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Geometric spacetime split

  1. Aug 6, 2008 #1
    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 lets 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.
     
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  3. Aug 6, 2008 #2

    robphy

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    Hi, heafnerj.

    Given a 4-velocity vector t, the electric-field seen by that observer is (up to sign) [tex]F_{ab}t^b[/tex]... 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 [tex]\epsilon[/tex] or the spatial metric [tex]h_{ab}[/tex] 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
     
    Last edited: Aug 6, 2008
  4. Aug 6, 2008 #3
    Doran and Lasenby use bivectors with a minkowski -+++ metric:

    [itex]\sigma_i = \gamma_i \gamma_0[/itex] as a spacial 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:

    [tex]
    F = \sum E^i \sigma_i + I B^i \sigma_i
    [/tex]

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

    [tex]
    E^i = F \cdot \sigma_i
    [/tex]
    [tex]
    B^i = (-I F) \cdot \sigma_i
    [/tex]

    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.
     
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