User:Muphrid15/Electromagnetic tensor draft

In Geometric Algebra

The electromagnetic tensor is also described as a bivector--an oriented, planar subspace in four-dimensional spacetime. As mentioned earlier, the electromagnetic tensor has only six unique components. These correspond to the six linearly-independent planes that can be constructed in spacetime. Thus, the electromagnetic tensor can be written more compactly as an explicit bivector:

${\vec {F}}=F^{01}\gamma _{01}+F^{02}\gamma _{02}+F^{03}\gamma _{03}+F^{23}\gamma _{23}+F^{31}\gamma _{31}+F^{12}\gamma _{12}$

where $F^{\alpha \beta }=-F^{\alpha \beta },\;F^{0m}=-E_{m}/c$, and $F^{lm}=-\epsilon _{lmn}B^{n}$, where Greek indices range from 0 to 3 and Latin indices range from 1 to 3, $\gamma _{0}$ is the timelike basis vector, and $\gamma _{\alpha \beta }$ represents a unit bivector in spacetime.

In the framework of geometric algebra and calculus, the main properties of the Faraday bivector in vacuum can be rephrased as

$\nabla {\vec {F}}=\mu _{0}{\vec {J}}\implies \nabla \cdot {\vec {F}}=\mu _{0}{\vec {J}}{\text{ and }}\nabla \wedge {\vec {F}}=0$

where $\nabla =\gamma ^{\alpha }\partial _{\alpha }$is the four-derivative. This is a complete encapsulation of Maxwell's equations into a first-order differential equation: that the derivative of a bivector field on spacetime has a vector field for its source.

Because $\nabla \wedge {\vec {F}}=0$, it's possible to introduce the four-potential $\nabla \wedge {\vec {A}}={\vec {F}}$. The equation relating the four-potential to the source, the four-current, is

$\nabla \cdot (\nabla \wedge {\vec {A}})=\mu _{0}{\vec {J}}$

A substitution from the equation $\nabla ^{2}{\vec {A}}=\nabla \wedge (\nabla \cdot {\vec {A}})+\nabla \cdot (\nabla \wedge {\vec {A}})$ reduces this to

$\nabla ^{2}{\vec {A}}-\nabla \wedge (\nabla \cdot {\vec {A}})=\mu _{0}{\vec {J}}$

Setting $\nabla \cdot {\vec {A}}=0$ is the Lorenz gauge choice, and giving an extremely convenient relation between ${\vec {A}},{\vec {J}}$ (the wave equation).