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).