A criterion for compatibility of conformal and projective structures

TL;DR

This paper establishes necessary and sufficient conditions for conformal and projective structures in space-time to be compatible, ensuring they derive from a unique metric tensor up to scale, applicable across all dimensions and signatures.

Contribution

It provides a general theorem characterizing compatibility of conformal and projective structures in space-time, extending previous results to all dimensions and signatures.

Findings

Derived necessary and sufficient compatibility conditions

Applicable to all space-time dimensions and signatures

Ensures unique metric tensor up to a constant factor

Abstract

In a space-time, a conformal structure is defined by the distribution of light-cones. Geodesics are traced by freely falling particles, and the collection of all unparameterized geodesics determines the projective structure of the space-time. The article contains a formulation of the necessary and sufficient conditions for these structures to be compatible, i.e. to come from a metric tensor which is then unique up to a constant factor. The theorem applies to all dimensions and signatures.