On the General Projective Theory of Matter and Gravitation

TL;DR

This paper develops a comprehensive gauge theory of gravity incorporating non-metricity, torsion, and projective invariance, unifying various geometric frameworks and introducing new topological invariants and spinor formulations.

Contribution

It introduces a generalized projective gauge theory of gravity with novel connections, topological invariants, and a new formulation of spinors ensuring local Lorentz covariance.

Findings

Defined projective symmetric teleparallel connections

Discovered a new topological invariant in projective Pontrjagin density

Formulated projective spinors with a Hermitian action leading to a chiral mass

Abstract

We develop a generalized projective gauge theory of gravity and spinorial matter, incorporating both non-metricity and torsion. The work is divided into three parts. Part I provides a thorough review of General Relativity, Metric-Affine gauge theory and Thomas-Whitehead (TW) Gravity. Part II constructs a gauge gravitational theory based on the Projective General Linear Group, wherein the newly found projective symmetric teleparallel (PST) connections are defined. Generalizing to non-inertial frames, and using nonlinear gauge symmetry realizations to implement local Lorentz symmetry, we construct a general geometric framework that unifies TW and Metric-Affine gravity with projectively invariant spacetime torsion. We introduce projective generalized Higgs fields and show how certain gauge choices reduce these to fundamental projective fields, and how they may be used to define fundamental geometric objects such as the projective 2-frame and spacetime connection. A Lovelock-inspired action is shown to support only curvature (metric) dynamics, implying a topologically constrained Schouten field via the Pontrjagin density. The projective Pontrjagin density is shown to contain a new topological invariant, not present in the literature. Part III formulates projective spinors by defining gamma matrices via the projective linear group metric, then employing nonlinear gauge symmetry realizations to ensure local Lorentz covariance. A general spinor metric introduces a complex phase of redundancy. Requiring a real, Hermitian action leads to a projective Einstein-Cartan-type theory. An induced chiral mass emerges without CP violation.