Unimodular conformal and projective relativity

TL;DR

This paper introduces Unimodular Conformal and Projective Relativity (UCPR), extending unimodular relativity by emphasizing conformal and projective structures, and explores its implications for gravitational theories and quantum gravity.

Contribution

It develops the UCPR framework, decomposes spacetime structures under SL(4,R), and proposes modifications to Einstein's equations with potential quantum gravity implications.

Findings

Decomposition of metric into volume and conformal parts.

Separation of affine connection into one-form and projective connection.

Potential for new gravitational theories via Lagrangian modifications.

Abstract

We outline unimodular conformal and projective relativity (UCPR), an extension of unimodular relativity in which the conformal and projective structures play central roles. Under $SL(4,\mathbb{R})$ symmetry group, the pseudo-Riemannian metric naturally decomposes into a four-volume element and a conformal metric; and the affine connection decomposes into a one-form and a trace-free projective connection. In UCPR, these four space-time structures are treated as independent fields that have clear physical interpretations. A Palatini-type variational principle for the usual general relativity Lagrangian leads to a breakup of the Einstein field equations and the compatibility conditions between the metric and connection. We indicate how new gravitational theories may be generated by modifications of this Lagrangian and discuss two such cases. Finally, we discuss possible physical consequences of our results for quantum gravity.