Projective structure and holonomy in four-dimensional Lorentz manifolds

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.

Projective structure and holonomy in general relativity

“…Again the conclusions of the lemma hold on M and if (M, g) is projectively related to (M, g ), λ is homothetic on M. It follows from lemma 1 that either λ is identically zero on M (and hence ∇ = ∇ on M) or λ cannot vanish at any point of U (since Riem does not vanish at any point of U ). This argument improves on that in [9] which may now be used to show the following results [11]. In the event that (M, g) is of holonomy type R 11 each point of U admits a coordinate neighbourhood with coordinates u, v, x 3 , x 4 on which ∂/∂u is g-null and ∇-covariantly constant and g and g take the following forms.…”

“…In this paper a brief description of the completion of this study is given and where it will be shown that the problem can essentially be solved for all holonomy types except the most general one. The full details are somewhat lengthy and may be of limited interest to relativists and will be published elsewhere ( [9] and [11]) but summarised here in a form which it is hoped will be useful to those working in Einsteins theory. The possible holonomy groups for a space-time (more precisely for its connection) may be classified into 15 mutually exclusive and exhaustive types and are labelled, following [10], by the symbols R 1 ,...R 15 with R 1 being the trivial flat space-time case and R 15 the most general case (and R 5 cannot occur for a space-time).…”

“…The possible holonomy groups for a space-time (more precisely for its connection) may be classified into 15 mutually exclusive and exhaustive types and are labelled, following [10], by the symbols R 1 ,...R 15 with R 1 being the trivial flat space-time case and R 15 the most general case (and R 5 cannot occur for a space-time). Since full details of this classification scheme are available [1] (and a summary has been given in [9,11]) no further review need be given here. The holonomy type of a space-time is a global statement about it and it is useful to introduce a supplementary (pointwise) classification in M determined by the curvature tensor Riem with components R a bcd .…”

“…The holonomy type of a space-time is a global statement about it and it is useful to introduce a supplementary (pointwise) classification in M determined by the curvature tensor Riem with components R a bcd . This classification has also been given in detail in [1] (and summarised in [9,11]) but will be partially repeated here in order to clarify notation. The remainder of the paper is devoted to a summary of the main results and techniques regarding projective relatedness, to demonstrating the (close) relationship between the holonomy types of projectively related metrics, to a uniqueness theorem which exists in the more complicated R 9 and R 14 cases of section 4, to a "rigidity" result for projective relatedness which exists for what might be called "generic" space-times and to an application of these results to the study of projective symmetry in space-times.…”

“…ab , which is a global metric on M, and ψ together satisfy (11) and constitute the required solution of (8) …”