On the general projective theory of matter and gravitation

We propose a novel approach to gravity and spinning particles (spinors) that allows additional twisting and stretching effects in spacetime, beyond the curving that is captured by Einstein s theory alone. These extra geometric features beyond simple curvature might help us solve some long-standing puzzles in physics, such as the cause of dark energy or how certain particles gain their mass. In particular, we include two types of geometric deformations often called torsion (twisting spacetime) and non-metricity (changes in how spacetime distances are measured) and treat them as part of a unified framework tied to projective transformations. These transformations are simply different descriptions of material motions that are all equivalent. The theory is presented in three parts.

Part I lays out essential background. We explain how spacetime geometry can be split into familiar features from Einstein s work and pieces that capture the extra twisting and stretching. We show how standard results, such as Einstein s field equations, emerge and then extend them to include dark energy and matter. We also introduce standard topological, or shape-preserving terms that may have interesting physical consequences yet do not affect everyday observations. We then reconstruct all of this in a format more suitable for the gauge, or locally redundant treatment of gravity. The purpose of this is to uncover how gravity fits in to our Standard Model of particle physics.

Part II develops the general projective gauge viewpoint in detail. Here, we treat the geometry of spacetime as living in a higher-dimensional setting, allowing us to interpret certain fields linked to material path reparameterizations as fundamental components. This leads to a richer picture of gravity, in which this new field acquires mass much like a Higgs field, produces a cosmological constant (dark energy), and leads to additional possibilities for how gravitational effects propagate throughout spacetime.

Part III constructs and incorporates material fields, or spinning matter fields (spinors). We discover that the twisting deformation of the geometry can directly influence material spin, while the stretching part despite other attempts has no material influence whatsoever. Interestingly, a chiral mass arises that can be used to provide neutrinos mass, linking small neutrino masses to higher-scale physics within this unified projective geometric framework.

Altogether, this thesis shows how an expanded picture of spacetime with new geometric fields and symmetries can both reproduce the successes of Einstein s gravity and offer novel insights into dark energy, mass generation, and the behavior of spinning particles. This lays the groundwork for future studies of how these new geometric features might shape cosmology, high-energy physics, and possibly the nature of existence.