# GR and quaternions

1. May 4, 2008

### mhill

my question is if we define the quaternion

$$U= dt-idx-jdy-kdz$$

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

$$U.U^{*}=(ds)^{2} =(dt)^{2}-(dx)^{2}-(dy)^{2}-(dz)^{2}$$ Minkowsky metric

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

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

$$ijdxdxy=-jidxdy$$ would hold

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

$$U(x,y,z,t)= f(x,y,z,t)dt-idxg(x,y,z,t)-jh(x,y,z,t)-kW(x,y,z,t)$$

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