Projective structure and holonomy in general relativity

Abstract

Abstract. This paper presents a study of the situation when two space-times admit the same (unparametrised) geodesics, that is, when they are projectively related. The solution is based on the curvature class and the holonomy type of a space-time and it transpires that all holonomy possibilities can be solved except the most general one and that the consequence of two space-times being projectively related leads, in many cases, to their associated Levi-Civita connections being identical. Some results are also given regarding the general case. It is also shown that the holonomy types of projectively related space-times are very closely related. The theory is then applied, with Einsteins principle of equivalence in mind, to "generic" space-times.