Geometric Algebra and Spacetime Split: What are the Applications?

[heafnerj]

I'm having difficulty understanding the geometry of a spacetime split as it applies to geometric algebra. I understand vectors, bivectors, and trivectors and I understand geometric products and wedge products. My impression is that a spacetime split let's you decompose an electromagnetic field into an E and B for a given inertial frame. Is this correct? Can someone clue me in on the geometry here? I've been studying from Hestenes and Doran & Lasenby.

 

[robphy]

Hi, heafnerj.

Given a 4-velocity vector t, the electric-field seen by that observer is (up to sign) $$F_{ab}t^b$$... essentially picking out a column or row from F (in t's coordinates). The magnetic field can be expressed similarly using the Hodge-dual *F (or F and either the Levi-Civita tensor $$\epsilon$$ or the spatial metric $$h_{ab}$$ with respect to t)... essentially picking out the spatial submatrix from F (in t's coordinates).

I'm not yet familiar enough with Geometric Algebra to use their terminology.

Check out section 2.3 of Malament's review of GR
http://arxiv.org/abs/gr-qc/0506065

 

[Peeter]

Doran and Lasenby use bivectors with a minkowski -+++ metric:

$\sigma_i = \gamma_i \gamma_0$ as a spatial basis. These have a positive square, like a set of euclidian orthonormal vectors, and thus behave vector like for all intents and purposes.

They write the electromagnetic field as the following bivector:

$$F = \sum E^i \sigma_i + I B^i \sigma_i$$

So, one can recover the electric and magnetic field components with:

$$E^i = F \cdot \sigma_i$$
$$B^i = (-I F) \cdot \sigma_i$$

Other than being a cool way to condense maxwell's equations into one equation, I haven't gotten as far as figuring out how to actually utilize all this in calculations. I've had to go back to basics and understand SR fundamentals a lot better before I can understand their applications of GA to SR, and E&M. The two days that we did SR in first year E&M back in my undergrad engineering days wasn't enough to to able to understand chapter 5 or 7 of that book;) I'm starting to get a handle on some of the SR basics but still haven't gotten back to the GA side of things.

fwiw, I've collected some notes on both the Hestenes and D/L texts. Both are too dense for me and I found I had to spend a lot of time mulling over details to be able to understand things sufficiently. I recently organized these notes a bit and dumped them here:

http://www.geocities.com/peeter_joot/

Perhaps some of it would be of it would be of interest to you ... I wrote it up to explain things to myself, because if I couldn't do that I found I didn't understand something.