In special relativity, the Minkowski metric is used to calculate the  interval between two spacetime intervals for inertial observers.  Einstein  recognized that inertial observes were "special", a unique class.  Therefore he set out to understand what was the most general notion for  transformations and metrics.  This lead to his study of Riemannian geometry, and eventually to general relativity.  In this post I shall start from the Lorentz invariant interval using quaternions, then try to generalize this approach using a different way which might prove compatible with quantum mechanics.

For the physics of gravity, general relativity (GR) makes the right  predictions of all experimental tests conducted to date.  For the physics  of atoms, quantum mechanics (QM) makes the right predictions to an even  high degree of precision.  The problem of building a quantum theory of  gravity (QG) hides between general relativity and quantum mechanics.  General relativity deals with the measurements of intervals in curved  spacetime, special relativity (SR)  being adapted to work in flat space.  Quantum mechanics is used to calculate the norms of wave functions in  a flat linear space.  A quantum gravity theory will be used to calculate  norms of wave functions in curved space.

                   measurement
                 interval  norm
  diff.  flat       SR      QM
  geo.   curved     GR      QG

This chart suggests that the form of measurement (interval/norm) should   be  independent of differential geometry (flat/curved).  That will  be the explicit goal of this post.

Quaternions come with a metric, a means of taking 4 numbers and returning a scalar.  Hamilton defined the roles like so:

The scalar result of squaring a differential quaternion in the interval of special relativity:

How can this be generalized?  It might seem natural to explore variations  on Hamilton's rules shown above.  Riemannian geometry uses that strategy.  When working with a field like quaternions, that approach bothers me  because Hamilton's rules are fundamental to the very definition of a  quaternion.  Change these rules and it may not be valid to compare physics  done with different metrics.  It may cause a compatibility problem.

Here is a different approach which generalizes the scalar of the square while being consistent with Hamilton's rules.

If g is the identity matrix.  Then then result is the flat Minkowski  interval.  The quaternion g could be anything.  What if g = i?  (what  would you guess, I was surprised :-)

Now the special direction x plays the same role as time!  Does this make  sense physically?  Here is one interpretation.  When g=1, a time-like  interval is being measured with a wristwatch.  When g=i, a space-like  interval along the x axis is being measured with a meter stick along the  x axis.

Examine the most general case, where small letters are scalar, and capital letters are 3-vectors:

In component form...

This has the same combination of ten differential terms found in the  Riemannian approach.  The difference is that Hamilton's rule impose an  additional structure.

I have not yet figured out how to represent the stress tensor, so  there are no field equations to be solved.  We can figure out some of the  properties of a static, spherically-symmetric metric.  Since it is static,  there will be no terms with the deferential element dt dx, dt dy, or dt  dz.  Since it is spherically symmetric, there will be no terms of the  form dx dy, dx dz, or dy dz.  These constraints can both be achieved if Gx  = Gy = Gz = 0.  This leaves four differential equations.

Here I will have to stop.  In time, I should be able to figure out  quaternion field equations that do the same work as Einstein's field  equations. I bet it will contain the Schwarzschild solution too :-)  Then it will be easy to create a Hilbert space with a non-Euclidean norm,  a norm that is determined by the distribution of mass-energy.  What sort  of calculation to do is a mystery to me, but someone will get to that  bridge...