One of the lessons coming out of Prof. Dennis Lehmkuhl‘s work is that geometrization of any (field) theory is just the “skin” or the “user interface” of the complete product – it is beautiful and easy to use but it hides the complexities and the specifications of the underlying physics.
This is profound and I argue this is the reason why the geometrical interpretation of GR can be linked to thermodynamics and information theory (see for example the concept of “Entropic Gravity” and the work of Thanu Padmanabhan): those theories are so general that they are the theories of everything and at the same time the theories of nothing.
In fact:
- The equation of state of thermodynamics and all the transformations of thermodynamic potentials do not say anything about the specific gas they are referring to.
- Information theory does not say anything about the physics of the hardware on which the information is codified via conventionally defined bits.
- The differential geometry of a 3+1 pseudo-Riemanninan abstract space does not say anything about the specific microscopic physics of the substrate-matter interaction.
Geometrization must come at the end of of theory development and it must “package” everything at once, not just one part of physics (as it is now with gravity).
I have no doubt that quaternions and geometric algebra will play a role in this (see for example Mendel Sachs‘s unified field theory), but again, they will serve only as a “user interface” once the underlying physics is understood in its basic mechanisms.
Thermodynamics was for some time just a phenomenological theory without a deeper mechanism explanation (“phainomena” = “that which appears/is observed”) alongside the well-developed mechanics. The two branches were unified when Boltzmann created statistical mechanics (mechanics applied to a multitude of microscopic objects) and proved that for N >> 1 one obtains the well-known thermodynamic relations.
In order to unify gravity with electromagnetism and matter, a similar approach must be followed. The “thermodynamic/superficial” level must emerge from the underlying “mechanical-statistical/deep” level and THEN (if one wants) everything can be geometrized.