my question is if we define the quaternion(adsbygoogle = window.adsbygoogle || []).push({});

[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

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# GR and quaternions

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