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Comment by jperras | Hacker News Reader

When I took a graduate class in General Relativity under Robert Brandenberger, one of the more unconventional (and brilliant) approaches he took with his students was to introduce the formalism of geometric algebra quite early on, along with tetrads and spinors. While confusing at first, it proved to be extremely enlightening when Maxwell's equations were brought into play.

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Would you mind expanding on this? With as much detail as you feel like writing? It's incredibly interesting.

Were Maxwell's equations still condensed to 4 equations (Heaviside's equations in Gibb's vector notation), or something less concise, as I believe they were with quanternions?

I believe they become Maxwell's Equation, and it's super simple.

Expressing Maxwell's equations using the six-component electromagnetic tensor F, they become

  dF  = μJ
  d*F = 0

where μ is the magnetic permeability of the vacuum and J is the electromagnetic 4-current. The operator 'd' is the differential operator from exterior algebra, and the '*' is the Hodge dual.

Using the bivector field F = E + iB from geometric algebra, they become

  DF = μJ

where μ and J are as before, and D is the covector derivative.

For comparison, using the traditional Gibbs/Heaviside notation Maxwell's equations are

  ∇ . E = ρ/ε
  ∇ . B = 0
  ∇ x E = -∂B/∂t
  ∇ x B = μ(J + ε ∂E/∂t)

I don't think it's that uncommon. The advanced GR course I took used tetrads and spinors. It seems to me that this is rather standard for a graduate GR course.

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