spacetime split of $F$ into an electric (relative vector) part $E$ and a magnetic (relative bivector) part $i B$ in the $e_{0}$ -system is achieved by separating $F$ into parts which anti-commute and commute with $e_{0}$ . Thus $F = E + i B$ , where

9.17

E : = (F \cdot e_{0}) e_{0} = \frac{1}{2} (F + F^{†})

is the part of $F$ that anti-commutes with $e_{0}$ , and

9.18

i B : = (F \land e_{0}) e_{0} = \frac{1}{2} (F - F^{†})

is the part that commutes; the quantity $F^{†} = E - i B$ is called the conjugated field. Importantly, this type of conjugation is not frame-independent, but is relative to the $e_{0}$ -frame.

The geometric properties of $i$ keep $F$ reference-frame-independent, even though the decomposition into relative fields $E$ and $B$ still implicitly depends upon the chosen time axis.
In STA electromagnetic theory, the geometric pseudo-scalar $i$ plays the role of the scalar imaginary unit $i$ . There is no need for the introduction of any additional complex entities.