Projective Structure and Holonomy in 4-dimensional Lorentz Manifolds

TL;DR

This paper explores the conditions under which two 4-dimensional Lorentz manifolds, representing space-times, share the same unparametrised geodesics, revealing that in many cases their Levi-Civita connections are identical.

Contribution

It provides a comprehensive analysis of projective relations in Lorentz manifolds by examining each holonomy type, advancing understanding of space-time geometries.

Findings

Most holonomy types can be fully analyzed.

Projectively related space-times often have identical Levi-Civita connections.

The study advances classification of space-times based on geodesic structures.

Abstract

This paper studies the situation when two 4-dimensional Lorentz manifolds (that is, space-times) admit the same (unparametrised) geodesics, that is, when they are projectively related. A review of some known results is given and then the problem is considered further by treating each holonomy type in turn for the space-time connection. It transpires that all holonomy possibilities can be dealt with completely except the most general one and that the consequences of two space-times being projectively related leads, in many cases, to their associated Levi-Civita connections being identical.