A prominent feature in relativistic physics is the Minkowski metric tensor \[\eta = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{bmatrix}\]
On probing where this comes from it was postulated that the matrix could be the ’Real’ (non-quaternion) part of the outer product of two unit quaternions \(Q=1+i+j+k\), \[\eta_{\mu \nu} = Q \otimes Q =Re \left( \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j\\ j & -k & -1 & i\\ k & j & -i & -1 \end{bmatrix} \right)\]
The implications of carrying through the physics made with this tensor without taking the real part were considered. The creation of an electromagnetic tensor is considered.
When being used for a metric in the form \[ds^2= \begin{bmatrix} dt & dx & dy & dz \end{bmatrix} \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j & -i & -1 \end{bmatrix} \begin{bmatrix} dt \\ dx \\ dy \\ dz \end{bmatrix}\]
Then from explicit calculation it can be shown that an equaivalent matrix gives \[ds^2= \begin{bmatrix} dt & dx & dy & dz \end{bmatrix} \begin{bmatrix} 1 & i & j & k \\ i & -1 & 0 & 0 \\ j & 0 & -1 & 0 \\ k & 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} dt \\ dx \\ dy \\ dz \end{bmatrix}\]
This would be equaivalent to a new type of number with the rules \[i \cdot i =-1 \\ j \cdot j =-1 \\ k \cdot k =-1 \\ i\cdot j=j\cdot i=0 \\ i\cdot k=k\cdot i=0 \\ j\cdot k=k\cdot j=0\]
This is similar to having 3 independant imaginary numbers or basis vectors. If this were an inner product then they are orthoganal but antiparallell with themselves. They form a NON-ASSOCIATIVE ’group’ under a product with elements \( {0,1,i,j,k,-1,-i,-j,-k} \), this is an Abelian relationship as the non-Abelian properties of the quaternions was removed with the cross interactions. For example, \[(i \cdot i) \cdot j = -1 \cdot j = -j \\ i \cdot ( i \cdot j ) = i \cdot 0 = 0 \\ (a \cdot b) \cdot c \ne a \cdot (b \cdot c)\]