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GR and quaternions

  1. May 4, 2008 #1
    my question is if we define the quaternion

    [tex] U= dt-idx-jdy-kdz [/tex]

    where i,j,k are the complex part of the quaternion , and we define 'U' so

    [tex] U.U^{*}=(ds)^{2} =(dt)^{2}-(dx)^{2}-(dy)^{2}-(dz)^{2} [/tex] Minkowsky metric

    or in the most general case we define the quaternion depending on x,y,z and t so we had the metric

    [tex] g_{ab}(x,y,z,t) [/tex] is a hypercomplex number , the problem is that Quaternion do not commute so the equality

    [tex] ijdxdxy=-jidxdy [/tex] would hold

    my question is if we could define space time , or an event on space time to be a quaternion, for example

    [tex] U(x,y,z,t)= f(x,y,z,t)dt-idxg(x,y,z,t)-jh(x,y,z,t)-kW(x,y,z,t) [/tex]

    with the axioms:

    - any event on space time is a quaternion defined by (a,b,c,d) a time coordinate and 3 spatial ones

    - the geommetry of space-time is defined by the quaternion group.

    i'm not very skilled in math, so forgive my errors please if anyone can give some information, thanks
  2. jcsd
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